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<!DOCTYPE html>
<html xml:lang="en" xmlns="http://www.w3.org/1999/xhtml"><head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>MultiMath Blu1</title>
<script type="text/javascript" src="./mathjax/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<link rel="stylesheet" href="stylesheets/foundation.css">
<link rel="stylesheet" href="stylesheets/foundation-icons.css">
<link rel="stylesheet" href="stylesheets/font.css">
<link rel="stylesheet" href="stylesheets/style.css">
<link rel="stylesheet" href="http://fonts.googleapis.com/css?family=Noto+Serif:400,700,400italic,700italic&amp;subset=latin,latin-ext,greek-ext">
<link rel="stylesheet" type="text/css" href="stylesheet.css">
<meta name="bsmart-page-names" content="3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39">
<script type="text/javascript" src="js/modernizr.js"></script><script type="text/javascript" src="js/jquery.js"></script><script type="text/javascript" src="js/fastclick.js"></script><script type="text/javascript" src="js/foundation.min.js"></script>
<style id="MathJax_SVG_styles">.MathJax_SVG_Display {text-align: center; margin: 1em 0em; position: relative; display: block!important; text-indent: 0; max-width: none; max-height: none; min-width: 0; min-height: 0; width: 100%}
.MathJax_SVG .MJX-monospace {font-family: monospace}
.MathJax_SVG .MJX-sans-serif {font-family: sans-serif}
.MathJax_SVG {display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 100%; font-size-adjust: none; text-indent: 0; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; padding: 0; margin: 0}
.MathJax_SVG * {transition: none; -webkit-transition: none; -moz-transition: none; -ms-transition: none; -o-transition: none}
.mjx-svg-href {fill: blue; stroke: blue}
</style></head>
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<div data-page-container="3" id="page-3" class="row chapters-content"><div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">3</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<h1 class="title" id="unit1">
<span class="pagebreak" epub:type="pagebreak" title="3" id="page3"></span>Capitolo 1<br>Numeri naturali e numeri interi relativi</h1>
<ul class="list1">
<li><p class="noindent"><span class="fronc">L’insieme dei numeri naturali</span></p></li>
<li><p class="noindent"><span class="fronc">Le quattro operazioni aritmetiche con i numeri naturali</span></p></li>
<li><p class="noindent"><span class="fronc">Potenze in ℕ e loro proprietà</span></p></li>
<li><p class="noindent"><span class="fronc">Espressioni con i numeri naturali</span></p></li>
<li><p class="noindent"><span class="fronc">Divisibilità e numeri primi</span></p></li>
<li><p class="noindent"><span class="fronc">Massimo comune divisore e minimo comune multiplo</span></p></li>
<li><p class="noindent"><span class="fronc">Sistemi di numerazione</span></p></li>
<li><p class="noindent"><span class="fronc">L’insieme dei numeri interi relativi</span></p></li>
<li><p class="noindent"><span class="fronc">Le quattro operazioni aritmetiche con i numeri interi relativi</span></p></li>
<li><p class="noindent"><span class="fronc">Potenza di un numero intero relativo</span></p></li>
</ul>
<div class="problem">
<h3 class="sec_title">Calciatori e automobili</h3>
<p class="noindent"><b>Zapping domenicale</b></p>
<p class="noindent">Al 48° giro è in testa Alonso con la Ferrari numero 3. Segue a 15 secondi Pérez con la McLaren numero 6... Al 12° minuto del secondo tempo Abate riceve il pallone sulla fascia destra, corre fino alla linea di fondo, crossa al centro dove il numero 21 Pirlo colpisce di testa... traversa!... A 20 minuti dal termine le due squadre sono ancora sullo 0 a 0...</p>
<p class="noindent1">Numeri, continuamente numeri: il numero dei giri, il numero della macchina, il distacco, il tempo trascorso, il tempo che manca, il numero sulla maglia, il punteggio...</p>
<p class="noindenttop"><b>Servono veramente tutti questi numeri? Sono essenziali o se ne potrebbe fare a meno?</b></p>
<div class="figure">
<p class="img" id="ch1.fg1"><img src="images/c01u01f01.jpg" alt="Image"></p>
<p class="figcap">FIGURA 1</p>
</div>
<div class="figure">
<p class="img" id="ch1.fg2"><img src="images/c01u01f02.jpg" alt="Image"></p>
<p class="figcap">FIGURA 2</p>
</div>
</div>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"></div>
</div></div>
<div data-page-container="4" id="page-4" class="row chapters-content">
<div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">4</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<h2 class="para_title" id="par01">
<span class="pagebreak" epub:type="pagebreak" title="4" id="page4"></span>L’insieme dei numeri naturali<a id="ind1"></a><!--<?"insieme|dei numeri|naturali N",4,0,2>-->
</h2>
<h3 class="sec_title" id="sec1">1. I numeri naturali e il loro ordinamento<a id="ind2"></a><!--<?"ordinamento|dei numeri naturali",4,0,2>-->
</h3>
<p class="noindent">I <b>numeri naturali</b>,<a id="ind3"></a><!--<?"numeri|naturali",4,0,2>--> cioè 0, 1, 2, 3, 4, ..., formano un insieme che viene detto <i>insieme dei numeri naturali</i> e che si indica con il simbolo ℕ:<a id="ind4"></a><!--<?"N|insieme dei numeri naturali",4,0,2>--></p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="43.667ex" height="4.333ex" viewBox="0 -1173.4 18791.1 1846.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJAMS-4E"></use><use x="1004" y="0" xlink:href="#MJMAIN-3D"></use><use x="2065" y="-1" xlink:href="#MJSZ2-7B"></use><use x="2737" y="0" xlink:href="#MJMAIN-30"></use><g transform="translate(3242,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="3852" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(4302,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(4912,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="5522" y="0" xlink:href="#MJMAIN-31"></use><g transform="translate(6027,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="6637" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(7086,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(7696,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(8306,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="8916" y="0" xlink:href="#MJMAIN-32"></use><g transform="translate(9421,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="10031" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(10481,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(11091,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(11701,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="12311" y="0" xlink:href="#MJMAIN-33"></use><g transform="translate(12816,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="13425" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(13875,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(14485,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(15095,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="15705" y="0" xlink:href="#MJMAIN-34"></use><g transform="translate(16210,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="16820" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(17270,0)"><use xlink:href="#MJMAIN-2E"></use><use x="283" y="0" xlink:href="#MJMAIN-2E"></use><use x="566" y="0" xlink:href="#MJMAIN-2E"></use></g><use x="18119" y="-1" xlink:href="#MJSZ2-7D"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ℕ</mi><mo>=</mo><mo>{</mo><mn>0</mn><mtext> </mtext><mo>;</mo><mtext> </mtext><mtext> </mtext><mn>1</mn><mtext> </mtext><mo>;</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>2</mn><mtext> </mtext><mo>;</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>3</mn><mtext> </mtext><mo>;</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>4</mn><mtext> </mtext><mo>;</mo><mn>...</mn><mo>}</mo></math></script></p>
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<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<p class="noindent">Poiché tali numeri nascono dall’attività del contare, sono detti <i>naturali</i>.</p>
<p class="noindent1"><i>Contare</i> significa passare da un numero al suo successivo (<a href="#ch1.fg3"><span class="fron">FIGURA 3</span></a>).</p>
<p class="noindent">Si costruisce così la <i>successione dei numeri naturali</i>.<a id="ind5"></a><!--<?"successione dei numeri naturali",4,0,2>--></p>
<div title_dea="Numeri naturali" key_dea="numeri naturali, naturali">
<div class="figure">
<p class="img" id="ch1.fg3"><img src="images/c01u01f03.jpg" alt="Image"></p>
<p class="figcap">FIGURA 3</p>
</div>
</div>
<p class="noindent">La <a href="#ch1.fg3"><span class="fron">FIGURA 3</span></a> ci suggerisce alcune proprietà dell’insieme ℕ.<a id="ind6"></a><!--<?"propriet&#x00E0;|dell&#x2019;insieme N",4,0,2>--></p>
<ul class="blist">
<li><p class="noindent">L’insieme dei numeri naturali è <b>infinito</b>.</p></li>
<li><p class="noindent">Ogni numero naturale <b>ha un successivo</b>.</p></li>
<li><p class="noindent">Ogni numero naturale, eccetto lo zero, <b>ha un precedente</b>.</p></li>
<li><p class="noindent">Lo zero è l’elemento minimo dell’insieme dei numeri naturali.</p></li>
<li><p class="noindent">L’insieme dei numeri naturali <b>non ha un elemento massimo</b>.</p></li>
</ul>
<p class="noindent">Per indicare che due numeri <i>a</i> e <i>b</i> sono <b>uguali</b>, useremo il simbolo = e scriveremo <span class="blue"><i>a</i> = <i>b</i></span> leggendo «<i>a</i> è uguale a <i>b</i>»; ad esempio 17 = 17.</p>
<p class="noindent1">La <b>relazione di uguaglianza</b><a id="ind7"></a><!--<?"uguaglianza|relazione di",4,0,2>--> tra due numeri naturali gode delle proprietà</p>
<ul class="blist">
<li><p class="noindent"><b>riflessiva</b>:<a id="ind8"></a><!--<?"riflessiva, propriet&#x00E0;",4,0,2>--> ogni numero è uguale a se stesso <span class="blue">(<i>a</i> = <i>a</i>)</span>;</p></li>
<li><p class="noindent"><b>simmetrica</b>:<a id="ind9"></a><!--<?"simmetrica, propriet&#x00E0;",4,0,2>--> <span class="blue">se <i>a</i> = <i>b</i> <b>allora</b> <i>b</i> = <i>a</i></span>;</p></li>
<li><p class="noindent"><b>transitiva</b>:<a id="ind10"></a><!--<?"transitiva, propriet&#x00E0;",4,0,2>--> <span class="blue">se <i>a</i> = <i>b</i> e <i>b</i> = <i>c</i> <b>allora</b> <i>a</i> = <i>c</i></span>.</p></li>
</ul>
<p class="noindent">In generale, nella scrittura</p>
<p class="img"><img src="images/fig1.jpg" alt="Image"></p>
<p class="noindent"><img src="images/fig2.jpg" alt="Image"> è il <b>primo membro</b> dell’uguaglianza e <img src="images/fig3.jpg" alt="Image"> è il <b>secondo membro</b> dell’uguaglianza.</p>
<p class="noindent">Per indicare che <i>a</i> e <i>b</i> <b>non</b> <i>sono uguali</i>, useremo il simbolo ≠ e scriveremo <span class="blue"><i>a</i> ≠ <i>b</i></span> leggendo «<i>a</i> è diverso da <i>b</i>»; ad esempio 6 ≠ 8 e 4 ≠ 0.</p>
<p class="noindent1">I numeri naturali hanno un <b>ordine</b>, cioè, dati due numeri naturali, diversi tra loro, è sempre possibile confrontarli stabilendo tra essi una <b>relazione di disuguaglianza</b>:<a id="ind11"></a><!--<?"disuguaglianza|relazione di",4,0,2>--> se nella successione dei numeri naturali un numero <i>a</i> precede un numero <i>b</i>, si dice che <b><i>a</i> è minore di <i>b</i></b> e si scrive <b><i>a</i></b> &lt; <b><i>b</i></b>; se invece <i>a</i> segue <i>b</i>, si dice che <b><i>a</i> è maggiore di <i>b</i></b> e si scrive <b><i>a</i></b> &gt; <b><i>b</i></b>.</p>
<p class="noindent">Ad esempio:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="60.833ex" height="3ex" viewBox="0 -875 26196.9 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-33"></use><g transform="translate(505,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><use x="609" y="0" xlink:href="#MJMAIN-3C"></use><g transform="translate(1392,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="2507" y="0" xlink:href="#MJMAIN-35"></use><g transform="translate(3012,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="4232" y="0" xlink:href="#MJMAIN-35"></use><g transform="translate(4737,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><use x="609" y="0" xlink:href="#MJMAIN-3E"></use><g transform="translate(1392,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="6740" y="0" xlink:href="#MJMAIN-33"></use><g transform="translate(7245,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(8465,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use></g><g transform="translate(9475,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><use x="609" y="0" xlink:href="#MJMAIN-3E"></use><g transform="translate(1392,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(11478,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><g transform="translate(12488,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(13708,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><g transform="translate(14718,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><use x="609" y="0" xlink:href="#MJMAIN-3C"></use><g transform="translate(1392,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(16721,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use></g><g transform="translate(17731,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="18951" y="0" xlink:href="#MJMAIN-30"></use><g transform="translate(19456,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><use x="609" y="0" xlink:href="#MJMAIN-3C"></use><g transform="translate(1392,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="21459" y="0" xlink:href="#MJMAIN-34"></use><g transform="translate(21964,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="23184" y="0" 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xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>3</mn><mtext> &lt; </mtext><mn>5</mn><mtext>  </mtext><mn>5</mn><mtext> &gt; </mtext><mn>3</mn><mtext>  </mtext><mn>20</mn><mtext> &gt; </mtext><mn>12</mn><mtext>  </mtext><mn>12</mn><mtext> &lt; </mtext><mn>20</mn><mtext>  </mtext><mn>0</mn><mtext> &lt; </mtext><mn>4</mn><mtext>  </mtext><mn>7</mn><mtext> &gt; </mtext><mn>0</mn></mrow></math></script></p>
<p class="noindent">Per indicare le relazioni d’ordine si usano anche altri due simboli: ≤, ≥.</p>
<p class="noindent">Il simbolo ≤ significa «minore o uguale»:</p>
<p class="math"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="41.5ex" height="3ex" viewBox="0 -875 17880.5 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-61"></use><g transform="translate(534,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="1421" y="0" xlink:href="#MJMAIN-2264"></use><g transform="translate(2482,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="3092" y="0" xlink:href="#MJMATHI-62"></use><g transform="translate(3526,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="4414" y="0" xlink:href="#MJMAIN-2194"></use><g transform="translate(5696,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="6306" y="0" xlink:href="#MJMATHI-61"></use><g transform="translate(6840,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><use x="609" y="0" xlink:href="#MJMAIN-3C"></use><g transform="translate(1392,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="8843" y="0" xlink:href="#MJMATHI-62"></use><g transform="translate(9277,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(10054,0)"><use xlink:href="#MJMAINB-6F"></use><use x="580" y="0" xlink:href="#MJMAINB-70"></use><use x="1224" y="0" xlink:href="#MJMAINB-70"></use><use x="1868" y="0" xlink:href="#MJMAINB-75"></use><use x="2512" y="0" xlink:href="#MJMAINB-72"></use><use x="2991" y="0" xlink:href="#MJMAINB-65"></use></g><g transform="translate(13744,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="14354" y="0" xlink:href="#MJMATHI-61"></use><g transform="translate(14888,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="15775" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(16836,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="17446" y="0" xlink:href="#MJMATHI-62"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>a</mi><mtext> </mtext><mo>≤</mo><mtext> </mtext><mi>b</mi><mtext> </mtext><mo>↔</mo><mtext> </mtext><mi>a</mi><mtext> &lt; </mtext><mi>b</mi><mtext> </mtext><mi mathvariant="bold">oppure</mi><mtext> </mtext><mi>a</mi><mtext> </mtext><mo>=</mo><mtext> </mtext><mi>b</mi></mrow></math></script></span></p>
<p class="noindent">Ad esempio, la scrittura <i>x</i> ≤ 5 significa che il numero <i>x</i> può essere minore di 5 o anche il numero 5 stesso. Si può anche scrivere 3 ≤ 5 perché delle due relazioni 3 &lt; 5 e 3 = 5 si verifica la prima.</p>
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<p class="noindent">L’insieme dei numeri naturali privato dello zero si indica con ℕ* (si legge «enne star»):</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="36.833ex" height="4.333ex" viewBox="0 -1173.4 15853.7 1846.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJAMS-4E"></use><use transform="scale(0.707)" x="1028" y="513" xlink:href="#MJMAIN-2217"></use><use x="1461" y="0" xlink:href="#MJMAIN-3D"></use><use x="2522" y="-1" xlink:href="#MJSZ2-7B"></use><use x="3194" y="0" xlink:href="#MJMAIN-31"></use><g transform="translate(3699,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="4309" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(4759,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(5369,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="5979" y="0" xlink:href="#MJMAIN-32"></use><g transform="translate(6484,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="7094" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(7543,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(8153,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="8763" y="0" xlink:href="#MJMAIN-33"></use><g transform="translate(9268,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="9878" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(10328,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(10938,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="11548" y="0" xlink:href="#MJMAIN-34"></use><g transform="translate(12053,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="12663" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(13112,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(13722,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(14332,0)"><use xlink:href="#MJMAIN-2E"></use><use x="283" y="0" xlink:href="#MJMAIN-2E"></use><use x="566" y="0" xlink:href="#MJMAIN-2E"></use></g><use x="15181" y="-1" xlink:href="#MJSZ2-7D"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ℕ</mi><mo>*</mo></msup><mo>=</mo><mo>{</mo><mn>1</mn><mtext> </mtext><mo>;</mo><mtext> </mtext><mtext> </mtext><mn>2</mn><mtext> </mtext><mo>;</mo><mtext> </mtext><mtext> </mtext><mn>3</mn><mtext> </mtext><mo>;</mo><mtext> </mtext><mtext> </mtext><mn>4</mn><mtext> </mtext><mo>;</mo><mtext> </mtext><mtext> </mtext><mn>...</mn><mo>}</mo></math></script></p>
</div></div>
</div>
</div>
<div data-page-container="5" id="page-5" class="row chapters-content"><div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">5</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<p class="noindent"><span class="pagebreak" epub:type="pagebreak" title="5" id="page5"></span>Il simbolo ≥ significa «maggiore o uguale»:</p>
<p class="math"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="44.333ex" height="3ex" viewBox="0 -875 19100.4 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-61"></use><g transform="translate(534,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="1421" y="0" xlink:href="#MJMAIN-2265"></use><g transform="translate(2482,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="3092" y="0" xlink:href="#MJMATHI-62"></use><g transform="translate(3526,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="4414" y="0" xlink:href="#MJMAIN-2194"></use><g transform="translate(5696,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="6306" y="0" xlink:href="#MJMATHI-61"></use><g transform="translate(6840,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><use x="609" y="0" xlink:href="#MJMAIN-3E"></use><g transform="translate(1392,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="8843" y="0" xlink:href="#MJMATHI-62"></use><g transform="translate(9277,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(9887,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(10664,0)"><use xlink:href="#MJMAINB-6F"></use><use x="580" y="0" xlink:href="#MJMAINB-70"></use><use x="1224" y="0" xlink:href="#MJMAINB-70"></use><use x="1868" y="0" xlink:href="#MJMAINB-75"></use><use x="2512" y="0" xlink:href="#MJMAINB-72"></use><use x="2991" y="0" xlink:href="#MJMAINB-65"></use></g><g transform="translate(14354,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(14964,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="15573" y="0" xlink:href="#MJMATHI-61"></use><g transform="translate(16107,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="16995" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(18056,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="18666" y="0" xlink:href="#MJMATHI-62"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>a</mi><mtext> </mtext><mo>≥</mo><mtext> </mtext><mi>b</mi><mtext> </mtext><mo>↔</mo><mtext> </mtext><mi>a</mi><mtext> &gt; </mtext><mi>b</mi><mtext> </mtext><mtext> </mtext><mi mathvariant="bold">oppure</mi><mtext> </mtext><mtext> </mtext><mi>a</mi><mtext> </mtext><mo>=</mo><mtext> </mtext><mi>b</mi></mrow></math></script></span></p>
<p class="noindent">Ad esempio, la scrittura <i>x</i> ≥ 7 indica che il numero <i>x</i> può essere uguale a 7 oppure maggiore di 7. Si può anche scrivere 7 ≥ 7 perché delle due relazioni 7 &gt; 7 e 7 = 7 si verifica la seconda.</p>
<p class="noindent">Una scrittura del tipo 2 &lt; 5, oppure 7 &gt; 1 o 4 ≤ 6 o 20 ≥ 18, è una <b>disuguaglianza</b>. In generale, nella scrittura</p>
<p class="img"><img src="images/fig4.jpg" alt="Image"></p>
<ul class="blist">
<li><p class="noindent"><img src="images/fig2.jpg" alt="Image"> è il <b>primo membro</b> (della disuguaglianza);</p></li>
<li><p class="noindent"><img src="images/fig3.jpg" alt="Image"> è il <b>secondo membro</b> (della disuguaglianza);</p></li>
<li><p class="noindent">&lt; è il simbolo del verso della disuguaglianza; impropriamente, diremo spesso che «&lt; è il verso della disuguaglianza».</p></li>
</ul>
<p class="noindent">Osserviamo ora la <a href="#ch1.fg4"><span class="fron">FIGURA 4</span></a>: è evidente che se 2 &lt; 4 e 4 &lt; 7, allora è anche 2 &lt; 7.</p>
<p class="noindent">Questa osservazione si può generalizzare:</p>
<p class="math"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="42.667ex" height="3ex" viewBox="0 -875 18398 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAINB-73"></use><use x="459" y="0" xlink:href="#MJMAINB-65"></use><g transform="translate(1157,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="1767" y="0" xlink:href="#MJMATHI-61"></use><g transform="translate(2301,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><use x="609" y="0" xlink:href="#MJMAIN-3C"></use></g><g transform="translate(3694,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="4304" y="0" xlink:href="#MJMATHI-62"></use><g transform="translate(4738,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="5348" y="0" xlink:href="#MJMAINB-65"></use><g transform="translate(5880,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="6490" y="0" xlink:href="#MJMATHI-62"></use><g transform="translate(6924,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><use x="609" y="0" xlink:href="#MJMAIN-3C"></use></g><g transform="translate(8317,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(8927,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="9537" y="0" xlink:href="#MJMATHI-63"></use><use x="9975" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(10424,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(11034,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(11811,0)"><use xlink:href="#MJMAINB-61"></use><use x="564" y="0" xlink:href="#MJMAINB-6C"></use><use x="888" y="0" xlink:href="#MJMAINB-6C"></use><use x="1212" y="0" xlink:href="#MJMAINB-6F"></use><use x="1792" y="0" xlink:href="#MJMAINB-72"></use><use x="2271" y="0" xlink:href="#MJMAINB-61"></use></g><g transform="translate(14813,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="15423" y="0" xlink:href="#MJMATHI-61"></use><g transform="translate(15957,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><use x="609" y="0" xlink:href="#MJMAIN-3C"></use></g><g transform="translate(17350,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="17960" y="0" xlink:href="#MJMATHI-63"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">se</mi><mtext> </mtext><mi>a</mi><mtext> &lt;</mtext><mtext> </mtext><mi>b</mi><mtext> </mtext><mi mathvariant="bold">e</mi><mtext> </mtext><mi>b</mi><mtext> &lt;</mtext><mtext> </mtext><mtext> </mtext><mi>c</mi><mo>,</mo><mtext> </mtext><mtext> </mtext><mi mathvariant="bold">allora</mi><mtext> </mtext><mi>a</mi><mtext> &lt;</mtext><mtext> </mtext><mi>c</mi></mrow></math></script></span></p>
<div class="figure">
<p class="img" id="ch1.fg4"><img src="images/c01u01f04.jpg" alt="Image"></p>
<p class="figcap">FIGURA 4</p>
</div>
<p class="noindent">Analogamente:</p>
<p class="math"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="42.833ex" height="3ex" viewBox="0 -875 18444 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAINB-73"></use><use x="459" y="0" xlink:href="#MJMAINB-65"></use><g transform="translate(1157,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="1767" y="0" xlink:href="#MJMATHI-70"></use><g transform="translate(2275,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><use x="609" y="0" xlink:href="#MJMAIN-3E"></use></g><g transform="translate(3668,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="4278" y="0" xlink:href="#MJMATHI-71"></use><g transform="translate(4743,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="5353" y="0" xlink:href="#MJMAINB-65"></use><g transform="translate(5885,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="6495" y="0" xlink:href="#MJMATHI-71"></use><g transform="translate(6960,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><use x="609" y="0" xlink:href="#MJMAIN-3E"></use></g><g transform="translate(8353,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(8963,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="9573" y="0" xlink:href="#MJMATHI-72"></use><use x="10029" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(10478,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(11088,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(11865,0)"><use xlink:href="#MJMAINB-61"></use><use x="564" y="0" xlink:href="#MJMAINB-6C"></use><use x="888" y="0" xlink:href="#MJMAINB-6C"></use><use x="1212" y="0" xlink:href="#MJMAINB-6F"></use><use x="1792" y="0" xlink:href="#MJMAINB-72"></use><use x="2271" y="0" xlink:href="#MJMAINB-61"></use></g><g transform="translate(14867,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="15477" y="0" xlink:href="#MJMATHI-70"></use><g transform="translate(15985,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><use x="609" y="0" xlink:href="#MJMAIN-3E"></use></g><g transform="translate(17378,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="17988" y="0" xlink:href="#MJMATHI-72"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">se</mi><mtext> </mtext><mi>p</mi><mtext> &gt;</mtext><mtext> </mtext><mi>q</mi><mtext> </mtext><mi mathvariant="bold">e</mi><mtext> </mtext><mi>q</mi><mtext> &gt;</mtext><mtext> </mtext><mtext> </mtext><mi>r</mi><mo>,</mo><mtext> </mtext><mtext> </mtext><mi mathvariant="bold">allora</mi><mtext> </mtext><mi>p</mi><mtext> &gt;</mtext><mtext> </mtext><mi>r</mi></mrow></math></script></span></p>
<p class="noindent">Le relazioni precedenti esprimono la <b>proprietà transitiva della disuguaglianza</b>.<a id="ind12"></a><!--<?"disuguaglianza|propriet&#x00E0; transitiva della",4,0,2>--></p>
<p class="noindent1">Possiamo rappresentare i numeri naturali su una <b>semiretta orientata</b>,<a id="ind13"></a><!--<?"semiretta orientata",4,0,2>--> cioè su una semiretta su cui sia fissato un <b>verso</b> di percorrenza,<a id="ind14"></a><!--<?"verso|di percorrenza",4,0,2>--> indicato convenzionalmente da una punta di freccia. Per semplicità, supponiamo che la semiretta sia disposta orizzontalmente e orientata da sinistra a destra.</p>
<p class="noindent">Fissiamo un segmento di lunghezza <b><i>u</i></b> come <b>unità di misura</b><a id="ind15"></a><!--<?"unit&#x00E0;|di misura",4,0,2>--> delle lunghezze e associamo all’<b>origine</b> <i>O</i> della semiretta il numero 0. A partire da <i>O</i>, individuiamo il punto <i>A</i> la cui distanza da <i>O</i> è <b><i>u</i></b> e associamo a esso il numero 1; poi, partendo da <i>A</i> e compiendo un altro «passo» della stessa lunghezza verso destra, individuiamo il punto <i>B</i>, a cui associamo il numero 2. Continuando in questo modo, si trovano gli altri punti cui associare i numeri naturali 3, 4, 5, ...</p>
<div class="figure">
<p class="img" id="ch1.fg5"><img src="images/c01u01f05.jpg" alt="Image"></p>
<p class="figcap">FIGURA 5</p>
</div>
<p class="noindent">Ad esempio, in <a href="#ch1.fg5"><span class="fron">FIGURA 5</span></a> al numero naturale 4 corrisponde il punto <i>D</i> e, viceversa, al punto <i>D</i> corrisponde il numero 4. È poi evidente che a ogni numero naturale corrisponde un punto della semiretta, ma che non è vero il viceversa: esistono punti della semiretta ai quali non corrispondono numeri naturali.</p>
<p class="noindent1">Dalla <a href="#ch1.fg5"><span class="fron">FIGURA 5</span></a> è evidente che l’insieme ℕ dei numeri naturali, oltre a essere infinito e ordinato, gode della seguente proprietà: tra due numeri naturali consecutivi (cioè tra un numero naturale e il suo successivo) non vi sono altri numeri naturali. Si dice perciò che ℕ è un <b>insieme discreto</b>.<a id="ind16"></a><!--<?"insieme|discreto",4,0,2>--></p>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"></div>
</div></div>
<div data-page-container="6" id="page-6" class="row chapters-content"><div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">6</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<h2 class="para_title" id="par02">
<span class="pagebreak" epub:type="pagebreak" title="6" id="page6"></span>Le quattro operazioni aritmetiche con i numeri naturali<a id="ind17"></a><!--<?"operazioni|con i numeri|naturali",4,0,2>-->
</h2>
<h3 class="sec_title" id="sec2">2. Addizione<a id="ind18"></a><!--<?"addizione",4,0,2>--> e sue proprietà<a id="ind19"></a><!--<?"addizione|propriet&#x00E0; dell&#x2019;",4,0,2>-->
</h3>
<p class="noindent">L’<b>addizione</b>, che si indica con il segno +, è un’operazione che si esegue tra due numeri, detti <b>addendi</b>.</p>
<p class="noindent">Il risultato dell’addizione si chiama <b>somma</b> (<a href="#ch1.fg6"><span class="fron">FIGURA 6</span></a>).</p>
<div class="figure">
<p class="img" id="ch1.fg6"><img src="images/c01u01f06.jpg" alt="Image"></p>
<p class="figcap">FIGURA 6</p>
</div>
<div class="definition" title_dea="Somma di due numeri naturali" key_dea="somma di due numeri naturali, somma, addizione, numeri naturali">
<h4 class="noindent">
<span class="ash">DEFINIZIONE</span> <span class="orangeb">SOMMA DI DUE NUMERI NATURALI</span>
</h4>
<p class="noindentin">La somma di due numeri naturali<a id="ind20"></a><!--<?"somma|di due numeri|naturali",4,0,2>--> è il numero naturale che si ottiene contando di seguito al primo tutte le unità del secondo.</p>
</div>
<p class="noindent1">L’addizione gode di alcune importanti proprietà.</p>
<ul class="blist">
<li>
<p class="noindent"><b>Proprietà commutativa</b>:<a id="ind21"></a><!--<?"commutativa, propriet&#x00E0;",4,0,2>--> cambiando l’ordine degli addendi, la somma non cambia:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="33.833ex" height="3ex" viewBox="0 -875 14592.6 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMATHI-61"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(756,0)"><use xlink:href="#MJMAIN-2B"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1761,0)"><use xlink:href="#MJMATHI-62"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(2473,0)"><use xlink:href="#MJMAIN-3D"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(3534,0)"><use xlink:href="#MJMATHI-62"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(4190,0)"><use xlink:href="#MJMAIN-2B"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(5195,0)"><use xlink:href="#MJMATHI-61"></use></g></g></g><g transform="translate(5729,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1219,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1829,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(2439,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="8779" y="0" xlink:href="#MJMAIN-35"></use><use x="9506" y="0" xlink:href="#MJMAIN-2B"></use><use x="10511" y="0" xlink:href="#MJMAIN-38"></use><use x="11294" y="0" xlink:href="#MJMAIN-3D"></use><use x="12355" y="0" xlink:href="#MJMAIN-38"></use><use x="13082" y="0" xlink:href="#MJMAIN-2B"></use><use x="14087" y="0" xlink:href="#MJMAIN-35"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mstyle color="#00aef0"><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mi>b</mi><mo>+</mo><mi>a</mi></mstyle><mi>     </mi><mn>5</mn><mo>+</mo><mn>8</mn><mo>=</mo><mn>8</mn><mo>+</mo><mn>5</mn></mrow></math></script></p>
</li>
<li>
<p class="noindent"><b>Proprietà associativa</b>:<a id="ind22"></a><!--<?"associativa, propriet&#x00E0;",4,0,2>--> la somma di tre numeri non cambia se a due addendi consecutivi si sostituisce la loro somma:</p>
<p class="math"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="24.667ex" height="2.5ex" viewBox="0 -773.9 10636.3 1047.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMATHI-61"></use><use x="1150" y="0" xlink:href="#MJMAIN-2B"></use><use x="2155" y="0" xlink:href="#MJMATHI-62"></use><use x="2589" y="0" xlink:href="#MJMAIN-29"></use><use x="3205" y="0" xlink:href="#MJMAIN-2B"></use><use x="4210" y="0" xlink:href="#MJMATHI-63"></use><use x="4926" y="0" xlink:href="#MJMAIN-3D"></use><use x="5987" y="0" xlink:href="#MJMATHI-61"></use><use x="6743" y="0" xlink:href="#MJMAIN-2B"></use><use x="7748" y="0" xlink:href="#MJMAIN-28"></use><use x="8142" y="0" xlink:href="#MJMATHI-62"></use><use x="8799" y="0" xlink:href="#MJMAIN-2B"></use><use x="9804" y="0" xlink:href="#MJMATHI-63"></use><use x="10242" y="0" xlink:href="#MJMAIN-29"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>+</mo><mi>c</mi><mo>=</mo><mi>a</mi><mo>+</mo><mo stretchy="false">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></math></script></span></p>
<p class="noindent">Ad esempio, la somma 2 + 3 + 7 si può calcolare in due modi diversi:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -2.5ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="44.667ex" height="6ex" viewBox="0 -1539.2 19234.4 2578.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-32"></use><use x="727" y="0" xlink:href="#MJMAIN-2B"></use><use x="1732" y="0" xlink:href="#MJMAIN-33"></use><use x="2459" y="0" xlink:href="#MJMAIN-2B"></use><use x="3464" y="0" xlink:href="#MJMAIN-37"></use><use x="4247" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(5308,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(6085,0)"><g transform="translate(-11,0)"><use x="0" y="725" xlink:href="#MJMAIN-2197"></use><use x="0" y="-766" xlink:href="#MJMAIN-2198"></use></g><g transform="translate(1794,0)"><g transform="translate(0,725)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-32"></use><use x="1121" y="0" xlink:href="#MJMAIN-2B"></use><use x="2126" y="0" xlink:href="#MJMAIN-33"></use><use x="2631" y="0" xlink:href="#MJMAIN-29"></use><use x="3247" y="0" xlink:href="#MJMAIN-2B"></use><use x="4252" y="0" xlink:href="#MJMAIN-37"></use><use x="5035" y="0" xlink:href="#MJMAIN-3D"></use><use x="6096" y="0" xlink:href="#MJMAIN-35"></use><use x="6823" y="0" xlink:href="#MJMAIN-2B"></use><use x="7828" y="0" xlink:href="#MJMAIN-37"></use><use x="8611" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(9672,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g></g><g transform="translate(0,-766)"><use xlink:href="#MJMAIN-32"></use><use x="727" y="0" xlink:href="#MJMAIN-2B"></use><use x="1732" y="0" xlink:href="#MJMAIN-28"></use><use x="2126" y="0" xlink:href="#MJMAIN-33"></use><use x="2853" y="0" xlink:href="#MJMAIN-2B"></use><use x="3858" y="0" xlink:href="#MJMAIN-37"></use><use x="4363" y="0" xlink:href="#MJMAIN-29"></use><use x="5035" y="0" xlink:href="#MJMAIN-3D"></use><use x="6096" y="0" xlink:href="#MJMAIN-32"></use><use x="6823" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(7828,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use></g><use x="9116" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(10177,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>2</mn><mo>+</mo><mn>3</mn><mo>+</mo><mn>7</mn><mo>=</mo><mtext> </mtext><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mo>↗</mo></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo><mo>+</mo><mn>7</mn><mo>=</mo><mn>5</mn><mo>+</mo><mn>7</mn><mo>=</mo><mn color="#00aef0">12</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mo>↘</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><mo>+</mo><mo stretchy="false">(</mo><mn>3</mn><mo>+</mo><mn>7</mn><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mo>+</mo><mn>10</mn><mo>=</mo><mn color="#00aef0">12</mn></mrow></mtd></mtr></mtable></mrow></math></script></p>
</li>
</ul>
<p class="noindent1">Grazie alle proprietà commutativa e associativa possiamo calcolare la somma di tre o più addendi cambiando l’ordine degli addendi e associandoli a piacimento.</p>
<ul class="blist">
<li>
<p class="noindent"><b>Elemento neutro</b><a id="ind23"></a><!--<?"elemento neutro",4,0,2>--> dell’addizione: è il numero zero. Ciò significa che addizionando zero a qualsiasi numero si ottiene il numero dato:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="41.5ex" height="3ex" viewBox="0 -875 17840.8 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMATHI-61"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(756,0)"><use xlink:href="#MJMAIN-2B"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1761,0)"><use xlink:href="#MJMAIN-30"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(2544,0)"><use xlink:href="#MJMAIN-3D"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(3605,0)"><use xlink:href="#MJMAIN-30"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(4332,0)"><use xlink:href="#MJMAIN-2B"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(5337,0)"><use xlink:href="#MJMATHI-61"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(6149,0)"><use xlink:href="#MJMAIN-3D"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(7210,0)"><use xlink:href="#MJMATHI-61"></use></g></g></g><g transform="translate(7744,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1219,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1829,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="10183" y="0" xlink:href="#MJMAIN-36"></use><use x="10911" y="0" xlink:href="#MJMAIN-2B"></use><use x="11916" y="0" xlink:href="#MJMAIN-30"></use><use x="12699" y="0" xlink:href="#MJMAIN-3D"></use><use x="13759" y="0" xlink:href="#MJMAIN-30"></use><use x="14487" y="0" xlink:href="#MJMAIN-2B"></use><use x="15492" y="0" xlink:href="#MJMAIN-36"></use><use x="16275" y="0" xlink:href="#MJMAIN-3D"></use><use x="17335" y="0" xlink:href="#MJMAIN-36"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mstyle color="#00aef0"><mi>a</mi><mo>+</mo><mn>0</mn><mo>=</mo><mn>0</mn><mo>+</mo><mi>a</mi><mo>=</mo><mi>a</mi></mstyle><mi>    </mi><mn>6</mn><mo>+</mo><mn>0</mn><mo>=</mo><mn>0</mn><mo>+</mo><mn>6</mn><mo>=</mo><mn>6</mn></mrow></math></script></p>
</li>
</ul>
<h3 class="sec_title" id="sec3">3. Sottrazione e sue proprietà</h3>
<p class="noindent">La <b>sottrazione</b>,<a id="ind24"></a><!--<?"sottrazione",4,0,2>--> che si indica con il segno −, è un’operazione che si esegue tra due numeri, considerati nell’ordine, il primo detto <b>minuendo</b> e il secondo <b>sottraendo</b>. Il risultato della sottrazione si chiama <b>differenza</b>.</p>
<div class="definition" title_dea="Differenza di due numeri naturali" key_dea="differenza, sottrazione, numeri naturali, differenza di due numeri naturali">
<h4 class="noindent">
<span class="ash">DEFINIZIONE</span> <span class="orangeb">DIFFERENZA DI DUE NUMERI NATURALI</span>
</h4>
<p class="noindentin">La differenza di due numeri naturali<a id="ind25"></a><!--<?"differenza|di due numeri|naturali",4,0,2>--> è il numero naturale, se esiste, che addizionato al sottraendo dà come somma il minuendo (<a href="#ch1.fg7"><span class="fron">FIGURA 7</span></a>).</p>
</div>
<p class="noindent">Ad esempio, è <span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="9.5ex" height="1.667ex" viewBox="0 -689.9 4081 735.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-35"></use><use x="727" y="0" xlink:href="#MJMAIN-2212"></use><use x="1732" y="0" xlink:href="#MJMAIN-33"></use><use x="2515" y="0" xlink:href="#MJMAIN-3D"></use><use x="3576" y="0" xlink:href="#MJMAIN-32"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>5</mn><mo>-</mo><mn>3</mn><mo>=</mo><mn>2</mn></mrow></math></script> perché <span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.333ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="9.5ex" height="1.833ex" viewBox="0 -689.9 4081 795.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-35"></use><use x="782" y="0" xlink:href="#MJMAIN-3D"></use><use x="1843" y="0" xlink:href="#MJMAIN-33"></use><use x="2570" y="0" xlink:href="#MJMAIN-2B"></use><use x="3576" y="0" xlink:href="#MJMAIN-32"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>5</mn><mo>=</mo><mn>3</mn><mo>+</mo><mn>2</mn></mrow></math></script> (5 è il minuendo, 3 è il sottraendo, 2 è la differenza).</p>
<div title_dea="Termini della sottrazione" key_dea="termini della differenza, termini della sottrazione, minuendo, sottraendo">
<div class="figure">
<p class="img" id="ch1.fg7"><img src="images/c01u01f07.jpg" alt="Image"></p>
<p class="figcap">FIGURA 7</p>
</div>
</div>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"></div>
</div></div>
<div data-page-container="7" id="page-7" class="row chapters-content">
<div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">7</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<p class="noindent"><span class="pagebreak" epub:type="pagebreak" title="7" id="page7"></span>La sottrazione 4 − 6 non si può eseguire in ℕ perché non esiste alcun numero naturale che, sommato a 6, dia 4.</p>
<p class="noindent">La sottrazione, nell’insieme dei numeri naturali, si può eseguire <b>solo se</b> il minuendo è maggiore o uguale al sottraendo:</p>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"></div>
</div>
<div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<p class="math"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="39.333ex" height="3ex" viewBox="0 -875 16934.2 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-61"></use><use x="756" y="0" xlink:href="#MJMAIN-2212"></use><use x="1761" y="0" xlink:href="#MJMATHI-62"></use><use x="2473" y="0" xlink:href="#MJMAIN-3D"></use><use x="3534" y="0" xlink:href="#MJMATHI-63"></use><g transform="translate(3972,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="4859" y="0" xlink:href="#MJMAIN-2194"></use><g transform="translate(6142,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="6752" y="0" xlink:href="#MJMATHI-61"></use><use x="7564" y="0" xlink:href="#MJMAIN-3D"></use><use x="8625" y="0" xlink:href="#MJMATHI-62"></use><use x="9281" y="0" xlink:href="#MJMAIN-2B"></use><use x="10286" y="0" xlink:href="#MJMATHI-63"></use><g transform="translate(10724,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(12111,0)"><use xlink:href="#MJMAINB-63"></use><use x="516" y="0" xlink:href="#MJMAINB-6F"></use><use x="1096" y="0" xlink:href="#MJMAINB-6E"></use></g><g transform="translate(14017,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="14627" y="0" xlink:href="#MJMATHI-61"></use><use x="15439" y="0" xlink:href="#MJMAIN-2265"></use><use x="16500" y="0" xlink:href="#MJMATHI-62"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>a</mi><mo>-</mo><mi>b</mi><mo>=</mo><mi>c</mi><mi> </mi><mo>↔</mo><mi> </mi><mi>a</mi><mo>=</mo><mi>b</mi><mo>+</mo><mi>c</mi><mi>  </mi><mi mathvariant="bold">con</mi><mtext> </mtext><mi>a</mi><mo>≥</mo><mi>b</mi></mrow></math></script></span></p>
<p class="noindent"><b>Casi particolari</b></p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -3ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="23.167ex" height="7ex" viewBox="0 -1770.9 9971.5 3041.7"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(167,0)"><g transform="translate(-11,0)"><g transform="translate(0,895)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMATHI-61"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(756,0)"><use xlink:href="#MJMAIN-2212"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1761,0)"><use xlink:href="#MJMATHI-61"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(2573,0)"><use xlink:href="#MJMAIN-3D"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(3634,0)"><use xlink:href="#MJMAINB-30"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(4214,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g fill="#00aef0" stroke="#00aef0" transform="translate(5433,0)"><use xlink:href="#MJMATHI-61"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(6190,0)"><use xlink:href="#MJMAIN-2212"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(7195,0)"><use xlink:href="#MJMAINB-30"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(8053,0)"><use xlink:href="#MJMAIN-3D"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(9113,0)"><use xlink:href="#MJMATHI-61"></use></g></g></g></g></g><g transform="translate(133,-827)"><use xlink:href="#MJMAIN-36"></use><use x="727" y="0" xlink:href="#MJMAIN-2212"></use><use x="1732" y="0" xlink:href="#MJMAIN-36"></use><use x="2515" y="0" xlink:href="#MJMAIN-3D"></use><use x="3576" y="0" xlink:href="#MJMAIN-30"></use><g transform="translate(4081,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="5300" y="0" xlink:href="#MJMAIN-36"></use><use x="6028" y="0" xlink:href="#MJMAIN-2212"></use><use x="7033" y="0" xlink:href="#MJMAIN-30"></use><use x="7816" y="0" xlink:href="#MJMAIN-3D"></use><use x="8876" y="0" xlink:href="#MJMAIN-36"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable align="left"><mtr><mtd columnalign="center" columnspan="1"><mstyle color="#00aef0"><mrow><mi>a</mi><mo>-</mo><mi>a</mi><mo>=</mo><mn mathvariant="bold-italic">0</mn><mi>  </mi><mi>a</mi><mo>-</mo><mn mathvariant="bold-italic">0</mn><mo>=</mo><mi>a</mi></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="center" columnspan="1"><mrow><mn>6</mn><mo>-</mo><mn>6</mn><mo>=</mo><mn>0</mn><mi>  </mi><mn>6</mn><mo>-</mo><mn>0</mn><mo>=</mo><mn>6</mn></mrow></mtd></mtr></mtable></mrow></math></script></p>
<ul class="blist">
<li>
<p class="noindent">La sottrazione gode della <b>proprietà invariantiva</b>:<a id="ind26"></a><!--<?"invariantiva, propriet&#x00E0;",4,0,2>--> se si somma o si sottrae uno stesso numero sia al minuendo sia al sottraendo, la differenza non cambia:</p>
<p class="math"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="57.833ex" height="3ex" viewBox="0 -875 24932.4 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-61"></use><use x="756" y="0" xlink:href="#MJMAIN-2212"></use><use x="1761" y="0" xlink:href="#MJMATHI-62"></use><use x="2473" y="0" xlink:href="#MJMAIN-3D"></use><use x="3534" y="0" xlink:href="#MJMAIN-28"></use><use x="3928" y="0" xlink:href="#MJMATHI-61"></use><use x="4684" y="0" xlink:href="#MJMAIN-2B"></use><use x="5689" y="0" xlink:href="#MJMATHI-63"></use><use x="6127" y="0" xlink:href="#MJMAIN-29"></use><use x="6743" y="0" xlink:href="#MJMAIN-2212"></use><use x="7748" y="0" xlink:href="#MJMAIN-28"></use><use x="8142" y="0" xlink:href="#MJMATHI-62"></use><use x="8799" y="0" xlink:href="#MJMAIN-2B"></use><use x="9804" y="0" xlink:href="#MJMATHI-63"></use><use x="10242" y="0" xlink:href="#MJMAIN-29"></use><g transform="translate(10636,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1219,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(12466,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1219,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="14296" y="0" xlink:href="#MJMATHI-61"></use><use x="15052" y="0" xlink:href="#MJMAIN-2212"></use><use x="16057" y="0" xlink:href="#MJMATHI-62"></use><use x="16769" y="0" xlink:href="#MJMAIN-3D"></use><use x="17830" y="0" xlink:href="#MJMAIN-28"></use><use x="18224" y="0" xlink:href="#MJMATHI-61"></use><use x="18980" y="0" xlink:href="#MJMAIN-2212"></use><use x="19985" y="0" xlink:href="#MJMATHI-63"></use><use x="20423" y="0" xlink:href="#MJMAIN-29"></use><use x="21039" y="0" xlink:href="#MJMAIN-2212"></use><use x="22044" y="0" xlink:href="#MJMAIN-28"></use><use x="22438" y="0" xlink:href="#MJMATHI-62"></use><use x="23095" y="0" xlink:href="#MJMAIN-2212"></use><use x="24100" y="0" xlink:href="#MJMATHI-63"></use><use x="24538" y="0" xlink:href="#MJMAIN-29"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>a</mi><mo>-</mo><mi>b</mi><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>c</mi><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo stretchy="false">)</mo><mi>   </mi><mi>   </mi><mi>a</mi><mo>-</mo><mi>b</mi><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo>-</mo><mi>c</mi><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>b</mi><mo>-</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></math></script></span></p>
</li>
</ul>
<p class="noindent">Grazie alla proprietà invariantiva possiamo eseguire rapidamente alcune sottrazioni:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -3ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="74.167ex" height="7ex" viewBox="0 -1770.9 31911.7 3041.7"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(167,0)"><g transform="translate(-11,0)"><g transform="translate(0,895)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-39"></use><use x="1010" y="0" xlink:href="#MJMAIN-38"></use><g transform="translate(1515,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="2347" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(3352,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(3962,0)"><use xlink:href="#MJMAIN-34"></use><use x="505" y="0" xlink:href="#MJMAIN-38"></use></g><g transform="translate(4972,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="5860" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(6920,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="7530" y="0" xlink:href="#MJMAIN-28"></use><g transform="translate(7924,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-39"></use><use x="1010" y="0" xlink:href="#MJMAIN-38"></use></g><g transform="translate(9439,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="10271" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(11277,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="11887" y="0" xlink:href="#MJMAIN-32"></use><use x="12392" y="0" xlink:href="#MJMAIN-29"></use><g transform="translate(12786,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="13618" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(14623,0)"><text font-family="STIXGeneral,'Arial Unicode 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xlink:href="#MJMAIN-32"></use><use x="1010" y="0" xlink:href="#MJMAIN-32"></use><g transform="translate(1515,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="2347" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(3352,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(3962,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-37"></use></g><g transform="translate(4972,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="5860" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(6920,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="7530" y="0" xlink:href="#MJMAIN-28"></use><g transform="translate(7924,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use><use x="1010" y="0" xlink:href="#MJMAIN-32"></use></g><g transform="translate(9439,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="10271" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(11277,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(11887,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="12897" y="0" xlink:href="#MJMAIN-29"></use><g transform="translate(13291,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="14123" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(15128,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="15738" y="0" xlink:href="#MJMAIN-28"></use><g transform="translate(16132,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-37"></use></g><g transform="translate(17142,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="17974" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(18979,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(19589,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="20599" y="0" xlink:href="#MJMAIN-29"></use><g transform="translate(20993,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="21881" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(22942,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(23552,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use><use x="1010" y="0" xlink:href="#MJMAIN-30"></use></g><g transform="translate(25067,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="25899" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(26904,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="27514" y="0" xlink:href="#MJMAIN-35"></use><g transform="translate(28019,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="28907" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(29968,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(30578,0)"><use xlink:href="#MJMAIN-39"></use><use x="505" y="0" xlink:href="#MJMAIN-35"></use></g></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd><mrow><mn>198</mn><mtext> </mtext><mo>−</mo><mtext> </mtext><mn>48</mn><mtext> </mtext><mo>=</mo><mtext> </mtext><mo stretchy="false">(</mo><mn>198</mn><mtext> </mtext><mo>+</mo><mtext> </mtext><mn>2</mn><mo stretchy="false">)</mo><mtext> </mtext><mo>−</mo><mtext> </mtext><mo stretchy="false">(</mo><mn>48</mn><mtext> </mtext><mo>+</mo><mtext> </mtext><mn>2</mn><mo stretchy="false">)</mo><mtext> </mtext><mo>=</mo><mtext> </mtext><mn>200</mn><mtext> </mtext><mo>−</mo><mtext> </mtext><mn>50</mn><mtext> </mtext><mo>=</mo><mtext> </mtext><mn>150</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>122</mn><mtext> </mtext><mo>−</mo><mtext> </mtext><mn>27</mn><mtext> </mtext><mo>=</mo><mtext> </mtext><mo stretchy="false">(</mo><mn>122</mn><mtext> </mtext><mo>−</mo><mtext> </mtext><mn>22</mn><mo stretchy="false">)</mo><mtext> </mtext><mo>−</mo><mtext> </mtext><mo stretchy="false">(</mo><mn>27</mn><mtext> </mtext><mo>−</mo><mtext> </mtext><mn>22</mn><mo stretchy="false">)</mo><mtext> </mtext><mo>=</mo><mtext> </mtext><mn>100</mn><mtext> </mtext><mo>−</mo><mtext> </mtext><mn>5</mn><mtext> </mtext><mo>=</mo><mtext> </mtext><mn>95</mn></mrow></mtd></mtr></mtable></mrow></math></script></p>
<p class="noindent">La sottrazione <b>non</b> gode della <i>proprietà commutativa</i> (7 − 5 = 2 ma 5 − 7 non è calcolabile in ℕ) e <b>non</b> ammette <i>elemento neutro</i> (2 − 0 = 2 ma 0 − 2 non si può eseguire in ℕ).</p>
<p class="noindent">La sottrazione <b>non</b> gode neppure della <i>proprietà associativa</i>. Ad esempio per calcolare 15 − 4 − 2 è necessario <b>eseguire le sottrazioni nell’ordine indicato</b>:</p>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"><div class="sidebox">
<p class="noindentf"><span class="red"><b>SIMBOLI</b></span></p>
<ul class="ulist">
<li><p class="noindentf">→ si legge «implica»</p></li>
<li><p class="noindentf">↔ si legge «equivale a» o «se e solo se»</p></li>
</ul>
</div></div>
</div>
<div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -3.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="14.667ex" height="5.167ex" viewBox="0 -700.9 6318.4 2244.6"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-35"></use><use x="1232" y="0" xlink:href="#MJMAIN-2212"></use><use x="2237" y="0" xlink:href="#MJMAIN-34"></use><g transform="translate(12,-555)"><use x="19" y="0" xlink:href="#MJSZ4-E152"></use><g transform="translate(489.14550264550263,0) scale(1.029100529100529,1)"><use xlink:href="#MJSZ4-E154"></use></g><g transform="translate(916,0)"><use xlink:href="#MJSZ4-E151"></use><use x="455" y="0" xlink:href="#MJSZ4-E150"></use></g><g transform="translate(1831.3677248677247,0) scale(1.029100529100529,1)"><use xlink:href="#MJSZ4-E154"></use></g><use x="2263" y="0" xlink:href="#MJSZ4-E153"></use></g><g transform="translate(1014,-1423)"><use transform="scale(0.707)" xlink:href="#MJMAIN-31"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-31"></use></g><use x="2964" y="0" xlink:href="#MJMAIN-2212"></use><use x="3969" y="0" xlink:href="#MJMAIN-32"></use><use x="4752" y="0" xlink:href="#MJMAIN-3D"></use><use x="5813" y="0" xlink:href="#MJMAIN-39"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><munder><munder><mrow><mn>15</mn><mo>−</mo><mn>4</mn></mrow><mo stretchy="true">︸</mo></munder><mrow><mn>11</mn></mrow></munder><mo>−</mo><mn>2</mn><mo>=</mo><mn>9</mn></mrow></math></script></p>
<h3 class="sec_title" id="sec4">4. Moltiplicazione e sue proprietà<a id="ind27"></a><!--<?"moltiplicazione|propriet&#x00E0; della",4,0,2>-->
</h3>
<p class="noindent">La <b>moltiplicazione</b>, che si indica con il segno × o più frequentemente con un puntino <b>·</b>, è un’operazione che si esegue tra due numeri, detti <b>fattori</b>.<a id="ind28"></a><!--<?"fattori|di un prodotto",4,0,2>--></p>
<p class="noindent">Il risultato della moltiplicazione si chiama <b>prodotto</b> (<a href="#ch1.fg8"><span class="fron">FIGURA 8</span></a>).</p>
<div class="definition" title_dea="Prodotto di due numeri naturali" key_dea="prodotto di due numeri naturali, moltiplicazione di due numeri naturali, prodotto, moltiplicazione, numeri naturali">
<h4 class="noindent">
<span class="ash">DEFINIZIONE</span> <span class="orangeb">PRODOTTO DI DUE NUMERI NATURALI</span>
</h4>
<p class="noindentin">Il prodotto di due numeri naturali<a id="ind29"></a><!--<?"prodotto|di due numeri naturali",4,0,2>--> è la somma di tanti addendi uguali al primo fattore quante sono le unità indicate dal secondo (<a href="#ch1.fg9"><span class="fron">FIGURA 9</span></a>):</p>
<div class="exp_imp" title_dea="Formula del prodotto di due numeri naturali" key_dea="formula del prodotto di due numeri naturali, moltiplicazione, numeri naturali">
<p class="math"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -5.5ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="22.5ex" height="7.667ex" viewBox="0 -992.9 9717.3 3299.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(275,0)"><use xlink:href="#MJMATHI-61"></use><use x="756" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1261" y="0" xlink:href="#MJMATHI-62"></use><use x="1973" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(3034,0)"><use xlink:href="#MJMATHI-61"></use><use x="756" y="0" xlink:href="#MJMAIN-2B"></use><use x="1761" y="0" xlink:href="#MJMATHI-61"></use><use x="2517" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(3522,0)"><use xlink:href="#MJMAIN-2E"></use><use x="283" y="0" xlink:href="#MJMAIN-2E"></use><use x="566" y="0" xlink:href="#MJMAIN-2E"></use></g><use x="4594" y="0" xlink:href="#MJMAIN-2B"></use><use x="5599" y="0" xlink:href="#MJMATHI-61"></use><g transform="translate(12,-615)"><use x="19" y="0" xlink:href="#MJSZ4-E152"></use><g transform="translate(509.32936507936506,0) scale(5.0658730158730165,1)"><use xlink:href="#MJSZ4-E154"></use></g><g transform="translate(2611,0)"><use xlink:href="#MJSZ4-E151"></use><use x="455" y="0" xlink:href="#MJSZ4-E150"></use></g><g transform="translate(3546.9960317460323,0) scale(5.0658730158730165,1)"><use xlink:href="#MJSZ4-E154"></use></g><use x="5654" y="0" xlink:href="#MJSZ4-E153"></use></g><g transform="translate(1251,-1614)"><use transform="scale(0.707)" xlink:href="#MJMATHI-62"></use><g transform="translate(306,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(738,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1169,0)"><use transform="scale(0.707)" xlink:href="#MJMAIN-61"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-64"></use><use transform="scale(0.707)" x="1066" y="0" xlink:href="#MJMAIN-64"></use><use transform="scale(0.707)" x="1627" y="0" xlink:href="#MJMAIN-65"></use><use transform="scale(0.707)" x="2076" y="0" xlink:href="#MJMAIN-6E"></use><use transform="scale(0.707)" x="2637" y="0" xlink:href="#MJMAIN-64"></use><use transform="scale(0.707)" x="3198" y="0" xlink:href="#MJMAIN-69"></use></g></g></g></g><line stroke-linecap="square" stroke="black" stroke-width="75" y2="37" y1="37" x1="37" x2="9679" transform="translate(0,904)"></line><line stroke-linecap="square" stroke="black" stroke-width="75" y1="37" x2="37" x1="37" y2="3234" transform="translate(9642,-2294)"></line><line stroke-linecap="square" stroke="black" stroke-width="75" y2="37" y1="37" x1="37" x2="9679" transform="translate(0,-2294)"></line><line stroke-linecap="square" stroke="black" stroke-width="75" y1="37" x2="37" x1="37" y2="3234" transform="translate(0,-2294)"></line></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><menclose notation="box"><mrow><mi>a</mi><mo>⋅</mo><mi>b</mi><mo>=</mo><munder><munder><mrow><mi>a</mi><mo>+</mo><mi>a</mi><mo>+</mo><mn>...</mn><mo>+</mo><mi>a</mi></mrow><mo stretchy="true">︸</mo></munder><mrow><mi>b</mi><mtext> </mtext><mtext> </mtext><mtext>addendi</mtext></mrow></munder></mrow></menclose></mrow></math></script></span></p>
</div>
</div>
<div title_dea="Termini del prodotto di due numeri naturali" key_dea="prodotto, moltiplicazione, numeri naturali, termini del prodotto di due numeri naturali, fattore, fattori">
<div class="figure">
<p class="img" id="ch1.fg8"><img src="images/c01u01f08.jpg" alt="Image"></p>
<p class="figcap">FIGURA 8</p>
</div>
<div class="figure">
<p class="img" id="ch1.fg9"><img src="images/c01u01f09.jpg" alt="Image"></p>
<p class="figcap">FIGURA 9</p>
</div>
</div>
<p class="noindent">La moltiplicazione<a id="ind30"></a><!--<?"moltiplicazione",4,0,2>--> gode di alcune importanti proprietà, analoghe a quelle dell’addizione.</p>
<ul class="blist">
<li>
<p class="noindent"><b>Proprietà commutativa</b>:<a id="ind31"></a><!--<?"commutativa, propriet&#x00E0;",4,0,2>--> cambiando l’ordine dei fattori, il prodotto non cambia:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="29.167ex" height="3ex" viewBox="0 -875 12592.6 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMATHI-61"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(756,0)"><use xlink:href="#MJMAIN-22C5"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1261,0)"><use xlink:href="#MJMATHI-62"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1973,0)"><use xlink:href="#MJMAIN-3D"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(3034,0)"><use xlink:href="#MJMATHI-62"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(3690,0)"><use xlink:href="#MJMAIN-22C5"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(4195,0)"><use xlink:href="#MJMATHI-61"></use></g></g></g><g transform="translate(4729,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1219,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1829,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(2439,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="7779" y="0" xlink:href="#MJMAIN-35"></use><use x="8506" y="0" xlink:href="#MJMAIN-22C5"></use><use x="9011" y="0" xlink:href="#MJMAIN-38"></use><use x="9794" y="0" xlink:href="#MJMAIN-3D"></use><use x="10855" y="0" xlink:href="#MJMAIN-38"></use><use x="11582" y="0" xlink:href="#MJMAIN-22C5"></use><use x="12087" y="0" xlink:href="#MJMAIN-35"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mstyle color="#00aef0"><mi>a</mi><mo>·</mo><mi>b</mi><mo>=</mo><mi>b</mi><mo>·</mo><mi>a</mi></mstyle><mi>     </mi><mn>5</mn><mo>·</mo><mn>8</mn><mo>=</mo><mn>8</mn><mo>·</mo><mn>5</mn></mrow></math></script></p>
</li>
</ul>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"><div class="sidebox">
<p class="noindentf">Se si eseguisse prima la seconda sottrazione, 4 − 2, si otterrebbe 13, che è un risultato errato.</p>
</div></div>
</div>
</div>
<div data-page-container="8" id="page-8" class="row chapters-content"><div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">8</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<ul class="blist">
<li>
<p class="noindent"><span class="pagebreak" epub:type="pagebreak" title="8" id="page8"></span><b>Proprietà associativa</b>:<a id="ind32"></a><!--<?"associativa, propriet&#x00E0;",4,0,2>--> il prodotto di tre numeri non cambia se a due fattori consecutivi si sostituisce il loro prodotto:</p>
<p class="math"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="20ex" height="2.5ex" viewBox="0 -773.9 8636.3 1047.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMATHI-61"></use><use x="1150" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1655" y="0" xlink:href="#MJMATHI-62"></use><use x="2089" y="0" xlink:href="#MJMAIN-29"></use><use x="2705" y="0" xlink:href="#MJMAIN-22C5"></use><use x="3210" y="0" xlink:href="#MJMATHI-63"></use><use x="3926" y="0" xlink:href="#MJMAIN-3D"></use><use x="4987" y="0" xlink:href="#MJMATHI-61"></use><use x="5743" y="0" xlink:href="#MJMAIN-22C5"></use><use x="6248" y="0" xlink:href="#MJMAIN-28"></use><use x="6642" y="0" xlink:href="#MJMATHI-62"></use><use x="7299" y="0" xlink:href="#MJMAIN-22C5"></use><use x="7804" y="0" xlink:href="#MJMATHI-63"></use><use x="8242" y="0" xlink:href="#MJMAIN-29"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>·</mo><mi>b</mi><mo stretchy="false">)</mo><mo>·</mo><mi>c</mi><mo>=</mo><mi>a</mi><mo>·</mo><mo stretchy="false">(</mo><mi>b</mi><mo>·</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></math></script></span></p>
<p class="noindent">Ad esempio, il prodotto 2 · 3 · 7 si può calcolare in due modi diversi:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -2.5ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="38.833ex" height="6ex" viewBox="0 -1539.2 16734.4 2578.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-32"></use><use x="727" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1232" y="0" xlink:href="#MJMAIN-33"></use><use x="1959" y="0" xlink:href="#MJMAIN-22C5"></use><use x="2464" y="0" xlink:href="#MJMAIN-37"></use><use x="3247" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(4308,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(5085,0)"><g transform="translate(-11,0)"><use x="0" y="725" xlink:href="#MJMAIN-2197"></use><use x="0" y="-766" xlink:href="#MJMAIN-2198"></use></g><g transform="translate(1794,0)"><g transform="translate(0,725)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-32"></use><use x="1121" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1626" y="0" xlink:href="#MJMAIN-33"></use><use x="2131" y="0" xlink:href="#MJMAIN-29"></use><use x="2747" y="0" xlink:href="#MJMAIN-22C5"></use><use x="3252" y="0" xlink:href="#MJMAIN-37"></use><use x="4035" y="0" xlink:href="#MJMAIN-3D"></use><use x="5096" y="0" xlink:href="#MJMAIN-36"></use><use x="5823" y="0" xlink:href="#MJMAIN-22C5"></use><use x="6328" y="0" xlink:href="#MJMAIN-37"></use><use x="7111" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(8172,0)"><use xlink:href="#MJMAIN-34"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g></g><g transform="translate(0,-766)"><use xlink:href="#MJMAIN-32"></use><use x="727" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1232" y="0" xlink:href="#MJMAIN-28"></use><use x="1626" y="0" xlink:href="#MJMAIN-33"></use><use x="2353" y="0" xlink:href="#MJMAIN-22C5"></use><use x="2858" y="0" xlink:href="#MJMAIN-37"></use><use x="3363" y="0" xlink:href="#MJMAIN-29"></use><use x="4035" y="0" xlink:href="#MJMAIN-3D"></use><use x="5096" y="0" xlink:href="#MJMAIN-32"></use><use x="5823" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(6328,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-31"></use></g><use x="7616" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(8677,0)"><use xlink:href="#MJMAIN-34"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>2</mn><mo>⋅</mo><mn>3</mn><mo>⋅</mo><mn>7</mn><mo>=</mo><mtext> </mtext><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mo>↗</mo></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>⋅</mo><mn>3</mn><mo stretchy="false">)</mo><mo>⋅</mo><mn>7</mn><mo>=</mo><mn>6</mn><mo>⋅</mo><mn>7</mn><mo>=</mo><mn color="#00aef0">42</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mo>↘</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><mo>⋅</mo><mo stretchy="false">(</mo><mn>3</mn><mo>⋅</mo><mn>7</mn><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mo>⋅</mo><mn>21</mn><mo>=</mo><mn color="#00aef0">42</mn></mrow></mtd></mtr></mtable></mrow></math></script></p>
</li>
</ul>
<p class="noindent">Grazie alle proprietà commutativa e associativa possiamo calcolare il prodotto di tre o più fattori cambiando l’ordine dei fattori e associandoli a piacimento.</p>
<ul class="blist">
<li>
<p class="noindent"><b>Proprietà distributiva</b><a id="ind33"></a><!--<?"distributiva, propriet&#x00E0;",4,0,2>--></p>
<ul class="blist">
<li>
<p class="noindent"><b>della moltiplicazione rispetto all’addizione</b>: per moltiplicare un numero per una somma si può moltiplicare quel numero per ciascun addendo e sommare i prodotti ottenuti:</p>
<p class="math" id="ch1.eq1"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="23.5ex" height="2.5ex" viewBox="0 -773.9 10109.8 1047.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-61"></use><use x="756" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1261" y="0" xlink:href="#MJMAIN-28"></use><use x="1655" y="0" xlink:href="#MJMATHI-62"></use><use x="2311" y="0" xlink:href="#MJMAIN-2B"></use><use x="3316" y="0" xlink:href="#MJMATHI-63"></use><use x="3754" y="0" xlink:href="#MJMAIN-29"></use><use x="4426" y="0" xlink:href="#MJMAIN-3D"></use><use x="5487" y="0" xlink:href="#MJMATHI-61"></use><use x="6243" y="0" xlink:href="#MJMAIN-22C5"></use><use x="6748" y="0" xlink:href="#MJMATHI-62"></use><use x="7405" y="0" xlink:href="#MJMAIN-2B"></use><use x="8410" y="0" xlink:href="#MJMATHI-61"></use><use x="9166" y="0" xlink:href="#MJMAIN-22C5"></use><use x="9671" y="0" xlink:href="#MJMATHI-63"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mo>⋅</mo><mi>b</mi><mo>+</mo><mi>a</mi><mo>⋅</mo><mi>c</mi></math></script><span class="eqnr">1</span></span></p>
<p class="noindent">Ad esempio:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -5.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="31ex" height="6.833ex" viewBox="0 -773.9 13340.8 2973.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-33"></use><use x="727" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1232" y="0" xlink:href="#MJMAIN-28"></use><g transform="translate(1626,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use><use x="1010" y="0" xlink:href="#MJMAIN-30"></use></g><use x="3363" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(4368,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-35"></use></g><use x="5378" y="0" xlink:href="#MJMAIN-29"></use><g transform="translate(12,-783)"><use x="19" y="0" xlink:href="#MJSZ4-E152"></use><g transform="translate(507.18386243386243,0) scale(4.636772486772487,1)"><use xlink:href="#MJSZ4-E154"></use></g><g transform="translate(2431,0)"><use xlink:href="#MJSZ4-E151"></use><use x="455" y="0" xlink:href="#MJSZ4-E150"></use></g><g transform="translate(3364.628306878307,0) scale(4.636772486772487,1)"><use xlink:href="#MJSZ4-E154"></use></g><use x="5293" y="0" xlink:href="#MJSZ4-E153"></use></g><g transform="translate(397,-1782)"><use transform="scale(0.707)" xlink:href="#MJMAIN-33"></use><g transform="translate(357,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><use transform="scale(0.707)" x="1114" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(988,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1419,0)"><use transform="scale(0.707)" xlink:href="#MJMAIN-31"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="1010" y="0" xlink:href="#MJMAIN-35"></use></g><g transform="translate(2491,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><use transform="scale(0.707)" x="4132" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(3476,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(3907,0)"><use transform="scale(0.707)" xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-37"></use><use transform="scale(0.707)" x="1010" y="0" xlink:href="#MJMAIN-35"></use></g></g><use x="6050" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(7111,0)"><use xlink:href="#MJMAIN-33"></use><use x="727" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(1232,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use><use x="1010" y="0" xlink:href="#MJMAIN-30"></use></g><use x="2969" y="0" xlink:href="#MJMAIN-2B"></use><use x="3974" y="0" xlink:href="#MJMAIN-33"></use><use x="4702" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(5207,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-35"></use></g><g transform="translate(12,-615)"><use x="19" y="0" xlink:href="#MJSZ4-E152"></use><g transform="translate(509.82936507936506,0) scale(5.165873015873016,1)"><use xlink:href="#MJSZ4-E154"></use></g><g transform="translate(2653,0)"><use xlink:href="#MJSZ4-E151"></use><use x="455" y="0" xlink:href="#MJSZ4-E150"></use></g><g transform="translate(3589.4960317460323,0) scale(5.165873015873016,1)"><use xlink:href="#MJSZ4-E154"></use></g><use x="5738" y="0" xlink:href="#MJSZ4-E153"></use></g><g transform="translate(264,-1614)"><use transform="scale(0.707)" xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-30"></use><use transform="scale(0.707)" x="1010" y="0" xlink:href="#MJMAIN-30"></use><g transform="translate(1071,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><use transform="scale(0.707)" x="2124" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(2056,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(2487,0)"><use transform="scale(0.707)" xlink:href="#MJMAIN-37"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-35"></use></g><g transform="translate(3201,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><use transform="scale(0.707)" x="5137" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(4186,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(4617,0)"><use transform="scale(0.707)" xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-37"></use><use transform="scale(0.707)" x="1010" y="0" xlink:href="#MJMAIN-35"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><munder><mrow><mn>3</mn><mo>⋅</mo><mo stretchy="false">(</mo><mn>100</mn><mo>+</mo><mn>25</mn><mo stretchy="false">)</mo></mrow><mo stretchy="true">︸</mo></munder><mrow><mn>3</mn><mtext> </mtext><mo>⋅</mo><mtext> </mtext><mn>125</mn><mtext> </mtext><mo>=</mo><mtext> </mtext><mn>375</mn></mrow></munder><mo>=</mo><munder><munder><mrow><mn>3</mn><mo>⋅</mo><mn>100</mn><mo>+</mo><mn>3</mn><mo>⋅</mo><mn>25</mn></mrow><mo stretchy="true">︸</mo></munder><mrow><mn>300</mn><mtext> </mtext><mo>+</mo><mtext> </mtext><mn>75</mn><mtext> </mtext><mo>=</mo><mtext> </mtext><mn>375</mn></mrow></munder></math></script></p>
</li>
<li>
<p class="noindent"><b>della moltiplicazione rispetto alla sottrazione</b>: per moltiplicare un numero per una differenza si può moltiplicare quel numero per il minuendo e per il sottraendo e poi eseguire la sottrazione tra i prodotti ottenuti:</p>
<p class="math" id="ch1.eq2"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="23.5ex" height="2.5ex" viewBox="0 -773.9 10109.8 1047.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-61"></use><use x="756" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1261" y="0" xlink:href="#MJMAIN-28"></use><use x="1655" y="0" xlink:href="#MJMATHI-62"></use><use x="2311" y="0" xlink:href="#MJMAIN-2212"></use><use x="3316" y="0" xlink:href="#MJMATHI-63"></use><use x="3754" y="0" xlink:href="#MJMAIN-29"></use><use x="4426" y="0" xlink:href="#MJMAIN-3D"></use><use x="5487" y="0" xlink:href="#MJMATHI-61"></use><use x="6243" y="0" xlink:href="#MJMAIN-22C5"></use><use x="6748" y="0" xlink:href="#MJMATHI-62"></use><use x="7405" y="0" xlink:href="#MJMAIN-2212"></use><use x="8410" y="0" xlink:href="#MJMATHI-61"></use><use x="9166" y="0" xlink:href="#MJMAIN-22C5"></use><use x="9671" y="0" xlink:href="#MJMATHI-63"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>b</mi><mo>−</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mo>⋅</mo><mi>b</mi><mo>−</mo><mi>a</mi><mo>⋅</mo><mi>c</mi></math></script><span class="eqnr">2</span></span></p>
<p class="noindent">Ad esempio:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -5.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="26.333ex" height="6.833ex" viewBox="0 -773.9 11320.8 2973.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-33"></use><use x="727" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1232" y="0" xlink:href="#MJMAIN-28"></use><g transform="translate(1626,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="2858" y="0" xlink:href="#MJMAIN-2212"></use><use x="3863" y="0" xlink:href="#MJMAIN-34"></use><use x="4368" y="0" xlink:href="#MJMAIN-29"></use><g transform="translate(12,-783)"><use x="19" y="0" xlink:href="#MJSZ4-E152"></use><g transform="translate(501.17195767195767,0) scale(3.4343915343915343,1)"><use xlink:href="#MJSZ4-E154"></use></g><g transform="translate(1926,0)"><use xlink:href="#MJSZ4-E151"></use><use x="455" y="0" xlink:href="#MJSZ4-E150"></use></g><g transform="translate(2853.6164021164022,0) scale(3.4343915343915343,1)"><use xlink:href="#MJSZ4-E154"></use></g><use x="4283" y="0" xlink:href="#MJSZ4-E153"></use></g><g transform="translate(427,-1782)"><use transform="scale(0.707)" xlink:href="#MJMAIN-33"></use><g transform="translate(357,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><use transform="scale(0.707)" x="1114" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(988,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><use transform="scale(0.707)" x="2007" y="0" xlink:href="#MJMAIN-38"></use><g transform="translate(1776,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><use transform="scale(0.707)" x="3122" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(2761,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(3193,0)"><use transform="scale(0.707)" xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-34"></use></g></g><use x="5040" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(6101,0)"><use xlink:href="#MJMAIN-33"></use><use x="727" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(1232,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="2464" y="0" xlink:href="#MJMAIN-2212"></use><use x="3469" y="0" xlink:href="#MJMAIN-33"></use><use x="4197" y="0" xlink:href="#MJMAIN-22C5"></use><use x="4702" y="0" xlink:href="#MJMAIN-34"></use><g transform="translate(12,-555)"><use x="19" y="0" xlink:href="#MJSZ4-E152"></use><g transform="translate(503.8174603174603,0) scale(3.963492063492064,1)"><use xlink:href="#MJSZ4-E154"></use></g><g transform="translate(2148,0)"><use xlink:href="#MJSZ4-E151"></use><use x="455" y="0" xlink:href="#MJSZ4-E150"></use></g><g transform="translate(3078.484126984127,0) scale(3.963492063492064,1)"><use xlink:href="#MJSZ4-E154"></use></g><use x="4728" y="0" xlink:href="#MJSZ4-E153"></use></g><g transform="translate(116,-1554)"><use transform="scale(0.707)" xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-36"></use><g transform="translate(714,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><use transform="scale(0.707)" x="1619" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(1699,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(2130,0)"><use transform="scale(0.707)" xlink:href="#MJMAIN-31"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-32"></use></g><g transform="translate(2844,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><use transform="scale(0.707)" x="4632" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(3829,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(4260,0)"><use transform="scale(0.707)" xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-34"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><munder><mrow><mn>3</mn><mo>⋅</mo><mo stretchy="false">(</mo><mn>12</mn><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><mo stretchy="true">︸</mo></munder><mrow><mn>3</mn><mtext> </mtext><mo>⋅</mo><mtext> </mtext><mn>8</mn><mtext> </mtext><mo>=</mo><mtext> </mtext><mn>24</mn></mrow></munder><mo>=</mo><munder><munder><mrow><mn>3</mn><mo>⋅</mo><mn>12</mn><mo>−</mo><mn>3</mn><mo>⋅</mo><mn>4</mn></mrow><mo stretchy="true">︸</mo></munder><mrow><mn>36</mn><mtext> </mtext><mo>−</mo><mtext> </mtext><mn>12</mn><mtext> </mtext><mo>=</mo><mtext> </mtext><mn>24</mn></mrow></munder></math></script></p>
</li>
</ul>
</li>
<li>
<p class="noindent"><b>Raccoglimento a fattor comune</b><a id="ind34"></a><!--<?"raccoglimento|a fattor comune",4,0,2>--></p>
<p class="noindent">Le uguaglianze <a href="#ch1.eq1"><span class="red">1</span></a> e <a href="#ch1.eq2"><span class="red">2</span></a>, lette da destra a sinistra, costituiscono la regola del <b>raccoglimento a fattor comune</b>: se in una somma tutti gli addendi hanno un fattore in comune, esso può essere <i>raccolto</i> e moltiplicato per la somma degli altri termini. Si procede analogamente nel caso di una differenza. Ad esempio:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -5.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="55.167ex" height="7.167ex" viewBox="0 -875 23728.7 3075"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(136,0)"><use xlink:href="#MJMAIN-33"></use><use x="727" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1232" y="0" xlink:href="#MJMAIN-37"></use><use x="1959" y="0" xlink:href="#MJMAIN-2B"></use><use x="2964" y="0" xlink:href="#MJMAIN-33"></use><use x="3692" y="0" xlink:href="#MJMAIN-22C5"></use><use x="4197" y="0" xlink:href="#MJMAIN-35"></use><g transform="translate(12,-615)"><use x="19" y="0" xlink:href="#MJSZ4-E152"></use><g transform="translate(500.81150793650795,0) scale(3.362301587301587,1)"><use xlink:href="#MJSZ4-E154"></use></g><g transform="translate(1896,0)"><use xlink:href="#MJSZ4-E151"></use><use x="455" y="0" xlink:href="#MJSZ4-E150"></use></g><g transform="translate(2822.9781746031745,0) scale(3.362301587301587,1)"><use xlink:href="#MJSZ4-E154"></use></g><use x="4223" y="0" xlink:href="#MJSZ4-E153"></use></g></g><g transform="translate(0,-1614)"><use transform="scale(0.707)" xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-31"></use><g transform="translate(714,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><use transform="scale(0.707)" x="1619" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(1699,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(2130,0)"><use transform="scale(0.707)" xlink:href="#MJMAIN-31"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-35"></use></g><g transform="translate(2844,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><use transform="scale(0.707)" x="4632" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(3829,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(4260,0)"><use transform="scale(0.707)" xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-36"></use></g></g><use x="5252" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(6313,0)"><g transform="translate(3,0)"><use xlink:href="#MJMAIN-33"></use><use x="727" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1232" y="0" xlink:href="#MJMAIN-28"></use><use x="1626" y="0" xlink:href="#MJMAIN-37"></use><use x="2353" y="0" xlink:href="#MJMAIN-2B"></use><use x="3358" y="0" xlink:href="#MJMAIN-35"></use><use x="3863" y="0" xlink:href="#MJMAIN-29"></use><g transform="translate(12,-783)"><use x="19" y="0" xlink:href="#MJSZ4-E152"></use><g transform="translate(498.16600529100526,0) scale(2.833201058201058,1)"><use xlink:href="#MJSZ4-E154"></use></g><g transform="translate(1673,0)"><use xlink:href="#MJSZ4-E151"></use><use x="455" y="0" xlink:href="#MJSZ4-E150"></use></g><g transform="translate(2598.1104497354495,0) scale(2.833201058201058,1)"><use xlink:href="#MJSZ4-E154"></use></g><use x="3778" y="0" xlink:href="#MJSZ4-E153"></use></g></g><g transform="translate(0,-1782)"><use transform="scale(0.707)" xlink:href="#MJMAIN-33"></use><g transform="translate(357,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><use transform="scale(0.707)" x="1114" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(988,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1419,0)"><use transform="scale(0.707)" xlink:href="#MJMAIN-31"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-32"></use></g><g transform="translate(2133,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><use transform="scale(0.707)" x="3627" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(3118,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(3550,0)"><use transform="scale(0.707)" xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-36"></use></g></g></g><g transform="translate(10578,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(11797,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(12407,0)"><use xlink:href="#MJMAIN-38"></use><use x="727" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(1232,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="2464" y="0" xlink:href="#MJMAIN-2212"></use><use x="3469" y="0" xlink:href="#MJMAIN-38"></use><use x="4197" y="0" xlink:href="#MJMAIN-22C5"></use><use x="4702" y="0" xlink:href="#MJMAIN-39"></use><g transform="translate(12,-555)"><use x="19" y="0" xlink:href="#MJSZ4-E152"></use><g transform="translate(503.8174603174603,0) scale(3.963492063492064,1)"><use xlink:href="#MJSZ4-E154"></use></g><g transform="translate(2148,0)"><use xlink:href="#MJSZ4-E151"></use><use x="455" y="0" xlink:href="#MJSZ4-E150"></use></g><g transform="translate(3078.484126984127,0) scale(3.963492063492064,1)"><use xlink:href="#MJSZ4-E154"></use></g><use x="4728" y="0" xlink:href="#MJSZ4-E153"></use></g><g transform="translate(116,-1554)"><use transform="scale(0.707)" xlink:href="#MJMAIN-39"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-36"></use><g transform="translate(714,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><use transform="scale(0.707)" x="1619" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(1699,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(2130,0)"><use transform="scale(0.707)" xlink:href="#MJMAIN-37"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-32"></use></g><g transform="translate(2844,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><use transform="scale(0.707)" x="4632" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(3829,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(4260,0)"><use transform="scale(0.707)" xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-34"></use></g></g></g><use x="17893" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(18953,0)"><use xlink:href="#MJMAIN-38"></use><use x="727" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1232" y="0" xlink:href="#MJMAIN-28"></use><g transform="translate(1626,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="2858" y="0" xlink:href="#MJMAIN-2212"></use><use x="3863" y="0" xlink:href="#MJMAIN-39"></use><use x="4368" y="0" xlink:href="#MJMAIN-29"></use><g transform="translate(12,-783)"><use x="19" y="0" xlink:href="#MJSZ4-E152"></use><g transform="translate(501.17195767195767,0) scale(3.4343915343915343,1)"><use xlink:href="#MJSZ4-E154"></use></g><g transform="translate(1926,0)"><use xlink:href="#MJSZ4-E151"></use><use x="455" y="0" xlink:href="#MJSZ4-E150"></use></g><g transform="translate(2853.6164021164022,0) scale(3.4343915343915343,1)"><use xlink:href="#MJSZ4-E154"></use></g><use x="4283" y="0" xlink:href="#MJSZ4-E153"></use></g><g transform="translate(427,-1782)"><use transform="scale(0.707)" xlink:href="#MJMAIN-38"></use><g transform="translate(357,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><use transform="scale(0.707)" x="1114" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(988,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><use transform="scale(0.707)" x="2007" y="0" xlink:href="#MJMAIN-33"></use><g transform="translate(1776,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><use transform="scale(0.707)" x="3122" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(2761,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(3193,0)"><use transform="scale(0.707)" xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-34"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><munder><mrow><mn>3</mn><mo>⋅</mo><mn>7</mn><mo>+</mo><mn>3</mn><mo>⋅</mo><mn>5</mn></mrow><mo stretchy="true">︸</mo></munder><mrow><mn>21</mn><mtext> </mtext><mo>+</mo><mtext> </mtext><mn>15</mn><mtext> </mtext><mo>=</mo><mtext> </mtext><mn>36</mn></mrow></munder><mo>=</mo><munder><munder><mrow><mn>3</mn><mo>⋅</mo><mo stretchy="false">(</mo><mn>7</mn><mo>+</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><mo stretchy="true">︸</mo></munder><mrow><mn>3</mn><mtext> </mtext><mo>⋅</mo><mtext> </mtext><mn>12</mn><mtext> </mtext><mo>=</mo><mtext> </mtext><mn>36</mn></mrow></munder><mtext>  </mtext><mtext> </mtext><munder><munder><mrow><mn>8</mn><mo>⋅</mo><mn>12</mn><mo>−</mo><mn>8</mn><mo>⋅</mo><mn>9</mn></mrow><mo stretchy="true">︸</mo></munder><mrow><mn>96</mn><mtext> </mtext><mo>−</mo><mtext> </mtext><mn>72</mn><mtext> </mtext><mo>=</mo><mtext> </mtext><mn>24</mn></mrow></munder><mo>=</mo><munder><munder><mrow><mn>8</mn><mo>⋅</mo><mo stretchy="false">(</mo><mn>12</mn><mo>−</mo><mn>9</mn><mo stretchy="false">)</mo></mrow><mo stretchy="true">︸</mo></munder><mrow><mn>8</mn><mtext> </mtext><mo>⋅</mo><mtext> </mtext><mn>3</mn><mtext> </mtext><mo>=</mo><mtext> </mtext><mn>24</mn></mrow></munder></math></script></p>
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<p class="noindent"><b>Elemento neutro</b><a id="ind35"></a><!--<?"elemento neutro",4,0,2>--> della moltiplicazione: è il numero 1. Ciò significa che moltiplicando qualsiasi numero per 1 si ottiene il numero dato:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="38.167ex" height="3ex" viewBox="0 -875 16450.8 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMATHI-61"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(756,0)"><use xlink:href="#MJMAIN-22C5"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1261,0)"><use xlink:href="#MJMAIN-31"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(2044,0)"><use xlink:href="#MJMAIN-3D"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(3105,0)"><use xlink:href="#MJMAIN-31"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(3832,0)"><use xlink:href="#MJMAIN-22C5"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(4337,0)"><use xlink:href="#MJMATHI-61"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(5149,0)"><use xlink:href="#MJMAIN-3D"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(6210,0)"><use xlink:href="#MJMATHI-61"></use></g></g></g><g transform="translate(6744,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1219,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1829,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(2439,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="9793" y="0" xlink:href="#MJMAIN-36"></use><use x="10520" y="0" xlink:href="#MJMAIN-22C5"></use><use x="11026" y="0" xlink:href="#MJMAIN-31"></use><use x="11808" y="0" xlink:href="#MJMAIN-3D"></use><use x="12869" y="0" xlink:href="#MJMAIN-31"></use><use x="13596" y="0" xlink:href="#MJMAIN-22C5"></use><use x="14102" y="0" xlink:href="#MJMAIN-36"></use><use x="14884" y="0" xlink:href="#MJMAIN-3D"></use><use x="15945" y="0" xlink:href="#MJMAIN-36"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mstyle color="#00aef0"><mi>a</mi><mo>·</mo><mn>1</mn><mo>=</mo><mn>1</mn><mo>·</mo><mi>a</mi><mo>=</mo><mi>a</mi></mstyle><mi>     </mi><mn>6</mn><mo>·</mo><mn>1</mn><mo>=</mo><mn>1</mn><mo>·</mo><mn>6</mn><mo>=</mo><mn>6</mn></mrow></math></script></p>
</li>
<li>
<p class="noindent"><b>Elemento annullatore</b><a id="ind36"></a><!--<?"annullatore della moltiplicazione, elemento",4,0,2>--> della moltiplicazione: è il numero 0. Questo significa che moltiplicando qualsiasi numero per 0 si ottiene 0:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="38.167ex" height="3ex" viewBox="0 -875 16421.8 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMATHI-61"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(756,0)"><use xlink:href="#MJMAIN-22C5"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1261,0)"><use xlink:href="#MJMAIN-30"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(2044,0)"><use xlink:href="#MJMAIN-3D"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(3105,0)"><use xlink:href="#MJMAIN-30"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(3832,0)"><use xlink:href="#MJMAIN-22C5"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(4337,0)"><use xlink:href="#MJMATHI-61"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(5149,0)"><use xlink:href="#MJMAIN-3D"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(6210,0)"><use xlink:href="#MJMAIN-30"></use></g></g></g><g transform="translate(6715,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1219,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1829,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(2439,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="9764" y="0" xlink:href="#MJMAIN-36"></use><use x="10491" y="0" xlink:href="#MJMAIN-22C5"></use><use x="10997" y="0" xlink:href="#MJMAIN-30"></use><use x="11779" y="0" xlink:href="#MJMAIN-3D"></use><use x="12840" y="0" xlink:href="#MJMAIN-30"></use><use x="13567" y="0" xlink:href="#MJMAIN-22C5"></use><use x="14073" y="0" xlink:href="#MJMAIN-36"></use><use x="14855" y="0" xlink:href="#MJMAIN-3D"></use><use x="15916" y="0" xlink:href="#MJMAIN-30"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mstyle color="#00aef0"><mi>a</mi><mo>·</mo><mn>0</mn><mo>=</mo><mn>0</mn><mo>·</mo><mi>a</mi><mo>=</mo><mn>0</mn></mstyle><mi>     </mi><mn>6</mn><mo>·</mo><mn>0</mn><mo>=</mo><mn>0</mn><mo>·</mo><mn>6</mn><mo>=</mo><mn>0</mn></mrow></math></script></p>
</li>
<li>
<p class="noindent"><b>Legge di annullamento del prodotto</b>.<a id="ind37"></a><!--<?"annullamento del prodotto, legge di",4,0,2>--> Se il prodotto di due numeri è 0, allora almeno uno dei fattori è 0:</p>
<p class="math"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="58.667ex" height="3ex" viewBox="0 -875 25277.2 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-61"></use><use x="756" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1261" y="0" xlink:href="#MJMATHI-62"></use><use x="1973" y="0" xlink:href="#MJMAIN-3D"></use><use x="3034" y="0" xlink:href="#MJMAIN-30"></use><g transform="translate(3539,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="4426" y="0" xlink:href="#MJMAIN-2192"></use><g transform="translate(5709,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="6319" y="0" xlink:href="#MJMATHI-61"></use><use x="7131" y="0" xlink:href="#MJMAIN-3D"></use><use x="8192" y="0" xlink:href="#MJMAIN-30"></use><g transform="translate(8697,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(9473,0)"><use xlink:href="#MJMAINB-6F"></use><use x="580" y="0" xlink:href="#MJMAINB-70"></use><use x="1224" y="0" xlink:href="#MJMAINB-70"></use><use x="1868" y="0" xlink:href="#MJMAINB-75"></use><use x="2512" y="0" xlink:href="#MJMAINB-72"></use><use x="2991" y="0" xlink:href="#MJMAINB-65"></use></g><g transform="translate(13163,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="13773" y="0" xlink:href="#MJMATHI-62"></use><use x="14485" y="0" xlink:href="#MJMAIN-3D"></use><use x="15545" y="0" xlink:href="#MJMAIN-30"></use><g transform="translate(16050,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(16827,0)"><use xlink:href="#MJMAINB-6F"></use><use x="580" y="0" xlink:href="#MJMAINB-70"></use><use x="1224" y="0" xlink:href="#MJMAINB-70"></use><use x="1868" y="0" xlink:href="#MJMAINB-75"></use><use x="2512" y="0" xlink:href="#MJMAINB-72"></use><use x="2991" y="0" xlink:href="#MJMAINB-65"></use></g><g transform="translate(20517,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="21127" y="0" xlink:href="#MJMATHI-61"></use><use x="21938" y="0" xlink:href="#MJMAIN-3D"></use><use x="22999" y="0" xlink:href="#MJMATHI-62"></use><use x="23711" y="0" xlink:href="#MJMAIN-3D"></use><use x="24772" y="0" xlink:href="#MJMAIN-30"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>a</mi><mo>·</mo><mi>b</mi><mo>=</mo><mn>0</mn><mi> </mi><mo>→</mo><mi> </mi><mi>a</mi><mo>=</mo><mn>0</mn><mi> </mi><mi mathvariant="bold">oppure</mi><mi> </mi><mi>b</mi><mo>=</mo><mn>0</mn><mi> </mi><mi mathvariant="bold">oppure</mi><mi> </mi><mi>a</mi><mo>=</mo><mi>b</mi><mo>=</mo><mn>0</mn></mrow></math></script></span></p>
</li>
</ul>
</div></div>
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<div data-page-container="9" id="page-9" class="row chapters-content">
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<div class="small-12 medium-1 columns"><div class="pagenumber">9</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<h3 class="sec_title" id="sec5">
<span class="pagebreak" epub:type="pagebreak" title="9" id="page9"></span>5. Divisione e sue proprietà<a id="ind38"></a><!--<?"divisione|propriet&#x00E0; della",4,0,2>-->
</h3>
<p class="noindent">La <b>divisione</b>,<a id="ind39"></a><!--<?"divisione",4,0,2>--> che si indica con il segno :, è un’operazione che si esegue tra due numeri, considerati nell’ordine, il primo detto <b>dividendo</b> e il secondo, diverso da zero, detto <b>divisore</b>.<a id="ind40"></a><!--<?"divisore",4,0,2>--></p>
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<p class="noindent">Il risultato della divisione si chiama <b>quoziente</b>.</p>
<div class="definition" title_dea="Quoziente tra due numeri naturali" key_dea="quoziente, divisione, numeri naturali, quoziente tra due numeri naturali">
<h4 class="noindent">
<span class="ash">DEFINIZIONE</span> <span class="orangeb">QUOZIENTE TRA DUE NUMERI NATURALI</span>
</h4>
<p class="noindentin">Il quoziente tra due numeri naturali,<a id="ind41"></a><!--<?"quoziente|tra due numeri naturali",4,0,2>--> il secondo dei quali diverso da 0, è quel numero naturale, se esiste, che moltiplicato per il divisore dà come prodotto il dividendo (<a href="#ch1.fg10"><span class="fron">FIGURA 10</span></a>):</p>
<div class="exp_imp" title_dea="Formula del quoziente tra due numeri naturali" key_dea="quoziente, divisione, numeri naturali, formula del quoziente tra due numeri naturali">
<p class="math"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="37.833ex" height="4.333ex" viewBox="0 -1150 16308 1869.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(275,0)"><use xlink:href="#MJMATHI-61"></use><use x="811" y="0" xlink:href="#MJMAIN-3A"></use><use x="1372" y="0" xlink:href="#MJMATHI-62"></use><use x="2084" y="0" xlink:href="#MJMAIN-3D"></use><use x="3145" y="0" xlink:href="#MJMATHI-63"></use><g transform="translate(3583,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(4803,0)"><use xlink:href="#MJMAINB-63"></use><use x="516" y="0" xlink:href="#MJMAINB-6F"></use><use x="1096" y="0" xlink:href="#MJMAINB-6E"></use></g><g transform="translate(6543,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="7152" y="0" xlink:href="#MJMATHI-62"></use><use x="7864" y="0" xlink:href="#MJMAIN-2260"></use><use x="8925" y="0" xlink:href="#MJMAINB-30"></use><g transform="translate(9505,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="10393" y="0" xlink:href="#MJMAIN-2194"></use><g transform="translate(11676,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="12285" y="0" xlink:href="#MJMATHI-61"></use><use x="13097" y="0" xlink:href="#MJMAIN-3D"></use><use x="14158" y="0" xlink:href="#MJMATHI-62"></use><use x="14814" y="0" xlink:href="#MJMAIN-22C5"></use><use x="15319" y="0" xlink:href="#MJMATHI-63"></use></g><line stroke-linecap="square" stroke="black" stroke-width="75" y2="37" y1="37" x1="37" x2="16270" transform="translate(0,1061)"></line><line stroke-linecap="square" stroke="black" stroke-width="75" y1="37" x2="37" x1="37" y2="1804" transform="translate(16232,-706)"></line><line stroke-linecap="square" stroke="black" stroke-width="75" y2="37" y1="37" x1="37" x2="16270" transform="translate(0,-706)"></line><line stroke-linecap="square" stroke="black" stroke-width="75" y1="37" x2="37" x1="37" y2="1804" transform="translate(0,-706)"></line></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><menclose notation="box"><mrow><mi>a</mi><mo>:</mo><mi>b</mi><mo>=</mo><mi>c</mi><mtext>  </mtext><mtext mathvariant="bold">con</mtext><mtext> </mtext><mi>b</mi><mo>≠</mo><mn mathvariant="bold">0</mn><mtext> </mtext><mo>↔</mo><mtext> </mtext><mi>a</mi><mo>=</mo><mi>b</mi><mo>⋅</mo><mi>c</mi></mrow></menclose></math></script></span></p>
</div>
</div>
<p class="noindent">Ad esempio, 24 : 6 = 4 perché 24 = 6 · 4.</p>
<div title_dea="Termini della divisione" key_dea="quoziente, divisione, numeri naturali, termini della divisione, dividendo, divisore">
<div class="figure">
<p class="img" id="ch1.fg10"><img src="images/c01u01f10.jpg" alt="Image"></p>
<p class="figcap">FIGURA 10</p>
</div>
</div>
<p class="noindent">Se la divisione <i>a</i> : <i>b</i> si può eseguire in ℕ, si dice che <i>a</i> è <b>divisibile</b> per <i>b</i> o anche che <i>a</i> è <b>multiplo</b><a id="ind42"></a><!--<?"multiplo",4,0,2>--> di <i>b</i>. Ad esempio, 24 è divisibile per 6 e 24 è multiplo di 6.</p>
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<p class="noindentf">I segni + − · : sono i simboli delle quattro operazioni.</p>
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<p class="noindent">La divisione 19 : 5 invece non si può eseguire in ℕ perché non esiste alcun numero naturale che, moltiplicato per 5, dia 19.</p>
<div class="box">
<p class="noindent"><span class="red"><b>NON È POSSIBILE DIVIDERE UN NUMERO PER ZERO</b></span></p>
<p class="noindent1">La divisione 7 : 0 non si può eseguire. Il quoziente di tale divisione dovrebbe essere un numero <i>q</i> tale che <i>q</i> · 0 = 7. Ma, moltiplicando qualunque numero per zero, si ottiene come prodotto zero. Perciò <i>non esiste</i> il quoziente richiesto. Ciò è vero anche se, al posto di 7, si prende come dividendo qualunque numero diverso da zero.</p>
<p class="noindent1">Se invece si vuole eseguire la divisione 0 : 0, si trovano infiniti numeri <i>q</i> che, moltiplicati per il divisore 0, danno come prodotto il dividendo 0. Perciò pure nel caso in cui il dividendo sia 0, non è possibile determinare in modo unico il quoziente, se anche il divisore è 0.</p>
<p class="noindent">Si conclude che non è mai possibile dividere un numero per zero, cioè <b>la divisione per zero<a id="ind45"></a><!--<?"divisione|per zero",4,0,2>--> non è definita</b>. Pertanto scritture del tipo 7 : 0 o 0 : 0 <b>non hanno significato</b>.</p>
</div>
<p class="noindent"><b>Casi particolari</b></p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -3ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="56.167ex" height="7ex" viewBox="0 -1770.9 24151.7 3041.7"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(167,0)"><g transform="translate(-11,0)"><g transform="translate(0,895)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMATHI-61"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(811,0)"><use xlink:href="#MJMAIN-3A"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1372,0)"><use xlink:href="#MJMAIN-31"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(2155,0)"><use 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xlink:href="#MJMAIN-3A"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1372,0)"><use xlink:href="#MJMATHI-61"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(2184,0)"><use xlink:href="#MJMAIN-3D"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(3245,0)"><use xlink:href="#MJMAIN-31"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(3750,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g fill="#00aef0" stroke="#00aef0" transform="translate(4360,0)"><use xlink:href="#MJMAINB-73"></use><use x="459" y="0" xlink:href="#MJMAINB-65"></use><g transform="translate(991,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="bold" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g fill="#00aef0" stroke="#00aef0" transform="translate(5961,0)"><use xlink:href="#MJMATHI-61"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(6772,0)"><use xlink:href="#MJMAIN-2260"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(7833,0)"><use xlink:href="#MJMAIN-30"></use></g></g></g></g></g><g transform="translate(0,-827)"><use xlink:href="#MJMAIN-36"></use><use x="782" y="0" xlink:href="#MJMAIN-3A"></use><use x="1343" y="0" xlink:href="#MJMAIN-36"></use><use x="2126" y="0" xlink:href="#MJMAIN-3D"></use><use x="3187" y="0" xlink:href="#MJMAIN-31"></use></g></g><g transform="translate(14288,0)"><g transform="translate(0,895)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g 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xlink:href="#MJMAIN-30"></use><use x="1392" y="0" xlink:href="#MJMAIN-3A"></use><use x="1953" y="0" xlink:href="#MJMAIN-36"></use><use x="2736" y="0" xlink:href="#MJMAIN-3D"></use><use x="3797" y="0" xlink:href="#MJMAIN-30"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mstyle color="#00aef0"><mrow><mi>a</mi><mo>:</mo><mn>1</mn><mo>=</mo><mi>a</mi><mtext> </mtext></mrow></mstyle></mtd><mtd columnalign="left"><mstyle color="#00aef0"><mrow><mi>a</mi><mo>:</mo><mi>a</mi><mo>=</mo><mn>1</mn><mtext> </mtext><mtext mathvariant="bold">se </mtext><mi>a</mi><mo>≠</mo><mn>0</mn></mrow></mstyle></mtd><mtd columnalign="left"><mstyle color="#00aef0"><mrow><mtext> </mtext><mn>0</mn><mo>:</mo><mi>a</mi><mo>=</mo><mn>0</mn><mtext> </mtext><mtext> </mtext><mtext mathvariant="bold">se </mtext><mi>a</mi><mo>≠</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mn>6</mn><mo>:</mo><mn>1</mn><mo>=</mo><mn>6</mn></mrow></mtd><mtd columnalign="left"><mrow><mn>6</mn><mo>:</mo><mn>6</mn><mo>=</mo><mn>1</mn></mrow></mtd><mtd columnalign="left"><mrow><mtext> </mtext><mn>0</mn><mo>:</mo><mn>6</mn><mo>=</mo><mn>0</mn></mrow></mtd></mtr></mtable></math></script></p>
<p class="noindent">La divisione gode delle seguenti proprietà, che si possono applicare solo se è possibile eseguire le divisioni indicate.</p>
<ul class="blist">
<li>
<p class="noindent"><b>Proprietà invariantiva</b>:<a id="ind46"></a><!--<?"invariantiva, propriet&#x00E0;",4,0,2>--> se si moltiplicano o si dividono sia il dividendo sia il divisore per uno stesso numero diverso da zero, il quoziente non cambia:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -3ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="65ex" height="7ex" viewBox="0 -1770.9 27951.5 3041.7"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(167,0)"><g transform="translate(-11,0)"><g transform="translate(0,895)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMATHI-61"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(811,0)"><use xlink:href="#MJMAIN-3A"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1372,0)"><use xlink:href="#MJMATHI-62"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(2084,0)"><use 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stretchy="false">)</mo><mo>=</mo><mn>130</mn><mo>:</mo><mn>10</mn><mo>=</mo><mn>13</mn></mrow></mtd></mtr><mtr><mtd><mstyle color="#00aef0"><mrow><mi>a</mi><mo>:</mo><mi>b</mi><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo>:</mo><mi>c</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><mi>b</mi><mo>:</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mrow><mtext> </mtext><mn>72</mn><mo>:</mo><mn>12</mn><mo>=</mo><mo stretchy="false">(</mo><mn>72</mn><mo>:</mo><mn>2</mn><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><mn>12</mn><mo>:</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mn>36</mn><mo>:</mo><mn>6</mn><mo>=</mo><mn>6</mn></mrow></mtd></mtr></mtable></math></script></p>
</li>
</ul>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"><div class="sidebox">
<p class="noindentf">Se la divisione tra due numeri naturali si può eseguire in ℕ, si parla anche di <b>divisione esatta</b><a id="ind43"></a><!--<?"divisione|esatta",4,0,2>--> e il quoziente è anche chiamato <b>quoto</b>.<a id="ind44"></a><!--<?"quoto",4,0,2>--></p>
</div></div>
</div>
</div>
<div data-page-container="10" id="page-10" class="row chapters-content">
<div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">10</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<ul class="blist">
<li>
<p class="noindent"><span class="pagebreak" epub:type="pagebreak" title="10" id="page10"></span><b>Proprietà distributiva</b><a id="ind47"></a><!--<?"distributiva, propriet&#x00E0;",4,0,2>--></p>
<ul class="blist">
<li>
<p class="noindent"><b>della divisione rispetto all’addizione</b>: per dividere una somma per un numero si può dividere ciascun addendo per quel numero e quindi sommare i quozienti:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -2.5ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="42.667ex" height="6ex" viewBox="0 -1539.2 18395.2 2578.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(167,0)"><g transform="translate(-11,0)"><g transform="translate(0,725)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-28"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(394,0)"><use xlink:href="#MJMATHI-61"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1150,0)"><use xlink:href="#MJMAIN-2B"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(2155,0)"><use xlink:href="#MJMATHI-62"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(2589,0)"><use xlink:href="#MJMAIN-29"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(3261,0)"><use xlink:href="#MJMAIN-3A"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(3822,0)"><use xlink:href="#MJMATHI-63"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(4537,0)"><use xlink:href="#MJMAIN-3D"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(5598,0)"><use xlink:href="#MJMATHI-61"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(6410,0)"><use xlink:href="#MJMAIN-3A"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(6971,0)"><use xlink:href="#MJMATHI-63"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(7631,0)"><use xlink:href="#MJMAIN-2B"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(8636,0)"><use xlink:href="#MJMATHI-62"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(9348,0)"><use xlink:href="#MJMAIN-3A"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(9909,0)"><use xlink:href="#MJMATHI-63"></use></g></g></g></g></g><g transform="translate(0,-766)"><use xlink:href="#MJMAIN-28"></use><g transform="translate(394,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="1626" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(2631,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-35"></use></g><use x="3641" y="0" xlink:href="#MJMAIN-29"></use><use x="4313" y="0" xlink:href="#MJMAIN-3A"></use><use x="4873" y="0" xlink:href="#MJMAIN-33"></use><use x="5656" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(6717,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="8005" y="0" xlink:href="#MJMAIN-3A"></use><use x="8566" y="0" xlink:href="#MJMAIN-33"></use><use x="9293" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(10298,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-35"></use></g><use x="11586" y="0" xlink:href="#MJMAIN-3A"></use><use x="12147" y="0" xlink:href="#MJMAIN-33"></use><use x="12929" y="0" xlink:href="#MJMAIN-3D"></use><use x="13990" y="0" xlink:href="#MJMAIN-34"></use><use x="14717" y="0" xlink:href="#MJMAIN-2B"></use><use x="15723" y="0" xlink:href="#MJMAIN-35"></use><use x="16505" y="0" xlink:href="#MJMAIN-3D"></use><use x="17566" y="0" xlink:href="#MJMAIN-39"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mstyle color="#00aef0"><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>:</mo><mi>c</mi><mo>=</mo><mi>a</mi><mo>:</mo><mi>c</mi><mo>+</mo><mi>b</mi><mo>:</mo><mi>c</mi></mrow></mstyle></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mn>12</mn><mo>+</mo><mn>15</mn><mo stretchy="false">)</mo><mo>:</mo><mn>3</mn><mo>=</mo><mn>12</mn><mo>:</mo><mn>3</mn><mo>+</mo><mn>15</mn><mo>:</mo><mn>3</mn><mo>=</mo><mn>4</mn><mo>+</mo><mn>5</mn><mo>=</mo><mn>9</mn></mrow></mtd></mtr></mtable></mrow></math></script></p>
</li>
<li>
<p class="noindent"><b>della divisione rispetto alla sottrazione</b>: per dividere una differenza per un numero si possono dividere sia il minuendo sia il sottraendo per quel numero e quindi eseguire la sottrazione tra i quozienti:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -2.5ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="42.667ex" height="6ex" viewBox="0 -1539.2 18395.2 2578.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(167,0)"><g transform="translate(-11,0)"><g transform="translate(0,725)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-28"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(394,0)"><use xlink:href="#MJMATHI-61"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1150,0)"><use xlink:href="#MJMAIN-2212"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(2155,0)"><use xlink:href="#MJMATHI-62"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(2589,0)"><use xlink:href="#MJMAIN-29"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(3261,0)"><use xlink:href="#MJMAIN-3A"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(3822,0)"><use xlink:href="#MJMATHI-63"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(4537,0)"><use xlink:href="#MJMAIN-3D"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(5598,0)"><use xlink:href="#MJMATHI-61"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(6410,0)"><use xlink:href="#MJMAIN-3A"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(6971,0)"><use xlink:href="#MJMATHI-63"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(7631,0)"><use xlink:href="#MJMAIN-2212"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(8636,0)"><use xlink:href="#MJMATHI-62"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(9348,0)"><use xlink:href="#MJMAIN-3A"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(9909,0)"><use xlink:href="#MJMATHI-63"></use></g></g></g></g></g><g transform="translate(0,-766)"><use xlink:href="#MJMAIN-28"></use><g transform="translate(394,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-34"></use></g><use x="1626" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(2631,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-38"></use></g><use x="3641" y="0" xlink:href="#MJMAIN-29"></use><use x="4313" y="0" xlink:href="#MJMAIN-3A"></use><use x="4873" y="0" xlink:href="#MJMAIN-36"></use><use x="5656" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(6717,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-34"></use></g><use x="8005" y="0" xlink:href="#MJMAIN-3A"></use><use x="8566" y="0" xlink:href="#MJMAIN-36"></use><use x="9293" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(10298,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-38"></use></g><use x="11586" y="0" xlink:href="#MJMAIN-3A"></use><use x="12147" y="0" xlink:href="#MJMAIN-36"></use><use x="12929" y="0" xlink:href="#MJMAIN-3D"></use><use x="13990" y="0" xlink:href="#MJMAIN-34"></use><use x="14717" y="0" xlink:href="#MJMAIN-2212"></use><use x="15723" y="0" xlink:href="#MJMAIN-33"></use><use x="16505" y="0" xlink:href="#MJMAIN-3D"></use><use x="17566" y="0" xlink:href="#MJMAIN-31"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mstyle color="#00aef0"><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>−</mo><mi>b</mi><mo stretchy="false">)</mo><mo>:</mo><mi>c</mi><mo>=</mo><mi>a</mi><mo>:</mo><mi>c</mi><mo>−</mo><mi>b</mi><mo>:</mo><mi>c</mi></mrow></mstyle></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mn>24</mn><mo>−</mo><mn>18</mn><mo stretchy="false">)</mo><mo>:</mo><mn>6</mn><mo>=</mo><mn>24</mn><mo>:</mo><mn>6</mn><mo>−</mo><mn>18</mn><mo>:</mo><mn>6</mn><mo>=</mo><mn>4</mn><mo>−</mo><mn>3</mn><mo>=</mo><mn>1</mn></mrow></mtd></mtr></mtable></mrow></math></script></p>
</li>
</ul>
</li>
</ul>
<p class="noindent">La proprietà distributiva della divisione può essere applicata per eseguire più rapidamente alcune divisioni:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="58.833ex" height="2.5ex" viewBox="0 -773.9 25298.8 1047.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-36"></use><use x="505" y="0" xlink:href="#MJMAIN-31"></use><use x="1010" y="0" xlink:href="#MJMAIN-32"></use><use x="1792" y="0" xlink:href="#MJMAIN-3A"></use><use x="2353" y="0" xlink:href="#MJMAIN-36"></use><use x="3136" y="0" xlink:href="#MJMAIN-3D"></use><use x="4197" y="0" xlink:href="#MJMAIN-28"></use><g transform="translate(4591,0)"><use xlink:href="#MJMAIN-36"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use><use x="1010" y="0" xlink:href="#MJMAIN-30"></use></g><use x="6328" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(7333,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="8343" y="0" xlink:href="#MJMAIN-29"></use><use x="9015" y="0" xlink:href="#MJMAIN-3A"></use><use x="9576" y="0" xlink:href="#MJMAIN-36"></use><use x="10358" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(11419,0)"><use xlink:href="#MJMAIN-36"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use><use x="1010" y="0" xlink:href="#MJMAIN-30"></use></g><use x="13212" y="0" xlink:href="#MJMAIN-3A"></use><use x="13773" y="0" xlink:href="#MJMAIN-36"></use><use x="14500" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(15505,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="16793" y="0" xlink:href="#MJMAIN-3A"></use><use x="17354" y="0" xlink:href="#MJMAIN-36"></use><use x="18136" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(19197,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use><use x="1010" y="0" xlink:href="#MJMAIN-30"></use></g><use x="20934" y="0" xlink:href="#MJMAIN-2B"></use><use x="21940" y="0" xlink:href="#MJMAIN-32"></use><use x="22722" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(23783,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use><use x="1010" y="0" xlink:href="#MJMAIN-32"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>612</mn><mo>:</mo><mn>6</mn><mo>=</mo><mo stretchy="false">(</mo><mn>600</mn><mo>+</mo><mn>12</mn><mo stretchy="false">)</mo><mo>:</mo><mn>6</mn><mo>=</mo><mn>600</mn><mo>:</mo><mn>6</mn><mo>+</mo><mn>12</mn><mo>:</mo><mn>6</mn><mo>=</mo><mn>100</mn><mo>+</mo><mn>2</mn><mo>=</mo><mn>102</mn></mrow></math></script></p>
<p class="noindent"><b>Non</b> esistono proprietà distributive per dividere un numero per una somma o per una differenza:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="48.333ex" height="3ex" viewBox="0 -875 20816.1 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-61"></use><use x="811" y="0" xlink:href="#MJMAIN-3A"></use><use x="1372" y="0" xlink:href="#MJMAIN-28"></use><use x="1766" y="0" xlink:href="#MJMATHI-62"></use><use x="2422" y="0" xlink:href="#MJMAIN-2B"></use><use x="3428" y="0" xlink:href="#MJMATHI-63"></use><use x="3866" y="0" xlink:href="#MJMAIN-29"></use><g transform="translate(4260,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(4869,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(5479,0)"><use xlink:href="#MJMAINB-6E"></use><use x="644" y="0" xlink:href="#MJMAINB-6F"></use><use x="1224" y="0" xlink:href="#MJMAINB-6E"></use><g transform="translate(1868,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="bold" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(2477,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="bold" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)">è</text></g></g><g transform="translate(8567,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(9177,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(9787,0)"><use xlink:href="#MJMAINB-75"></use><use x="644" y="0" xlink:href="#MJMAINB-67"></use><use x="1224" y="0" xlink:href="#MJMAINB-75"></use><use x="1868" y="0" xlink:href="#MJMAINB-61"></use><use x="2432" y="0" xlink:href="#MJMAINB-6C"></use><use x="2756" y="0" xlink:href="#MJMAINB-65"></use></g><g transform="translate(13075,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(13685,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(14295,0)"><use xlink:href="#MJMAIN-61"></use><use x="505" y="0" xlink:href="#MJMAIN-64"></use></g><g transform="translate(15361,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="15971" y="0" xlink:href="#MJMATHI-61"></use><use x="16783" y="0" xlink:href="#MJMAIN-3A"></use><use x="17344" y="0" xlink:href="#MJMATHI-62"></use><use x="18000" y="0" xlink:href="#MJMAIN-2B"></use><use x="19005" y="0" xlink:href="#MJMATHI-61"></use><use x="19817" y="0" xlink:href="#MJMAIN-3A"></use><use x="20378" y="0" xlink:href="#MJMATHI-63"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>a</mi><mo>:</mo><mo stretchy="false">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo stretchy="false">)</mo><mtext> </mtext><mtext> </mtext><mstyle mathvariant="bold" mathsize="normal"><mi>n</mi><mi>o</mi><mi>n</mi><mi> </mi><mo>è</mo></mstyle><mtext> </mtext><mtext> </mtext><mstyle mathvariant="bold" mathsize="normal"><mi>u</mi><mi>g</mi><mi>u</mi><mi>a</mi><mi>l</mi><mi>e</mi></mstyle><mtext> </mtext><mtext> </mtext><mtext>ad</mtext><mtext> </mtext><mi>a</mi><mo>:</mo><mi>b</mi><mo>+</mo><mi>a</mi><mo>:</mo><mi>c</mi></mrow></math></script></p>
<p class="noindent">La divisione <b>non</b> gode della <i>proprietà commutativa</i> (15 : 3 = 5 ma 3 : 15 non si può eseguire nell’insieme ℕ) e <b>non</b> ammette <i>elemento neutro</i> (3 : 1 = 3 ma 1 : 3 non è calcolabile in ℕ).</p>
<p class="noindent">La divisione non gode neppure della <i>proprietà associativa.</i> Ad esempio, per calcolare 48 : 12 : 2 è necessario <b>eseguire le divisioni nell’ordine indicato</b>:</p>
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<div class="small-12 medium-8 columns"><div class="row">
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<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -3.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="14ex" height="5.167ex" viewBox="0 -700.9 6045.7 2252.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-34"></use><use x="505" y="0" xlink:href="#MJMAIN-38"></use><use x="1287" y="0" xlink:href="#MJMAIN-3A"></use><g transform="translate(1848,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><g transform="translate(12,-555)"><use x="19" y="0" xlink:href="#MJSZ4-E152"></use><g transform="translate(489.8366402116402,0) scale(1.1673280423280425,1)"><use xlink:href="#MJSZ4-E154"></use></g><g transform="translate(974,0)"><use xlink:href="#MJSZ4-E151"></use><use x="455" y="0" xlink:href="#MJSZ4-E150"></use></g><g transform="translate(1890.1144179894181,0) scale(1.1673280423280425,1)"><use xlink:href="#MJSZ4-E154"></use></g><use x="2379" y="0" xlink:href="#MJSZ4-E153"></use></g><use transform="scale(0.707)" x="1768" y="-2023" xlink:href="#MJMAIN-34"></use><use x="3136" y="0" xlink:href="#MJMAIN-3A"></use><use x="3697" y="0" xlink:href="#MJMAIN-32"></use><use x="4479" y="0" xlink:href="#MJMAIN-3D"></use><use x="5540" y="0" xlink:href="#MJMAIN-32"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><munder><munder><mrow><mn>48</mn><mo>:</mo><mn>12</mn></mrow><mo stretchy="true">︸</mo></munder><mn>4</mn></munder><mo>:</mo><mn>2</mn><mo>=</mo><mn>2</mn></mrow></math></script></p>
<h4 class="h4">Divisione approssimata (divisione con resto)<a id="ind48"></a><!--<?"resto|divisione con",4,0,2>-->
</h4>
<p class="noindent1">Se il dividendo non è multiplo del divisore, la divisione esatta non si può eseguire. Si può ricorrere allora alla <b>divisione approssimata</b><a id="ind49"></a><!--<?"divisione|approssimata",4,0,2>--> che associa al dividendo e al divisore due numeri naturali, detti rispettivamente <i>quoziente</i> e <i>resto</i>.</p>
<ul class="blist">
<li><p class="noindent">Il <b>quoziente</b><a id="ind50"></a><!--<?"invariantiva, propriet&#x00E0;",4,0,2>--> della divisione approssimata è il più grande numero naturale che, moltiplicato per il divisore, dà un prodotto minore o uguale al dividendo.</p></li>
<li><p class="noindent">Il <b>resto</b> è la differenza tra il dividendo e tale prodotto. Il resto risulta sempre minore del divisore.</p></li>
</ul>
<p class="noindent">In simboli</p>
<p class="math"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="53.5ex" height="3ex" viewBox="0 -875 23001.7 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-61"></use><use x="811" y="0" xlink:href="#MJMAIN-3A"></use><use x="1372" y="0" xlink:href="#MJMATHI-62"></use><use x="2084" y="0" xlink:href="#MJMAIN-3D"></use><use x="3145" y="0" xlink:href="#MJMATHI-71"></use><g transform="translate(3610,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(4220,0)"><use xlink:href="#MJMAINB-63"></use><use x="516" y="0" xlink:href="#MJMAINB-6F"></use><use x="1096" y="0" xlink:href="#MJMAINB-6E"></use></g><g transform="translate(5960,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(6570,0)"><use xlink:href="#MJMAINB-72"></use><use x="479" y="0" xlink:href="#MJMAINB-65"></use><use x="1011" y="0" xlink:href="#MJMAINB-73"></use><use x="1470" y="0" xlink:href="#MJMAINB-74"></use><use x="1922" y="0" xlink:href="#MJMAINB-6F"></use></g><g transform="translate(9072,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="9681" y="0" xlink:href="#MJMATHI-72"></use><g transform="translate(10137,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="11025" y="0" xlink:href="#MJMAIN-2194"></use><g transform="translate(12308,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="12918" y="0" xlink:href="#MJMATHI-72"></use><use x="13652" y="0" xlink:href="#MJMAIN-3D"></use><use x="14712" y="0" xlink:href="#MJMATHI-61"></use><use x="15469" y="0" xlink:href="#MJMAIN-2212"></use><use x="16474" y="0" xlink:href="#MJMATHI-62"></use><use x="17130" y="0" xlink:href="#MJMAIN-22C5"></use><use x="17635" y="0" xlink:href="#MJMATHI-71"></use><g transform="translate(18100,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="19320" y="0" xlink:href="#MJMAIN-28"></use><use x="19714" y="0" xlink:href="#MJMATHI-72"></use><g transform="translate(20170,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><use x="609" y="0" xlink:href="#MJMAIN-3C"></use></g><g transform="translate(21563,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="22173" y="0" xlink:href="#MJMATHI-62"></use><use x="22607" y="0" xlink:href="#MJMAIN-29"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>a</mi><mo>:</mo><mi>b</mi><mo>=</mo><mi>q</mi><mtext> </mtext><mstyle mathvariant="bold" mathsize="normal"><mi>c</mi><mi>o</mi><mi>n</mi></mstyle><mtext> </mtext><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi><mi>e</mi><mi>s</mi><mi>t</mi><mi>o</mi></mstyle><mtext> </mtext><mi>r</mi><mtext> </mtext><mo>↔</mo><mtext> </mtext><mi>r</mi><mo>=</mo><mi>a</mi><mo>−</mo><mi>b</mi><mo>⋅</mo><mi>q</mi><mtext>  </mtext><mo stretchy="false">(</mo><mi>r</mi><mtext> &lt;</mtext><mtext> </mtext><mi>b</mi><mo stretchy="false">)</mo></mrow></math></script></span></p>
<p class="noindent">o anche</p>
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<p class="noindentf">Se si eseguisse prima la seconda divisione 12 : 2 = 6 si otterrebbe il risultato 8 che è errato.</p>
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<div class="exp_imp" title_dea="Formula della divisione approssimata (o con resto)" key_dea="divisione approssimata, divisione con resto, resto, formula della divisione approssimata (o con resto)">
<p class="math"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="54.667ex" height="4.333ex" viewBox="0 -1150 23551.7 1869.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(275,0)"><use xlink:href="#MJMATHI-61"></use><use x="811" y="0" xlink:href="#MJMAIN-3A"></use><use x="1372" y="0" xlink:href="#MJMATHI-62"></use><use x="2084" y="0" xlink:href="#MJMAIN-3D"></use><use x="3145" y="0" xlink:href="#MJMATHI-71"></use><g transform="translate(3610,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(4220,0)"><use 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mathsize="normal"><mi>r</mi><mi>e</mi><mi>s</mi><mi>t</mi><mi>o</mi></mstyle><mtext> </mtext><mi>r</mi><mtext> </mtext><mo>↔</mo><mtext> </mtext><mi>a</mi><mo>=</mo><mi>b</mi><mo>⋅</mo><mi>q</mi><mo>+</mo><mi>r</mi><mtext>  </mtext><mo stretchy="false">(</mo><mi>r</mi><mtext> &lt;</mtext><mtext> </mtext><mi>b</mi><mo stretchy="false">)</mo></mrow></menclose></mrow></math></script></span></p>
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<p class="noindentf">Se <i>r</i> = 0, allora l’uguaglianza <i>a</i> = <i>b</i> · <i>q</i> + <i>r</i> diviene</p>
<p class="math"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.5ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="8.167ex" height="2.167ex" viewBox="0 -717.9 3499 935.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-61"></use><use x="811" y="0" xlink:href="#MJMAIN-3D"></use><use x="1872" y="0" xlink:href="#MJMATHI-62"></use><use x="2528" y="0" xlink:href="#MJMAIN-22C5"></use><use x="3034" y="0" xlink:href="#MJMATHI-71"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>a</mi><mo>=</mo><mi>b</mi><mo>⋅</mo><mi>q</mi></mrow></math></script></span></p>
<p class="noindentf">e la divisione risulta esatta.</p>
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<div class="small-12 medium-1 columns"><div class="pagenumber">11</div></div>
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<span class="pagebreak" epub:type="pagebreak" title="11" id="page11"></span><span class="gep"><span class="sgr">ESEMPIO</span></span>
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<p class="noindent">Calcoliamo 19 : 5. Poiché non esiste alcun numero che moltiplicato per 5 dia 19, la divisione esatta non si può eseguire. Per calcolare il quoziente della divisione approssimata osserviamo che</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -3ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="48.833ex" height="7ex" viewBox="0 -1770.9 21057 3041.7"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(167,0)"><g transform="translate(-11,0)"><g transform="translate(0,895)"><use xlink:href="#MJMAIN-35"></use><use x="727" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1232" y="0" xlink:href="#MJMAIN-30"></use><use x="2015" y="0" xlink:href="#MJMAIN-3D"></use><use x="3076" y="0" xlink:href="#MJMAIN-30"></use><use x="3858" y="0" xlink:href="#MJMAIN-3C"></use><g transform="translate(4919,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-39"></use></g><g transform="translate(5929,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="7149" y="0" xlink:href="#MJMAIN-35"></use><use x="7876" y="0" xlink:href="#MJMAIN-22C5"></use><use x="8381" y="0" xlink:href="#MJMAIN-31"></use><use x="9164" y="0" xlink:href="#MJMAIN-3D"></use><use x="10225" y="0" xlink:href="#MJMAIN-35"></use><use x="11008" y="0" xlink:href="#MJMAIN-3C"></use><g transform="translate(12069,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-39"></use></g><g transform="translate(13079,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="14298" y="0" xlink:href="#MJMAIN-35"></use><use x="15026" y="0" xlink:href="#MJMAIN-22C5"></use><use x="15531" y="0" xlink:href="#MJMAIN-32"></use><use x="16314" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(17374,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use></g><use x="18662" y="0" xlink:href="#MJMAIN-3C"></use><g transform="translate(19723,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-39"></use></g></g><g transform="translate(3322,-827)"><use xlink:href="#MJMAIN-35"></use><use x="727" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1232" y="0" xlink:href="#MJMAIN-33"></use><use x="2015" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(3076,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-35"></use></g><use x="4363" y="0" xlink:href="#MJMAIN-3C"></use><g transform="translate(5424,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-39"></use></g><g transform="translate(6434,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="7654" y="0" xlink:href="#MJMAIN-35"></use><use x="8381" y="0" xlink:href="#MJMAIN-22C5"></use><use x="8886" y="0" xlink:href="#MJMAIN-34"></use><use x="9669" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(10730,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use></g><use x="12018" y="0" xlink:href="#MJMAIN-3E"></use><g transform="translate(13079,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-39"></use></g></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd><mrow><mn>5</mn><mo>⋅</mo><mn>0</mn><mo>=</mo><mn>0</mn><mo>&lt;</mo><mn>19</mn><mtext>  </mtext><mn>5</mn><mo>⋅</mo><mn>1</mn><mo>=</mo><mn>5</mn><mo>&lt;</mo><mn>19</mn><mtext>  </mtext><mn>5</mn><mo>⋅</mo><mn>2</mn><mo>=</mo><mn>10</mn><mo>&lt;</mo><mn>19</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>5</mn><mo>⋅</mo><mn>3</mn><mo>=</mo><mn>15</mn><mo>&lt;</mo><mn>19</mn><mtext>  </mtext><mn>5</mn><mo>⋅</mo><mn>4</mn><mo>=</mo><mn>20</mn><mo>&gt;</mo><mn>19</mn></mrow></mtd></mtr></mtable></mrow></math></script></p>
<p class="noindent">Dunque 3 è il più grande numero naturale che, moltiplicato per il divisore 5, dà un prodotto minore o uguale al dividendo 19.</p>
<p class="noindent">Quindi risulta</p>
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<p class="noindent"><b>Proprietà invariantiva della divisione approssimata</b>: se si moltiplicano o si dividono sia il dividendo sia il divisore per uno stesso numero <i>diverso da zero</i> il quoziente non cambia, mentre il resto risulta moltiplicato o diviso per il numero dato:</p>
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stretchy="false">(</mo><mi>a</mi><mtext> </mtext><mo>:</mo><mtext> </mtext><mi>c</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><mi>b</mi><mtext> </mtext><mo>:</mo><mtext> </mtext><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mi>q</mi><mtext> </mtext><mstyle mathvariant="bold" mathsize="normal"><mi>c</mi><mi>o</mi><mi>n</mi></mstyle><mtext> </mtext><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi><mi>e</mi><mi>s</mi><mi>t</mi><mi>o</mi></mstyle><mtext> </mtext><mi>r</mi><mtext> </mtext><mo>:</mo><mtext> </mtext><mi>c</mi></mrow></mtd></mtr></mtable></mrow></math></script></span></p>
<p class="noindent">Ad esempio:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -5.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="66.667ex" height="12.333ex" viewBox="0 -2889.2 28727.1 5278.3"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-33"></use><use x="505" y="0" xlink:href="#MJMAIN-34"></use><use x="1287" y="0" xlink:href="#MJMAIN-3A"></use><use x="1848" y="0" xlink:href="#MJMAIN-36"></use><use x="2631" y="0" xlink:href="#MJMAIN-3D"></use><use x="3692" y="0" xlink:href="#MJMAIN-35"></use><g transform="translate(4197,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><use x="609" y="0" xlink:href="#MJMAIN-63"></use><use x="1058" y="0" xlink:href="#MJMAIN-6F"></use><use x="1563" y="0" xlink:href="#MJMAIN-6E"></use><g transform="translate(2124,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="2734" y="0" xlink:href="#MJMAIN-72"></use><use x="3131" y="0" xlink:href="#MJMAIN-65"></use><use x="3580" y="0" xlink:href="#MJMAIN-73"></use><use x="3979" y="0" xlink:href="#MJMAIN-74"></use><use x="4373" y="0" xlink:href="#MJMAIN-6F"></use><g transform="translate(4878,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="9685" y="0" xlink:href="#MJMAIN-34"></use><g transform="translate(10190,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: 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xlink:href="#MJMAIN-3A"></use><use x="1343" y="0" xlink:href="#MJMAIN-32"></use><g transform="translate(12,-555)"><use x="19" y="0" xlink:href="#MJSZ4-E152"></use><g transform="translate(469,0)"><use xlink:href="#MJSZ4-E151"></use><use x="455" y="0" xlink:href="#MJSZ4-E150"></use></g><use x="1369" y="0" xlink:href="#MJSZ4-E153"></use></g><use transform="scale(0.707)" x="1054" y="-2011" xlink:href="#MJMAIN-33"></use></g><use x="6222" y="0" xlink:href="#MJMAIN-29"></use><use x="6894" y="0" xlink:href="#MJMAIN-3D"></use><use x="7955" y="0" xlink:href="#MJMAIN-35"></use><g transform="translate(8460,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><use x="609" y="0" xlink:href="#MJMAIN-63"></use><use x="1058" y="0" xlink:href="#MJMAIN-6F"></use><use x="1563" y="0" xlink:href="#MJMAIN-6E"></use><g transform="translate(2124,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="2734" y="0" xlink:href="#MJMAIN-72"></use><use x="3131" y="0" xlink:href="#MJMAIN-65"></use><use x="3580" y="0" xlink:href="#MJMAIN-73"></use><use x="3979" y="0" xlink:href="#MJMAIN-74"></use><use x="4373" y="0" xlink:href="#MJMAIN-6F"></use><g transform="translate(4878,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(13949,0)"><use xlink:href="#MJMAIN-34"></use><use x="782" y="0" xlink:href="#MJMAIN-3A"></use><use x="1343" y="0" xlink:href="#MJMAIN-32"></use><g transform="translate(12,-533)"><use x="19" y="0" xlink:href="#MJSZ4-E152"></use><g transform="translate(469,0)"><use xlink:href="#MJSZ4-E151"></use><use x="455" y="0" xlink:href="#MJSZ4-E150"></use></g><use x="1369" y="0" xlink:href="#MJSZ4-E153"></use></g><use transform="scale(0.707)" x="1054" y="-1981" xlink:href="#MJMAIN-32"></use></g></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>34</mn><mo>:</mo><mn>6</mn><mo>=</mo><mn>5</mn><mtext> con resto </mtext><mn>4</mn><mtext> </mtext><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mo>↗</mo></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><munder><munder><mrow><mn>34</mn><mo>⋅</mo><mn>2</mn></mrow><mo stretchy="true">︸</mo></munder><mrow><mn>68</mn></mrow></munder><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><munder><munder><mrow><mn>6</mn><mo>⋅</mo><mn>2</mn></mrow><mo stretchy="true">︸</mo></munder><mrow><mn>12</mn></mrow></munder><mo stretchy="false">)</mo><mo>=</mo><mn>5</mn><mtext> con resto </mtext><munder><munder><mrow><mn>4</mn><mo>⋅</mo><mn>2</mn></mrow><mo stretchy="true">︸</mo></munder><mn>8</mn></munder></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mo>↘</mo></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><munder><munder><mrow><mn>34</mn><mo>:</mo><mn>2</mn></mrow><mo stretchy="true">︸</mo></munder><mrow><mn>17</mn></mrow></munder><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><munder><munder><mrow><mn>6</mn><mo>:</mo><mn>2</mn></mrow><mo stretchy="true">︸</mo></munder><mn>3</mn></munder><mo stretchy="false">)</mo><mo>=</mo><mn>5</mn><mtext> con resto </mtext><munder><munder><mrow><mn>4</mn><mo>:</mo><mn>2</mn></mrow><mo stretchy="true">︸</mo></munder><mn>2</mn></munder></mrow></mtd></mtr></mtable></mrow></math></script></p>
</li>
</ul>
<div class="box">
<p class="noindent"><span class="red"><b>OSSERVAZIONI SUL CONCETTO DI OPERAZIONE</b><a id="ind51"></a><!--<?"operazione|concetto di",4,0,2>--></span></p>
<ul class="blist">
<li>
<p class="noindent">Ciascuna delle operazioni di addizione, sottrazione, moltiplicazione e divisione nell’insieme ℕ associa a due numeri naturali <i>a</i> e <i>b</i>, <i>termini</i> dell’operazione, considerati nell’ordine, un terzo numero naturale <i>c</i>, <i>risultato</i> dell’operazione.</p>
<p class="noindent">Nel caso dell’addizione e della moltiplicazione, comunque scegliamo i termini dell’operazione, il risultato <i>c</i> esiste ed è unico. Diremo perciò che <i>l’addizione e la moltiplicazione sono ovunque definite in</i> ℕ,<a id="ind52"></a><!--<?"operazioni|ovunque definite in N",4,0,2>--> ossia sono <b>operazioni interne</b> a ℕ.<a id="ind53"></a><!--<?"operazioni|interne a N",4,0,2>--></p>
<p class="noindent">Invece, nel caso della sottrazione e della divisione, può accadere che, considerati i termini <i>a</i> e <i>b</i>, non sia possibile determinare il risultato nell’insieme ℕ. Diremo quindi che <i>la sottrazione e la divisione sono operazioni non ovunque definite in</i> ℕ.<a id="ind54"></a><!--<?"operazioni|non ovunque definite in N",4,0,2>--></p>
<p class="noindent">Poiché il <b>risultato</b> di un’operazione, se esiste, deve essere <b>unico</b>, la <i>divisione approssimata</i> tra numeri naturali è una «operazione» in senso improprio. Si tratta, in realtà, di una doppia operazione: una permette di associare a due numeri naturali il quoziente e l’altra di determinare il resto.</p>
</li>
<li><p class="noindent">Il concetto di operazione non riguarda solo l’aritmetica o l’algebra, ma ogni campo della matematica. Ad esempio, nel seguito, avrai modo di incontrare l’<i>operazione di unione tra insiemi</i> oppure l’<i>operazione di congiunzione logica tra enunciati</i>.</p></li>
</ul>
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<div class="small-12 medium-1 columns"><div class="pagenumber">12</div></div>
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<h2 class="para_title" id="par03">
<span class="pagebreak" epub:type="pagebreak" title="12" id="page12"></span>Potenze in ℕ<a id="ind55"></a><!--<?"potenze|in N",4,0,2>--> e loro proprietà</h2>
<h3 class="sec_title" id="sec6">6. Definizione di potenza</h3>
<p class="noindent">L’<b>elevamento a potenza</b><a id="ind56"></a><!--<?"elevamento a potenza",4,0,2>--> è un’operazione che si esegue tra due numeri naturali, il primo detto <b>base</b><a id="ind57"></a><!--<?"base|di una potenza",4,0,2>--> e il secondo <b>esponente</b>.</p>
<div class="definition" title_dea="Potenza di un numero naturale" key_dea="potenza, elevamento a potenza, base, esponente, numero naturale, numeri naturali, potenza di un numero naturale">
<h4 class="noindent">
<span class="ash">DEFINIZIONE</span> <span class="orangeb">POTENZA DI UN NUMERO NATURALE</span>
</h4>
<p class="noindentin">La potenza di base <i>a</i> ed esponente <i>n</i> si indica con <i>a<sup>n</sup></i> ed è uguale al prodotto di <i>n</i> fattori uguali ad <i>a</i>:</p>
<div class="exp_imp" title_dea="Formula di elevamento a potenza di un numero naturale" key_dea="potenza, elevamento a potenza, numero naturale, numeri naturali, formula di elevamento a potenza di un numero naturale">
<p class="math"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -5.333ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="17.667ex" height="7.5ex" viewBox="0 -972.1 7583.7 3207.1"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(275,0)"><use xlink:href="#MJMATHI-61"></use><use transform="scale(0.707)" x="755" y="513" xlink:href="#MJMATHI-6E"></use><use x="1339" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(2400,0)"><use xlink:href="#MJMATHI-61"></use><use x="756" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1261" y="0" xlink:href="#MJMATHI-61"></use><use x="2017" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(2522,0)"><use xlink:href="#MJMAIN-2E"></use><use x="283" y="0" xlink:href="#MJMAIN-2E"></use><use x="566" y="0" xlink:href="#MJMAIN-2E"></use></g><use x="3594" y="0" xlink:href="#MJMAIN-22C5"></use><use x="4099" y="0" xlink:href="#MJMATHI-61"></use><g transform="translate(12,-543)"><use x="19" y="0" xlink:href="#MJSZ4-E152"></use><g transform="translate(500.4007936507937,0) scale(3.2801587301587296,1)"><use xlink:href="#MJSZ4-E154"></use></g><g transform="translate(1861,0)"><use xlink:href="#MJSZ4-E151"></use><use x="455" y="0" xlink:href="#MJSZ4-E150"></use></g><g transform="translate(2788.0674603174602,0) scale(3.2801587301587296,1)"><use xlink:href="#MJSZ4-E154"></use></g><use x="4154" y="0" xlink:href="#MJSZ4-E153"></use></g><g transform="translate(901,-1542)"><use transform="scale(0.707)" xlink:href="#MJMATHI-6E"></use><g transform="translate(427,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(859,0)"><use transform="scale(0.707)" xlink:href="#MJMAIN-66"></use><use transform="scale(0.707)" x="311" y="0" xlink:href="#MJMAIN-61"></use><use transform="scale(0.707)" x="816" y="0" xlink:href="#MJMAIN-74"></use><use transform="scale(0.707)" x="1210" y="0" xlink:href="#MJMAIN-74"></use><use transform="scale(0.707)" x="1604" y="0" xlink:href="#MJMAIN-6F"></use><use transform="scale(0.707)" x="2109" y="0" xlink:href="#MJMAIN-72"></use><use transform="scale(0.707)" x="2506" y="0" xlink:href="#MJMAIN-69"></use></g></g></g></g><line stroke-linecap="square" stroke="black" stroke-width="75" y2="37" y1="37" x1="37" x2="7546" transform="translate(0,883)"></line><line stroke-linecap="square" stroke="black" stroke-width="75" y1="37" x2="37" x1="37" y2="3141" transform="translate(7508,-2222)"></line><line stroke-linecap="square" stroke="black" stroke-width="75" y2="37" y1="37" x1="37" x2="7546" transform="translate(0,-2222)"></line><line stroke-linecap="square" stroke="black" stroke-width="75" y1="37" x2="37" x1="37" y2="3141" transform="translate(0,-2222)"></line></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><menclose notation="box"><mrow><msup><mi>a</mi><mi>n</mi></msup><mo>=</mo><munder><munder><mrow><mi>a</mi><mo>⋅</mo><mi>a</mi><mo>⋅</mo><mn>...</mn><mo>⋅</mo><mi>a</mi></mrow><mo stretchy="true">︸</mo></munder><mrow><mi>n</mi><mtext> </mtext><mtext>fattori</mtext></mrow></munder></mrow></menclose></mrow></math></script></span></p>
</div>
</div>
<p class="noindent">Ad esempio:</p>
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<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -4.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="20.5ex" height="6.667ex" viewBox="0 -901.7 8851.5 2873.6"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="714" y="568" xlink:href="#MJMAIN-34"></use><use x="1239" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(2300,0)"><use xlink:href="#MJMAIN-33"></use><use x="727" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1232" y="0" xlink:href="#MJMAIN-33"></use><use x="1959" y="0" xlink:href="#MJMAIN-22C5"></use><use x="2464" y="0" xlink:href="#MJMAIN-33"></use><use x="3192" y="0" xlink:href="#MJMAIN-22C5"></use><use x="3697" y="0" xlink:href="#MJMAIN-33"></use><g transform="translate(12,-555)"><use x="19" y="0" xlink:href="#MJSZ4-E152"></use><g transform="translate(497.83531746031747,0) scale(2.7670634920634916,1)"><use xlink:href="#MJSZ4-E154"></use></g><g transform="translate(1646,0)"><use xlink:href="#MJSZ4-E151"></use><use x="455" y="0" xlink:href="#MJSZ4-E150"></use></g><g transform="translate(2570.001984126984,0) scale(2.7670634920634916,1)"><use xlink:href="#MJSZ4-E154"></use></g><use x="3723" y="0" xlink:href="#MJSZ4-E153"></use></g><g transform="translate(720,-1554)"><use transform="scale(0.707)" xlink:href="#MJMAIN-34"></use><g transform="translate(357,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(788,0)"><use transform="scale(0.707)" xlink:href="#MJMAIN-66"></use><use transform="scale(0.707)" x="311" y="0" xlink:href="#MJMAIN-61"></use><use transform="scale(0.707)" x="816" y="0" xlink:href="#MJMAIN-74"></use><use transform="scale(0.707)" x="1210" y="0" xlink:href="#MJMAIN-74"></use><use transform="scale(0.707)" x="1604" y="0" xlink:href="#MJMAIN-6F"></use><use transform="scale(0.707)" x="2109" y="0" xlink:href="#MJMAIN-72"></use><use transform="scale(0.707)" x="2506" y="0" xlink:href="#MJMAIN-69"></use></g></g></g><use x="6780" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(7841,0)"><use xlink:href="#MJMAIN-38"></use><use x="505" y="0" xlink:href="#MJMAIN-31"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mn>3</mn><mn>4</mn></msup><mo>=</mo><munder><munder><mrow><mn>3</mn><mo>⋅</mo><mn>3</mn><mo>⋅</mo><mn>3</mn><mo>⋅</mo><mn>3</mn></mrow><mo stretchy="true">︸</mo></munder><mrow><mn>4</mn><mtext> </mtext><mtext>fattori</mtext></mrow></munder><mo>=</mo><mn>81</mn></mrow></math></script></p>
<p class="noindent">Si conviene che:</p>
<ul class="blist">
<li>
<p class="noindent">la potenza con esponente 1<a id="ind58"></a><!--<?"potenze|con esponente|1",4,0,2>--> di un qualsiasi numero naturale sia il numero stesso:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="41.333ex" height="3.167ex" viewBox="0 -905.9 17780.1 1350.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMATHI-61"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(534,362)"><use transform="scale(0.707)" xlink:href="#MJMAIN-31"></use></g></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1268,0)"><use xlink:href="#MJMAIN-3D"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(2329,0)"><use xlink:href="#MJMATHI-61"></use></g></g></g><g transform="translate(2863,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(4083,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(4693,0)"><use xlink:href="#MJMAIN-35"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-31"></use></g><use x="5933" y="0" xlink:href="#MJMAIN-3D"></use><use x="6994" y="0" xlink:href="#MJMAIN-35"></use><g transform="translate(7499,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(8109,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(9328,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-34"></use><use transform="scale(0.707)" x="1428" y="585" xlink:href="#MJMAIN-31"></use></g><use x="11073" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(12134,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-34"></use></g><g transform="translate(13144,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(14364,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(14974,0)"><use xlink:href="#MJMAIN-30"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-31"></use></g><use x="16214" y="0" xlink:href="#MJMAIN-3D"></use><use x="17275" y="0" xlink:href="#MJMAIN-30"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mstyle color="#00aef0"><msup><mi>a</mi><mn>1</mn></msup><mo>=</mo><mi>a</mi></mstyle><mtext>  </mtext><mtext> </mtext><msup><mn>5</mn><mn>1</mn></msup><mo>=</mo><mn>5</mn><mtext> </mtext><mtext>  </mtext><msup><mrow><mn>24</mn></mrow><mn>1</mn></msup><mo>=</mo><mn>24</mn><mtext>  </mtext><mtext> </mtext><msup><mn>0</mn><mn>1</mn></msup><mo>=</mo><mn>0</mn></mrow></math></script></p>
</li>
<li>
<p class="noindent">la potenza con esponente 0<a id="ind59"></a><!--<?"potenze|con esponente|0",4,0,2>--> di un qualsiasi numero naturale diverso da 0 sia 1:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="52.333ex" height="3.167ex" viewBox="0 -905.9 22498.6 1350.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMATHI-61"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(534,362)"><use transform="scale(0.707)" xlink:href="#MJMAIN-30"></use></g></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1268,0)"><use xlink:href="#MJMAIN-3D"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(2329,0)"><use xlink:href="#MJMAIN-31"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(2834,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g fill="#00aef0" stroke="#00aef0" transform="translate(3444,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAINB-70"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(644,0)"><use xlink:href="#MJMAINB-65"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1176,0)"><use xlink:href="#MJMAINB-72"></use></g></g></g><g fill="#00aef0" stroke="#00aef0" transform="translate(5099,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g fill="#00aef0" stroke="#00aef0" transform="translate(5709,0)"><use xlink:href="#MJMATHI-61"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(6521,0)"><use xlink:href="#MJMAIN-2260"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(7582,0)"><use xlink:href="#MJMAIN-30"></use></g></g></g><g transform="translate(8087,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(9307,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(9916,0)"><use xlink:href="#MJMAIN-35"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-30"></use></g><use x="11156" y="0" xlink:href="#MJMAIN-3D"></use><use x="12217" y="0" xlink:href="#MJMAIN-31"></use><g transform="translate(12722,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(13942,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(14552,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-34"></use><use transform="scale(0.707)" x="1428" y="585" xlink:href="#MJMAIN-30"></use></g><use x="16297" y="0" xlink:href="#MJMAIN-3D"></use><use x="17358" y="0" xlink:href="#MJMAIN-31"></use><g transform="translate(17863,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(19083,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(19692,0)"><use xlink:href="#MJMAIN-31"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-30"></use></g><use x="20932" y="0" xlink:href="#MJMAIN-3D"></use><use x="21993" y="0" xlink:href="#MJMAIN-31"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mstyle color="#00aef0"><msup><mi>a</mi><mn>0</mn></msup><mo>=</mo><mn>1</mn><mtext> </mtext><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi><mi>e</mi><mi>r</mi></mstyle><mtext> </mtext><mi>a</mi><mo>≠</mo><mn>0</mn></mstyle><mtext>  </mtext><mtext> </mtext><msup><mn>5</mn><mn>0</mn></msup><mo>=</mo><mn>1</mn><mtext>  </mtext><mtext> </mtext><msup><mrow><mn>24</mn></mrow><mn>0</mn></msup><mo>=</mo><mn>1</mn><mtext>  </mtext><mtext> </mtext><msup><mn>1</mn><mn>0</mn></msup><mo>=</mo><mn>1</mn></mrow></math></script></p>
</li>
<li><p class="noindent">la potenza <span class="cyan"><b>0<sup>0</sup></b></span>, cioè con base 0 ed esponente 0, <span class="cyan"><b>non ha significato</b></span>.</p></li>
</ul>
<p class="noindent"><b>Casi particolari</b>:</p>
<p class="math"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="45.167ex" height="3ex" viewBox="0 -875 19457.2 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-31"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMATHI-6E"></use><use x="1310" y="0" xlink:href="#MJMAIN-3D"></use><use x="2371" y="0" xlink:href="#MJMAIN-31"></use><g transform="translate(2876,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(3486,0)"><use xlink:href="#MJMAINB-70"></use><use x="644" y="0" xlink:href="#MJMAINB-65"></use><use x="1176" y="0" xlink:href="#MJMAINB-72"></use><g transform="translate(1655,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="bold" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="2264" y="0" xlink:href="#MJMAINB-6F"></use><use x="2844" y="0" xlink:href="#MJMAINB-67"></use><use x="3424" y="0" xlink:href="#MJMAINB-6E"></use><use x="4068" y="0" xlink:href="#MJMAINB-69"></use></g><g transform="translate(7879,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="8489" y="0" xlink:href="#MJMATHI-6E"></use><g transform="translate(9094,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1219,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(10924,0)"><use xlink:href="#MJMAIN-30"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMATHI-6E"></use></g><use x="12234" y="0" xlink:href="#MJMAIN-3D"></use><use x="13295" y="0" xlink:href="#MJMAIN-30"></use><g transform="translate(13800,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(14577,0)"><use xlink:href="#MJMAINB-70"></use><use x="644" y="0" xlink:href="#MJMAINB-65"></use><use x="1176" y="0" xlink:href="#MJMAINB-72"></use></g><g transform="translate(16398,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="17008" y="0" xlink:href="#MJMATHI-6E"></use><use x="17891" y="0" xlink:href="#MJMAIN-2260"></use><use x="18952" y="0" xlink:href="#MJMAIN-30"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><mn>1</mn><mi> </mi><mi mathvariant="bold">per ogni</mi><mi> </mi><mi>n</mi><mi>   </mi><msup><mrow><mn>0</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><mn>0</mn><mi> </mi><mi mathvariant="bold">per</mi><mi> </mi><mi>n</mi><mo>≠</mo><mn>0</mn></mrow></math></script></span></p>
<h3 class="sec_title" id="sec7">7. Proprietà delle potenze<a id="ind60"></a><!--<?"potenze|propriet&#x00E0; delle",4,0,2>-->
</h3>
<ul class="blist">
<li>
<p class="noindent"><b>Il prodotto di potenze<a id="ind61"></a><!--<?"prodotto|di potenze",4,0,2>--> con la stessa base</b> è una potenza che ha per base la stessa base e per esponente la somma degli esponenti:</p>
<div class="exp_imp" title_dea="Prodotto di potenze con la stessa base" key_dea="potenza, potenze, stessa base, uguale base, prodotto di potenze, numeri naturali, prodotto di potenze con la stessa base, somma di esponenti">
<p class="math"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.833ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="16.667ex" height="3.167ex" viewBox="0 -1071.1 7176 1380.1"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(275,0)"><use xlink:href="#MJMATHI-61"></use><use transform="scale(0.707)" x="755" y="513" xlink:href="#MJMATHI-6D"></use><use x="1480" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(1985,0)"><use xlink:href="#MJMATHI-61"></use><use transform="scale(0.707)" x="755" y="513" xlink:href="#MJMATHI-6E"></use></g><use x="3325" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(4386,0)"><use xlink:href="#MJMATHI-61"></use><g transform="translate(534,362)"><use transform="scale(0.707)" xlink:href="#MJMATHI-6D"></use><use transform="scale(0.707)" x="883" y="0" xlink:href="#MJMAIN-2B"></use><use transform="scale(0.707)" x="1666" y="0" xlink:href="#MJMATHI-6E"></use></g></g></g><line stroke-linecap="square" stroke="black" stroke-width="75" y2="37" y1="37" x1="37" x2="7138" transform="translate(0,982)"></line><line stroke-linecap="square" stroke="black" stroke-width="75" y1="37" x2="37" x1="37" y2="1314" transform="translate(7101,-295)"></line><line stroke-linecap="square" stroke="black" stroke-width="75" y2="37" y1="37" x1="37" x2="7138" transform="translate(0,-295)"></line><line stroke-linecap="square" stroke="black" stroke-width="75" y1="37" x2="37" x1="37" y2="1314" transform="translate(0,-295)"></line></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><menclose notation="box"><mrow><msup><mi>a</mi><mi>m</mi></msup><mo>⋅</mo><msup><mi>a</mi><mi>n</mi></msup><mo>=</mo><msup><mi>a</mi><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup></mrow></menclose></mrow></math></script></span></p>
</div>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="52.667ex" height="3.167ex" viewBox="0 -901 22691.6 1345.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="714" y="568" xlink:href="#MJMAIN-35"></use><use x="1184" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(1689,0)"><use xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="714" y="568" xlink:href="#MJMAIN-37"></use></g><use x="2929" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(3990,0)"><use xlink:href="#MJMAIN-33"></use><g transform="translate(505,401)"><use transform="scale(0.707)" xlink:href="#MJMAIN-35"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-2B"></use><use transform="scale(0.707)" x="1288" y="0" xlink:href="#MJMAIN-37"></use></g></g><use x="6140" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(7201,0)"><use xlink:href="#MJMAIN-33"></use><g transform="translate(505,401)"><use transform="scale(0.707)" xlink:href="#MJMAIN-31"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-32"></use></g></g><g transform="translate(8520,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1219,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1829,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(2439,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(11570,0)"><use xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-33"></use></g><use x="12754" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(13260,0)"><use xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-32"></use></g><use x="14444" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(14949,0)"><use xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-35"></use></g><use x="16189" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(17250,0)"><use xlink:href="#MJMAIN-32"></use><g transform="translate(505,402)"><use transform="scale(0.707)" xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-2B"></use><use transform="scale(0.707)" x="1288" y="0" xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="1793" y="0" xlink:href="#MJMAIN-2B"></use><use transform="scale(0.707)" x="2576" y="0" xlink:href="#MJMAIN-35"></use></g></g><use x="20311" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(21372,0)"><use xlink:href="#MJMAIN-32"></use><g transform="translate(505,402)"><use transform="scale(0.707)" xlink:href="#MJMAIN-31"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-30"></use></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mn>3</mn></mrow><mrow><mn>5</mn></mrow></msup><mo>·</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>7</mn></mrow></msup><mo>=</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>5</mn><mo>+</mo><mn>7</mn></mrow></msup><mo>=</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>12</mn></mrow></msup><mi>     </mi><msup><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></msup><mo>·</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>·</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>5</mn></mrow></msup><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>3</mn><mo>+</mo><mn>2</mn><mo>+</mo><mn>5</mn></mrow></msup><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>10</mn></mrow></msup></mrow></math></script></p>
</li>
<li>
<p class="noindent"><b>Il quoziente di due potenze<a id="ind62"></a><!--<?"quoziente|di due potenze",4,0,2>--> con la stessa base</b> è una potenza che ha per base la stessa base e per esponente la differenza degli esponenti:</p>
<div class="exp_imp" title_dea="Quoziente di due potenze con la stessa base" key_dea="potenza, potenze, stessa base, uguale base, quoziente di potenze, numeri naturali, quoziente di due potenze con la stessa base, differenza degli esponenti">
<p class="math"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="34.667ex" height="4.333ex" viewBox="0 -1150 14948.8 1869.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(275,0)"><use xlink:href="#MJMATHI-61"></use><use transform="scale(0.707)" x="755" y="513" xlink:href="#MJMATHI-6D"></use><use x="1536" y="0" xlink:href="#MJMAIN-3A"></use><g transform="translate(2096,0)"><use xlink:href="#MJMATHI-61"></use><use transform="scale(0.707)" x="755" y="513" xlink:href="#MJMATHI-6E"></use></g><use x="3436" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(4497,0)"><use xlink:href="#MJMATHI-61"></use><g transform="translate(534,362)"><use transform="scale(0.707)" xlink:href="#MJMATHI-6D"></use><use transform="scale(0.707)" x="883" y="0" xlink:href="#MJMAIN-2212"></use><use transform="scale(0.707)" x="1666" y="0" xlink:href="#MJMATHI-6E"></use></g></g><g transform="translate(6737,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="7347" y="0" xlink:href="#MJMAIN-28"></use><use x="7741" y="0" xlink:href="#MJMATHI-61"></use><use x="8552" y="0" xlink:href="#MJMAIN-2260"></use><use x="9613" y="0" xlink:href="#MJMAIN-30"></use><use x="10118" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(10568,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="11178" y="0" xlink:href="#MJMATHI-6D"></use><use x="12339" y="0" xlink:href="#MJMAIN-2265"></use><use x="13399" y="0" xlink:href="#MJMATHI-6E"></use><use x="14004" y="0" xlink:href="#MJMAIN-29"></use></g><line stroke-linecap="square" stroke="black" stroke-width="75" y2="37" y1="37" x1="37" x2="14911" transform="translate(0,1061)"></line><line stroke-linecap="square" stroke="black" stroke-width="75" y1="37" x2="37" x1="37" y2="1804" transform="translate(14873,-706)"></line><line stroke-linecap="square" stroke="black" stroke-width="75" y2="37" y1="37" x1="37" x2="14911" transform="translate(0,-706)"></line><line stroke-linecap="square" stroke="black" stroke-width="75" y1="37" x2="37" x1="37" y2="1804" transform="translate(0,-706)"></line></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><menclose notation="box"><mrow><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>:</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi><mo>-</mo><mi>n</mi></mrow></msup><mi> </mi><mo stretchy="false">(</mo><mi>a</mi><mo>≠</mo><mn>0</mn><mo>,</mo><mi> </mi><mi>m</mi><mo>≥</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></menclose></mrow></math></script></span></p>
</div>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="19.167ex" height="2.167ex" viewBox="0 -893.9 8274.8 939.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="714" y="568" xlink:href="#MJMAIN-38"></use><use x="1239" y="0" xlink:href="#MJMAIN-3A"></use><g transform="translate(1800,0)"><use xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="714" y="568" xlink:href="#MJMAIN-33"></use></g><use x="3040" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(4101,0)"><use xlink:href="#MJMAIN-33"></use><g transform="translate(505,401)"><use transform="scale(0.707)" xlink:href="#MJMAIN-38"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-2212"></use><use transform="scale(0.707)" x="1288" y="0" xlink:href="#MJMAIN-33"></use></g></g><use x="6251" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(7312,0)"><use xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="714" y="568" xlink:href="#MJMAIN-35"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mn>3</mn></mrow><mrow><mn>8</mn></mrow></msup><mo>:</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>3</mn></mrow></msup><mo>=</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>8</mn><mo>-</mo><mn>3</mn></mrow></msup><mo>=</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>5</mn></mrow></msup></mrow></math></script></p>
</li>
<li>
<p class="noindent"><b>La potenza di una potenza</b><a id="ind63"></a><!--<?"potenza|di una potenza",4,0,2>--> è una potenza che ha per base la stessa base e per esponente il prodotto degli esponenti:</p>
<div class="exp_imp" title_dea="Potenza di una potenza" key_dea="potenza, potenze, potenza di potenza, prodotto degli esponenti">
<p class="math"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.333ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="15ex" height="3.667ex" viewBox="0 -1048.9 6426.2 1597.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(275,0)"><use xlink:href="#MJMAIN-28"></use><g transform="translate(394,0)"><use xlink:href="#MJMATHI-61"></use><use transform="scale(0.707)" x="755" y="513" xlink:href="#MJMATHI-6D"></use></g><use x="1652" y="0" xlink:href="#MJMAIN-29"></use><use x="2046" y="0" xlink:href="#MJMATHI-6E"></use><use x="2929" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(3989,0)"><use xlink:href="#MJMATHI-61"></use><g transform="translate(534,362)"><use transform="scale(0.707)" xlink:href="#MJMATHI-6D"></use><use transform="scale(0.707)" x="883" y="0" xlink:href="#MJMAIN-22C5"></use><use transform="scale(0.707)" x="1166" y="0" xlink:href="#MJMATHI-6E"></use></g></g></g><line stroke-linecap="square" stroke="black" stroke-width="75" y2="37" y1="37" x1="37" x2="6388" transform="translate(0,960)"></line><line stroke-linecap="square" stroke="black" stroke-width="75" y1="37" x2="37" x1="37" y2="1532" transform="translate(6351,-535)"></line><line stroke-linecap="square" stroke="black" stroke-width="75" y2="37" y1="37" x1="37" x2="6388" transform="translate(0,-535)"></line><line stroke-linecap="square" stroke="black" stroke-width="75" y1="37" x2="37" x1="37" y2="1532" transform="translate(0,-535)"></line></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><menclose notation="box"><mrow><mo stretchy="false">(</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi><mo>·</mo><mi>n</mi></mrow></msup></mrow></menclose></mrow></math></script></span></p>
</div>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="18ex" height="2.667ex" viewBox="0 -902.7 7770.7 1176.6"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><g transform="translate(394,0)"><use xlink:href="#MJMAIN-35"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-34"></use></g><use x="1356" y="0" xlink:href="#MJMAIN-29"></use><use x="1750" y="0" xlink:href="#MJMAIN-36"></use><use x="2532" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(3593,0)"><use xlink:href="#MJMAIN-35"></use><g transform="translate(505,402)"><use transform="scale(0.707)" xlink:href="#MJMAIN-34"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-22C5"></use><use transform="scale(0.707)" x="788" y="0" xlink:href="#MJMAIN-36"></use></g></g><use x="5390" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(6451,0)"><use xlink:href="#MJMAIN-35"></use><g transform="translate(505,402)"><use transform="scale(0.707)" xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-34"></use></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msup><mrow><mn>5</mn></mrow><mrow><mn>4</mn></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mn>6</mn></mrow></msup><mo>=</mo><msup><mrow><mn>5</mn></mrow><mrow><mn>4</mn><mo>·</mo><mn>6</mn></mrow></msup><mo>=</mo><msup><mrow><mn>5</mn></mrow><mrow><mn>24</mn></mrow></msup></mrow></math></script></p>
</li>
</ul>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"><div class="sidebox">
<p class="noindentf">3<sup>4</sup> si legge tre elevato alla quarta potenza, o anche semplicemente tre alla quarta; 3 è la base, 4 è l’esponente.</p>
</div></div>
</div>
</div>
<div data-page-container="13" id="page-13" class="row chapters-content">
<div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">13</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<ul class="blist">
<li>
<p class="noindent"><span class="pagebreak" epub:type="pagebreak" title="13" id="page13"></span><b>Il prodotto di potenze di uguale esponente</b> è una potenza con lo stesso esponente, che ha per base il prodotto delle basi:</p>
<div class="exp_imp" title_dea="Prodotto di potenze di uguale esponente" key_dea="potenza, potenze, stesso esponente, uguale esponente, prodotto di potenze, prodotto di potenze di uguale esponente">
<p class="math"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.333ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="17.833ex" height="3.833ex" viewBox="0 -1096.2 7650.8 1645.2"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(275,0)"><use xlink:href="#MJMATHI-61"></use><use transform="scale(0.707)" x="755" y="513" xlink:href="#MJMATHI-6E"></use><use x="1284" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(1789,0)"><use xlink:href="#MJMATHI-62"></use><use transform="scale(0.707)" x="613" y="513" xlink:href="#MJMATHI-6E"></use></g><use x="3028" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(4089,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMATHI-61"></use><use x="1150" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1655" y="0" xlink:href="#MJMATHI-62"></use><use x="2089" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="3512" y="688" xlink:href="#MJMATHI-6E"></use></g></g><line stroke-linecap="square" stroke="black" stroke-width="75" y2="37" y1="37" x1="37" x2="7613" transform="translate(0,1007)"></line><line stroke-linecap="square" stroke="black" stroke-width="75" y1="37" x2="37" x1="37" y2="1579" transform="translate(7575,-535)"></line><line stroke-linecap="square" stroke="black" stroke-width="75" y2="37" y1="37" x1="37" x2="7613" transform="translate(0,-535)"></line><line stroke-linecap="square" stroke="black" stroke-width="75" y1="37" x2="37" x1="37" y2="1579" transform="translate(0,-535)"></line></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><menclose notation="box"><mrow><msup><mi>a</mi><mi>n</mi></msup><mo>⋅</mo><msup><mi>b</mi><mi>n</mi></msup><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>⋅</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><mi>n</mi></msup></mrow></menclose></mrow></math></script></span></p>
</div>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="29.667ex" height="2.667ex" viewBox="0 -905.9 12748.2 1179.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="714" y="568" xlink:href="#MJMAIN-35"></use><use x="1184" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(1689,0)"><use xlink:href="#MJMAIN-37"></use><use transform="scale(0.707)" x="714" y="584" xlink:href="#MJMAIN-35"></use></g><use x="2873" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(3379,0)"><use xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-35"></use></g><use x="4618" y="0" xlink:href="#MJMAIN-3D"></use><use x="5679" y="0" xlink:href="#MJMAIN-28"></use><use x="6073" y="0" xlink:href="#MJMAIN-33"></use><use x="6800" y="0" xlink:href="#MJMAIN-22C5"></use><use x="7306" y="0" xlink:href="#MJMAIN-37"></use><use x="8033" y="0" xlink:href="#MJMAIN-22C5"></use><use x="8538" y="0" xlink:href="#MJMAIN-32"></use><use x="9043" y="0" xlink:href="#MJMAIN-29"></use><use x="9437" y="0" xlink:href="#MJMAIN-35"></use><use x="10220" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(11281,0)"><use xlink:href="#MJMAIN-34"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="1428" y="585" xlink:href="#MJMAIN-35"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mn>3</mn></mrow><mrow><mn>5</mn></mrow></msup><mo>·</mo><msup><mrow><mn>7</mn></mrow><mrow><mn>5</mn></mrow></msup><mo>·</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>5</mn></mrow></msup><mo>=</mo><mo stretchy="false">(</mo><mn>3</mn><mo>·</mo><mn>7</mn><mo>·</mo><mn>2</mn><mo stretchy="false">)</mo><msup><mrow><mn>5</mn></mrow></msup><mo>=</mo><msup><mrow><mn>42</mn></mrow><mrow><mn>5</mn></mrow></msup></mrow></math></script></p>
</li>
<li>
<p class="noindent"><b>Il quoziente di potenze di uguale esponente</b> è una potenza con lo stesso esponente, che ha per base il quoziente delle basi:</p>
<div class="exp_imp" title_dea="Quoziente di potenze di uguale esponente" key_dea="potenza, potenze, stesso esponente, uguale esponente, quoziente di potenze, quoziente di potenze di uguale esponente">
<p class="math"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.5ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="28.167ex" height="4ex" viewBox="0 -1125 12122.1 1750.1"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(167,0)"><g transform="translate(389,0)"><g transform="translate(0,34)"><use xlink:href="#MJMATHI-61"></use><use transform="scale(0.707)" x="755" y="513" xlink:href="#MJMATHI-6E"></use><use x="1339" y="0" xlink:href="#MJMAIN-3A"></use><g transform="translate(1900,0)"><use xlink:href="#MJMATHI-62"></use><use transform="scale(0.707)" x="613" y="513" xlink:href="#MJMATHI-6E"></use></g><use x="3139" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(4200,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMATHI-61"></use><use x="1205" y="0" xlink:href="#MJMAIN-3A"></use><use x="1766" y="0" xlink:href="#MJMATHI-62"></use><use x="2200" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="3669" y="688" xlink:href="#MJMATHI-6E"></use></g><g transform="translate(7323,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="7933" y="0" xlink:href="#MJMAIN-28"></use><use x="8327" y="0" xlink:href="#MJMATHI-62"></use><use x="9038" y="0" xlink:href="#MJMAIN-2260"></use><use x="10099" y="0" xlink:href="#MJMAIN-30"></use><use x="10604" y="0" xlink:href="#MJMAIN-29"></use></g></g><line stroke-linecap="square" stroke-width="20.9" y2="10" y1="10" x1="10" x2="11777" transform="translate(0,1090)"></line><line stroke-linecap="square" stroke-width="20.9" y2="10" y1="10" x1="10" x2="11777" transform="translate(0,-612)"></line><line stroke-linecap="square" stroke-width="20.9" y1="10" x2="10" x1="10" y2="1711" transform="translate(0,-612)"></line><line stroke-linecap="square" stroke-width="20.9" y1="10" x2="10" x1="10" y2="1711" transform="translate(11767,-612)"></line></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mtable frame="solid"><mrow><msup><mi>a</mi><mi>n</mi></msup><mo>:</mo><msup><mi>b</mi><mi>n</mi></msup><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>:</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><mi>n</mi></msup><mtext> </mtext><mo stretchy="false">(</mo><mi>b</mi><mo>≠</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mtable></mrow></math></script></span></p>
</div>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="24.5ex" height="2.667ex" viewBox="0 -905.9 10553.5 1179.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-34"></use><use transform="scale(0.707)" x="1428" y="585" xlink:href="#MJMAIN-35"></use><use x="1744" y="0" xlink:href="#MJMAIN-3A"></use><g transform="translate(2305,0)"><use xlink:href="#MJMAIN-36"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-35"></use></g><use x="3545" y="0" xlink:href="#MJMAIN-3D"></use><use x="4606" y="0" xlink:href="#MJMAIN-28"></use><g transform="translate(5000,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-34"></use></g><use x="6288" y="0" xlink:href="#MJMAIN-3A"></use><use x="6848" y="0" xlink:href="#MJMAIN-36"></use><use x="7353" y="0" xlink:href="#MJMAIN-29"></use><use x="7747" y="0" xlink:href="#MJMAIN-35"></use><use x="8530" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(9591,0)"><use xlink:href="#MJMAIN-34"></use><use transform="scale(0.707)" x="714" y="585" xlink:href="#MJMAIN-35"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mn>24</mn></mrow><mrow><mn>5</mn></mrow></msup><mo>:</mo><msup><mrow><mn>6</mn></mrow><mrow><mn>5</mn></mrow></msup><mo>=</mo><mo stretchy="false">(</mo><mn>24</mn><mo>:</mo><mn>6</mn><mo stretchy="false">)</mo><msup><mrow><mn>5</mn></mrow></msup><mo>=</mo><msup><mrow><mn>4</mn></mrow><mrow><mn>5</mn></mrow></msup></mrow></math></script></p>
</li>
</ul>
<div class="box">
<p class="noindent"><span class="red"><b>ATTENZIONE!</b></span></p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="36.833ex" height="3ex" viewBox="0 -875 15869.9 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMATHI-61"></use><use x="1150" y="0" xlink:href="#MJMAIN-2B"></use><use x="2155" y="0" xlink:href="#MJMATHI-62"></use><use x="2589" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="4219" y="688" xlink:href="#MJMATHI-6E"></use><g transform="translate(3511,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(4121,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(4731,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="bold" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)">è</text></g><g transform="translate(5341,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(5951,0)"><use xlink:href="#MJMAINB-64"></use><use x="644" y="0" xlink:href="#MJMAINB-69"></use><use x="968" y="0" xlink:href="#MJMAINB-76"></use><use x="1580" y="0" xlink:href="#MJMAINB-65"></use><use x="2112" y="0" xlink:href="#MJMAINB-72"></use><use x="2591" y="0" xlink:href="#MJMAINB-73"></use><use x="3050" y="0" xlink:href="#MJMAINB-6F"></use></g><g transform="translate(9581,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(10191,0)"><use xlink:href="#MJMAINB-64"></use><use x="644" y="0" xlink:href="#MJMAINB-61"></use></g><g transform="translate(11399,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(12008,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(12618,0)"><use xlink:href="#MJMATHI-61"></use><use transform="scale(0.707)" x="755" y="513" xlink:href="#MJMATHI-6E"></use></g><use x="13902" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(14908,0)"><use xlink:href="#MJMATHI-62"></use><use transform="scale(0.707)" x="613" y="513" xlink:href="#MJMATHI-6E"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><mi>n</mi></msup><mtext> </mtext><mtext> </mtext><mstyle mathvariant="bold" mathsize="normal"><mi>è</mi></mstyle><mtext> </mtext><mstyle mathvariant="bold" mathsize="normal"><mi>d</mi><mi>i</mi><mi>v</mi><mi>e</mi><mi>r</mi><mi>s</mi><mi>o</mi></mstyle><mtext> </mtext><mstyle mathvariant="bold" mathsize="normal"><mi>d</mi><mi>a</mi></mstyle><mtext> </mtext><mtext> </mtext><msup><mi>a</mi><mi>n</mi></msup><mo>+</mo><msup><mi>b</mi><mi>n</mi></msup></mrow></math></script></p>
<p class="noindent">Ad esempio: (3 + 4)<sup>2</sup> = 49 e 3<sup>2</sup> + 4<sup>2</sup> = 25.</p>
</div>
<h2 class="para_title" id="par04">Espressioni con i numeri naturali<a id="ind64"></a><!--<?"espressioni con i numeri naturali",4,0,2>-->
</h2>
<h3 class="sec_title" id="sec8">8. Priorità delle operazioni<a id="ind65"></a><!--<?"operazioni|priorit&#x00E0; delle",4,0,2>-->
</h3>
<p class="noindent">Se in un’espressione sono indicate diverse operazioni, queste devono essere eseguite rispettando il loro <b>grado di priorità</b>.</p>
<p class="noindent">In un’espressione si devono eseguire <i>prima</i> gli <b>elevamenti a potenza</b>, <i>poi</i> le <b>moltiplicazioni</b> e le <b>divisioni</b>, <i>infine</i> le <b>addizioni</b> e le <b>sottrazioni</b>.</p>
<p class="noindent">Se ad esempio vogliamo calcolare 5 + 7 · 3, occorre eseguire prima la moltiplicazione 7 · 3 = 21 e poi l’addizione 5 + 21 = 26:</p>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"></div>
</div>
<div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.333ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="23ex" height="1.833ex" viewBox="0 -699.9 9899.4 805.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-35"></use><use x="727" y="0" xlink:href="#MJMAIN-2B"></use><use x="1732" y="0" xlink:href="#MJMAIN-37"></use><use x="2459" y="0" xlink:href="#MJMAIN-22C5"></use><use x="2964" y="0" xlink:href="#MJMAIN-33"></use><use x="3747" y="0" xlink:href="#MJMAIN-3D"></use><use x="4808" y="0" xlink:href="#MJMAIN-35"></use><use x="5535" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(6540,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-31"></use></g><use x="7828" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(8889,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-36"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>5</mn><mo>+</mo><mn>7</mn><mo>·</mo><mn>3</mn><mo>=</mo><mn>5</mn><mo>+</mo><mn>21</mn><mo>=</mo><mn>26</mn></mrow></math></script></p>
<p class="noindent">Nel caso siano indicate di seguito <i>diverse operazioni con lo stesso grado di priorità</i>, esse vanno eseguite nell’ordine dato. Ad esempio per calcolare 168 : 12 · 3 : 2 si deve eseguire</p>
<ol class="olist">
<li><p class="noindent">innanzitutto la divisione&nbsp;&nbsp;168 : 12 = 14</p></li>
<li><p class="noindent">poi la moltiplicazione&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;14 · 3 = 42</p></li>
<li><p class="noindent">e infine la divisione&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;42 : 2 = 21</p></li>
</ol>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="39.333ex" height="1.667ex" viewBox="0 -700.9 16904.8 746.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-36"></use><use x="1010" y="0" xlink:href="#MJMAIN-38"></use><use x="1792" y="0" xlink:href="#MJMAIN-3A"></use><g transform="translate(2353,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="3585" y="0" xlink:href="#MJMAIN-22C5"></use><use x="4091" y="0" xlink:href="#MJMAIN-33"></use><use x="4873" y="0" xlink:href="#MJMAIN-3A"></use><use x="5434" y="0" xlink:href="#MJMAIN-32"></use><use x="6217" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(7278,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-34"></use></g><use x="8510" y="0" xlink:href="#MJMAIN-22C5"></use><use x="9015" y="0" xlink:href="#MJMAIN-33"></use><use x="9798" y="0" xlink:href="#MJMAIN-3A"></use><use x="10359" y="0" xlink:href="#MJMAIN-32"></use><use x="11141" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(12202,0)"><use xlink:href="#MJMAIN-34"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="13490" y="0" xlink:href="#MJMAIN-3A"></use><use x="14051" y="0" xlink:href="#MJMAIN-32"></use><use x="14833" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(15894,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-31"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>168</mn><mo>:</mo><mn>12</mn><mo>·</mo><mn>3</mn><mo>:</mo><mn>2</mn><mo>=</mo><mn>14</mn><mo>·</mo><mn>3</mn><mo>:</mo><mn>2</mn><mo>=</mo><mn>42</mn><mo>:</mo><mn>2</mn><mo>=</mo><mn>21</mn></mrow></math></script></p>
<h3 class="sec_title" id="sec9">9. Le parentesi<a id="ind66"></a><!--<?"parentesi",4,0,2>-->
</h3>
<p class="noindent">Per indicare che le operazioni si devono eseguire in un ordine diverso da quello dato dal loro grado di priorità, si utilizzano le parentesi. Le parentesi, in un’espressione, devono sempre comparire in coppie: a ogni parentesi aperta deve corrispondere una parentesi chiusa. Si devono eseguire per prime le operazioni indicate nelle coppie di parentesi più interne, ossia in quelle coppie, formate da una parentesi aperta e una chiusa, all’interno delle quali non vi siano altre parentesi. Tali coppie di parentesi devono quindi essere sostituite con i risultati rispettivamente ottenuti. Si prosegue in questo modo fino a quando non vi sono più parentesi.</p>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"><div class="sidebox">
<p class="noindent"><span class="red"><b>NON FARLO!</b></span></p>
<p class="noindentf">Sarebbe un <b>grave errore</b> il passaggio</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.333ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="16.333ex" height="1.833ex" viewBox="0 -699.9 7050.9 805.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-35"></use><use x="727" y="0" xlink:href="#MJMAIN-2B"></use><use x="1732" y="0" xlink:href="#MJMAIN-37"></use><use x="2459" y="0" xlink:href="#MJMAIN-22C5"></use><use x="2964" y="0" xlink:href="#MJMAIN-33"></use><use x="3747" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(4808,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="6040" y="0" xlink:href="#MJMAIN-22C5"></use><use x="6545" y="0" xlink:href="#MJMAIN-33"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>5</mn><mo>+</mo><mn>7</mn><mo>⋅</mo><mn>3</mn><mo>=</mo><mn>12</mn><mo>⋅</mo><mn>3</mn></mrow></math></script></p>
</div></div>
</div>
</div>
<div data-page-container="14" id="page-14" class="row chapters-content"><div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">14</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<div class="example">
<h4 class="noindent">
<span class="pagebreak" epub:type="pagebreak" title="14" id="page14"></span><span class="gep"><span class="sgr">ESEMPIO</span></span>
</h4>
<p class="noindent">Per calcolare il valore dell’espressione</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="49.167ex" height="4.333ex" viewBox="0 -1173.4 21133.7 1846.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJSZ2-7B"></use><use x="672" y="0" xlink:href="#MJMAIN-28"></use><use x="1066" y="0" xlink:href="#MJMAIN-32"></use><use x="1793" y="0" xlink:href="#MJMAIN-2B"></use><use x="2798" y="0" xlink:href="#MJMAIN-33"></use><use x="3303" y="0" xlink:href="#MJMAIN-29"></use><use x="3919" y="0" xlink:href="#MJMAIN-22C5"></use><use x="4424" y="0" xlink:href="#MJMAIN-5B"></use><use x="4707" y="0" xlink:href="#MJMAIN-33"></use><use x="5435" y="0" xlink:href="#MJMAIN-2B"></use><use x="6440" y="0" xlink:href="#MJMAIN-28"></use><use x="6834" y="0" xlink:href="#MJMAIN-34"></use><use x="7561" y="0" xlink:href="#MJMAIN-2B"></use><use x="8566" y="0" xlink:href="#MJMAIN-32"></use><use x="9071" y="0" xlink:href="#MJMAIN-29"></use><use x="9688" y="0" xlink:href="#MJMAIN-22C5"></use><use x="10193" y="0" xlink:href="#MJMAIN-37"></use><use x="10698" y="0" xlink:href="#MJMAIN-5D"></use><use x="11259" y="0" xlink:href="#MJMAIN-3A"></use><use x="11819" y="0" xlink:href="#MJMAIN-39"></use><use x="12547" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(13552,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-35"></use></g><use x="14562" y="-1" xlink:href="#MJSZ2-7D"></use><use x="15456" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(15961,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-36"></use><use x="1121" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1626" y="0" xlink:href="#MJMAIN-33"></use><use x="2353" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(3358,0)"><use xlink:href="#MJMAIN-34"></use><use transform="scale(0.707)" x="714" y="585" xlink:href="#MJMAIN-32"></use></g><use x="4320" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="6667" y="875" xlink:href="#MJMAIN-32"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo>{</mo><mo stretchy="false">(</mo><mn>2</mn><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo><mo>·</mo><mo stretchy="false">[</mo><mn>3</mn><mo>+</mo><mo stretchy="false">(</mo><mn>4</mn><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mo>·</mo><mn>7</mn><mo stretchy="false">]</mo><mo>:</mo><mn>9</mn><mo>+</mo><mn>15</mn><mo>}</mo><mo>·</mo><msup><mrow><mo stretchy="false">(</mo><mn>6</mn><mo>·</mo><mn>3</mn><mo>−</mo><msup><mn>4</mn><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></math></script></p>
<p class="noindent">si segue il procedimento schematizzato in <a href="#ch1.fg11"><span class="fron">FIGURA 11</span></a>.</p>
<div class="figure">
<p class="img" id="ch1.fg11"><img src="images/c01u01f11.jpg" alt="Image"></p>
<p class="figcap">FIGURA 11</p>
</div>
</div>
<p class="noindent">L’uso di parentesi di tipo diverso, come le tonde, le quadre e le graffe, non è strettamente necessario. In matematica è comune anche usare un solo tipo di parentesi, solitamente le tonde. L’espressione dell’esempio precedente si può scrivere utilizzando solo le parentesi tonde in questo modo:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -5.833ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="48.333ex" height="8.333ex" viewBox="0 -1110.9 20799.7 3622.8"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><g transform="translate(394,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-32"></use><use x="1121" y="0" xlink:href="#MJMAIN-2B"></use><use x="2126" y="0" xlink:href="#MJMAIN-33"></use><use x="2631" y="0" xlink:href="#MJMAIN-29"></use><g transform="translate(12,-783)"><use x="19" y="0" xlink:href="#MJSZ4-E152"></use><g transform="translate(490.8300264550264,0) scale(1.366005291005291,1)"><use xlink:href="#MJSZ4-E154"></use></g><g transform="translate(1057,0)"><use 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xlink:href="#MJMAIN-39"></use><use x="12491" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(13496,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-35"></use></g><use x="14506" y="0" xlink:href="#MJMAIN-29"></use><g transform="translate(12,-2275)"><use x="19" y="0" xlink:href="#MJSZ4-E152"></use><g transform="translate(561.5132275132275,0) scale(15.502645502645503,1)"><use xlink:href="#MJSZ4-E154"></use></g><g transform="translate(6995,0)"><use xlink:href="#MJSZ4-E151"></use><use x="455" y="0" xlink:href="#MJSZ4-E150"></use></g><g transform="translate(7982.624338624339,0) scale(15.502645502645503,1)"><use xlink:href="#MJSZ4-E154"></use></g><use x="14421" y="0" xlink:href="#MJSZ4-E153"></use></g><use x="15122" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(15627,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-36"></use><use x="1121" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1626" y="0" xlink:href="#MJMAIN-33"></use><use x="2353" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(3358,0)"><use xlink:href="#MJMAIN-34"></use><use transform="scale(0.707)" x="714" y="585" xlink:href="#MJMAIN-32"></use></g><use x="4320" y="0" xlink:href="#MJMAIN-29"></use><g transform="translate(12,-783)"><use x="19" y="0" xlink:href="#MJSZ4-E152"></use><g transform="translate(500.88677269873864,0) scale(3.3773545397477327,1)"><use xlink:href="#MJSZ4-E154"></use></g><g transform="translate(1902,0)"><use xlink:href="#MJSZ4-E151"></use><use x="455" y="0" xlink:href="#MJSZ4-E150"></use></g><g transform="translate(2829.3756793927864,0) scale(3.3773545397477327,1)"><use xlink:href="#MJSZ4-E154"></use></g><use x="4235" y="0" xlink:href="#MJSZ4-E153"></use></g><use transform="scale(0.707)" x="6667" y="875" xlink:href="#MJMAIN-32"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><munder><munder><mrow><mo stretchy="false">(</mo><munder><munder><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><mo stretchy="true">︸</mo></munder></munder><mo>⋅</mo><munder><munder><mrow><mo stretchy="false">(</mo><mn>3</mn><mo>+</mo><munder><munder><mrow><mo stretchy="false">(</mo><mn>4</mn><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mo stretchy="true">︸</mo></munder></munder><mo>⋅</mo><mn>7</mn><mo stretchy="false">)</mo></mrow><mo stretchy="true">︸</mo></munder></munder><mo>:</mo><mn>9</mn><mo>+</mo><mn>15</mn><mo stretchy="false">)</mo></mrow><mo stretchy="true">︸</mo></munder></munder><mo>⋅</mo><msup><mrow><munder><munder><mrow><mo stretchy="false">(</mo><mn>6</mn><mo>⋅</mo><mn>3</mn><mo>−</mo><msup><mn>4</mn><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><mo stretchy="true">︸</mo></munder></munder></mrow><mn>2</mn></msup></mrow></math></script></p>
<h3 class="sec_title" id="sec10">10. Altre proprietà delle operazioni</h3>
<p class="noindent">Oltre alle proprietà studiate nei precedenti paragrafi, ve ne sono altre che possono risultare utili per semplificare il calcolo delle espressioni.</p>
<ul class="blist">
<li>
<p class="noindent"><b>Per dividere un prodotto per un numero</b>, si può dividere uno solo dei fattori per quel numero:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -2.5ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="46.5ex" height="6ex" viewBox="0 -1539.2 20037 2578.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(167,0)"><g transform="translate(-11,0)"><g transform="translate(4172,725)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMATHI-61"></use><use x="1150" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1655" y="0" xlink:href="#MJMATHI-62"></use><use x="2311" y="0" xlink:href="#MJMAIN-22C5"></use><use x="2816" y="0" xlink:href="#MJMATHI-63"></use><use x="3254" y="0" xlink:href="#MJMAIN-29"></use><use x="3926" y="0" xlink:href="#MJMAIN-3A"></use><use x="4487" y="0" xlink:href="#MJMATHI-64"></use><use x="5293" y="0" xlink:href="#MJMAIN-3D"></use><use x="6353" y="0" xlink:href="#MJMATHI-61"></use><use x="7110" y="0" xlink:href="#MJMAIN-22C5"></use><use x="7615" y="0" xlink:href="#MJMAIN-28"></use><use x="8009" y="0" xlink:href="#MJMATHI-62"></use><use x="8721" y="0" xlink:href="#MJMAIN-3A"></use><use x="9281" y="0" xlink:href="#MJMATHI-64"></use><use x="9809" y="0" xlink:href="#MJMAIN-29"></use><use x="10426" y="0" xlink:href="#MJMAIN-22C5"></use><use x="10931" y="0" xlink:href="#MJMATHI-63"></use></g><g transform="translate(0,-766)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-37"></use><use x="1121" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(1626,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-35"></use></g><use x="2858" y="0" xlink:href="#MJMAIN-22C5"></use><use x="3363" y="0" xlink:href="#MJMAIN-36"></use><use x="3868" y="0" xlink:href="#MJMAIN-29"></use><use x="4540" y="0" xlink:href="#MJMAIN-3A"></use><use x="5101" y="0" xlink:href="#MJMAIN-35"></use><use x="5884" y="0" xlink:href="#MJMAIN-3D"></use><use x="6944" y="0" xlink:href="#MJMAIN-37"></use><use x="7672" y="0" xlink:href="#MJMAIN-22C5"></use><use x="8177" y="0" xlink:href="#MJMAIN-28"></use><g transform="translate(8571,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-35"></use></g><use x="9859" y="0" xlink:href="#MJMAIN-3A"></use><use x="10419" y="0" xlink:href="#MJMAIN-35"></use><use x="10924" y="0" xlink:href="#MJMAIN-29"></use><use x="11541" y="0" xlink:href="#MJMAIN-22C5"></use><use x="12046" y="0" xlink:href="#MJMAIN-36"></use><use x="12829" y="0" xlink:href="#MJMAIN-3D"></use><use x="13889" y="0" xlink:href="#MJMAIN-37"></use><use x="14617" y="0" xlink:href="#MJMAIN-22C5"></use><use x="15122" y="0" xlink:href="#MJMAIN-33"></use><use x="15849" y="0" xlink:href="#MJMAIN-22C5"></use><use x="16354" y="0" xlink:href="#MJMAIN-36"></use><use x="17137" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(18198,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use><use x="1010" y="0" xlink:href="#MJMAIN-36"></use></g></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>⋅</mo><mi>b</mi><mo>⋅</mo><mi>c</mi><mo stretchy="false">)</mo><mo>:</mo><mi>d</mi><mo>=</mo><mi>a</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>b</mi><mo>:</mo><mi>d</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>c</mi></mrow></mtd></mtr><mtr><mtd><mrow><mo stretchy="false">(</mo><mn>7</mn><mo>⋅</mo><mn>15</mn><mo>⋅</mo><mn>6</mn><mo stretchy="false">)</mo><mo>:</mo><mn>5</mn><mo>=</mo><mn>7</mn><mo>⋅</mo><mo stretchy="false">(</mo><mn>15</mn><mo>:</mo><mn>5</mn><mo stretchy="false">)</mo><mo>⋅</mo><mn>6</mn><mo>=</mo><mn>7</mn><mo>⋅</mo><mn>3</mn><mo>⋅</mo><mn>6</mn><mo>=</mo><mn>126</mn></mrow></mtd></mtr></mtable></mrow></math></script></p>
<p class="noindent">In particolare, <b>per dividere un prodotto per uno dei suoi fattori</b>, è sufficiente sopprimere quel fattore:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -2.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="31.333ex" height="6.5ex" viewBox="0 -1638.2 13459.5 2776.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(167,0)"><g transform="translate(-11,0)"><g transform="translate(2588,824)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMATHI-61"></use><use x="1150" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1655" y="0" xlink:href="#MJMATHI-62"></use><use x="2311" y="0" xlink:href="#MJMAIN-22C5"></use><use x="2816" y="0" xlink:href="#MJMATHI-63"></use><use x="3254" y="0" xlink:href="#MJMAIN-29"></use><use x="3926" y="0" xlink:href="#MJMAIN-3A"></use><use x="4487" y="0" xlink:href="#MJMATHI-62"></use><use x="5199" y="0" xlink:href="#MJMAIN-3D"></use><use x="6259" y="0" xlink:href="#MJMATHI-61"></use><use x="7016" y="0" xlink:href="#MJMAIN-22C5"></use><use x="7521" y="0" xlink:href="#MJMATHI-63"></use></g><g transform="translate(0,-818)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-37"></use><use x="1121" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(1848,0)"><g transform="translate(275,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-35"></use></g><line fill="none" stroke="black" stroke-width="75" x1="37" y1="-270" x2="1522" y2="913"></line></g><use x="3630" y="0" xlink:href="#MJMAIN-22C5"></use><use x="4136" y="0" xlink:href="#MJMAIN-36"></use><use x="4641" y="0" xlink:href="#MJMAIN-29"></use><use x="5312" y="0" xlink:href="#MJMAIN-3A"></use><g transform="translate(6151,0)"><g transform="translate(275,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-35"></use></g><line fill="none" stroke="black" stroke-width="75" x1="37" y1="-270" x2="1522" y2="913"></line></g><use x="7989" y="0" xlink:href="#MJMAIN-3D"></use><use x="9050" y="0" xlink:href="#MJMAIN-37"></use><use x="9777" y="0" xlink:href="#MJMAIN-22C5"></use><use x="10282" y="0" xlink:href="#MJMAIN-36"></use><use x="11065" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(12126,0)"><use xlink:href="#MJMAIN-34"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>⋅</mo><mi>b</mi><mo>⋅</mo><mi>c</mi><mo stretchy="false">)</mo><mo>:</mo><mi>b</mi><mo>=</mo><mi>a</mi><mo>⋅</mo><mi>c</mi></mrow></mtd></mtr><mtr><mtd><mrow><mo stretchy="false">(</mo><mn>7</mn><mo>⋅</mo><menclose notation="updiagonalstrike"><mrow><mn>15</mn></mrow></menclose><mo>⋅</mo><mn>6</mn><mo stretchy="false">)</mo><mo>:</mo><menclose notation="updiagonalstrike"><mrow><mn>15</mn></mrow></menclose><mo>=</mo><mn>7</mn><mo>⋅</mo><mn>6</mn><mo>=</mo><mn>42</mn></mrow></mtd></mtr></mtable></mrow></math></script></p>
</li>
<li>
<p class="noindent"><b>Per dividere un numero per un prodotto</b>, si può dividere successivamente quel numero per ciascun fattore:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -2.5ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="37.333ex" height="6ex" viewBox="0 -1539.2 16056.9 2578.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(167,0)"><g transform="translate(-11,0)"><g transform="translate(3381,725)"><use xlink:href="#MJMATHI-61"></use><use x="811" y="0" xlink:href="#MJMAIN-3A"></use><use x="1372" y="0" xlink:href="#MJMAIN-28"></use><use x="1766" y="0" xlink:href="#MJMATHI-62"></use><use x="2422" y="0" xlink:href="#MJMAIN-22C5"></use><use x="2928" y="0" xlink:href="#MJMATHI-63"></use><use x="3366" y="0" xlink:href="#MJMAIN-29"></use><use x="4037" y="0" xlink:href="#MJMAIN-3D"></use><use x="5098" y="0" xlink:href="#MJMAIN-28"></use><use x="5492" y="0" xlink:href="#MJMATHI-61"></use><use x="6304" y="0" xlink:href="#MJMAIN-3A"></use><use x="6865" y="0" xlink:href="#MJMATHI-62"></use><use x="7299" y="0" xlink:href="#MJMAIN-29"></use><use x="7970" y="0" xlink:href="#MJMAIN-3A"></use><use x="8531" y="0" xlink:href="#MJMATHI-63"></use></g><g transform="translate(0,-766)"><use xlink:href="#MJMAIN-34"></use><use x="505" y="0" xlink:href="#MJMAIN-38"></use><use x="1287" y="0" xlink:href="#MJMAIN-3A"></use><use x="1848" y="0" xlink:href="#MJMAIN-28"></use><use x="2242" y="0" xlink:href="#MJMAIN-32"></use><use x="2969" y="0" xlink:href="#MJMAIN-22C5"></use><use x="3475" y="0" xlink:href="#MJMAIN-33"></use><use x="3980" y="0" xlink:href="#MJMAIN-29"></use><use x="4651" y="0" xlink:href="#MJMAIN-3D"></use><use x="5712" y="0" xlink:href="#MJMAIN-28"></use><g transform="translate(6106,0)"><use xlink:href="#MJMAIN-34"></use><use x="505" y="0" xlink:href="#MJMAIN-38"></use></g><use x="7394" y="0" xlink:href="#MJMAIN-3A"></use><use x="7955" y="0" xlink:href="#MJMAIN-32"></use><use x="8460" y="0" xlink:href="#MJMAIN-29"></use><use x="9131" y="0" xlink:href="#MJMAIN-3A"></use><use x="9692" y="0" xlink:href="#MJMAIN-33"></use><use x="10475" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(11536,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-34"></use></g><use x="12823" y="0" xlink:href="#MJMAIN-3A"></use><use x="13384" y="0" xlink:href="#MJMAIN-33"></use><use x="14167" y="0" xlink:href="#MJMAIN-3D"></use><use x="15228" y="0" xlink:href="#MJMAIN-38"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd><mrow><mi>a</mi><mo>:</mo><mo stretchy="false">(</mo><mi>b</mi><mo>⋅</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo>:</mo><mi>b</mi><mo stretchy="false">)</mo><mo>:</mo><mi>c</mi></mrow></mtd></mtr><mtr><mtd><mrow><mn>48</mn><mo>:</mo><mo stretchy="false">(</mo><mn>2</mn><mo>⋅</mo><mn>3</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>48</mn><mo>:</mo><mn>2</mn><mo stretchy="false">)</mo><mo>:</mo><mn>3</mn><mo>=</mo><mn>24</mn><mo>:</mo><mn>3</mn><mo>=</mo><mn>8</mn></mrow></mtd></mtr></mtable></mrow></math></script></p>
</li>
<li>
<p class="noindent"><b>Per moltiplicare un numero per un quoziente</b>, si può moltiplicare il numero per il dividendo e poi dividere il prodotto ottenuto per il divisore, oppure dividere il numero per il divisore e successivamente moltiplicare il risultato per il dividendo:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -2.5ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="72.167ex" height="6ex" viewBox="0 -1539.2 31072.7 2578.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-61"></use><use x="756" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1261" y="0" xlink:href="#MJMAIN-28"></use><use x="1655" y="0" xlink:href="#MJMATHI-62"></use><use x="2367" y="0" xlink:href="#MJMAIN-3A"></use><use x="2928" y="0" xlink:href="#MJMATHI-63"></use><use x="3366" y="0" xlink:href="#MJMAIN-29"></use><use x="4037" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(5265,0)"><g transform="translate(-11,0)"><use x="0" y="725" xlink:href="#MJMAIN-2197"></use><use x="0" y="-766" xlink:href="#MJMAIN-2198"></use></g><g 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xlink:href="#MJMAIN-34"></use></g></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>a</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>b</mi><mo>:</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mo>↗</mo></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>⋅</mo><mi>b</mi><mo stretchy="false">)</mo><mo>:</mo><mi>c</mi></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mo>↘</mo></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>:</mo><mi>c</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>b</mi></mrow></mtd></mtr></mtable><mtext>  </mtext><mn>6</mn><mo>⋅</mo><mo stretchy="false">(</mo><mn>12</mn><mo>:</mo><mn>3</mn><mo stretchy="false">)</mo><mo>=</mo><mtext> </mtext><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mo>↗</mo></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mn>6</mn><mo>⋅</mo><mn>12</mn><mo stretchy="false">)</mo><mo>:</mo><mn>3</mn><mo>=</mo><mn>72</mn><mo>:</mo><mn>3</mn><mo>=</mo><mn color="#00aef0">24</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mo>↘</mo></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mn>6</mn><mo>:</mo><mn>3</mn><mo stretchy="false">)</mo><mo>⋅</mo><mn>12</mn><mo>=</mo><mn>2</mn><mo>⋅</mo><mn>12</mn><mo>=</mo><mn color="#00aef0">24</mn></mrow></mtd></mtr></mtable></mrow></math></script></p>
</li>
</ul>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"></div>
</div></div>
<div data-page-container="15" id="page-15" class="row chapters-content">
<div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">15</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<ul class="blist">
<li>
<p class="noindent"><span class="pagebreak" epub:type="pagebreak" title="15" id="page15"></span><b>Per dividere un numero per un quoziente</b>, si può dividere il numero per il dividendo e poi moltiplicare il risultato ottenuto per il divisore, oppure moltiplicare il numero per il divisore e successivamente dividere il risultato per il dividendo:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -2.5ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="75ex" height="6ex" viewBox="0 -1539.2 32305 2578.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-61"></use><use x="811" y="0" xlink:href="#MJMAIN-3A"></use><use x="1372" y="0" xlink:href="#MJMAIN-28"></use><use x="1766" y="0" xlink:href="#MJMATHI-62"></use><use x="2478" y="0" xlink:href="#MJMAIN-3A"></use><use x="3039" y="0" xlink:href="#MJMATHI-63"></use><use x="3477" y="0" xlink:href="#MJMAIN-29"></use><use x="4148" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(5376,0)"><g transform="translate(-11,0)"><use x="0" y="725" xlink:href="#MJMAIN-2197"></use><use x="0" y="-766" xlink:href="#MJMAIN-2198"></use></g><g 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xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(10414,0)"><use xlink:href="#MJMAIN-36"></use></g></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>a</mi><mo>:</mo><mo stretchy="false">(</mo><mi>b</mi><mo>:</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mo>↗</mo></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>:</mo><mi>b</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>c</mi></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mo>↘</mo></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>⋅</mo><mi>c</mi><mo stretchy="false">)</mo><mo>:</mo><mi>b</mi></mrow></mtd></mtr></mtable><mtext> </mtext><mtext> </mtext><mn>24</mn><mo>:</mo><mo stretchy="false">(</mo><mn>12</mn><mo>:</mo><mn>3</mn><mo stretchy="false">)</mo><mo>=</mo><mtext> </mtext><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mo>↗</mo></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mn>24</mn><mo>:</mo><mn>12</mn><mo stretchy="false">)</mo><mo>⋅</mo><mn>3</mn><mo>=</mo><mn>2</mn><mo>⋅</mo><mn>3</mn><mo>=</mo><mn color="#00aef0">6</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mo>↘</mo></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mn>24</mn><mo>⋅</mo><mn>3</mn><mo stretchy="false">)</mo><mo>:</mo><mn>12</mn><mo>=</mo><mn>72</mn><mo>:</mo><mn>12</mn><mo>=</mo><mn color="#00aef0">6</mn></mrow></mtd></mtr></mtable></mrow></math></script></p>
</li>
</ul>
<p class="noindent">Naturalmente queste proprietà si possono applicare solo se le divisioni da eseguire sono possibili.</p>
<h2 class="para_title" id="par05">Divisibilità e numeri primi</h2>
<h3 class="sec_title" id="sec11">11. Multipli e divisori</h3>
<div class="definition" title_dea="Multiplo" key_dea="multiplo, multipli, numero naturale, multiplo di un numero naturale">
<h4 class="noindent">
<span class="ash">DEFINIZIONE</span> <span class="orangeb">MULTIPLO<a id="ind67"></a><!--<?"multiplo",4,0,2>--></span>
</h4>
<p class="noindentin">Dati due numeri naturali <i>a</i> e <i>b</i>, con <i>b</i> ≠ 0, il numero <i>a</i> è multiplo del numero <i>b</i> se <i>a</i> è il prodotto di <i>b</i> per un numero naturale <i>n</i>, ossia se esiste un numero naturale <i>n</i> tale che <i>a</i> = <i>b</i> · <i>n</i>.</p>
</div>
<p class="noindent">I <b>multipli</b> di un numero<a id="ind68"></a><!--<?"multipli|di un numero",4,0,2>--> <i><b>b</b></i> sono quindi tutti i numeri che si ottengono moltiplicando <i><b>b</b></i> per 0, 1, 2, 3, ...</p>
<p class="noindent">Perciò</p>
<ul class="blist">
<li><p class="noindent">36 è multiplo di 9, perché 36 = 9 · 4;</p></li>
<li>
<p class="noindent">i multipli di 6 sono:</p>
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<p class="noindent">cioè</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="62.833ex" height="3ex" viewBox="0 -875 27077.4 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-30"></use><use x="505" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(954,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="1564" y="0" xlink:href="#MJMAIN-36"></use><use x="2069" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(2519,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" 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xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>0</mn><mo>,</mo><mi> </mi><mn>6</mn><mo>,</mo><mi> </mi><mn>12</mn><mo>,</mo><mi> </mi><mn>18</mn><mo>,</mo><mi> </mi><mn>24</mn><mo>,</mo><mi> </mi><mn>30</mn><mo>,</mo><mi> </mi><mn>36</mn><mo>,</mo><mi> </mi><mn>42</mn><mo>,</mo><mi> </mi><mn>48</mn><mo>,</mo><mi> </mi><mn>54</mn><mo>,</mo><mi> </mi><mn>60</mn><mo>,</mo><mi> </mi><mn>66</mn><mo>,</mo><mi> </mi><mn>72</mn><mo>,</mo><mi> </mi><mo>.</mo><mo>.</mo><mo>.</mo></mrow></math></script></p>
</li>
</ul>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"></div>
</div>
<div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<div class="definition" title_dea="Divisibilità" key_dea="divisibilità, divisibile">
<h4 class="noindent">
<span class="ash">DEFINIZIONE</span> <span class="orangeb">DIVISIBILITÀ<a id="ind69"></a><!--<?"divisibilit&#x00E0;",4,0,2>--></span>
</h4>
<p class="noindentin">Dati due numeri naturali <i>a</i> e <i>b</i>, con <i>b</i> ≠ 0, il numero <i>a</i> è divisibile per <i>b</i> (e <i>b</i> è divisore<a id="ind70"></a><!--<?"divisore",4,0,2>--> di <i>a</i>) se si può eseguire la divisione esatta <i>a</i> : <i>b</i>.</p>
</div>
<p class="noindent">Quindi, ad esempio,</p>
<ul class="blist">
<li><p class="noindent">36 è divisibile per 9, perché si può eseguire la divisione esatta 36 : 9 = 4;</p></li>
<li><p class="noindent">36 non è divisibile per 7, perché la divisione approssimata 36 : 7 dà 5 con resto 1, diverso da 0;</p></li>
<li><p class="noindent">i divisori di 36 sono 1, 2, 3, 4, 6, 9, 12, 18, 36.</p></li>
</ul>
<p class="noindent">I numeri divisibili per 2 si dicono <b>pari</b><a id="ind71"></a><!--<?"numeri|pari",4,0,2>-->. I numeri che <i>non</i> sono divisibili per 2 si dicono <b>dispari</b>.<a id="ind72"></a><!--<?"numeri|dispari",4,0,2>--> Lo zero, essendo divisibile per 2, è considerato un numero pari.</p>
<h3 class="sec_title" id="sec12">12. Criteri di divisibilità<a id="ind73"></a><!--<?"criteri di divisibilit&#x00E0;",4,0,2>-->
</h3>
<ul class="blist">
<li><p class="noindent"><b>Divisibilità per 2</b>. Un numero è divisibile per 2 se la sua ultima cifra è pari.</p></li>
<li><p class="noindent"><b>Divisibilità per 3</b>. Un numero è divisibile per 3 se la somma delle sue cifre è divisibile per 3.</p></li>
</ul>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"><div class="sidebox">
<p class="noindentf"><span class="red"><b>IMPORTANTE!</b></span></p>
<p class="noindent">Dati due numeri naturali <i>a</i> e <i>b</i>, con <i>b</i> ≠ 0, è la stessa cosa dire che</p>
<ul class="ulist">
<li><p class="noindentf"><i>a</i> è divisibile per <i>b</i></p></li>
<li><p class="noindentf"><i>a</i> è multiplo di <i>b</i></p></li>
<li><p class="noindentf"><i>b</i> è divisore di <i>a</i></p></li>
<li><p class="noindentf">si può eseguire la divisione esatta <i>a</i> : <i>b</i></p></li>
<li><p class="noindentf">la divisione approssimata <i>a</i> : <i>b</i> dà come resto 0</p></li>
</ul>
</div></div>
</div>
</div>
<div data-page-container="16" id="page-16" class="row chapters-content"><div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">16</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<ul class="blist">
<li><p class="noindent"><span class="pagebreak" epub:type="pagebreak" title="16" id="page16"></span><b>Divisibilità per 4</b>. Un numero è divisibile per 4 se il numero formato dalle sue ultime due cifre è divisibile per 4 oppure se le sue ultime due cifre sono due zeri.</p></li>
<li><p class="noindent"><b>Divisibilità per 5</b>. Un numero è divisibile per 5 se la sua ultima cifra è 0 o 5.</p></li>
<li><p class="noindent"><b>Divisibilità per 9</b>. Un numero è divisibile per 9 se la somma delle sue cifre è divisibile per 9.</p></li>
<li><p class="noindent"><b>Divisibilità per 10</b>. Un numero è divisibile per 10 se la sua ultima cifra è 0.</p></li>
<li><p class="noindent"><b>Divisibilità per 11</b>. Un numero è divisibile per 11 se lo è la differenza tra la somma delle sue cifre di posto dispari (contandole per esempio da destra a sinistra), eventualmente aumentata di un multiplo di 11, e la somma delle cifre di posto pari.</p></li>
<li><p class="noindent"><b>Divisibilità per 25</b>. Un numero è divisibile per 25 se le sue ultime due cifre sono 00 o 25 o 50 o 75.</p></li>
</ul>
<div class="example">
<h4 class="noindent"><span class="gep"><span class="sgr">ESEMPIO</span></span></h4>
<p class="noindent">Consideriamo il numero 1716.</p>
<ul class="blist">
<li><p class="noindent">L’ultima cifra, 6, è pari → 1716 <span class="cyan">è divisibile per 2</span>.</p></li>
<li><p class="noindent">La somma delle cifre 1 + 7 + 1 + 6 = 15 è divisibile per 3 → 1716 <span class="cyan">è divisibile per 3</span>.</p></li>
<li><p class="noindent">Il numero formato dalle ultime due cifre, 16, è divisibile per 4 → 1716 è <span class="cyan">divisibile per 4</span>.</p></li>
<li><p class="noindent">L’ultima cifra non è né 0 né 5 → 1716 <span class="cyan">non è divisibile per 5</span>.</p></li>
<li><p class="noindent">La somma delle cifre 1 + 7 + 1 + 6 = 15 non è divisibile per 9 → 1716 <span class="cyan">non è divisibile per 9</span>.</p></li>
<li><p class="noindent">L’ultima cifra non è 0 → 1716 <span class="cyan">non è divisibile per 10</span>.</p></li>
<li><p class="noindent">La somma delle cifre di posto dispari è 6 + 7 = 13, la somma delle cifre di posto pari è 1 + 1 = 2 e si ha 13 − 2 = 11, che è divisibile per 11 → 1716 <span class="cyan">è divisibile per 11</span>.</p></li>
<li><p class="noindent">Il numero formato dalle ultime due cifre è 16 → 1716 <span class="cyan">non è divisibile per 25</span>.</p></li>
</ul>
</div>
<h3 class="sec_title" id="sec13">13. Scomposizione in fattori primi<a id="ind74"></a><!--<?"scomposizione|in fattori primi",4,0,2>-->
</h3>
<div class="definition" title_dea="Numero primo" key_dea="numero primo, numeri primi">
<h4 class="noindent">
<span class="ash">DEFINIZIONE</span> <span class="orangeb">NUMERO PRIMO<a id="ind75"></a><!--<?"numero|primo",4,0,2>--></span>
</h4>
<p class="noindentin">Un numero naturale si dice primo se è divisibile solo per se stesso e per 1. Il numero 1, per convenzione, <i>non</i> si considera un numero primo.</p>
</div>
<p class="noindent">Ad esempio, il numero 7 è divisibile solo per 1 e per 7, quindi 7 è un numero primo. Il numero 6 è divisibile, oltre che per 1 e per 6, anche per 2 e per 3, quindi 6 non è un numero primo.</p>
<p class="noindent"><b>Ogni numero naturale, diverso da 0, che non sia primo si può esprimere, in un solo modo, come prodotto di fattori primi</b><a id="ind76"></a><!--<?"prodotto|di fattori primi",4,0,2>-->.</p>
<p class="noindent">Scomporre in fattori primi un numero naturale significa determinare tali fattori.</p>
<p class="noindent">Per scomporre un numero in fattori primi<a id="ind77"></a><!--<?"fattori|primi",4,0,2>--> si cercano i suoi divisori utilizzando i criteri di divisibilità, partendo dal primo numero della successione dei numeri primi, cioè 2, e procedendo in ordine crescente. Si esegue la divisione del numero dato per il più piccolo numero primo che risulta suo divisore. Si divide il quoziente ottenuto per il suo divisore primo più piccolo e si continua, ripetendo il procedimento, finché il quoziente risulta uguale a 1.</p>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"></div>
</div></div>
<div data-page-container="17" id="page-17" class="row chapters-content"><div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">17</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<div class="example">
<h4 class="noindent">
<span class="pagebreak" epub:type="pagebreak" title="17" id="page17"></span><span class="gep"><span class="sgr">ESEMPIO</span></span>
</h4>
<p class="noindent">Il procedimento per scomporre in fattori primi il numero 6552 può essere schematizzato come in <a href="#ch1.fg12"><span class="fron">FIGURA 12</span></a>. Pertanto</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="20.833ex" height="2.167ex" viewBox="0 -894.2 8980.1 940.1"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-36"></use><use x="505" y="0" xlink:href="#MJMAIN-35"></use><use x="1010" y="0" xlink:href="#MJMAIN-35"></use><use x="1515" y="0" xlink:href="#MJMAIN-32"></use><use x="2297" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(3358,0)"><use xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-33"></use></g><use x="4542" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(5048,0)"><use xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="714" y="568" xlink:href="#MJMAIN-32"></use></g><use x="6232" y="0" xlink:href="#MJMAIN-22C5"></use><use x="6737" y="0" xlink:href="#MJMAIN-37"></use><use x="7464" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(7970,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-33"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>6552</mn><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></msup><mo>·</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>·</mo><mn>7</mn><mo>·</mo><mn>13</mn></mrow></math></script></p>
<div class="figure">
<p class="img" id="ch1.fg12"><img src="images/c01u01f12.jpg" alt="Image"></p>
<p class="figcap">FIGURA 12</p>
</div>
</div>
<h2 class="para_title" id="par06">Massimo comune divisore e minimo comune multiplo</h2>
<h3 class="sec_title" id="sec14">14. Massimo comune divisore</h3>
<p class="noindent">Consideriamo due numeri naturali, ad esempio 24 e 36.</p>
<p class="noindent">I divisori di 24 sono</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="29ex" height="3ex" viewBox="0 -875 12467.3 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(954,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="1564" y="0" xlink:href="#MJMAIN-32"></use><use x="2069" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(2519,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="3129" y="0" xlink:href="#MJMAIN-33"></use><use x="3634" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(4083,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="4693" y="0" xlink:href="#MJMAIN-34"></use><use x="5198" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(5648,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="6258" y="0" xlink:href="#MJMAIN-36"></use><use x="6763" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(7213,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="7823" y="0" xlink:href="#MJMAIN-38"></use><use x="8328" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(8777,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(9387,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="10397" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(10847,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(11457,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-34"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn><mo>,</mo><mtext> </mtext><mn>2</mn><mo>,</mo><mtext> </mtext><mn>3</mn><mo>,</mo><mtext> </mtext><mn>4</mn><mo>,</mo><mtext> </mtext><mn>6</mn><mo>,</mo><mtext> </mtext><mn>8</mn><mo>,</mo><mtext> </mtext><mn>12</mn><mo>,</mo><mtext> </mtext><mn>24</mn></mrow></math></script></p>
<p class="noindent">e i divisori di 36 sono</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="33.833ex" height="3ex" viewBox="0 -875 14536.9 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(954,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="1564" y="0" xlink:href="#MJMAIN-32"></use><use x="2069" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(2519,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="3129" y="0" xlink:href="#MJMAIN-33"></use><use x="3634" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(4083,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="4693" y="0" xlink:href="#MJMAIN-34"></use><use x="5198" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(5648,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="6258" y="0" xlink:href="#MJMAIN-36"></use><use x="6763" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(7213,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="7823" y="0" xlink:href="#MJMAIN-39"></use><use x="8328" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(8777,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(9387,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="10397" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(10847,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(11457,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-38"></use></g><use x="12467" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(12916,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(13526,0)"><use xlink:href="#MJMAIN-33"></use><use x="505" y="0" xlink:href="#MJMAIN-36"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn><mo>,</mo><mtext> </mtext><mn>2</mn><mo>,</mo><mtext> </mtext><mn>3</mn><mo>,</mo><mtext> </mtext><mn>4</mn><mo>,</mo><mtext> </mtext><mn>6</mn><mo>,</mo><mtext> </mtext><mn>9</mn><mo>,</mo><mtext> </mtext><mn>12</mn><mo>,</mo><mtext> </mtext><mn>18</mn><mo>,</mo><mtext> </mtext><mn>36</mn></mrow></math></script></p>
<p class="noindent">I numeri 1, 2, 3, 4, 6, 12 sono divisori sia di 24 sia di 36; per questo motivo vengono detti <i>divisori comuni</i><a id="ind78"></a><!--<?"divisori comuni",4,0,2>--> di 24 e 36. Il più grande di essi, ossia 12, è il <b>massimo comune divisore</b> di 24 e 36.</p>
<div class="definition" title_dea="Massimo comune divisore" key_dea="massimo comune divisore, MCD">
<h4 class="noindent">
<span class="ash">DEFINIZIONE</span> <span class="orangeb">MASSIMO COMUNE DIVISORE</span>
</h4>
<p class="noindentin">Il massimo comune divisore (<i>MCD</i>) di due o più numeri naturali,<a id="ind79"></a><!--<?"MCD|di numeri naturali",4,0,2>--> diversi da zero, è il più grande dei loro divisori comuni.</p>
</div>
<p class="noindent">Il <i>MCD</i> di due numeri <i>a</i> e <i>b</i> si indica con la scrittura <i>MCD</i>(<i>a</i> ; <i>b</i>), quello di tre numeri <i>a</i>, <i>b</i>, <i>c</i> si indica con <i>MCD</i>(<i>a</i> ; <i>b</i> ; <i>c</i>) e così via.</p>
<div class="rule" title_dea="Regola per determinare il massimo comune divisore" key_dea="calcolo massimo comune divisore, massimo comune divisore, calcolo MCD, MCD, regola per determinare il massimo comune divisore">
<h4 class="noindent"><span class="red"><b>REGOLA</b></span></h4>
<p class="noindent">Per determinare il <i>MCD</i> di due o più numeri naturali</p>
<ol class="olist">
<li><p class="noindent">si scompongono in fattori primi i numeri dati;</p></li>
<li><p class="noindent">si moltiplicano fra loro tutti i fattori primi comuni ai numeri dati, presi una sola volta, ciascuno con l’esponente minore con cui figura.</p></li>
</ol>
</div>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"></div>
</div></div>
<div data-page-container="18" id="page-18" class="row chapters-content"><div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">18</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<div class="example">
<h4 class="noindent">
<span class="pagebreak" epub:type="pagebreak" title="18" id="page18"></span><span class="gep"><span class="sgr">ESEMPIO</span></span>
</h4>
<p class="hang"><b>1</b>&nbsp;&nbsp;Determiniamo <i>MCD</i>(126; 360; 216).</p>
<ol class="olist">
<li>
<p class="noindent">Scomponiamo in fattori primi i numeri dati:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="57.167ex" height="3.167ex" viewBox="0 -894.2 24622.8 1338.7"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use><use x="1010" y="0" xlink:href="#MJMAIN-36"></use><use x="1792" y="0" xlink:href="#MJMAIN-3D"></use><use x="2853" y="0" xlink:href="#MJMAIN-32"></use><use x="3580" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(4086,0)"><use xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="714" y="568" xlink:href="#MJMAIN-32"></use></g><use x="5270" y="0" xlink:href="#MJMAIN-22C5"></use><use x="5775" y="0" xlink:href="#MJMAIN-37"></use><g transform="translate(6280,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1219,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1829,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(2439,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(9330,0)"><use xlink:href="#MJMAIN-33"></use><use x="505" y="0" xlink:href="#MJMAIN-36"></use><use x="1010" y="0" xlink:href="#MJMAIN-30"></use></g><use x="11123" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(12183,0)"><use xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-33"></use></g><use x="13368" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(13873,0)"><use xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="714" y="568" xlink:href="#MJMAIN-32"></use></g><use x="15057" y="0" xlink:href="#MJMAIN-22C5"></use><use x="15562" y="0" xlink:href="#MJMAIN-35"></use><g transform="translate(16067,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1219,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1829,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(2439,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(19117,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-31"></use><use x="1010" y="0" xlink:href="#MJMAIN-36"></use></g><use x="20910" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(21971,0)"><use xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-33"></use></g><use x="23155" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(23660,0)"><use xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="714" y="568" xlink:href="#MJMAIN-33"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>126</mn><mo>=</mo><mn>2</mn><mo>·</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>·</mo><mn>7</mn><mi>     </mi><mn>360</mn><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></msup><mo>·</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>·</mo><mn>5</mn><mi>     </mi><mn>216</mn><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></msup><mo>·</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>3</mn></mrow></msup></mrow></math></script></p>
</li>
<li>
<p class="noindent">I fattori primi comuni a tutti i numeri dati sono 2 e 3. L’esponente più piccolo con cui compare il fattore 2 è 1, l’esponente più piccolo con cui compare il 3 è 2. Quindi è</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="41.333ex" height="3.167ex" viewBox="0 -893.9 17817.7 1338.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-4D"></use><use x="1056" y="0" xlink:href="#MJMATHI-43"></use><use x="1821" y="0" xlink:href="#MJMATHI-44"></use><g transform="translate(2654,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="3263" y="0" xlink:href="#MJMAIN-28"></use><g transform="translate(3657,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use><use x="1010" y="0" xlink:href="#MJMAIN-36"></use></g><g transform="translate(5172,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="5782" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(6232,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(6842,0)"><use xlink:href="#MJMAIN-33"></use><use x="505" y="0" xlink:href="#MJMAIN-36"></use><use x="1010" y="0" xlink:href="#MJMAIN-30"></use></g><g transform="translate(8357,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="8967" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(9417,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(10027,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-31"></use><use x="1010" y="0" xlink:href="#MJMAIN-36"></use></g><use x="11542" y="0" xlink:href="#MJMAIN-29"></use><use x="12213" y="0" xlink:href="#MJMAIN-3D"></use><use x="13274" y="0" xlink:href="#MJMAIN-32"></use><use x="14001" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(14507,0)"><use xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="714" y="568" xlink:href="#MJMAIN-32"></use></g><use x="15746" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(16807,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-38"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>M</mi><mi>C</mi><mi>D</mi><mtext> </mtext><mo stretchy="false">(</mo><mn>126</mn><mtext> </mtext><mo>;</mo><mtext> </mtext><mn>360</mn><mtext> </mtext><mo>;</mo><mtext> </mtext><mn>216</mn><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mo>⋅</mo><msup><mn>3</mn><mn>2</mn></msup><mo>=</mo><mn color="#00aef0">18</mn></mrow></math></script></p>
</li>
</ol>
</div>
<p class="noindent">Osserviamo che se <i>a</i> è un multiplo di <i>b</i>, ossia <i>b</i> è un divisore di <i>a</i>, allora <i>MCD</i>(<i>a</i> ; <i>b</i>) = <i>b</i>. Ad esempio, <i>MCD</i>(12 ; 4) = 4 e <i>MCD</i>(3 ; 18) = 3.</p>
<div class="definition" title_dea="Numeri primi tra loro" key_dea="numeri primi tra loro, primi tra loro, comprimi, numeri coprimi">
<h4 class="noindent">
<span class="ash">DEFINIZIONE</span> <span class="orangeb">NUMERI PRIMI TRA LORO</span>
</h4>
<p class="noindentin">Due o più numeri naturali sono primi tra loro<a id="ind80"></a><!--<?"numeri|primi tra loro",4,0,2>--> (o <i>coprimi</i>) se il loro massimo comune divisore è 1.</p>
</div>
<div class="example">
<h4 class="noindent"><span class="gep"><span class="sgr">ESEMPIO</span></span></h4>
<p class="hang"><b>2</b>&nbsp;&nbsp;Scomponiamo in fattori primi i numeri 50 e 63:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="28.167ex" height="3.167ex" viewBox="0 -894.9 12135.9 1339.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-35"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use><use x="1287" y="0" xlink:href="#MJMAIN-3D"></use><use x="2348" y="0" xlink:href="#MJMAIN-32"></use><use x="3075" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(3581,0)"><use xlink:href="#MJMAIN-35"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-32"></use></g><g transform="translate(4543,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1219,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1829,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(2439,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(7592,0)"><use xlink:href="#MJMAIN-36"></use><use x="505" y="0" xlink:href="#MJMAIN-33"></use></g><use x="8880" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(9941,0)"><use xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="714" y="568" xlink:href="#MJMAIN-32"></use></g><use x="11125" y="0" xlink:href="#MJMAIN-22C5"></use><use x="11630" y="0" xlink:href="#MJMAIN-37"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>50</mn><mo>=</mo><mn>2</mn><mo>·</mo><msup><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></msup><mi>     </mi><mn>63</mn><mo>=</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>·</mo><mn>7</mn></mrow></math></script></p>
<p class="hangg">I fattori primi di 50 sono 2 e 5; nessuno di essi è fattore primo di 63. Dunque 50 e 63 sono primi fra loro. Si ha quindi <i>MCD</i>(50 ; 63) = <span class="cyan">1</span>.</p>
</div>
<h3 class="sec_title" id="sec15">15. Minimo comune multiplo<a id="ind81"></a><!--<?"minimo comune multiplo",4,0,2>-->
</h3>
<p class="noindent">Consideriamo i multipli diversi da zero di due numeri naturali, ad esempio 8 e 12.</p>
<p class="noindent">I multipli di 8, diversi da 0, sono:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="58.5ex" height="3ex" viewBox="0 -875 25179.4 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-38"></use><use x="505" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(954,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1564,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-36"></use></g><use x="2574" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(3024,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(3634,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-34"></use></g><use x="4644" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(5093,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(5703,0)"><use xlink:href="#MJMAIN-33"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="6713" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(7163,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(7773,0)"><use xlink:href="#MJMAIN-34"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use></g><use x="8783" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(9233,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(9843,0)"><use xlink:href="#MJMAIN-34"></use><use x="505" y="0" xlink:href="#MJMAIN-38"></use></g><use x="10853" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(11302,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(11912,0)"><use xlink:href="#MJMAIN-35"></use><use x="505" y="0" xlink:href="#MJMAIN-36"></use></g><use x="12922" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(13372,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(13982,0)"><use xlink:href="#MJMAIN-36"></use><use x="505" y="0" xlink:href="#MJMAIN-34"></use></g><use x="14992" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(15441,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(16051,0)"><use xlink:href="#MJMAIN-37"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="17061" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(17511,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(18121,0)"><use xlink:href="#MJMAIN-38"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use></g><use x="19131" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(19581,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(20191,0)"><use xlink:href="#MJMAIN-38"></use><use x="505" y="0" xlink:href="#MJMAIN-38"></use></g><use x="21201" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(21650,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(22260,0)"><use xlink:href="#MJMAIN-39"></use><use x="505" y="0" xlink:href="#MJMAIN-36"></use></g><use x="23270" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(23720,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(24330,0)"><use xlink:href="#MJMAIN-2E"></use><use x="283" y="0" xlink:href="#MJMAIN-2E"></use><use x="566" y="0" xlink:href="#MJMAIN-2E"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>8</mn><mo>,</mo><mtext> </mtext><mn>16</mn><mo>,</mo><mtext> </mtext><mn>24</mn><mo>,</mo><mtext> </mtext><mn>32</mn><mo>,</mo><mtext> </mtext><mn>40</mn><mo>,</mo><mtext> </mtext><mn>48</mn><mo>,</mo><mtext> </mtext><mn>56</mn><mo>,</mo><mtext> </mtext><mn>64</mn><mo>,</mo><mtext> </mtext><mn>72</mn><mo>,</mo><mtext> </mtext><mn>80</mn><mo>,</mo><mtext> </mtext><mn>88</mn><mo>,</mo><mtext> </mtext><mn>96</mn><mo>,</mo><mtext> </mtext><mn>...</mn></mrow></math></script></p>
<p class="noindent">I multipli di 12, diversi da 0, sono:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="46.333ex" height="3ex" viewBox="0 -875 19980.6 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use><use x="1010" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(1459,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(2069,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-34"></use></g><use x="3079" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(3529,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(4139,0)"><use xlink:href="#MJMAIN-33"></use><use x="505" y="0" xlink:href="#MJMAIN-36"></use></g><use x="5149" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(5598,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(6208,0)"><use xlink:href="#MJMAIN-34"></use><use x="505" y="0" xlink:href="#MJMAIN-38"></use></g><use x="7218" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(7668,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(8278,0)"><use xlink:href="#MJMAIN-36"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use></g><use x="9288" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(9738,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(10348,0)"><use xlink:href="#MJMAIN-37"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="11358" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(11807,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(12417,0)"><use xlink:href="#MJMAIN-38"></use><use x="505" y="0" xlink:href="#MJMAIN-34"></use></g><use x="13427" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(13877,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(14487,0)"><use xlink:href="#MJMAIN-39"></use><use x="505" y="0" xlink:href="#MJMAIN-36"></use></g><use x="15497" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(15946,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(16556,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use><use x="1010" y="0" xlink:href="#MJMAIN-38"></use></g><use x="18071" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(18521,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(19131,0)"><use xlink:href="#MJMAIN-2E"></use><use x="283" y="0" xlink:href="#MJMAIN-2E"></use><use x="566" y="0" xlink:href="#MJMAIN-2E"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>12</mn><mo>,</mo><mtext> </mtext><mn>24</mn><mo>,</mo><mtext> </mtext><mn>36</mn><mo>,</mo><mtext> </mtext><mn>48</mn><mo>,</mo><mtext> </mtext><mn>60</mn><mo>,</mo><mtext> </mtext><mn>72</mn><mo>,</mo><mtext> </mtext><mn>84</mn><mo>,</mo><mtext> </mtext><mn>96</mn><mo>,</mo><mtext> </mtext><mn>108</mn><mo>,</mo><mtext> </mtext><mn>...</mn></mrow></math></script></p>
<p class="noindent">Osserviamo che, mentre i divisori di un numero sono finiti, i suoi multipli sono infiniti.</p>
<p class="noindent">I numeri 24, 48, 72, 96, ... sono multipli sia di 8 sia di 12. Per questo motivo vengono detti <i>multipli comuni</i><a id="ind82"></a><!--<?"multipli|comuni",4,0,2>--> di 8 e 12. Il più piccolo di essi, ossia 24, è il <b>minimo comune multiplo</b> di 8 e 12.</p>
<div class="definition" title_dea="Minimo comune multiplo" key_dea="minimo comune multiplo, mcm">
<h4 class="noindent">
<span class="ash">DEFINIZIONE</span> <span class="orangeb">MINIMO COMUNE MULTIPLO</span>
</h4>
<p class="noindentin">Il minimo comune multiplo (<i>mcm</i>) di due o più numeri naturali,<a id="ind83"></a><!--<?"mcm|di numeri naturali",4,0,2>--> diversi da zero, è il più piccolo dei loro multipli comuni diversi da zero.</p>
</div>
<p class="noindent">Il <i>mcm</i> di due numeri <i>a</i> e <i>b</i> si indica con <i>mcm</i>(<i>a</i> ; <i>b</i>), quello di tre numeri <i>a</i>, <i>b</i>, <i>c</i> si indica con <i>mcm</i>(<i>a</i> ; <i>b</i> ; <i>c</i>) e così via.</p>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"></div>
</div></div>
<div data-page-container="19" id="page-19" class="row chapters-content"><div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">19</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<div class="rule" title_dea="Regola per determinare il minimo comune multiplo" key_dea="calcolo del minimo comune multiplo, minimo comune multiplo, calcolo mcm, mcm, regola per determinare il minimo comune multiplo">
<h4 class="noindent">
<span class="pagebreak" epub:type="pagebreak" title="19" id="page19"></span><span class="red"><b>REGOLA</b></span>
</h4>
<p class="noindent">Per determinare il <i>mcm</i> di due o più numeri naturali</p>
<ol class="olist">
<li><p class="noindent">si scompongono in fattori primi i numeri dati;</p></li>
<li><p class="noindent">si moltiplicano fra loro tutti i fattori primi, comuni e non comuni, dei numeri dati, presi una sola volta, ciascuno con l’esponente maggiore con cui figura.</p></li>
</ol>
</div>
<div class="example">
<h4 class="noindent"><span class="gep"><span class="sgr">ESEMPIO</span></span></h4>
<p class="noindent">Determiniamo <i>mcm</i>(12 ; 15 ; 18).</p>
<ol class="olist">
<li>
<p class="noindent">Scomponiamo in fattori primi i numeri dati:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="44.833ex" height="3.167ex" viewBox="0 -894.9 19271.7 1339.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use><use x="1287" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(2348,0)"><use xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-32"></use></g><use x="3532" y="0" xlink:href="#MJMAIN-22C5"></use><use x="4038" y="0" xlink:href="#MJMAIN-33"></use><g transform="translate(4543,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1219,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1829,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(2439,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(7592,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-35"></use></g><use x="8880" y="0" xlink:href="#MJMAIN-3D"></use><use x="9941" y="0" xlink:href="#MJMAIN-33"></use><use x="10668" y="0" xlink:href="#MJMAIN-22C5"></use><use x="11173" y="0" xlink:href="#MJMAIN-35"></use><g transform="translate(11678,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1219,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1829,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(2439,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(14728,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-38"></use></g><use x="16016" y="0" xlink:href="#MJMAIN-3D"></use><use x="17077" y="0" xlink:href="#MJMAIN-32"></use><use x="17804" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(18309,0)"><use xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="714" y="568" xlink:href="#MJMAIN-32"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>12</mn><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>·</mo><mn>3</mn><mi>     </mi><mn>15</mn><mo>=</mo><mn>3</mn><mo>·</mo><mn>5</mn><mi>     </mi><mn>18</mn><mo>=</mo><mn>2</mn><mo>·</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></script></p>
</li>
<li>
<p class="noindent">I fattori primi, comuni e non comuni, di tutti i numeri dati sono 2, 3 e 5. L’esponente maggiore con cui compare il fattore 2 è 2, l’esponente maggiore con cui compare il 3 è 2, l’esponente maggiore con cui compare il 5 è 1. Quindi</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="41.833ex" height="3.167ex" viewBox="0 -894.9 18047.3 1339.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-6D"></use><use x="883" y="0" xlink:href="#MJMATHI-63"></use><use x="1321" y="0" xlink:href="#MJMATHI-6D"></use><g transform="translate(2204,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="2813" y="0" xlink:href="#MJMAIN-28"></use><g transform="translate(3207,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><g transform="translate(4217,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="4827" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(5277,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(5887,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-35"></use></g><g transform="translate(6897,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="7507" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(7957,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(8567,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-38"></use></g><use x="9577" y="0" xlink:href="#MJMAIN-29"></use><use x="10248" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(11309,0)"><use xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-32"></use></g><use x="12493" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(12999,0)"><use xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="714" y="568" xlink:href="#MJMAIN-32"></use></g><use x="14183" y="0" xlink:href="#MJMAIN-22C5"></use><use x="14688" y="0" xlink:href="#MJMAIN-35"></use><use x="15471" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(16532,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-38"></use><use x="1010" y="0" xlink:href="#MJMAIN-30"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>m</mi><mi>c</mi><mi>m</mi><mtext> </mtext><mo stretchy="false">(</mo><mn>12</mn><mtext> </mtext><mo>;</mo><mtext> </mtext><mn>15</mn><mtext> </mtext><mo>;</mo><mtext> </mtext><mn>18</mn><mo stretchy="false">)</mo><mo>=</mo><msup><mn>2</mn><mn>2</mn></msup><mo>⋅</mo><msup><mn>3</mn><mn>2</mn></msup><mo>⋅</mo><mn>5</mn><mo>=</mo><mn color="#00aef0">180</mn></mrow></math></script></p>
</li>
</ol>
</div>
<ul class="blist">
<li>
<p class="noindent">Se <i>a</i> è un multiplo di <i>b</i>, ossia <i>b</i> è un divisore di <i>a</i>, si ha <i>mcm</i>(<i>a</i> ; <i>b</i>) = <i>a</i>.</p>
<p class="noindent">Ad esempio <i>mcm</i>(12 ; 4) = 12 e <i>mcm</i>(3 ; 21) = 21.</p>
</li>
<li><p class="noindent">Il minimo comune multiplo di due numeri primi tra loro è il loro prodotto.</p></li>
</ul>
<h2 class="para_title" id="par07">Sistemi di numerazione<a id="ind84"></a><!--<?"numerazione, sistemi di",4,0,2>-->
</h2>
<h3 class="sec_title" id="sec16">16. Sistema decimale</h3>
<p class="noindent">I numeri naturali che consideriamo sono scritti in un sistema di numerazione detto <b>sistema decimale</b> o <b>sistema a base 10</b>. Ad esempio, il numero 725 contiene 7 centinaia, 2 decine e 5 unità e possiamo scrivere</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="68.333ex" height="3.167ex" viewBox="0 -894.9 29430.6 1339.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-37"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use><use x="1010" y="0" xlink:href="#MJMAIN-35"></use><use x="1792" y="0" xlink:href="#MJMAIN-3D"></use><use x="2853" y="0" xlink:href="#MJMAIN-37"></use><use x="3580" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(4086,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use><use x="1010" y="0" xlink:href="#MJMAIN-30"></use></g><use x="5823" y="0" xlink:href="#MJMAIN-2B"></use><use x="6828" y="0" xlink:href="#MJMAIN-32"></use><use x="7555" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(8060,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use></g><use x="9293" y="0" xlink:href="#MJMAIN-2B"></use><use x="10298" y="0" xlink:href="#MJMAIN-35"></use><g transform="translate(10803,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1219,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="12910" y="0" xlink:href="#MJMAIN-2192"></use><g transform="translate(14193,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1219,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(16023,0)"><use xlink:href="#MJMAIN-37"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use><use x="1010" y="0" xlink:href="#MJMAIN-35"></use></g><use x="17816" y="0" xlink:href="#MJMAIN-3D"></use><use x="18877" y="0" xlink:href="#MJMAIN-37"></use><use x="19604" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(20109,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use><use transform="scale(0.707)" x="1428" y="569" xlink:href="#MJMAIN-32"></use></g><use x="21798" y="0" xlink:href="#MJMAIN-2B"></use><use x="22804" y="0" xlink:href="#MJMAIN-32"></use><use x="23531" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(24036,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use><use transform="scale(0.707)" x="1428" y="569" xlink:href="#MJMAIN-31"></use></g><use x="25725" y="0" xlink:href="#MJMAIN-2B"></use><use x="26731" y="0" xlink:href="#MJMAIN-35"></use><use x="27458" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(27963,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use><use transform="scale(0.707)" x="1428" y="569" xlink:href="#MJMAIN-30"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>725</mn><mo>=</mo><mn>7</mn><mo>·</mo><mn>100</mn><mo>+</mo><mn>2</mn><mo>·</mo><mn>10</mn><mo>+</mo><mn>5</mn><mi>   </mi><mo>→</mo><mi>   </mi><mn>725</mn><mo>=</mo><mn>7</mn><mo>·</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><mo>·</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>1</mn></mrow></msup><mo>+</mo><mn>5</mn><mo>·</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>0</mn></mrow></msup></mrow></math></script></p>
<p class="noindent">Quest’ultima scrittura, che utilizza le potenze di 10, è detta <b>forma polinomiale</b>.<a id="ind85"></a><!--<?"polinomiale, forma",4,0,2>--></p>
<p class="noindent">In generale, un <b>sistema di numerazione</b><a id="ind86"></a><!--<?"sistema|di numerazione|a base 10",4,0,2>--> è un insieme di <i>simboli</i>, detti <b>cifre</b>, e di regole per combinarli, per mezzo del quale è possibile rappresentare un qualsiasi numero naturale.</p>
<p class="noindent">Le cifre del sistema decimale sono i primi 10 numeri naturali:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="33.833ex" height="3ex" viewBox="0 -875 14586.6 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-30"></use><use x="505" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(954,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="1564" y="0" xlink:href="#MJMAIN-31"></use><use x="2069" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(2519,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="3129" y="0" xlink:href="#MJMAIN-32"></use><use x="3634" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(4083,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="4693" y="0" xlink:href="#MJMAIN-33"></use><use x="5198" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(5648,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="6258" y="0" xlink:href="#MJMAIN-34"></use><use x="6763" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(7213,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="7823" y="0" xlink:href="#MJMAIN-35"></use><use x="8328" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(8777,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="9387" y="0" xlink:href="#MJMAIN-36"></use><use x="9892" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(10342,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="10952" y="0" xlink:href="#MJMAIN-37"></use><use x="11457" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(11906,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="12516" y="0" xlink:href="#MJMAIN-38"></use><use x="13021" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(13471,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="14081" y="0" xlink:href="#MJMAIN-39"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>0</mn><mo>,</mo><mi> </mi><mn>1</mn><mo>,</mo><mi> </mi><mn>2</mn><mo>,</mo><mi> </mi><mn>3</mn><mo>,</mo><mi> </mi><mn>4</mn><mo>,</mo><mi> </mi><mn>5</mn><mo>,</mo><mi> </mi><mn>6</mn><mo>,</mo><mi> </mi><mn>7</mn><mo>,</mo><mi> </mi><mn>8</mn><mo>,</mo><mi> </mi><mn>9</mn></mrow></math></script></p>
<p class="noindent1">Tutti gli altri numeri naturali si rappresentano mediante una sequenza di tali simboli; si chiama <b>ordine di una cifra</b><a id="ind87"></a><!--<?"ordine|di una cifra",4,0,2>--> il posto che essa occupa in tale sequenza, contando, a partire da zero, dall’ultima cifra a destra verso sinistra. Nel sistema decimale, 10 unità di un ordine formano un’unità dell’ordine immediatamente superiore.</p>
<p class="noindent">Un sistema di numerazione è <b>posizionale</b><a id="ind88"></a><!--<?"posizionale, sistema di numerazione",4,0,2>--> quando il valore numerico associato a ogni cifra varia a seconda della <i>posizione</i> che essa occupa nella scrittura del numero.</p>
<p class="noindent">Il sistema decimale è posizionale. Ad esempio, i numeri 5028 e 2085 sono ovviamente diversi: il 5 rappresenta 5 migliaia nel primo e 5 unità nel secondo.</p>
</div></div>
</div></div>
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</div></div>
<div data-page-container="20" id="page-20" class="row chapters-content">
<div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">20</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<div class="example">
<h4 class="noindent">
<span class="pagebreak" epub:type="pagebreak" title="20" id="page20"></span><span class="gep"><span class="sgr">ESEMPIO</span></span>
</h4>
<p class="noindent">Nel numero 5028</p>
<ul class="blist">
<li><p class="noindent">8 è la cifra di ordine 0 e rappresenta 8 unità (8 · 10<sup>0</sup>)</p></li>
<li><p class="noindent">2 è la cifra di ordine 1 e rappresenta 2 decine (2 · 10<sup>1</sup>)</p></li>
<li><p class="noindent">0 è la cifra di ordine 2 e rappresenta 0 centinaia (0 · 10<sup>2</sup>)</p></li>
<li><p class="noindent">5 è la cifra di ordine 3 e rappresenta 5 migliaia (5 · 10<sup>3</sup>)</p></li>
</ul>
<p class="noindent">Perciò il numero 5028 si può scrivere in <b>forma polinomiale</b> in questo modo:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.333ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="41.5ex" height="2.333ex" viewBox="0 -894.9 17839 1000.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-35"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use><use x="1010" y="0" xlink:href="#MJMAIN-32"></use><use x="1515" y="0" xlink:href="#MJMAIN-38"></use><use x="2297" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(3358,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-35"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(727,0)"><use xlink:href="#MJMAIN-22C5"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1232,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use></g></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1010,402)"><use transform="scale(0.707)" xlink:href="#MJMAIN-33"></use></g></g><g fill="#00aef0" stroke="#00aef0" transform="translate(2921,0)"><use xlink:href="#MJMAIN-2B"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(3926,0)"><use xlink:href="#MJMAIN-30"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(4654,0)"><use xlink:href="#MJMAIN-22C5"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(5159,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use></g></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1010,402)"><use transform="scale(0.707)" xlink:href="#MJMAIN-32"></use></g></g><g fill="#00aef0" stroke="#00aef0" transform="translate(6848,0)"><use xlink:href="#MJMAIN-2B"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(7853,0)"><use xlink:href="#MJMAIN-32"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(8581,0)"><use xlink:href="#MJMAIN-22C5"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(9086,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use></g></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1010,402)"><use transform="scale(0.707)" xlink:href="#MJMAIN-31"></use></g></g><g fill="#00aef0" stroke="#00aef0" transform="translate(10775,0)"><use xlink:href="#MJMAIN-2B"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(11780,0)"><use xlink:href="#MJMAIN-38"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(12508,0)"><use xlink:href="#MJMAIN-22C5"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(13013,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use></g></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1010,402)"><use transform="scale(0.707)" xlink:href="#MJMAIN-30"></use></g></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>5028</mn><mo>=</mo><mstyle color="#00aef0"><mn>5</mn><mo>⋅</mo><msup><mrow><mn>10</mn></mrow><mn>3</mn></msup><mo>+</mo><mn>0</mn><mo>⋅</mo><msup><mrow><mn>10</mn></mrow><mn>2</mn></msup><mo>+</mo><mn>2</mn><mo>⋅</mo><msup><mrow><mn>10</mn></mrow><mn>1</mn></msup><mo>+</mo><mn>8</mn><mo>⋅</mo><msup><mrow><mn>10</mn></mrow><mn>0</mn></msup></mstyle></mrow></math></script></p>
</div>
<h3 class="sec_title" id="sec17">17. Cambiamenti di base<a id="ind89"></a><!--<?"base|cambiamento di",4,0,2>-->
</h3>
<p class="noindent">Il sistema di numerazione decimale, cioè a base 10, è stato universalmente adottato per la sua grande praticità. Il problema della numerazione (che poi, in pratica, è quello di contare) potrebbe benissimo essere risolto anche se, invece del 10, si assumesse come base un qualsiasi altro numero naturale maggiore o uguale a 2. Si potrebbe, ad esempio, creare un sistema a base 4, nel quale</p>
<ul class="blist">
<li><p class="noindent">le cifre usate per scrivere tutti i numeri sono quattro: 0, 1, 2, 3;</p></li>
<li><p class="noindent">le unità si raggruppano a quattro a quattro;</p></li>
<li><p class="noindent">quattro unità di un certo ordine formano un’unità dell’ordine immediatamente superiore.</p></li>
</ul>
</div></div>
</div></div>
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</div>
<div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<p class="noindent">Quindi anche questo sistema è posizionale e in esso ogni numero può essere espresso in <b>forma polinomiale</b>.</p>
<p class="noindent">In passato, i sistemi di numerazione in base diversa da 10 trovarono applicazione principalmente nel ramo della <i>crittografia</i><a id="ind90"></a><!--<?"crittografia",4,0,2>-->, che studia i metodi per nascondere il significato di un messaggio a chi non è autorizzato a leggerlo.</p>
<p class="noindent">Con l’avvento dei calcolatori elettronici, in informatica fu ampiamente utilizzato il <b>sistema di numerazione a base 2</b>,<a id="ind91"></a><!--<?"sistema|di numerazione|a base 2",4,0,2>--> o <b>sistema binario</b>, perché il calcolatore riconosce <i>due</i> stati fondamentali, associati alle cifre 0 e 1.</p>
<h4 class="h4">Dalla base <i>b</i> alla base 10</h4>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"><div class="sidebox">
<p class="noindentf"><span class="red"><b>BASI MAGGIORI DI 10</b></span></p>
<p class="noindentf">Se la base del sistema di numerazione è maggiore di 10, oltre alle 10 cifre da 0 a 9, si usano altre cifre rappresentate, di solito, da lettere maiuscole. Ad esempio, nel sistema in base 12 le cifre sono</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="28.333ex" height="3ex" viewBox="0 -875 12179.8 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-30"></use><use x="505" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(954,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1564,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="2174" y="0" xlink:href="#MJMAIN-31"></use><use x="2679" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(3129,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(3739,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(4349,0)"><use xlink:href="#MJMAIN-2E"></use><use x="283" y="0" xlink:href="#MJMAIN-2E"></use><use x="566" y="0" xlink:href="#MJMAIN-2E"></use></g><use x="5198" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(5647,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(6257,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="6867" y="0" xlink:href="#MJMAIN-39"></use><use x="7372" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(7822,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(8432,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="9042" y="0" xlink:href="#MJMAIN-41"></use><use x="9797" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(10246,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(10856,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="11466" y="0" xlink:href="#MJMAIN-42"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>0</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mn>1</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mn>...</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mn>9</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mtext>A</mtext><mo>,</mo><mtext> </mtext><mtext> </mtext><mtext>B</mtext></mrow></math></script></p>
</div></div>
</div>
<div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<p class="noindent">Per convertire in base 10 un numero dato in base <i>b</i>, dobbiamo scrivere quest’ultimo in forma polinomiale. Poiché, in un sistema a base <i>b</i>, le unità sono raggruppate a <i>b</i> a <i>b</i>, alla cifra di ordine <i>n</i> è associato il valore numerico che, scritto nel sistema decimale, è il prodotto della cifra stessa per <i>b<sup>n</sup></i>.</p>
<div class="example">
<h4 class="noindent"><span class="gep"><span class="sgr">ESEMPI</span></span></h4>
<p class="hang"><b>1</b>&nbsp;&nbsp;Consideriamo il numero naturale che, in base 4, è scritto <span class="red">132</span>, cioè 132<sub>4</sub>.</p>
<ul class="blist">
<li>
<p class="noindent">Alla cifra 2, di ordine 0, è associato il valore numerico che, scritto nel sistema decimale, è 2 · 4<sup>0</sup>:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="24.167ex" height="3.167ex" viewBox="0 -905.9 10399.5 1350.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-32"></use><g transform="translate(505,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="1392" y="0" xlink:href="#MJMAIN-2192"></use><g transform="translate(2675,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="3285" y="0" xlink:href="#MJMAIN-32"></use><use x="4012" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(4517,0)"><use xlink:href="#MJMAIN-34"></use><use transform="scale(0.707)" x="714" y="585" xlink:href="#MJMAIN-30"></use></g><use x="5757" y="0" xlink:href="#MJMAIN-3D"></use><use x="6818" y="0" xlink:href="#MJMAIN-32"></use><use x="7545" y="0" xlink:href="#MJMAIN-22C5"></use><use x="8050" y="0" xlink:href="#MJMAIN-31"></use><use x="8833" y="0" xlink:href="#MJMAIN-3D"></use><use x="9894" y="0" xlink:href="#MJMAIN-32"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>2</mn><mtext> </mtext><mo>→</mo><mtext> </mtext><mn>2</mn><mo>⋅</mo><msup><mn>4</mn><mn>0</mn></msup><mo>=</mo><mn>2</mn><mo>⋅</mo><mn>1</mn><mo>=</mo><mn>2</mn></mrow></math></script></p>
</li>
<li>
<p class="noindent">Alla cifra 3, di ordine 1, è associato il valore numerico 3 · 4<sup>1</sup>:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="18.167ex" height="3.167ex" viewBox="0 -905.9 7828.5 1350.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-33"></use><g transform="translate(505,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="1392" y="0" xlink:href="#MJMAIN-2192"></use><g transform="translate(2675,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="3285" y="0" xlink:href="#MJMAIN-33"></use><use x="4012" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(4517,0)"><use xlink:href="#MJMAIN-34"></use><use transform="scale(0.707)" x="714" y="585" xlink:href="#MJMAIN-31"></use></g><use x="5757" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(6818,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>3</mn><mtext> </mtext><mo>→</mo><mtext> </mtext><mn>3</mn><mo>⋅</mo><msup><mn>4</mn><mn>1</mn></msup><mo>=</mo><mn>12</mn></mrow></math></script></p>
</li>
<li>
<p class="noindent">Alla cifra 1, di ordine 2, è associato il valore numerico 1 · 4<sup>2</sup>:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="18.167ex" height="3.167ex" viewBox="0 -905.9 7828.5 1350.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-31"></use><g transform="translate(505,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="1392" y="0" xlink:href="#MJMAIN-2192"></use><g transform="translate(2675,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="3285" y="0" xlink:href="#MJMAIN-31"></use><use x="4012" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(4517,0)"><use xlink:href="#MJMAIN-34"></use><use transform="scale(0.707)" x="714" y="585" xlink:href="#MJMAIN-32"></use></g><use x="5757" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(6818,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-36"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn><mtext> </mtext><mo>→</mo><mtext> </mtext><mn>1</mn><mo>⋅</mo><msup><mn>4</mn><mn>2</mn></msup><mo>=</mo><mn>16</mn></mrow></math></script></p>
</li>
</ul>
</div>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"><div class="sidebox">
<p class="noindentf"><span class="red"><b>ATTENZIONE!</b></span></p>
<p class="noindentf">Per evitare confusione, indicheremo la base in cui è espresso ciascun numero con un indice <i>espresso sempre nel sistema decimale</i>.</p>
</div></div>
</div>
</div>
<div data-page-container="21" id="page-21" class="row chapters-content"><div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">21</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<div class="example">
<p class="hangg"><span class="pagebreak" epub:type="pagebreak" title="21" id="page21"></span>Pertanto:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -4.833ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="52.167ex" height="6.833ex" viewBox="0 -905.9 22440.3 2937.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-33"></use><use x="1010" y="0" xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="2142" y="-213" xlink:href="#MJMAIN-34"></use><use x="2249" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(3310,0)"><use xlink:href="#MJMAIN-31"></use><use x="727" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(1232,0)"><use xlink:href="#MJMAIN-34"></use><use transform="scale(0.707)" x="714" y="585" xlink:href="#MJMAIN-32"></use></g><use x="2416" y="0" xlink:href="#MJMAIN-2B"></use><use x="3421" y="0" xlink:href="#MJMAIN-33"></use><use x="4149" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(4654,0)"><use xlink:href="#MJMAIN-34"></use><use transform="scale(0.707)" x="714" y="585" xlink:href="#MJMAIN-31"></use></g><use x="5838" y="0" xlink:href="#MJMAIN-2B"></use><use x="6843" y="0" xlink:href="#MJMAIN-32"></use><use x="7571" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(8076,0)"><use xlink:href="#MJMAIN-34"></use><use transform="scale(0.707)" x="714" y="585" xlink:href="#MJMAIN-30"></use></g><g transform="translate(12,-615)"><use x="19" y="0" xlink:href="#MJSZ4-E152"></use><g transform="translate(526.6219583078562,0) scale(8.524391661571244,1)"><use xlink:href="#MJSZ4-E154"></use></g><g transform="translate(4064,0)"><use xlink:href="#MJSZ4-E151"></use><use x="455" y="0" xlink:href="#MJSZ4-E150"></use></g><g transform="translate(5016.866456167779,0) scale(8.524391661571244,1)"><use xlink:href="#MJSZ4-E154"></use></g><use x="8559" y="0" xlink:href="#MJSZ4-E153"></use></g><g transform="translate(1612,-1614)"><use transform="scale(0.707)" xlink:href="#MJMAIN-66"></use><use transform="scale(0.707)" x="311" y="0" xlink:href="#MJMAIN-6F"></use><use transform="scale(0.707)" x="816" y="0" xlink:href="#MJMAIN-72"></use><use transform="scale(0.707)" x="1213" y="0" xlink:href="#MJMAIN-6D"></use><use transform="scale(0.707)" x="2051" y="0" xlink:href="#MJMAIN-61"></use><g transform="translate(1807,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(2238,0)"><use transform="scale(0.707)" xlink:href="#MJMAIN-70"></use><use transform="scale(0.707)" x="561" y="0" xlink:href="#MJMAIN-6F"></use><use transform="scale(0.707)" x="1066" y="0" xlink:href="#MJMAIN-6C"></use><use transform="scale(0.707)" x="1349" y="0" xlink:href="#MJMAIN-69"></use><use transform="scale(0.707)" x="1632" y="0" xlink:href="#MJMAIN-6E"></use><use transform="scale(0.707)" x="2193" y="0" xlink:href="#MJMAIN-6F"></use><use transform="scale(0.707)" x="2698" y="0" xlink:href="#MJMAIN-6D"></use><use transform="scale(0.707)" x="3536" y="0" xlink:href="#MJMAIN-69"></use><use transform="scale(0.707)" x="3819" y="0" xlink:href="#MJMAIN-61"></use><use transform="scale(0.707)" x="4324" y="0" xlink:href="#MJMAIN-6C"></use><use transform="scale(0.707)" x="4607" y="0" xlink:href="#MJMAIN-65"></use></g></g></g><use x="12626" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(13687,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-36"></use></g><use x="14919" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(15925,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="17157" y="0" xlink:href="#MJMAIN-2B"></use><use x="18162" y="0" xlink:href="#MJMAIN-32"></use><use x="18945" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(20006,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g fill="#00aef0" stroke="#00aef0" transform="translate(20616,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-33"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use></g></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1010,-150)"><g fill="#00aef0" stroke="#00aef0"><use transform="scale(0.707)" xlink:href="#MJMAIN-31"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-30"></use></g></g></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mn>132</mn></mrow><mn>4</mn></msub><mo>=</mo><munder><munder><mrow><mn>1</mn><mo>⋅</mo><msup><mn>4</mn><mn>2</mn></msup><mo>+</mo><mn>3</mn><mo>⋅</mo><msup><mn>4</mn><mn>1</mn></msup><mo>+</mo><mn>2</mn><mo>⋅</mo><msup><mn>4</mn><mn>0</mn></msup></mrow><mo stretchy="true">︸</mo></munder><mrow><mtext>forma</mtext><mtext> </mtext><mtext>polinomiale</mtext></mrow></munder><mo>=</mo><mn>16</mn><mo>+</mo><mn>12</mn><mo>+</mo><mn>2</mn><mo>=</mo><mtext> </mtext><mstyle color="#00aef0"><msub><mrow><mn>30</mn></mrow><mrow><mn>10</mn></mrow></msub></mstyle></mrow></math></script></p>
<p class="hang"><b>2</b>&nbsp;&nbsp;Rappresentiamo nel sistema decimale il numero che in base 5 si scrive 20413:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><span style="display: inline-block; white-space: nowrap; padding: 1px 0px;"><span style="display: inline-block; position: relative; vertical-align: -2.667ex; width: 53.333ex; height: 6.667ex;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="margin-left: 0ex; margin-right: 0ex; position: absolute; left: 0px;" width="53.333ex" height="6.667ex" viewBox="0 -1669.4 22989.9 2838.8"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(167,0)"><g transform="translate(-11,0)"><g transform="translate(0,766)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use><use x="1010" y="0" xlink:href="#MJMAIN-34"></use><use x="1515" y="0" xlink:href="#MJMAIN-31"></use><use x="2020" y="0" xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="3570" y="-213" 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<p class="hang"><b>3</b>&nbsp;&nbsp;Rappresentiamo nel sistema decimale il numero che in base 2 si scrive 1101:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.5ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="67ex" height="2.5ex" viewBox="0 -894.9 28828.3 1081.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-31"></use><use x="1010" y="0" xlink:href="#MJMAIN-30"></use><use x="1515" y="0" xlink:href="#MJMAIN-31"></use><use transform="scale(0.707)" x="2856" y="-213" xlink:href="#MJMAIN-32"></use><use x="2754" y="0" xlink:href="#MJMAIN-3D"></use><use x="3815" y="0" xlink:href="#MJMAIN-31"></use><use x="4542" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(5048,0)"><use xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-33"></use></g><use x="6232" y="0" xlink:href="#MJMAIN-2B"></use><use x="7237" y="0" xlink:href="#MJMAIN-31"></use><use x="7964" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(8470,0)"><use xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-32"></use></g><use x="9654" y="0" xlink:href="#MJMAIN-2B"></use><use x="10659" y="0" xlink:href="#MJMAIN-30"></use><use x="11386" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(11892,0)"><use xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-31"></use></g><use x="13076" y="0" xlink:href="#MJMAIN-2B"></use><use x="14081" y="0" xlink:href="#MJMAIN-31"></use><use x="14808" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(15314,0)"><use xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-30"></use></g><use x="16553" y="0" xlink:href="#MJMAIN-3D"></use><use x="17614" y="0" xlink:href="#MJMAIN-38"></use><use x="18341" y="0" xlink:href="#MJMAIN-2B"></use><use x="19347" y="0" xlink:href="#MJMAIN-34"></use><use x="20074" y="0" xlink:href="#MJMAIN-2B"></use><use x="21079" y="0" xlink:href="#MJMAIN-30"></use><use x="21806" y="0" xlink:href="#MJMAIN-2B"></use><use x="22812" y="0" xlink:href="#MJMAIN-31"></use><use x="23594" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(24655,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-33"></use></g><use x="25943" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(27004,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-33"></use></g></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1010,-150)"><g fill="#00aef0" stroke="#00aef0"><use transform="scale(0.707)" xlink:href="#MJMAIN-31"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-30"></use></g></g></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mn>1101</mn></mrow><mn>2</mn></msub><mo>=</mo><mn>1</mn><mo>⋅</mo><msup><mn>2</mn><mn>3</mn></msup><mo>+</mo><mn>1</mn><mo>⋅</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><mn>0</mn><mo>⋅</mo><msup><mn>2</mn><mn>1</mn></msup><mo>+</mo><mn>1</mn><mo>⋅</mo><msup><mn>2</mn><mn>0</mn></msup><mo>=</mo><mn>8</mn><mo>+</mo><mn>4</mn><mo>+</mo><mn>0</mn><mo>+</mo><mn>1</mn><mo>=</mo><mn>13</mn><mo>=</mo><mstyle color="#00aef0"><msub><mrow><mn>13</mn></mrow><mrow><mn>10</mn></mrow></msub></mstyle></mrow></math></script></p>
</div>
<h4 class="h4">Dalla base 10 alla base <i>b</i>
</h4>
<div class="boxr">
<p class="noindent"><span class="red"><b>ALGORITMO DELLE DIVISIONI SUCCESSIVE<a id="ind92"></a><!--<?"algoritmo|delle divisioni successive",4,0,2>--></b></span></p>
<p class="noindent">Per convertire in base <i>b</i> un numero <i>x</i>, rappresentato in base 10, si procede così:</p>
<ol class="olist">
<li><p class="noindent">si divide <i>x</i> per <i>b</i>; il resto ottenuto costituisce l’ultima cifra a destra del numero <i>x</i> espresso in base <i>b</i>;</p></li>
<li><p class="noindent">si divide nuovamente per <i>b</i> il quoziente ottenuto; il nuovo resto è la cifra, in base <i>b</i>, immediatamente a sinistra di quella ottenuta precedentemente;</p></li>
<li><p class="noindent">si ripete l’operazione <span class="red"><b>b.</b></span> fino a ottenere un quoziente nullo.</p></li>
</ol>
<p class="noindent">L’ultimo resto è la prima cifra a sinistra nel numero <i>x</i> scritto nella base <i>b</i>.</p>
</div>
<div class="rbox">
<p class="noindent"><span class="red"><b>ETIMOLOGIA</b></span></p>
<p class="noindent">Il termine <b>algoritmo</b><a id="ind93"></a><!--<?"algoritmo",4,0,2>--> deriva dal nome del matematico di cultura araba <i>Mohammed ibn-Musa al-Khuwarizmi</i>,<a id="ind94"></a><!--<?"al-Khuwarizmi",4,0,2>--> che visse a Baghdad nel IX secolo d.C.; egli ci tramandò non solo un importante libro di calcolo numerico, ma anche un libro di algebra sulle equazioni di primo e secondo grado che fu basilare per lo sviluppo dell’algebra stessa.</p>
<p class="noindent">La parola algoritmo indica un <i>procedimento di calcolo</i>. Esso consiste in una successione finita di operazioni elementari da eseguire una dopo l’altra in un ordine ben preciso e deve avere le seguenti caratteristiche: deve essere <i>finito</i> (cioè terminare dopo un numero finito di operazioni), <i>definito</i> (ossia essere conciso e non ambiguo), <i>completo</i> e deve <i>raggiungere il risultato</i> per il quale è stato progettato.</p>
</div>
<div class="example">
<h4 class="noindent"><span class="gep"><span class="sgr">ESEMPI</span></span></h4>
<p class="hang"><b>4</b>&nbsp;&nbsp;Esprimiamo il numero 26<sub>10</sub> nel sistema <span class="green">binario</span>.</p>
<p class="hangg">Applichiamo l’algoritmo delle divisioni successive:</p>
<p class="math"><img src="images/pg21.jpg" alt="Image"></p>
<p class="hangg">Quindi 26<sub>10</sub> = <span class="cyan">11010<sub>2</sub></span>.</p>
</div>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"></div>
</div></div>
<div data-page-container="22" id="page-22" class="row chapters-content">
<div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">22</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<div class="example">
<p class="hang"><span class="pagebreak" epub:type="pagebreak" title="22" id="page22"></span><b>5</b>&nbsp;&nbsp;Esprimiamo in base 8 il numero 4327<sub>10</sub>.</p>
<p class="hangg">Possiamo disporre il calcolo nel seguente modo:</p>
<div class="table">
<p class="math"><img src="images/pg22.jpg" alt="Image"></p>
<p class="hangg">Quindi 4327<sub>10</sub> = <span class="cyan">10347<sub>8</sub></span>.</p>
</div>
</div>
<h2 class="para_title" id="par08">L’insieme dei numeri interi relativi<a id="ind95"></a><!--<?"insieme|dei numeri|interi relativi",4,0,2>-->
</h2>
<h3 class="sec_title" id="sec18">18. I numeri interi relativi<a id="ind96"></a><!--<?"numeri|interi relativi",4,0,2>-->
</h3>
<p class="noindent">I numeri naturali non consentono di risolvere tutti i problemi. Ad esempio, nell’insieme dei numeri naturali non possiamo eseguire la sottrazione 8 − 12. In generale in ℕ non si può eseguire una sottrazione se il sottraendo è maggiore del minuendo.</p>
<p class="noindent">Eppure una simile sottrazione, in diverse situazioni, può avere significato. Ad esempio (<a href="#ch1.fg13"><span class="fron">FIGURA 13</span></a>): la temperatura ieri era di 8 gradi, ma oggi si è abbassata di 12 gradi. Qual è ora la temperatura?</p>
<p class="noindent">Per risolvere problemi come questo è necessario introdurre i <b>numeri interi relativi</b>.</p>
<div class="figure">
<p class="img" id="ch1.fg13"><img src="images/c01u01f13.jpg" alt="Image"></p>
<p class="figcap">FIGURA 13</p>
</div>
<div class="definition" title_dea="Numero intero relativo" key_dea="numero intero relativo, numeri interi relativi, interi relativi, intero relativo">
<h4 class="noindent">
<span class="ash">DEFINIZIONE</span> <span class="orangeb">NUMERO INTERO RELATIVO</span>
</h4>
<p class="noindentin">Un numero intero relativo, o semplicemente <i>numero intero</i>, è un numero naturale preceduto da un <i>segno</i> + o −.</p>
</div>
<p class="noindent">L’insieme dei numeri interi relativi<a id="ind97"></a><!--<?"Z, insieme dei numeri interi relativi",4,0,2>--> viene indicato con la lettera ℤ.</p>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"></div>
</div>
<div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<p class="noindent">I numeri interi preceduti dal segno + sono <b>positivi</b>,<a id="ind98"></a><!--<?"numeri|positivi",4,0,2>--> quelli preceduti dal segno − sono <b>negativi</b>.<a id="ind99"></a><!--<?"numeri|negativi",4,0,2>--></p>
<p class="noindent">Per convenzione si considerano uguali i numeri +0 e −0; per tale motivo lo zero si indica solitamente senza segno e non si considera né positivo né negativo:</p>
<p class="math"><span class="cyan">+ 0 = −0 = 0</span></p>
<p class="noindent">I <b>numeri positivi possono essere indicati senza segno</b>: il segno + può essere sottinteso. Ad esempio</p>
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Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(22992,0)"><use xlink:href="#MJMAIN-33"></use><use x="505" y="0" xlink:href="#MJMAIN-34"></use><use x="1010" y="0" xlink:href="#MJMAIN-35"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>+</mo><mn>3</mn><mi>  </mi><mo>+</mo><mn>100</mn><mi>  </mi><mo>+</mo><mn>12</mn><mi> </mi><mn>345</mn><mi>  </mi><mtext>o anche</mtext><mi>  </mi><mn>3</mn><mi>  </mi><mn>100</mn><mi>  </mi><mn>12</mn><mi> </mi><mn>345</mn></mrow></math></script></p>
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<ul class="ulist">
<li><p class="noindent"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="3ex" height="1.833ex" viewBox="0 -796.1 1325.7 821.1"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJAMS-5A"></use><use transform="scale(0.707)" x="950" y="513" xlink:href="#MJMAIN-2B"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>ℤ</mi><mo>+</mo></msup></mrow></math></script> è l’insieme dei numeri interi positivi.</p></li>
<li><p class="noindent"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="3ex" height="1.667ex" viewBox="0 -706.9 1325.7 731.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJAMS-5A"></use><use transform="scale(0.707)" x="950" y="513" xlink:href="#MJMAIN-2212"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>ℤ</mi><mo>-</mo></msup></mrow></math></script> è l’insieme dei numeri interi negativi.</p></li>
</ul>
</div></div>
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<div data-page-container="23" id="page-23" class="row chapters-content">
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<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">23</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<p class="noindent"><span class="pagebreak" epub:type="pagebreak" title="23" id="page23"></span>Due numeri interi<a id="ind100"></a><!--<?"concordi|numeri interi",4,0,2>--> con lo stesso segno sono <b>concordi</b>; due numeri interi con segni diversi sono <b>discordi</b><a id="ind101"></a><!--<?"discordi|numeri interi",4,0,2>-->. Ad esempio:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="57.333ex" height="3ex" viewBox="0 -875 24697.8 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2B"></use><use x="783" y="0" xlink:href="#MJMAIN-34"></use><g transform="translate(1288,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="1897" y="0" xlink:href="#MJMAIN-65"></use><g transform="translate(2346,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="2956" y="0" xlink:href="#MJMAIN-38"></use><g transform="translate(3461,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(4071,0)"><use xlink:href="#MJMAIN-73"></use><use x="399" y="0" xlink:href="#MJMAIN-6F"></use><use x="904" y="0" xlink:href="#MJMAIN-6E"></use><use x="1465" y="0" xlink:href="#MJMAIN-6F"></use></g><g transform="translate(6041,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(6818,0)"><use xlink:href="#MJMATHI-63"></use><use x="438" y="0" xlink:href="#MJMATHI-6F"></use><use x="928" y="0" xlink:href="#MJMATHI-6E"></use><use x="1533" y="0" xlink:href="#MJMATHI-63"></use><use x="1971" y="0" xlink:href="#MJMATHI-6F"></use><use x="2461" y="0" xlink:href="#MJMATHI-72"></use><use x="2917" y="0" xlink:href="#MJMATHI-64"></use><use x="3442" y="0" xlink:href="#MJMATHI-69"></use></g><g transform="translate(10777,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1219,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1829,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(2439,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="13826" y="0" xlink:href="#MJMAIN-35"></use><g transform="translate(14331,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="14941" y="0" xlink:href="#MJMAIN-65"></use><g transform="translate(15390,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="16223" y="0" xlink:href="#MJMAIN-2212"></use><use x="17228" y="0" xlink:href="#MJMAIN-39"></use><g transform="translate(17733,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(18343,0)"><use xlink:href="#MJMAIN-73"></use><use x="399" y="0" xlink:href="#MJMAIN-6F"></use><use x="904" y="0" xlink:href="#MJMAIN-6E"></use><use x="1465" y="0" xlink:href="#MJMAIN-6F"></use></g><g transform="translate(20313,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(21089,0)"><use xlink:href="#MJMATHI-64"></use><use x="525" y="0" xlink:href="#MJMATHI-69"></use><use x="875" y="0" xlink:href="#MJMATHI-73"></use><use x="1349" y="0" xlink:href="#MJMATHI-63"></use><use x="1787" y="0" xlink:href="#MJMATHI-6F"></use><use x="2277" y="0" xlink:href="#MJMATHI-72"></use><use x="2733" y="0" xlink:href="#MJMATHI-64"></use><use x="3258" y="0" xlink:href="#MJMATHI-69"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>+</mo><mn>4</mn><mtext> </mtext><mtext>e</mtext><mtext> </mtext><mn>8</mn><mtext> </mtext><mtext>sono</mtext><mtext> </mtext><mi mathvariant="italic">concordi</mi><mi>     </mi><mn>5</mn><mtext> </mtext><mtext>e</mtext><mtext> </mtext><mo>-</mo><mn>9</mn><mtext> </mtext><mtext>sono</mtext><mtext> </mtext><mi mathvariant="italic">discordi</mi></mrow></math></script></p>
<p class="noindent">I numeri interi relativi sono dunque</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="68.333ex" height="3ex" viewBox="0 -875 29450.8 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2E"></use><use x="283" y="0" xlink:href="#MJMAIN-2E"></use><use x="566" y="0" xlink:href="#MJMAIN-2E"></use><use x="849" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(1298,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="2130" y="0" xlink:href="#MJMAIN-2212"></use><use x="3136" y="0" xlink:href="#MJMAIN-35"></use><use x="3641" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(4090,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="4922" y="0" xlink:href="#MJMAIN-2212"></use><use x="5928" y="0" xlink:href="#MJMAIN-34"></use><use x="6433" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(6882,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="7714" y="0" xlink:href="#MJMAIN-2212"></use><use x="8720" y="0" xlink:href="#MJMAIN-33"></use><use x="9225" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(9674,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="10507" y="0" xlink:href="#MJMAIN-2212"></use><use x="11512" y="0" xlink:href="#MJMAIN-32"></use><use x="12017" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(12466,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="13299" y="0" xlink:href="#MJMAIN-2212"></use><use x="14304" y="0" xlink:href="#MJMAIN-31"></use><use x="14809" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(15258,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="15868" y="0" xlink:href="#MJMAIN-30"></use><use x="16373" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(16823,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="17655" y="0" xlink:href="#MJMAIN-2B"></use><use x="18660" y="0" xlink:href="#MJMAIN-31"></use><use x="19165" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(19615,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="20447" y="0" xlink:href="#MJMAIN-2B"></use><use x="21453" y="0" xlink:href="#MJMAIN-32"></use><use x="21958" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(22407,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="23239" y="0" xlink:href="#MJMAIN-2B"></use><use x="24245" y="0" xlink:href="#MJMAIN-33"></use><use x="24750" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(25199,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="26031" y="0" xlink:href="#MJMAIN-2B"></use><use x="27037" y="0" xlink:href="#MJMAIN-34"></use><use x="27542" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(27991,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(28601,0)"><use xlink:href="#MJMAIN-2E"></use><use x="283" y="0" xlink:href="#MJMAIN-2E"></use><use x="566" y="0" xlink:href="#MJMAIN-2E"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>...</mn><mo>;</mo><mtext> </mtext><mo>−</mo><mn>5</mn><mo>;</mo><mtext> </mtext><mo>−</mo><mn>4</mn><mo>;</mo><mtext> </mtext><mo>−</mo><mn>3</mn><mo>;</mo><mtext> </mtext><mo>−</mo><mn>2</mn><mo>;</mo><mtext> </mtext><mo>−</mo><mn>1</mn><mo>;</mo><mtext> </mtext><mn>0</mn><mo>;</mo><mtext> </mtext><mo>+</mo><mn>1</mn><mo>;</mo><mtext> </mtext><mo>+</mo><mn>2</mn><mo>;</mo><mtext> </mtext><mo>+</mo><mn>3</mn><mo>;</mo><mtext> </mtext><mo>+</mo><mn>4</mn><mo>;</mo><mtext> </mtext><mn>...</mn></mrow></math></script></p>
<p class="noindent">o anche</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="57ex" height="3ex" viewBox="0 -875 24541 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2E"></use><use x="283" y="0" xlink:href="#MJMAIN-2E"></use><use x="566" y="0" xlink:href="#MJMAIN-2E"></use><use x="849" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(1298,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="2130" y="0" xlink:href="#MJMAIN-2212"></use><use x="3136" y="0" xlink:href="#MJMAIN-35"></use><use x="3641" y="0" xlink:href="#MJMAIN-3B"></use><g 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xlink:href="#MJMAIN-2212"></use><use x="11512" y="0" xlink:href="#MJMAIN-32"></use><use x="12017" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(12466,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="13299" y="0" xlink:href="#MJMAIN-2212"></use><use x="14304" y="0" xlink:href="#MJMAIN-31"></use><use x="14809" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(15258,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="15868" y="0" xlink:href="#MJMAIN-30"></use><use x="16373" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(16823,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: 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transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="22127" y="0" xlink:href="#MJMAIN-34"></use><use x="22632" y="0" xlink:href="#MJMAIN-3B"></use><g transform="translate(23082,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(23692,0)"><use xlink:href="#MJMAIN-2E"></use><use x="283" y="0" xlink:href="#MJMAIN-2E"></use><use x="566" y="0" xlink:href="#MJMAIN-2E"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>...</mn><mo>;</mo><mtext> </mtext><mo>−</mo><mn>5</mn><mo>;</mo><mtext> </mtext><mo>−</mo><mn>4</mn><mo>;</mo><mtext> </mtext><mo>−</mo><mn>3</mn><mo>;</mo><mtext> </mtext><mo>−</mo><mn>2</mn><mo>;</mo><mtext> </mtext><mo>−</mo><mn>1</mn><mo>;</mo><mtext> </mtext><mn>0</mn><mo>;</mo><mtext> </mtext><mn>1</mn><mo>;</mo><mtext> </mtext><mn>2</mn><mo>;</mo><mtext> </mtext><mn>3</mn><mo>;</mo><mtext> </mtext><mn>4</mn><mo>;</mo><mtext> </mtext><mn>...</mn></mrow></math></script></p>
<p class="noindent">Il sottoinsieme <span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.833ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="3ex" height="2.833ex" viewBox="0 -842.7 1325.7 1192.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJAMS-5A"></use><use transform="scale(0.707)" x="950" y="579" xlink:href="#MJMAIN-2B"></use><use transform="scale(0.707)" x="950" y="-444" xlink:href="#MJMAIN-30"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>ℤ</mi><mn>0</mn><mo>+</mo></msubsup></mrow></math></script> di ℤ formato dai numeri interi positivi e dallo zero può essere identificato con l’insieme ℕ dei numeri naturali; quindi i numeri naturali possono essere considerati un sottoinsieme dei numeri interi<a id="ind102"></a><!--<?"N|sottoinsieme dei numeri interi relativi",4,0,2>--> relativi (<a href="#ch1.fg14"><span class="fron">FIGURA 14</span></a>). Per questo motivo si può dire, anche se impropriamente, che l’insieme ℤ dei numeri interi relativi è un <b>ampliamento</b> dell’insieme ℕ dei numeri naturali.</p>
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</div>
<div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<div title_dea="Rappresentazione insiemistica dei numeri interi relativi" key_dea="insieme dei numeri interi relativi, numero intero relativo, numeri interi relativi, interi relativi, intero relativo, numeri interi negativi, rappresentazione insiemistica dei numeri interi relativi">
<div class="figure">
<p class="img" id="ch1.fg14"><img src="images/c01u01f14.jpg" alt="Image"></p>
<p class="figcap">FIGURA 14</p>
</div>
</div>
<div class="box">
<p class="noindent"><span class="red"><b>APPROFONDIMENTO: ℤ COME AMPLIAMENTO DI ℕ</b></span></p>
<p class="noindent">Indichiamo con <span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.833ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="3ex" height="2.833ex" viewBox="0 -842.7 1325.7 1192.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJAMS-5A"></use><use transform="scale(0.707)" x="950" y="579" xlink:href="#MJMAIN-2B"></use><use transform="scale(0.707)" x="950" y="-444" xlink:href="#MJMAIN-30"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>ℤ</mi><mn>0</mn><mo>+</mo></msubsup></mrow></math></script> l’insieme formato dallo 0 e dai numeri interi positivi e con <span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="3ex" height="1.667ex" viewBox="0 -706.9 1325.7 731.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJAMS-5A"></use><use transform="scale(0.707)" x="950" y="513" xlink:href="#MJMAIN-2212"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ℤ</mi><mo>−</mo></msup></math></script> l’insieme dei numeri interi negativi.</p>
<p class="noindent">Associamo al numero 0 di <span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.833ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="3ex" height="2.833ex" viewBox="0 -842.7 1325.7 1192.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJAMS-5A"></use><use transform="scale(0.707)" x="950" y="579" xlink:href="#MJMAIN-2B"></use><use transform="scale(0.707)" x="950" y="-444" xlink:href="#MJMAIN-30"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>ℤ</mi><mn>0</mn><mo>+</mo></msubsup></mrow></math></script> lo 0 di ℕ (e viceversa), al numero intero relativo +1 il numero naturale 1 (e viceversa), ..., al numero positivo +5 il numero naturale 5 (e viceversa), ... In generale, associamo a ogni numero intero relativo positivo il numero naturale che si ottiene privando il numero relativo del segno + e viceversa: si dice che in questo modo si è stabilita una <i>corrispondenza biunivoca</i><a id="ind103"></a><!--<?"corrispondenza|biunivoca",4,0,2>--> tra <span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.833ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="3ex" height="2.833ex" viewBox="0 -842.7 1325.7 1192.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJAMS-5A"></use><use transform="scale(0.707)" x="950" y="579" xlink:href="#MJMAIN-2B"></use><use transform="scale(0.707)" x="950" y="-444" xlink:href="#MJMAIN-30"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>ℤ</mi><mn>0</mn><mo>+</mo></msubsup></mrow></math></script> e ℕ. Secondo tale corrispondenza possiamo pensare di identificare, ad esempio, il numero intero relativo +7 con il numero naturale 7, +16 con il numero naturale 16 e così via. Diremo quindi che ℤ<a id="ind104"></a><!--<?"ampliamento|di N",4,0,2>--> è un <b>ampliamento</b> di ℕ (<a href="#ch1.fg15"><span class="fron">FIGURA 15</span></a>): naturalmente nel nuovo insieme ℤ il confronto e le operazioni tra numeri dovranno essere definiti in modo che si conservino sia i risultati ottenuti in ℕ sia le stesse proprietà delle operazioni. A questo punto potremo considerare ℕ come un sottoinsieme di ℤ (come già visto in <a href="#ch1.fg14"><span class="fron">FIGURA 14</span></a>).</p>
<div class="figure" title_dea="I numeri interi relativi come ampliamento dei naturali" key_dea="numero intero relativo, numeri interi relativi, interi relativi, intero relativo, i numeri interi relativi come ampliamento dei naturali, ampliamento dei numeri naturali">
<p class="img" id="ch1.fg15"><img src="images/c01u01f15.jpg" alt="Image"></p>
<p class="figcap">FIGURA 15</p>
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<p class="noindent">La scrittura <span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="6.333ex" height="1.833ex" viewBox="0 -706.9 2737.6 771.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJAMS-4E"></use><use x="1004" y="0" xlink:href="#MJMAIN-2282"></use><use x="2065" y="0" xlink:href="#MJAMS-5A"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ℕ</mi><mo>⊂</mo><mi>ℤ</mi></mrow></math></script> si legge «ℕ è contenuto in ℤ».</p>
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<div class="small-12 medium-1 columns"><div class="pagenumber">24</div></div>
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<h4 class="h4">
<span class="pagebreak" epub:type="pagebreak" title="24" id="page24"></span>Valore assoluto e numeri opposti</h4>
<div class="definition" title_dea="Valore assoluto di un numero intero relativo" key_dea="valore assoluto, modulo, valore assoluto di un numero intero relativo">
<h4 class="noindent">
<span class="ash">DEFINIZIONE</span> <span class="orangeb">VALORE ASSOLUTO DI UN NUMERO</span><a id="ind105"></a><!--<?"valore|assoluto di un numero|intero relativo",4,0,2>--> <span class="orangeb">INTERO RELATIVO</span>
</h4>
<p class="noindentin">Il valore assoluto o <i>modulo</i> di un numero intero relativo, positivo o negativo, è il numero stesso privato del suo segno.</p>
</div>
<div class="definition" title_dea="Numeri opposti" key_dea="numeri opposti, opposti, opposto">
<h4 class="noindent">
<span class="ash">DEFINIZIONE</span> <span class="orangeb">NUMERI OPPOSTI<a id="ind106"></a><!--<?"numeri|opposti",4,0,2>--></span>
</h4>
<p class="noindentin">Due numeri<a id="ind107"></a><!--<?"opposti|numeri interi",4,0,2>--> interi relativi con lo stesso valore assoluto e con segni diversi sono opposti.</p>
</div>
<p class="noindent">Il valore assoluto di un numero <i>a</i> si indica scrivendo |<i>a</i>|:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="41.5ex" height="3.167ex" viewBox="0 -944.5 17863.6 1389"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(0,930)"><use x="0" y="-760" xlink:href="#MJMAIN-2223"></use><use x="0" y="-1103" xlink:href="#MJMAIN-2223"></use></g><use x="283" y="0" xlink:href="#MJMAIN-2B"></use><use x="1066" y="0" xlink:href="#MJMAIN-34"></use><g transform="translate(1571,930)"><use x="0" y="-760" xlink:href="#MJMAIN-2223"></use><use x="0" y="-1103" xlink:href="#MJMAIN-2223"></use></g><use x="2131" y="0" xlink:href="#MJMAIN-3D"></use><use x="3192" y="0" xlink:href="#MJMAIN-34"></use><g transform="translate(3697,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1219,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1829,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(2439,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(6747,930)"><use x="0" y="-760" xlink:href="#MJMAIN-2223"></use><use x="0" y="-1103" xlink:href="#MJMAIN-2223"></use></g><g transform="translate(7030,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use></g><g transform="translate(8040,930)"><use x="0" y="-760" xlink:href="#MJMAIN-2223"></use><use x="0" y="-1103" xlink:href="#MJMAIN-2223"></use></g><use x="8601" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(9661,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use></g><g transform="translate(10671,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1219,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(1829,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(2439,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(13721,930)"><use x="0" y="-760" xlink:href="#MJMAIN-2223"></use><use x="0" y="-1103" xlink:href="#MJMAIN-2223"></use></g><use x="14226" y="0" xlink:href="#MJMAIN-2212"></use><use x="15232" y="0" xlink:href="#MJMAIN-38"></use><g transform="translate(15737,930)"><use x="0" y="-760" xlink:href="#MJMAIN-2223"></use><use x="0" y="-1103" xlink:href="#MJMAIN-2223"></use></g><use x="16297" y="0" xlink:href="#MJMAIN-3D"></use><use x="17358" y="0" xlink:href="#MJMAIN-38"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>|</mo><mo>+</mo><mn>4</mn><mo>|</mo><mo>=</mo><mn>4</mn><mi>     </mi><mo>|</mo><mn>10</mn><mo>|</mo><mo>=</mo><mn>10</mn><mi>     </mi><mo>|</mo><mo>-</mo><mn>8</mn><mo>|</mo><mo>=</mo><mn>8</mn></mrow></math></script></p>
<p class="noindent">Anche se il numero 0 è privo di segno, si può estendere a esso la definizione di valore assoluto scrivendo |0| = 0.</p>
<p class="noindent">Risulta poi, per qualsiasi <i>a</i>, |<i>a</i>| ≥ 0.</p>
<p class="noindent">L’opposto di un numero <i>a</i>, positivo o negativo, si ottiene <b>cambiandogli il segno</b>: se <i>a</i> è positivo il suo opposto è negativo, mentre se <i>a</i> è negativo il suo opposto è positivo:</p>
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y="0" xlink:href="#MJMAIN-73"></use><use x="3097" y="0" xlink:href="#MJMAIN-74"></use><use x="3491" y="0" xlink:href="#MJMAIN-6F"></use><g transform="translate(3996,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(4605,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(5215,0)"><use xlink:href="#MJMAIN-64"></use><use x="561" y="0" xlink:href="#MJMAIN-69"></use></g><use x="6282" y="0" xlink:href="#MJMAIN-2B"></use><use x="7287" y="0" xlink:href="#MJMAIN-33"></use><g transform="translate(7792,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 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transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="7279" y="0" xlink:href="#MJMAIN-36"></use><g transform="translate(7784,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(8394,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(9004,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)">è</text></g><use x="9836" y="0" xlink:href="#MJMAIN-2212"></use><use x="10842" y="0" xlink:href="#MJMAIN-36"></use></g><g transform="translate(0,-1688)"><use xlink:href="#MJMAIN-6C"></use><use x="283" y="0" 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transform="translate(10232,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="11064" y="0" xlink:href="#MJMAIN-2B"></use><use x="12069" y="0" xlink:href="#MJMAIN-32"></use></g></g></g></g></svg></span></span></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>l’opposto</mtext><mtext> </mtext><mtext> </mtext><mtext>di</mtext><mo>+</mo><mn>3</mn><mtext> </mtext><mtext> </mtext><mtext>è</mtext><mo>−</mo><mn>3</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>l’opposto</mtext><mtext> </mtext><mtext> </mtext><mtext>di</mtext><mtext> </mtext><mtext> </mtext><mn>6</mn><mtext> </mtext><mtext> </mtext><mtext>è</mtext><mo>−</mo><mn>6</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>l’opposto</mtext><mtext> </mtext><mtext> </mtext><mtext>di</mtext><mo>−</mo><mn>2</mn><mtext> </mtext><mtext> </mtext><mtext>è</mtext><mtext> </mtext><mtext> </mtext><mo>+</mo><mn>2</mn></mrow></mtd></mtr></mtable></mrow></math></script></p>
<p class="noindent">In generale l’opposto di un numero relativo <i>a</i> si indica premettendo a esso il segno −, cioè scrivendo −<i>a</i>; inoltre, poiché per convenzione +0 = −0 = 0, si considera come <i>opposto di</i> 0 <i>il numero</i> 0 <i>stesso</i>. Inoltre l’opposto di −<i>a</i> è +<i>a</i>, cioè −(−<i>a</i>) = +<i>a</i>.</p>
<p class="noindent">In base alle considerazioni precedenti, possiamo esprimere sinteticamente la definizione di valore assoluto di un numero <i>a</i>:</p>
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font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)">è</text></g></g><g fill="#00aef0" stroke="#00aef0" transform="translate(3211,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g fill="#00aef0" stroke="#00aef0" transform="translate(3821,0)"><use xlink:href="#MJMAIN-6E"></use><use x="561" y="0" xlink:href="#MJMAIN-65"></use><use x="1010" y="0" xlink:href="#MJMAIN-67"></use><use x="1515" y="0" xlink:href="#MJMAIN-61"></use><use x="2020" y="0" xlink:href="#MJMAIN-74"></use><use x="2414" y="0" 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xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mstyle color="#00aef0"><mo stretchy="false">|</mo><mi>a</mi><mo stretchy="false">|</mo><mo>=</mo><mrow><mo>{</mo> <mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mi>a</mi></mtd><mtd columnalign="left"><mrow><mtext>se</mtext><mtext> </mtext><mi>a</mi><mtext> è</mtext><mtext> </mtext><mtext>positivo</mtext></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>−</mo><mi>a</mi></mrow></mtd><mtd columnalign="left"><mrow><mtext>se</mtext><mtext> </mtext><mi>a</mi><mtext> è</mtext><mtext> </mtext><mtext>negativo</mtext></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow><mtext>se</mtext><mtext> </mtext><mi>a</mi><mtext> è</mtext><mtext> </mtext><mtext>uguale</mtext><mtext> </mtext><mtext>a</mtext><mtext> </mtext><mn>0</mn></mrow></mtd></mtr></mtable></mrow></mrow></mstyle><mtext> </mtext><mtext> </mtext><mtext>o</mtext><mtext> </mtext><mtext>anche</mtext><mtext> </mtext><mstyle color="#00aef0"><mo stretchy="false">|</mo><mi>a</mi><mo stretchy="false">|</mo><mo>=</mo><mrow><mo>{</mo> <mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mi>a</mi></mtd><mtd columnalign="left"><mrow><mtext>se</mtext><mtext> </mtext><mi>a</mi><mo>≥</mo><mn>0</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>−</mo><mi>a</mi></mrow></mtd><mtd columnalign="left"><mrow><mtext>se</mtext><mtext> </mtext><mi>a</mi><mo>&lt;</mo><mn>0</mn></mrow></mtd></mtr></mtable></mrow></mrow></mstyle></mrow></math></script></p>
<p class="noindent">Ad esempio:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -4.833ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="47.333ex" height="6.833ex" viewBox="0 -875 20401.5 2907"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-7C"></use><g transform="translate(283,0)"><g transform="translate(242,0)"><g transform="translate(189,0)"><use xlink:href="#MJMAIN-2B"></use><use x="783" y="0" xlink:href="#MJMAIN-33"></use></g><g transform="translate(0,-615)"><use x="19" y="0" xlink:href="#MJSZ4-E152"></use><g transform="translate(390,0)"><use xlink:href="#MJSZ4-E151"></use><use x="455" y="0" xlink:href="#MJSZ4-E150"></use></g><use x="1211" y="0" xlink:href="#MJSZ4-E153"></use></g></g><g transform="translate(0,-1614)"><use transform="scale(0.707)" xlink:href="#MJMATHI-61"></use><g transform="translate(377,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><use transform="scale(0.707)" x="1143" y="0" xlink:href="#MJMAIN-3E"></use><g transform="translate(1362,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><use transform="scale(0.707)" x="2536" y="0" xlink:href="#MJMAIN-30"></use></g></g><use x="2433" y="0" xlink:href="#MJMAIN-7C"></use><use x="2994" y="0" xlink:href="#MJMAIN-3D"></use><use x="4055" y="0" xlink:href="#MJMAIN-2B"></use><use x="4838" y="0" xlink:href="#MJMAIN-33"></use><use x="5621" y="0" xlink:href="#MJMAIN-3D"></use><use x="6682" y="0" xlink:href="#MJMAIN-33"></use><g transform="translate(7187,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(7797,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="9016" y="0" xlink:href="#MJMAIN-7C"></use><g transform="translate(9299,0)"><g transform="translate(242,0)"><g transform="translate(189,0)"><use xlink:href="#MJMAIN-2212"></use><use x="783" y="0" xlink:href="#MJMAIN-37"></use></g><g transform="translate(0,-555)"><use x="19" y="0" xlink:href="#MJSZ4-E152"></use><g transform="translate(390,0)"><use xlink:href="#MJSZ4-E151"></use><use x="455" y="0" xlink:href="#MJSZ4-E150"></use></g><use x="1211" y="0" xlink:href="#MJSZ4-E153"></use></g></g><g transform="translate(0,-1554)"><use transform="scale(0.707)" xlink:href="#MJMATHI-61"></use><g transform="translate(377,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><use transform="scale(0.707)" x="1143" y="0" xlink:href="#MJMAIN-3C"></use><g transform="translate(1362,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><use transform="scale(0.707)" x="2536" y="0" xlink:href="#MJMAIN-30"></use></g></g><use x="11450" y="0" xlink:href="#MJMAIN-7C"></use><use x="12011" y="0" xlink:href="#MJMAIN-3D"></use><use x="13072" y="0" xlink:href="#MJMAIN-2212"></use><use x="13855" y="0" xlink:href="#MJMAIN-28"></use><use x="14249" y="0" xlink:href="#MJMAIN-2212"></use><use x="15032" y="0" xlink:href="#MJMAIN-37"></use><use x="15537" y="0" xlink:href="#MJMAIN-29"></use><use x="16209" y="0" xlink:href="#MJMAIN-3D"></use><use x="17269" y="0" xlink:href="#MJMAIN-2B"></use><use x="18052" y="0" xlink:href="#MJMAIN-37"></use><use x="18835" y="0" xlink:href="#MJMAIN-3D"></use><use x="19896" y="0" xlink:href="#MJMAIN-37"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">|</mo><munder><munder><mrow><mo>+</mo><mn>3</mn></mrow><mo stretchy="true">︸</mo></munder><mrow><mi>a</mi><mtext> </mtext><mo>&gt;</mo><mtext> </mtext><mn>0</mn></mrow></munder><mo stretchy="false">|</mo><mo>=</mo><mo>+</mo><mn>3</mn><mo>=</mo><mn>3</mn><mtext> </mtext><mtext>  </mtext><mo stretchy="false">|</mo><munder><munder><mrow><mo>−</mo><mn>7</mn></mrow><mo stretchy="true">︸</mo></munder><mrow><mi>a</mi><mtext> </mtext><mo>&lt;</mo><mtext> </mtext><mn>0</mn></mrow></munder><mo stretchy="false">|</mo><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mn>7</mn><mo stretchy="false">)</mo><mo>=</mo><mo>+</mo><mn>7</mn><mo>=</mo><mn>7</mn></mrow></math></script></p>
<h3 class="sec_title" id="sec19">19. Rappresentazione dei numeri interi relativi su una retta orientata<a id="ind108"></a><!--<?"rappresentazione|dei numeri|interi relativi su una retta orientata",4,0,2>-->
</h3>
<p class="noindent">Già sappiamo rappresentare i numeri naturali su una semiretta orientata. In modo analogo, possiamo rappresentare i numeri interi su una <b>retta orientata</b>.<a id="ind109"></a><!--<?"retta|orientata",4,0,2>--> Anche in questo caso, per semplicità, supponiamo che la retta sia disposta orizzontalmente e che il verso di percorrenza sia quello che va da sinistra a destra. Scegliamo un punto qualsiasi della retta (<b>origine</b>), al quale associamo il numero 0, e fissiamo un segmento di lunghezza <i><b>u</b></i> come <b>unità di misura</b><a id="ind110"></a><!--<?"unit&#x00E0;|di misura",4,0,2>--> delle lunghezze. Riportando <span class="pagebreak" epub:type="pagebreak" title="25" id="page25"></span>tale segmento a partire dall’origine verso destra, come abbiamo fatto per i numeri naturali, determineremo i punti ai quali associare i numeri interi positivi; per individuare i punti cui associare i numeri interi negativi procederemo allo stesso modo, ma spostandoci nel verso opposto, cioè verso sinistra.</p>
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<div class="small-12 medium-1 columns"><div class="pagenumber">25</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<p class="noindent">Nella rappresentazione dei numeri interi relativi su una retta, due numeri opposti sono sempre simmetrici rispetto al punto associato allo 0, ossia si trovano alla stessa distanza dallo 0, ma da parti opposte (<a href="#ch1.fg16"><span class="fron">FIGURA 16</span></a>).</p>
<div title_dea="Rappresentazione dei numeri interi relativi" key_dea="numero intero relativo, numeri interi relativi, interi relativi, intero relativo, rappresentazione dei numeri interi relativi su una retta, numeri opposti">
<div class="figure">
<p class="img" id="ch1.fg16"><img src="images/c01u01f16.jpg" alt="Image"></p>
<p class="figcap">FIGURA 16</p>
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</div>
<h3 class="sec_title" id="sec20">20. L’ordinamento nell’insieme dei numeri interi relativi<a id="ind111"></a><!--<?"ordinamento|nell&#x2019;insieme dei numeri interi relativi",4,0,2>-->
</h3>
<p class="noindent">La rappresentazione dei numeri interi relativi su una retta permette di comprendere l’ordinamento dell’insieme ℤ.</p>
<p class="noindent">Vediamo, ad esempio, che i numeri negativi precedono i numeri positivi: diremo perciò che, tra due numeri discordi, il numero negativo è sempre minore del numero positivo.</p>
<p class="noindent">Vediamo anche che +3 precede +5, e perciò +3 è minore di +5 (ossia +5 è maggiore di +3), mentre −5 precede −3 e perciò −5 è minore di −3 (ossia −3 è maggiore di −5) (<a href="#ch1.fg17"><span class="fron">FIGURA 17</span></a>).</p>
<div title_dea="Ordinamento dei numeri interi relativi" key_dea="numero intero relativo, numeri interi relativi, interi relativi, intero relativo, ordinamento, ordinamento dei numeri interi relativi">
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<p class="img" id="ch1.fg17"><img src="images/c01u01f17.jpg" alt="Image"></p>
<p class="figcap">FIGURA 17</p>
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</div>
<p class="noindent">Possiamo pertanto dire che</p>
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<ul class="blist">
<li><p class="noindent">tra due numeri discordi, il numero negativo è minore del numero positivo;</p></li>
<li><p class="noindent">lo zero è maggiore di qualsiasi numero negativo e minore di qualsiasi numero positivo;</p></li>
<li><p class="noindent">tra due numeri positivi il minore è quello che ha il minore valore assoluto;</p></li>
<li><p class="noindent">tra due numeri negativi il minore è quello che ha il maggiore valore assoluto.</p></li>
</ul>
<div class="example">
<h4 class="noindent"><span class="gep"><span class="sgr">ESEMPIO</span></span></h4>
<p class="noindent">Scriviamo in ordine crescente, ossia dal minore al maggiore, i seguenti numeri interi:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="47.167ex" height="3ex" viewBox="0 -875 20299.6 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2B"></use><use x="783" y="0" xlink:href="#MJMAIN-38"></use><g transform="translate(1288,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="2730" y="0" xlink:href="#MJMAIN-2212"></use><use x="3735" y="0" xlink:href="#MJMAIN-32"></use><g transform="translate(4240,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="5682" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(6687,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-31"></use></g><g transform="translate(7697,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="8917" y="0" xlink:href="#MJMAIN-30"></use><g transform="translate(9422,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="10864" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(11869,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use><use x="1010" y="0" xlink:href="#MJMAIN-30"></use></g><g transform="translate(13384,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="14827" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(15832,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use></g><g transform="translate(16842,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="18284" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(19289,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>+</mo><mn>8</mn><mi>  </mi><mo>-</mo><mn>2</mn><mi>  </mi><mo>-</mo><mn>21</mn><mi>  </mi><mn>0</mn><mi>  </mi><mo>+</mo><mn>100</mn><mi>  </mi><mo>-</mo><mn>10</mn><mi>  </mi><mo>+</mo><mn>12</mn></mrow></math></script></p>
<p class="noindent">I numeri negativi (−2; −21; −10) devono precedere lo zero e tra essi il minore è quello che ha il valore assoluto maggiore; dunque devono essere posti in quest’ordine: −21, −10, −2. A essi devono seguire 0 e poi i numeri positivi, nell’ordine dato dal loro valore assoluto, ossia +8, +12, +100.</p>
<p class="noindent">Quindi i numeri dati, scritti in ordine crescente, sono:</p>
<p class="math"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="47.167ex" height="3ex" viewBox="0 -875 20299.6 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2212"></use><g transform="translate(783,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-31"></use></g><g transform="translate(1793,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="3235" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(4240,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use></g><g transform="translate(5250,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="6692" y="0" xlink:href="#MJMAIN-2212"></use><use x="7697" y="0" xlink:href="#MJMAIN-32"></use><g transform="translate(8202,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="9422" y="0" xlink:href="#MJMAIN-30"></use><g transform="translate(9927,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="11369" y="0" xlink:href="#MJMAIN-2B"></use><use x="12374" y="0" xlink:href="#MJMAIN-38"></use><g transform="translate(12879,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="14322" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(15327,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><g transform="translate(16337,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><use x="17779" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(18784,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use><use x="1010" y="0" xlink:href="#MJMAIN-30"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>-</mo><mn>21</mn><mi>  </mi><mo>-</mo><mn>10</mn><mi>  </mi><mo>-</mo><mn>2</mn><mi>  </mi><mn>0</mn><mi>  </mi><mo>+</mo><mn>8</mn><mi>  </mi><mo>+</mo><mn>12</mn><mi>  </mi><mo>+</mo><mn>100</mn></mrow></math></script></span></p>
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<div class="small-12 medium-1 columns"><div class="pagenumber">26</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<h3 class="sec_title" id="sec21">
<span class="pagebreak" epub:type="pagebreak" title="26" id="page26"></span>21. Le quattro proprietà dell’insieme dei numeri interi relativi<a id="ind112"></a><!--<?"propriet&#x00E0;|dell&#x2019;insieme dei numeri|interi relativi",4,0,2>-->
</h3>
<p class="noindent">L’insieme ℤ dei numeri interi relativi gode delle seguenti proprietà.</p>
<ul class="blist">
<li><p class="noindent">L’insieme dei numeri interi è <b>ordinato</b> e <b>infinito</b>.</p></li>
<li><p class="noindent">Ogni numero intero <b>ha un successivo</b>.</p></li>
<li><p class="noindent">Ogni numero intero <b>ha un precedente</b>.</p></li>
<li><p class="noindent">L’insieme dei numeri interi <b>non ha un elemento minimo</b>.</p></li>
<li><p class="noindent">L’insieme dei numeri interi <b>non ha un elemento massimo</b>.</p></li>
<li><p class="noindent">L’insieme dei numeri interi è <b>discreto</b>.<a id="ind113"></a><!--<?"insieme|discreto",4,0,2>--></p></li>
</ul>
<h2 class="para_title" id="par09">Le quattro operazioni aritmetiche con i numeri interi relativi<a id="ind114"></a><!--<?"operazioni|con i numeri|interi relativi",4,0,2>-->
</h2>
<h3 class="sec_title" id="sec22">22. Addizione</h3>
<div class="definition" title_dea="Somma di due numeri interi relativi" key_dea="somma, addizione, numero intero relativo, numeri interi relativi, interi relativi, intero relativo, somma di due numeri interi relativi, somma di due numeri interi relativi concordi, somma di due numeri interi relativi discordi, somma di due numeri opposti">
<h4 class="noindent">
<span class="ash">DEFINIZIONE</span> <span class="orangeb">SOMMA DI DUE NUMERI INTERI RELATIVI</span>
</h4>
<p class="noindentin">La <b>somma di due numeri interi relativi concordi</b><a id="ind115"></a><!--<?"somma|di due numeri|concordi",4,0,2>--> è il numero intero relativo che ha per segno lo stesso segno degli addendi e per valore assoluto la somma dei valori assoluti degli addendi.</p>
<p class="noindentin">La <b>somma di due numeri interi relativi discordi</b><a id="ind116"></a><!--<?"somma|di due numeri|discordi",4,0,2>--> è il numero intero relativo che ha per segno il segno dell’addendo maggiore in valore assoluto e per valore assoluto la differenza tra il maggiore e il minore dei due valori assoluti.</p>
<p class="noindentin">La <b>somma di due numeri opposti</b><a id="ind117"></a><!--<?"somma|di due numeri|opposti",4,0,2>--> è zero.</p>
</div>
<div class="example">
<h4 class="noindent"><span class="gep"><span class="sgr">ESEMPI</span></span></h4>
<p class="hang"><b>1</b>&nbsp;&nbsp;(+12) + (+4)</p>
<p class="hangg">I due addendi sono concordi, perciò (+12) + (+4) = +(12 + 4) = <span class="cyan">+16</span>. Il segno della somma è quello dei due addendi, ossia +.</p>
<p class="hang"><b>2</b>&nbsp;&nbsp;(−12) + (−4)</p>
<p class="hangg">I due addendi sono concordi, perciò si ha (−12) + (−4) = −(12 + 4) = <span class="cyan">−16</span>. Il segno della somma è quello dei due addendi, ossia −.</p>
<p class="hang"><b>3</b>&nbsp;&nbsp;(+12) + (−4)</p>
<p class="hangg">I due addendi sono discordi, perciò si ha (+12) + (−4) = +(12 − 4) = <span class="cyan">+8</span>. Il segno della somma è +, ossia quello dell’addendo maggiore in valore assoluto.</p>
<p class="hang"><b>4</b>&nbsp;&nbsp;(−12) + (+4)</p>
<p class="hangg">I due addendi sono discordi, perciò (−12) + (+4) = −(12 − 4) = <span class="cyan">−8</span>. Il segno della somma è −, ossia quello dell’addendo maggiore in valore assoluto.</p>
<p class="hang"><b>5</b>&nbsp;&nbsp;(+15) + (−15)</p>
<p class="hangg">I due addendi sono opposti, quindi la loro somma è <span class="cyan">0</span>.</p>
</div>
<p class="noindent">L’addizione tra numeri interi relativi<a id="ind118"></a><!--<?"addizione|tra numeri interi relativi",4,0,2>--> gode delle stesse proprietà dell’addizione<a id="ind119"></a><!--<?"addizione|propriet&#x00E0; dell&#x2019;",4,0,2>--> tra numeri naturali: valgono la <b>proprietà commutativa</b><a id="ind120"></a><!--<?"commutativa, propriet&#x00E0;",4,0,2>--> e quella <b>associativa</b><a id="ind121"></a><!--<?"associativa, propriet&#x00E0;",4,0,2>--> e lo <b>0 è l’elemento neutro</b><a id="ind122"></a><!--<?"elemento neutro",4,0,2>--> (vedi <span class="fron">TEORIA.ZIP</span>).</p>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"></div>
</div></div>
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<div class="small-12 medium-1 columns"><div class="pagenumber">27</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<h3 class="sec_title" id="sec23">
<span class="pagebreak" epub:type="pagebreak" title="27" id="page27"></span>23. Sottrazione<a id="ind123"></a><!--<?"sottrazione",4,0,2>-->
</h3>
<div class="definition" title_dea="Differenza di due numeri interi relativi" key_dea="differenza, sottrazione, numero intero relativo, numeri interi relativi, interi relativi, intero relativo, differenza di due numeri interi relativi">
<h4 class="noindent">
<span class="ash">DEFINIZIONE</span> <span class="orangeb">DIFFERENZA DI DUE NUMERI INTERI RELATIVI</span>
</h4>
<p class="noindentin">La differenza di due numeri interi relativi<a id="ind124"></a><!--<?"differenza|di due numeri|interi relativi",4,0,2>--> è la somma del minuendo con l’opposto del sottraendo.</p>
</div>
<p class="noindent">Ad esempio:</p>
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stretchy="false">(</mo><mo>+</mo><mn>12</mn><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo><mo>=</mo><mo>+</mo><mn>8</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mo>+</mo><mn>12</mn><mo stretchy="false">)</mo><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mo>+</mo><mn>12</mn><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mo>=</mo><mo>+</mo><mn>16</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>12</mn><mo stretchy="false">)</mo><mo>−</mo><mo stretchy="false">(</mo><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><mn>12</mn><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mo>−</mo><mn>4</mn><mo 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<p class="noindent">Come accade nella sottrazione tra numeri naturali, anche per i numeri interi relativi <i>la differenza è quel numero che, addizionato al sottraendo</i>, dà per somma il minuendo.</p>
<p class="noindent">È importante notare che <b>nell’insieme</b> ℤ <b>la sottrazione si può sempre eseguire</b>: la sottrazione è un’operazione <b>interna</b> a ℤ.<a id="ind125"></a><!--<?"operazione|interna|a Z",4,0,2>--> La sottrazione tra numeri interi relativi gode della <b>proprietà invariantiva</b><a id="ind126"></a><!--<?"invariantiva, propriet&#x00E0;",4,0,2>-->, ma <b>non</b> gode delle proprietà <i>commutativa</i> e <i>associativa</i> e <b>non</b> ha <i>elemento neutro</i> (<i>a</i> − 0 ≠ 0 − <i>a</i>).</p>
<h3 class="sec_title" id="sec24">24. Addizione algebrica</h3>
<p class="noindent">Sappiamo che la sottrazione tra numeri interi relativi si può ricondurre all’addizione. Possiamo perciò parlare di addizione algebrica<a id="ind127"></a><!--<?"addizione|algebrica",4,0,2>-->, senza distinguere tra addizioni e sottrazioni.</p>
<p class="noindent">Questa osservazione permette di semplificare espressioni del tipo 5 + (−10) + 2, scrivendo 5 − 10 + 2. Quindi, per calcolare il valore di una somma algebrica, si può procedere in due modi.</p>
<p class="noindent1"><b>Primo metodo</b>. Si calcolano</p>
<ul class="blist">
<li><p class="noindent">la somma dei valori assoluti di tutti i termini positivi;</p></li>
<li><p class="noindent">la somma dei valori assoluti di tutti i termini negativi;</p></li>
<li><p class="noindent">la differenza tra la prima somma e la seconda.</p></li>
</ul>
<p class="noindent1"><b>Secondo metodo</b>. Si eseguono di seguito le addizioni indicate dai segni + e le sottrazioni indicate dai segni −, nell’ordine in cui esse si presentano.</p>
<div class="example">
<h4 class="noindent"><span class="gep"><span class="sgr">ESEMPIO</span></span></h4>
<p class="hang"><b>1</b>&nbsp;&nbsp;Calcoliamo con entrambi i metodi la somma algebrica <span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.333ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="17.333ex" height="1.833ex" viewBox="0 -699.9 7495.3 805.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2212"></use><use x="783" y="0" xlink:href="#MJMAIN-37"></use><use x="1510" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(2515,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="3747" y="0" xlink:href="#MJMAIN-2B"></use><use x="4752" y="0" xlink:href="#MJMAIN-35"></use><use x="5480" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(6485,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>-</mo><mn>7</mn><mo>+</mo><mn>12</mn><mo>+</mo><mn>5</mn><mo>-</mo><mn>20</mn></mrow></math></script> e verifichiamo che si ottiene lo stesso risultato.</p>
<p class="hangg"><b>Primo metodo</b></p>
<ul class="blist">
<li><p class="noindent">La somma dei valori assoluti dei termini positivi è 12 + 5 = 17</p></li>
<li><p class="noindent">la somma dei valori assoluti dei termini negativi è 7 + 20 = 27</p></li>
<li><p class="noindent">eseguiamo la sottrazione: 17 − 27 = <span class="cyan">−10</span>.</p></li>
</ul>
</div>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"></div>
</div></div>
<div data-page-container="28" id="page-28" class="row chapters-content">
<div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">28</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<div class="example">
<p class="hangg"><span class="pagebreak" epub:type="pagebreak" title="28" id="page28"></span><b>Secondo metodo</b></p>
<p class="hangg">Eseguiamo nell’ordine le addizioni e le sottrazioni indicate:</p>
<p class="img"><img src="images/pg46-1.jpg" alt="Image"></p>
</div>
<h4 class="h4">L’addizione algebrica e la proprietà commutativa</h4>
<p class="noindent">Le proprietà dell’addizione valgono anche nelle addizioni algebriche. Vale in particolare la proprietà commutativa;<a id="ind128"></a><!--<?"commutativa, propriet&#x00E0;",4,0,2>--> per applicarla si deve ricordare che il segno + davanti al primo termine può essere omesso.</p>
<div class="example">
<h4 class="noindent"><span class="gep"><span class="sgr">ESEMPI</span></span></h4>
<p class="hang"><b>2</b>&nbsp;&nbsp;<span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.333ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="29.667ex" height="1.833ex" viewBox="0 -700.9 12743.2 806.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-31"></use><use x="727" y="0" xlink:href="#MJMAIN-2B"></use><use x="1732" y="0" xlink:href="#MJMAIN-32"></use><use x="2459" y="0" xlink:href="#MJMAIN-2212"></use><use x="3464" y="0" xlink:href="#MJMAIN-33"></use><use x="4192" y="0" xlink:href="#MJMAIN-2212"></use><use x="5197" y="0" xlink:href="#MJMAIN-34"></use><use x="5980" y="0" xlink:href="#MJMAIN-3D"></use><use x="7040" y="0" xlink:href="#MJMAIN-31"></use><use x="7768" y="0" xlink:href="#MJMAIN-2212"></use><use x="8773" y="0" xlink:href="#MJMAIN-33"></use><use x="9500" y="0" xlink:href="#MJMAIN-2B"></use><use x="10505" y="0" xlink:href="#MJMAIN-32"></use><use x="11233" y="0" xlink:href="#MJMAIN-2212"></use><use x="12238" y="0" xlink:href="#MJMAIN-34"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mo>−</mo><mn>3</mn><mo>−</mo><mn>4</mn><mo>=</mo><mn>1</mn><mo>−</mo><mn>3</mn><mo>+</mo><mn>2</mn><mo>−</mo><mn>4</mn></mrow></math></script></p>
<p class="hangg">Abbiamo scambiato di posto il secondo e il terzo termine. Ciascuno dei due conserva il proprio segno.</p>
<p class="hang"><b>3</b>&nbsp;&nbsp;<img src="images/pg46-2.jpg" alt="Image"></p>
<p class="hangg">Il primo termine, −7, viene portato al terzo posto. Il secondo termine, +12, passa così al primo posto e può perciò essere scritto senza il segno +.</p>
<p class="hang"><b>4</b>&nbsp;&nbsp;<img src="images/pg46-3.jpg" alt="Image"></p>
<p class="hangg">Il primo termine, 2, viene portato all’ultimo posto e lo si scrive con il segno +. Il secondo termine, −5, passa così al primo posto, e conserva il segno − che non si può omettere.</p>
</div>
<h4 class="h4">L’addizione algebrica e le parentesi<a id="ind129"></a><!--<?"parentesi|addizione algebrica e le",4,0,2>-->
</h4>
<p class="noindent">Un termine di una somma algebrica può essere, a sua volta, una somma algebrica racchiusa tra parentesi:</p>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"></div>
</div>
<div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="69.167ex" height="3ex" viewBox="0 -875 29801.6 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use><use x="1232" y="0" xlink:href="#MJMAIN-2B"></use><use x="2237" y="0" xlink:href="#MJMAIN-28"></use><use x="2631" y="0" xlink:href="#MJMAIN-33"></use><use x="3358" y="0" xlink:href="#MJMAIN-2B"></use><use x="4363" y="0" xlink:href="#MJMAIN-35"></use><use x="5091" y="0" xlink:href="#MJMAIN-2212"></use><use x="6096" y="0" xlink:href="#MJMAIN-32"></use><use x="6601" y="0" xlink:href="#MJMAIN-29"></use><use x="7217" y="0" xlink:href="#MJMAIN-2212"></use><use x="8222" y="0" xlink:href="#MJMAIN-37"></use><use x="8950" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(9955,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-31"></use></g><g transform="translate(10965,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(12185,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="13017" y="0" xlink:href="#MJMAIN-2212"></use><use x="14022" y="0" xlink:href="#MJMAIN-37"></use><use x="14749" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(15754,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="16987" y="0" xlink:href="#MJMAIN-2212"></use><use x="17992" y="0" xlink:href="#MJMAIN-5B"></use><g transform="translate(18275,0)"><use xlink:href="#MJMAIN-34"></use><use x="505" y="0" xlink:href="#MJMAIN-31"></use></g><use x="19507" y="0" xlink:href="#MJMAIN-2212"></use><use x="20512" y="0" xlink:href="#MJMAIN-28"></use><use x="20906" y="0" xlink:href="#MJMAIN-2212"></use><use x="21689" y="0" xlink:href="#MJMAIN-39"></use><use x="22417" y="0" xlink:href="#MJMAIN-2212"></use><use x="23422" y="0" xlink:href="#MJMAIN-34"></use><use x="24149" y="0" xlink:href="#MJMAIN-2B"></use><use x="25154" y="0" xlink:href="#MJMAIN-32"></use><use x="25659" y="0" xlink:href="#MJMAIN-29"></use><use x="26275" y="0" xlink:href="#MJMAIN-2B"></use><use x="27281" y="0" xlink:href="#MJMAIN-35"></use><use x="27786" y="0" xlink:href="#MJMAIN-5D"></use><use x="28291" y="0" xlink:href="#MJMAIN-2B"></use><use x="29296" y="0" xlink:href="#MJMAIN-32"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>12</mn><mo>+</mo><mo stretchy="false">(</mo><mn>3</mn><mo>+</mo><mn>5</mn><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo>−</mo><mn>7</mn><mo>−</mo><mn>21</mn><mtext>  </mtext><mtext> </mtext><mo>−</mo><mn>7</mn><mo>+</mo><mn>12</mn><mo>−</mo><mo stretchy="false">[</mo><mn>41</mn><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mn>9</mn><mo>−</mo><mn>4</mn><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mo>+</mo><mn>5</mn><mo stretchy="false">]</mo><mo>+</mo><mn>2</mn></mrow></math></script></p>
<ul class="blist">
<li><p class="noindent">Se la parentesi è preceduta dal segno +, essa si può eliminare scrivendo i termini in essa contenuti con il loro segno;</p></li>
<li><p class="noindent">se invece la parentesi è preceduta dal segno −, la si può eliminare scrivendo i termini in essa contenuti con segno cambiato.</p></li>
</ul>
<p class="noindent">Liberiamo perciò dalle parentesi la prima delle due espressioni precedenti.</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="59.5ex" height="2.5ex" viewBox="0 -773.9 25612.6 1047.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use><use x="1232" y="0" xlink:href="#MJMAIN-2B"></use><use x="2237" y="0" xlink:href="#MJMAIN-28"></use><use x="2631" y="0" xlink:href="#MJMAIN-33"></use><use x="3358" y="0" xlink:href="#MJMAIN-2B"></use><use x="4363" y="0" xlink:href="#MJMAIN-35"></use><use x="5091" y="0" xlink:href="#MJMAIN-2212"></use><use x="6096" y="0" xlink:href="#MJMAIN-32"></use><use x="6601" y="0" xlink:href="#MJMAIN-29"></use><use x="7217" y="0" xlink:href="#MJMAIN-2212"></use><use x="8222" y="0" xlink:href="#MJMAIN-37"></use><use x="8950" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(9955,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-31"></use></g><use x="11243" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(12303,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="13536" y="0" xlink:href="#MJMAIN-2B"></use><use x="14541" y="0" xlink:href="#MJMAIN-33"></use><use x="15268" y="0" xlink:href="#MJMAIN-2B"></use><use x="16273" y="0" xlink:href="#MJMAIN-35"></use><use x="17000" y="0" xlink:href="#MJMAIN-2212"></use><use x="18006" y="0" xlink:href="#MJMAIN-32"></use><use x="18733" y="0" xlink:href="#MJMAIN-2212"></use><use x="19738" y="0" xlink:href="#MJMAIN-37"></use><use x="20465" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(21471,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-31"></use></g><use x="22758" y="0" xlink:href="#MJMAIN-3D"></use><use x="23819" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(24602,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>12</mn><mo>+</mo><mo stretchy="false">(</mo><mn>3</mn><mo>+</mo><mn>5</mn><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo>−</mo><mn>7</mn><mo>−</mo><mn>21</mn><mo>=</mo><mn>12</mn><mo>+</mo><mn>3</mn><mo>+</mo><mn>5</mn><mo>−</mo><mn>2</mn><mo>−</mo><mn>7</mn><mo>−</mo><mn>21</mn><mo>=</mo><mo>−</mo><mn>10</mn></mrow></math></script></p>
<p class="noindent1">Liberiamo dalle parentesi la seconda espressione.</p>
<p class="noindent">In questo caso occorrono due passaggi.</p>
<p class="noindent">Eliminiamo le parentesi tonde:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="76.333ex" height="2.5ex" viewBox="0 -773.9 32891.8 1047.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2212"></use><use x="783" y="0" xlink:href="#MJMAIN-37"></use><use x="1510" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(2515,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="3747" y="0" xlink:href="#MJMAIN-2212"></use><use x="4752" y="0" xlink:href="#MJMAIN-5B"></use><g transform="translate(5035,0)"><use xlink:href="#MJMAIN-34"></use><use x="505" y="0" xlink:href="#MJMAIN-31"></use></g><use x="6268" y="0" xlink:href="#MJMAIN-2212"></use><use x="7273" y="0" xlink:href="#MJMAIN-28"></use><use x="7667" y="0" xlink:href="#MJMAIN-2212"></use><use x="8450" y="0" xlink:href="#MJMAIN-39"></use><use x="9177" y="0" xlink:href="#MJMAIN-2212"></use><use x="10182" y="0" xlink:href="#MJMAIN-34"></use><use x="10910" y="0" xlink:href="#MJMAIN-2B"></use><use x="11915" y="0" xlink:href="#MJMAIN-32"></use><use x="12420" y="0" xlink:href="#MJMAIN-29"></use><use x="13036" y="0" xlink:href="#MJMAIN-2B"></use><use x="14041" y="0" xlink:href="#MJMAIN-35"></use><use x="14546" y="0" xlink:href="#MJMAIN-5D"></use><use x="15051" y="0" xlink:href="#MJMAIN-2B"></use><use x="16057" y="0" xlink:href="#MJMAIN-32"></use><use x="16839" y="0" xlink:href="#MJMAIN-3D"></use><use x="17900" y="0" xlink:href="#MJMAIN-2212"></use><use x="18683" y="0" xlink:href="#MJMAIN-37"></use><use x="19410" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(20416,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="21648" y="0" xlink:href="#MJMAIN-2212"></use><use x="22653" y="0" xlink:href="#MJMAIN-5B"></use><g transform="translate(22936,0)"><use xlink:href="#MJMAIN-34"></use><use x="505" y="0" xlink:href="#MJMAIN-31"></use></g><use x="24168" y="0" xlink:href="#MJMAIN-2B"></use><use x="25174" y="0" xlink:href="#MJMAIN-39"></use><use x="25901" y="0" xlink:href="#MJMAIN-2B"></use><use x="26906" y="0" xlink:href="#MJMAIN-34"></use><use x="27633" y="0" xlink:href="#MJMAIN-2212"></use><use x="28638" y="0" xlink:href="#MJMAIN-32"></use><use x="29366" y="0" xlink:href="#MJMAIN-2B"></use><use x="30371" y="0" xlink:href="#MJMAIN-35"></use><use x="30876" y="0" xlink:href="#MJMAIN-5D"></use><use x="31381" y="0" xlink:href="#MJMAIN-2B"></use><use x="32386" y="0" xlink:href="#MJMAIN-32"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>−</mo><mn>7</mn><mo>+</mo><mn>12</mn><mo>−</mo><mo stretchy="false">[</mo><mn>41</mn><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mn>9</mn><mo>−</mo><mn>4</mn><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mo>+</mo><mn>5</mn><mo stretchy="false">]</mo><mo>+</mo><mn>2</mn><mo>=</mo><mo>−</mo><mn>7</mn><mo>+</mo><mn>12</mn><mo>−</mo><mo stretchy="false">[</mo><mn>41</mn><mo>+</mo><mn>9</mn><mo>+</mo><mn>4</mn><mo>−</mo><mn>2</mn><mo>+</mo><mn>5</mn><mo stretchy="false">]</mo><mo>+</mo><mn>2</mn></mrow></math></script></p>
<p class="noindent1">Ora eliminiamo le parentesi quadre:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="78.667ex" height="2.5ex" viewBox="0 -773.9 33886.3 1047.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2212"></use><use x="783" y="0" xlink:href="#MJMAIN-37"></use><use x="1510" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(2515,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="3747" y="0" xlink:href="#MJMAIN-2212"></use><use x="4752" y="0" xlink:href="#MJMAIN-5B"></use><g transform="translate(5035,0)"><use xlink:href="#MJMAIN-34"></use><use x="505" y="0" xlink:href="#MJMAIN-31"></use></g><use x="6268" y="0" xlink:href="#MJMAIN-2B"></use><use x="7273" y="0" xlink:href="#MJMAIN-39"></use><use x="8000" y="0" xlink:href="#MJMAIN-2B"></use><use x="9005" y="0" xlink:href="#MJMAIN-34"></use><use x="9733" y="0" xlink:href="#MJMAIN-2212"></use><use x="10738" y="0" xlink:href="#MJMAIN-32"></use><use x="11465" y="0" xlink:href="#MJMAIN-2B"></use><use x="12470" y="0" xlink:href="#MJMAIN-35"></use><use x="12975" y="0" xlink:href="#MJMAIN-5D"></use><use x="13480" y="0" xlink:href="#MJMAIN-2B"></use><use x="14486" y="0" xlink:href="#MJMAIN-32"></use><use x="15268" y="0" xlink:href="#MJMAIN-3D"></use><use x="16329" y="0" xlink:href="#MJMAIN-2212"></use><use x="17112" y="0" xlink:href="#MJMAIN-37"></use><use x="17839" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(18845,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="20077" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(21082,0)"><use xlink:href="#MJMAIN-34"></use><use x="505" y="0" xlink:href="#MJMAIN-31"></use></g><use x="22314" y="0" xlink:href="#MJMAIN-2212"></use><use x="23320" y="0" xlink:href="#MJMAIN-39"></use><use x="24047" y="0" xlink:href="#MJMAIN-2212"></use><use x="25052" y="0" xlink:href="#MJMAIN-34"></use><use x="25779" y="0" xlink:href="#MJMAIN-2B"></use><use x="26784" y="0" xlink:href="#MJMAIN-32"></use><use x="27512" y="0" xlink:href="#MJMAIN-2212"></use><use x="28517" y="0" xlink:href="#MJMAIN-35"></use><use x="29244" y="0" xlink:href="#MJMAIN-2B"></use><use x="30249" y="0" xlink:href="#MJMAIN-32"></use><use x="31032" y="0" xlink:href="#MJMAIN-3D"></use><use x="32093" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(32876,0)"><use xlink:href="#MJMAIN-35"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>−</mo><mn>7</mn><mo>+</mo><mn>12</mn><mo>−</mo><mo stretchy="false">[</mo><mn>41</mn><mo>+</mo><mn>9</mn><mo>+</mo><mn>4</mn><mo>−</mo><mn>2</mn><mo>+</mo><mn>5</mn><mo stretchy="false">]</mo><mo>+</mo><mn>2</mn><mo>=</mo><mo>−</mo><mn>7</mn><mo>+</mo><mn>12</mn><mo>−</mo><mn>41</mn><mo>−</mo><mn>9</mn><mo>−</mo><mn>4</mn><mo>+</mo><mn>2</mn><mo>−</mo><mn>5</mn><mo>+</mo><mn>2</mn><mo>=</mo><mo>−</mo><mn>50</mn></mrow></math></script></p>
</div></div>
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<div class="small-12 medium-4 columns"><div class="sidebox">
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="20.833ex" height="2.5ex" viewBox="0 -773.9 8950.4 1047.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2B"></use><use x="783" y="0" xlink:href="#MJMAIN-28"></use><use x="1177" y="0" xlink:href="#MJMAIN-2212"></use><use x="1960" y="0" xlink:href="#MJMAIN-32"></use><use x="2687" y="0" xlink:href="#MJMAIN-2B"></use><use x="3692" y="0" xlink:href="#MJMAIN-35"></use><use x="4197" y="0" xlink:href="#MJMAIN-29"></use><use x="4869" y="0" xlink:href="#MJMAIN-3D"></use><use x="5929" y="0" xlink:href="#MJMAIN-2212"></use><use x="6712" y="0" xlink:href="#MJMAIN-32"></use><use x="7440" y="0" xlink:href="#MJMAIN-2B"></use><use x="8445" y="0" xlink:href="#MJMAIN-35"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>+</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo>+</mo><mn>5</mn><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>2</mn><mo>+</mo><mn>5</mn></mrow></math></script></p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="20.833ex" height="2.5ex" viewBox="0 -773.9 8950.4 1047.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2212"></use><use x="783" y="0" xlink:href="#MJMAIN-28"></use><use x="1177" y="0" xlink:href="#MJMAIN-2212"></use><use x="1960" y="0" xlink:href="#MJMAIN-33"></use><use x="2687" y="0" xlink:href="#MJMAIN-2B"></use><use x="3692" y="0" xlink:href="#MJMAIN-37"></use><use x="4197" y="0" xlink:href="#MJMAIN-29"></use><use x="4869" y="0" xlink:href="#MJMAIN-3D"></use><use x="5929" y="0" xlink:href="#MJMAIN-2B"></use><use x="6712" y="0" xlink:href="#MJMAIN-33"></use><use x="7440" y="0" xlink:href="#MJMAIN-2212"></use><use x="8445" y="0" xlink:href="#MJMAIN-37"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mn>3</mn><mo>+</mo><mn>7</mn><mo stretchy="false">)</mo><mo>=</mo><mo>+</mo><mn>3</mn><mo>−</mo><mn>7</mn></mrow></math></script></p>
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<div data-page-container="29" id="page-29" class="row chapters-content">
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<div class="small-12 medium-1 columns"><div class="pagenumber">29</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<h3 class="sec_title" id="sec25">
<span class="pagebreak" epub:type="pagebreak" title="29" id="page29"></span>25. Moltiplicazione<a id="ind130"></a><!--<?"moltiplicazione",4,0,2>-->
</h3>
<div class="definition" title_dea="Prodotto di due numeri interi relativi" key_dea="prodotto, moltiplicazione, numero intero relativo, numeri interi relativi, interi relativi, intero relativo">
<h4 class="noindent">
<span class="ash">DEFINIZIONE</span> <span class="orangeb">PRODOTTO DI DUE NUMERI INTERI RELATIVI</span>
</h4>
<p class="noindentin">Il prodotto di due numeri interi relativi<a id="ind131"></a><!--<?"prodotto|di numeri interi relativi",4,0,2>--> è il numero che ha per valore assoluto il prodotto dei valori assoluti dei due fattori e, per segno, il segno + se i due fattori sono concordi, il segno − se sono discordi.</p>
</div>
<div class="rule" title_dea="Regola dei segni" key_dea="regola dei segni, segno, segni">
<h4 class="noindent"><span class="red"><b>REGOLA</b></span></h4>
<p class="noindent">Si determina il segno di un prodotto mediante la <b>regola dei segni</b>,<a id="ind132"></a><!--<?"segno|di un prodotto mediante la regola dei segni",4,0,2>--> che riassumiamo nella tabella.</p>
<div class="table" title_dea="Tabella della regola dei segni" key_dea="tabella della regola dei segni, regola dei segni, segno del prodotto">
<p class="img"><img src="images/pg47.jpg" alt="Image"></p>
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<div class="row">
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<div class="small-12 medium-11 columns"><div class="ch1">
<div class="example">
<h4 class="noindent"><span class="gep"><span class="sgr">ESEMPI</span></span></h4>
<p class="hang"><b>1</b>&nbsp;&nbsp;(+3) · (+4)</p>
<p class="hangg">I due fattori sono concordi, quindi il segno del prodotto è +. Il prodotto dei valori assoluti dei due fattori è 3 · 4 = 12.</p>
<p class="hangg">Quindi:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="18.667ex" height="2.5ex" viewBox="0 -773.9 8011 1047.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2B"></use><use x="1177" y="0" xlink:href="#MJMAIN-33"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use x="2298" y="0" xlink:href="#MJMAIN-22C5"></use><use x="2803" y="0" xlink:href="#MJMAIN-28"></use><use x="3197" y="0" xlink:href="#MJMAIN-2B"></use><use x="3980" y="0" xlink:href="#MJMAIN-34"></use><use x="4485" y="0" xlink:href="#MJMAIN-29"></use><use x="5157" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(6217,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-2B"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(783,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mo>=</mo><mstyle color="#00aef0"><mo>+</mo><mn>12</mn></mstyle></mrow></math></script></p>
<p class="hang"><b>2</b>&nbsp;&nbsp;(−5) · (+2)</p>
<p class="hangg">I due fattori sono discordi, quindi il segno del prodotto è −. Il prodotto dei valori assoluti dei due fattori è 5 · 2 = 10.</p>
<p class="hangg">Quindi:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="18.667ex" height="2.5ex" viewBox="0 -773.9 8011 1047.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-35"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use x="2298" y="0" xlink:href="#MJMAIN-22C5"></use><use x="2803" y="0" xlink:href="#MJMAIN-28"></use><use x="3197" y="0" xlink:href="#MJMAIN-2B"></use><use x="3980" y="0" xlink:href="#MJMAIN-32"></use><use x="4485" y="0" xlink:href="#MJMAIN-29"></use><use x="5157" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(6217,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-2212"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(783,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>5</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mstyle color="#00aef0"><mo>−</mo><mn>10</mn></mstyle></mrow></math></script></p>
<p class="hang"><b>3</b>&nbsp;&nbsp;(−6) · (−3)</p>
<p class="hangg">I due fattori sono concordi, quindi il segno del prodotto è +. Il prodotto dei valori assoluti dei due fattori è 6 · 3 = 18.</p>
<p class="hangg">Quindi:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="18.667ex" height="2.5ex" viewBox="0 -773.9 8011 1047.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-36"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use x="2298" y="0" xlink:href="#MJMAIN-22C5"></use><use x="2803" y="0" xlink:href="#MJMAIN-28"></use><use x="3197" y="0" xlink:href="#MJMAIN-2212"></use><use x="3980" y="0" xlink:href="#MJMAIN-33"></use><use x="4485" y="0" xlink:href="#MJMAIN-29"></use><use x="5157" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(6217,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-2B"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(783,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-38"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>6</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo><mo>=</mo><mstyle color="#00aef0"><mo>+</mo><mn>18</mn></mstyle></mrow></math></script></p>
</div>
<p class="noindent">La moltiplicazione, nell’insieme dei numeri interi relativi, gode delle stesse proprietà che valgono nell’insieme dei numeri naturali (vedi <span class="fron">TEORIA.ZIP</span>).</p>
<h4 class="h4">Prodotto di tre o più fattori</h4>
<p class="noindent">La proprietà commutativa e la proprietà associativa della moltiplicazione ci permettono di calcolare il prodotto di tre o più fattori.</p>
<p class="noindent">La seguente regola permette di semplificare i calcoli.</p>
<div class="rule" title_dea="Regola dei segni: il prodotto di tre o più fattori" key_dea="regola dei segni, prodotto, più fattori, regola dei segni: il prodotto di tre o più fattori">
<h4 class="noindent"><span class="red"><b>REGOLA</b></span></h4>
<p class="noindent">Il prodotto di tre o più numeri interi relativi è il numero intero relativo che ha per valore assoluto il prodotto dei valori assoluti dei fattori e, per segno, il segno + se il numero dei fattori negativi è pari, il segno − se il numero dei fattori negativi è dispari.</p>
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<div class="small-12 medium-4 columns"><div class="sidebox">
<p class="noindentf">Il segno di moltiplicazione tra coppie di parentesi si può anche omettere:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -2.5ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="24.833ex" height="6ex" viewBox="0 -1539.2 10693.5 2578.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(167,0)"><g transform="translate(-11,0)"><g transform="translate(0,725)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2B"></use><use x="1177" y="0" xlink:href="#MJMAIN-33"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use x="2298" y="0" xlink:href="#MJMAIN-22C5"></use><use x="2803" y="0" xlink:href="#MJMAIN-28"></use><use x="3197" y="0" xlink:href="#MJMAIN-2B"></use><use x="3980" y="0" xlink:href="#MJMAIN-34"></use><use x="4485" y="0" xlink:href="#MJMAIN-29"></use><use x="5157" y="0" xlink:href="#MJMAIN-3D"></use><use x="6217" y="0" xlink:href="#MJMAIN-28"></use><use x="6611" y="0" xlink:href="#MJMAIN-2B"></use><use x="7394" y="0" xlink:href="#MJMAIN-33"></use><use x="7899" y="0" xlink:href="#MJMAIN-29"></use><use x="8294" y="0" xlink:href="#MJMAIN-28"></use><use x="8688" y="0" xlink:href="#MJMAIN-2B"></use><use x="9471" y="0" xlink:href="#MJMAIN-34"></use><use x="9976" y="0" xlink:href="#MJMAIN-29"></use></g><g transform="translate(1571,-766)"><use xlink:href="#MJMAIN-37"></use><use x="727" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1232" y="0" xlink:href="#MJMAIN-28"></use><use x="1626" y="0" xlink:href="#MJMAIN-2212"></use><use x="2409" y="0" xlink:href="#MJMAIN-38"></use><use x="2914" y="0" xlink:href="#MJMAIN-29"></use><use x="3586" y="0" xlink:href="#MJMAIN-3D"></use><use x="4647" y="0" xlink:href="#MJMAIN-37"></use><use x="5152" y="0" xlink:href="#MJMAIN-28"></use><use x="5546" y="0" xlink:href="#MJMAIN-2212"></use><use x="6329" y="0" xlink:href="#MJMAIN-38"></use><use x="6834" y="0" xlink:href="#MJMAIN-29"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd><mrow><mo stretchy="false">(</mo><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mn>7</mn><mo>⋅</mo><mo stretchy="false">(</mo><mo>−</mo><mn>8</mn><mo stretchy="false">)</mo><mo>=</mo><mn>7</mn><mo stretchy="false">(</mo><mo>−</mo><mn>8</mn><mo stretchy="false">)</mo></mrow></mtd></mtr></mtable></mrow></math></script></p>
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<div data-page-container="30" id="page-30" class="row chapters-content">
<div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">30</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<div class="example">
<h4 class="noindent">
<span class="pagebreak" epub:type="pagebreak" title="30" id="page30"></span><span class="gep"><span class="sgr">ESEMPI</span></span>
</h4>
<p class="hang"><b>4</b>&nbsp;&nbsp;Calcoliamo (−3) · (−5) · (+3) · (−10) · (−1) · (+2).</p>
<ul class="blist">
<li><p class="noindent">Il prodotto dei valori assoluti dei fattori è 3 · 5 · 3 · 10 · 1 · 2 = 900.</p></li>
<li><p class="noindent">I fattori negativi sono 4, in numero pari; perciò il segno del prodotto è +.</p></li>
<li>
<p class="noindent">Possiamo concludere che:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="47ex" height="2.5ex" viewBox="0 -773.9 20234.8 1047.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-33"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use x="2298" y="0" xlink:href="#MJMAIN-22C5"></use><use x="2803" y="0" xlink:href="#MJMAIN-28"></use><use x="3197" y="0" xlink:href="#MJMAIN-2212"></use><use x="3980" y="0" xlink:href="#MJMAIN-35"></use><use x="4485" y="0" xlink:href="#MJMAIN-29"></use><use x="5101" y="0" xlink:href="#MJMAIN-22C5"></use><use x="5606" y="0" xlink:href="#MJMAIN-28"></use><use x="6000" y="0" xlink:href="#MJMAIN-2B"></use><use x="6783" y="0" xlink:href="#MJMAIN-33"></use><use x="7288" y="0" xlink:href="#MJMAIN-29"></use><use x="7905" y="0" xlink:href="#MJMAIN-22C5"></use><use x="8410" y="0" xlink:href="#MJMAIN-28"></use><use x="8804" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(9587,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use></g><use x="10597" y="0" xlink:href="#MJMAIN-29"></use><use x="11213" y="0" xlink:href="#MJMAIN-22C5"></use><use x="11718" y="0" xlink:href="#MJMAIN-28"></use><use x="12112" y="0" xlink:href="#MJMAIN-2212"></use><use x="12895" y="0" xlink:href="#MJMAIN-31"></use><use x="13400" y="0" xlink:href="#MJMAIN-29"></use><use x="14017" y="0" xlink:href="#MJMAIN-22C5"></use><use x="14522" y="0" xlink:href="#MJMAIN-28"></use><use x="14916" y="0" xlink:href="#MJMAIN-2B"></use><use x="15699" y="0" xlink:href="#MJMAIN-32"></use><use x="16204" y="0" xlink:href="#MJMAIN-29"></use><use x="16876" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(17936,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-2B"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(783,0)"><use xlink:href="#MJMAIN-39"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use><use x="1010" y="0" xlink:href="#MJMAIN-30"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo>−</mo><mn>5</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo>−</mo><mn>10</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mstyle color="#00aef0"><mo>+</mo><mn>900</mn></mstyle></mrow></math></script></p>
</li>
</ul>
<p class="hang"><b>5</b>&nbsp;&nbsp;Calcoliamo (−2) · (+3) · (−3) · (+12) · (−10) · (−1) · (−20).</p>
<ul class="blist">
<li><p class="noindent">Il prodotto dei valori assoluti dei fattori è 2 · 3 · 3 · 12 · 10 · 1 · 20 = 43 200.</p></li>
<li><p class="noindent">I fattori negativi sono 5, in numero dispari; perciò il segno del prodotto è −.</p></li>
<li>
<p class="noindent">Concludendo:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="59.667ex" height="3ex" viewBox="0 -875 25668.2 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-32"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use x="2298" y="0" xlink:href="#MJMAIN-22C5"></use><use x="2803" y="0" xlink:href="#MJMAIN-28"></use><use x="3197" y="0" xlink:href="#MJMAIN-2B"></use><use x="3980" y="0" xlink:href="#MJMAIN-33"></use><use x="4485" y="0" xlink:href="#MJMAIN-29"></use><use x="5101" y="0" xlink:href="#MJMAIN-22C5"></use><use x="5606" y="0" xlink:href="#MJMAIN-28"></use><use x="6000" y="0" xlink:href="#MJMAIN-2212"></use><use x="6783" y="0" xlink:href="#MJMAIN-33"></use><use x="7288" y="0" xlink:href="#MJMAIN-29"></use><use x="7905" y="0" xlink:href="#MJMAIN-22C5"></use><use x="8410" y="0" xlink:href="#MJMAIN-28"></use><use x="8804" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(9587,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="10597" y="0" xlink:href="#MJMAIN-29"></use><use x="11213" y="0" xlink:href="#MJMAIN-22C5"></use><use x="11718" y="0" xlink:href="#MJMAIN-28"></use><use x="12112" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(12895,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use></g><use x="13905" y="0" xlink:href="#MJMAIN-29"></use><use x="14522" y="0" xlink:href="#MJMAIN-22C5"></use><use x="15027" y="0" xlink:href="#MJMAIN-28"></use><use x="15421" y="0" xlink:href="#MJMAIN-2212"></use><use x="16204" y="0" xlink:href="#MJMAIN-31"></use><use x="16709" y="0" xlink:href="#MJMAIN-29"></use><use x="17325" y="0" xlink:href="#MJMAIN-22C5"></use><use x="17830" y="0" xlink:href="#MJMAIN-28"></use><use x="18224" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(19007,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use></g><use x="20017" y="0" xlink:href="#MJMAIN-29"></use><use x="20689" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(21750,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-2212"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(783,0)"><use xlink:href="#MJMAIN-34"></use><use x="505" y="0" xlink:href="#MJMAIN-33"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1793,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g fill="#00aef0" stroke="#00aef0" transform="translate(2402,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use><use x="1010" y="0" xlink:href="#MJMAIN-30"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo>+</mo><mn>12</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo>−</mo><mn>10</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo>−</mo><mn>20</mn><mo stretchy="false">)</mo><mo>=</mo><mstyle color="#00aef0"><mo>−</mo><mn>43</mn><mtext> </mtext><mn>200</mn></mstyle></mrow></math></script></p>
</li>
</ul>
</div>
<h3 class="sec_title" id="sec26">26. Divisione<a id="ind133"></a><!--<?"divisione",4,0,2>-->
</h3>
<div class="definition" title_dea="Quoziente di due numeri interi relativi" key_dea="quoziente, divisione, numero intero relativo, numeri interi relativi, interi relativi, intero relativo, quoziente di due numeri interi relativi">
<h4 class="noindent">
<span class="ash">DEFINIZIONE</span> <span class="orangeb">QUOZIENTE DI DUE NUMERI INTERI RELATIVI</span>
</h4>
<p class="noindentin">Il quoziente di due numeri interi relativi,<a id="ind134"></a><!--<?"quoziente|di due numeri|interi relativi",4,0,2>--> il secondo dei quali diverso da zero, è il numero intero relativo, se esiste, che ha per valore assoluto il quoziente della divisione tra i valori assoluti del dividendo e del divisore e, per segno, il segno + se i due numeri sono concordi, il segno − se sono discordi.</p>
</div>
<p class="noindent"><a id="ind135"></a><!--<?"divisibilit&#x00E0;",4,0,2>-->Anche nell’insieme dei numeri interi relativi, <b>la divisione per zero<a id="ind136"></a><!--<?"divisione|per zero",4,0,2>--> non ha significato</b>.</p>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"></div>
</div>
<div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<p class="noindent">La regola, mediante la quale si determina il segno di un quoziente, è la stessa regola dei segni già vista a proposito della moltiplicazione.</p>
<div class="example">
<h4 class="noindent"><span class="gep"><span class="sgr">ESEMPI</span></span></h4>
<p class="hang"><b>1</b>&nbsp;&nbsp;(+8) : (+4)</p>
<p class="hangg">Dividendo e divisore sono concordi, perciò il segno del quoziente è +.</p>
<p class="hangg">Il quoziente dei valori assoluti è 8 : 4 = 2. Quindi</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="17.667ex" height="2.5ex" viewBox="0 -773.9 7617.1 1047.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2B"></use><use x="1177" y="0" xlink:href="#MJMAIN-38"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use x="2353" y="0" xlink:href="#MJMAIN-3A"></use><use x="2914" y="0" xlink:href="#MJMAIN-28"></use><use x="3308" y="0" xlink:href="#MJMAIN-2B"></use><use x="4091" y="0" xlink:href="#MJMAIN-34"></use><use x="4596" y="0" xlink:href="#MJMAIN-29"></use><use x="5268" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(6329,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-2B"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(783,0)"><use xlink:href="#MJMAIN-32"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mo>+</mo><mn>8</mn><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mo>=</mo><mstyle color="#00aef0"><mo>+</mo><mn>2</mn></mstyle></mrow></math></script></p>
<p class="hang"><b>2</b>&nbsp;&nbsp;(−15) : (+3)</p>
<p class="hangg">Dividendo e divisore sono discordi, perciò il segno del quoziente è −.</p>
<p class="hangg">Il quoziente dei valori assoluti è 15 : 3 = 5. Quindi</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="18.833ex" height="2.5ex" viewBox="0 -773.9 8122.1 1047.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(1177,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-35"></use></g><use x="2187" y="0" xlink:href="#MJMAIN-29"></use><use x="2858" y="0" xlink:href="#MJMAIN-3A"></use><use x="3419" y="0" xlink:href="#MJMAIN-28"></use><use x="3813" y="0" xlink:href="#MJMAIN-2B"></use><use x="4596" y="0" xlink:href="#MJMAIN-33"></use><use x="5101" y="0" xlink:href="#MJMAIN-29"></use><use x="5773" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(6834,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-2212"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(783,0)"><use xlink:href="#MJMAIN-35"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>15</mn><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo><mo>=</mo><mstyle color="#00aef0"><mo>−</mo><mn>5</mn></mstyle></mrow></math></script></p>
<p class="hang"><b>3</b>&nbsp;&nbsp;(−16): (−4)</p>
<p class="hangg">Dividendo e divisore sono concordi, perciò il segno del quoziente è +.</p>
<p class="hangg">Il quoziente dei valori assoluti è 16 : 4 = 4. Quindi</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="18.833ex" height="2.5ex" viewBox="0 -773.9 8122.1 1047.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(1177,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-36"></use></g><use x="2187" y="0" xlink:href="#MJMAIN-29"></use><use x="2858" y="0" xlink:href="#MJMAIN-3A"></use><use x="3419" y="0" xlink:href="#MJMAIN-28"></use><use x="3813" y="0" xlink:href="#MJMAIN-2212"></use><use x="4596" y="0" xlink:href="#MJMAIN-34"></use><use x="5101" y="0" xlink:href="#MJMAIN-29"></use><use x="5773" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(6834,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-2B"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(783,0)"><use xlink:href="#MJMAIN-34"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>16</mn><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo><mo>=</mo><mstyle color="#00aef0"><mo>+</mo><mn>4</mn></mstyle></mrow></math></script></p>
</div>
<p class="noindent">Come accade nella divisione tra numeri naturali, anche per i numeri interi relativi <b>il quoziente è quel numero che, moltiplicato per il divisore, dà per prodotto il dividendo</b>.</p>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"><div class="sidebox">
<p class="noindent">Le nozioni sulla divisibilità<a id="ind137"></a><!--<?"divisibilit&#x00E0;",4,0,2>--> tra numeri in ℕ si estendono anche ai numeri interi relativi.</p>
<p class="noindent">Ad esempio, −12 è divisibile per 4 perché |−12| = 12 è divisibile per 4 e risulta</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="16.167ex" height="3ex" viewBox="0 -875 6983 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2212"></use><g transform="translate(783,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><g transform="translate(1793,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="2680" y="0" xlink:href="#MJMAIN-3A"></use><g transform="translate(3241,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="3851" y="0" xlink:href="#MJMAIN-34"></use><use x="4634" y="0" xlink:href="#MJMAIN-3D"></use><use x="5695" y="0" xlink:href="#MJMAIN-2212"></use><use x="6478" y="0" xlink:href="#MJMAIN-33"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>−</mo><mn>12</mn><mtext> </mtext><mo>:</mo><mtext> </mtext><mn>4</mn><mo>=</mo><mo>−</mo><mn>3</mn></mrow></math></script></p>
</div></div>
</div>
</div>
<div data-page-container="31" id="page-31" class="row chapters-content"><div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">31</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<p class="noindent"><span class="pagebreak" epub:type="pagebreak" title="31" id="page31"></span>Come per i numeri naturali, anche in ℤ non sempre esiste il quoziente tra due numeri: ad esempio non è possibile determinare il quoziente tra (+8) e (−3).</p>
<p class="noindent">Inoltre la divisione in ℤ gode delle stesse proprietà che valgono in ℕ (vedi <span class="fron">TEORIA.ZIP</span>).</p>
<p class="noindent">La divisione <b>non</b> gode delle proprietà <i>commutativa</i> e <i>associativa</i> e <b>non</b> ha <i>elemento neutro</i> (<i>a</i> : 1 = <i>a</i> ma 1 : <i>a</i> non si può eseguire in ℤ se <i>a</i> ≠ ±1).</p>
<h2 class="para_title" id="par10">Potenza di un numero intero relativo<a id="ind138"></a><!--<?"potenza|di un numero intero relativo",4,0,2>-->
</h2>
<h3 class="sec_title" id="sec27">27. Le potenze nell’insieme dei numeri interi relativi</h3>
<p class="noindent">La definizione e le proprietà delle potenze in ℤ<a id="ind139"></a><!--<?"potenze|in Z",4,0,2>--> sono le stesse che valgono in ℕ (vedi <span class="fron">TEORIA.ZIP</span>). Occorre osservare che le potenze dei numeri interi relativi hanno significato in ℤ solo se l’<b>esponente</b> è un <b>numero naturale</b>.</p>
<div class="example">
<h4 class="noindent"><span class="gep"><span class="sgr">ESEMPI</span></span></h4>
<p class="hang"><b>1</b>&nbsp;&nbsp;<span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="41.833ex" height="2.833ex" viewBox="0 -978.2 18044.6 1252.1"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2B"></use><use x="1177" y="0" xlink:href="#MJMAIN-35"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-33"></use><use x="2810" y="0" xlink:href="#MJMAIN-3D"></use><use x="3871" y="0" xlink:href="#MJMAIN-28"></use><use x="4265" y="0" xlink:href="#MJMAIN-2B"></use><use x="5048" y="0" xlink:href="#MJMAIN-35"></use><use x="5553" y="0" xlink:href="#MJMAIN-29"></use><use x="6169" y="0" xlink:href="#MJMAIN-22C5"></use><use x="6675" y="0" xlink:href="#MJMAIN-28"></use><use x="7069" y="0" xlink:href="#MJMAIN-2B"></use><use x="7852" y="0" xlink:href="#MJMAIN-35"></use><use x="8357" y="0" xlink:href="#MJMAIN-29"></use><use x="8973" y="0" xlink:href="#MJMAIN-22C5"></use><use x="9478" y="0" xlink:href="#MJMAIN-28"></use><use x="9872" y="0" xlink:href="#MJMAIN-2B"></use><use x="10655" y="0" xlink:href="#MJMAIN-35"></use><use x="11160" y="0" xlink:href="#MJMAIN-29"></use><use x="11832" y="0" xlink:href="#MJMAIN-3D"></use><use x="12893" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(13676,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use><use x="1010" y="0" xlink:href="#MJMAIN-35"></use></g><use x="15468" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(16529,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use><use x="1010" y="0" xlink:href="#MJMAIN-35"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mo stretchy="false">(</mo><mo>+</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><mo>=</mo><mo stretchy="false">(</mo><mo>+</mo><mn>5</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo>+</mo><mn>5</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo>+</mo><mn>5</mn><mo stretchy="false">)</mo><mo>=</mo><mo>+</mo><mn>125</mn><mo>=</mo><mn color="#00aef0">125</mn></mrow></math></script></p>
<p class="hangg"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="47.333ex" height="2.833ex" viewBox="0 -978.2 20399.1 1252.1"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-32"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-33"></use><use x="2810" y="0" xlink:href="#MJMAIN-3D"></use><use x="3871" y="0" xlink:href="#MJMAIN-28"></use><use x="4265" y="0" xlink:href="#MJMAIN-2212"></use><use x="5048" y="0" xlink:href="#MJMAIN-32"></use><use x="5553" y="0" xlink:href="#MJMAIN-29"></use><use x="6169" y="0" xlink:href="#MJMAIN-22C5"></use><use x="6675" y="0" xlink:href="#MJMAIN-28"></use><use x="7069" y="0" xlink:href="#MJMAIN-2212"></use><use x="7852" y="0" xlink:href="#MJMAIN-32"></use><use x="8357" y="0" xlink:href="#MJMAIN-29"></use><use x="8973" y="0" xlink:href="#MJMAIN-22C5"></use><use x="9478" y="0" xlink:href="#MJMAIN-28"></use><use x="9872" y="0" xlink:href="#MJMAIN-2212"></use><use x="10655" y="0" xlink:href="#MJMAIN-32"></use><use x="11160" y="0" xlink:href="#MJMAIN-29"></use><use x="11832" y="0" xlink:href="#MJMAIN-3D"></use><use x="12893" y="0" xlink:href="#MJMAIN-28"></use><use x="13287" y="0" xlink:href="#MJMAIN-2B"></use><use x="14070" y="0" xlink:href="#MJMAIN-34"></use><use x="14575" y="0" xlink:href="#MJMAIN-29"></use><use x="15191" y="0" xlink:href="#MJMAIN-22C5"></use><use x="15696" y="0" xlink:href="#MJMAIN-28"></use><use x="16090" y="0" xlink:href="#MJMAIN-2212"></use><use x="16873" y="0" xlink:href="#MJMAIN-32"></use><use x="17378" y="0" xlink:href="#MJMAIN-29"></use><use x="18050" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(19111,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-2212"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(783,0)"><use xlink:href="#MJMAIN-38"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mstyle color="#00aef0"><mo>−</mo><mn>8</mn></mstyle></mrow></math></script></p>
<p class="hang"><b>2</b>&nbsp;&nbsp;<span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="71.333ex" height="3ex" viewBox="0 -986.7 30712.9 1260.6"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-32"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-34"></use><use x="2810" y="0" xlink:href="#MJMAIN-3D"></use><use x="3871" y="0" xlink:href="#MJMAIN-28"></use><use x="4265" y="0" xlink:href="#MJMAIN-2212"></use><use x="5048" y="0" xlink:href="#MJMAIN-32"></use><use x="5553" y="0" xlink:href="#MJMAIN-29"></use><use x="5947" y="0" xlink:href="#MJMAIN-28"></use><use x="6341" y="0" xlink:href="#MJMAIN-2212"></use><use x="7124" y="0" xlink:href="#MJMAIN-32"></use><use x="7629" y="0" xlink:href="#MJMAIN-29"></use><use x="8023" y="0" xlink:href="#MJMAIN-28"></use><use x="8417" y="0" xlink:href="#MJMAIN-2212"></use><use x="9200" y="0" xlink:href="#MJMAIN-32"></use><use x="9705" y="0" xlink:href="#MJMAIN-29"></use><use x="10099" y="0" xlink:href="#MJMAIN-28"></use><use x="10493" y="0" xlink:href="#MJMAIN-2212"></use><use x="11276" y="0" xlink:href="#MJMAIN-32"></use><use x="11781" y="0" xlink:href="#MJMAIN-29"></use><use x="12453" y="0" xlink:href="#MJMAIN-3D"></use><use x="13514" y="0" xlink:href="#MJMAIN-28"></use><use x="13908" y="0" xlink:href="#MJMAIN-2B"></use><use x="14691" y="0" xlink:href="#MJMAIN-34"></use><use x="15196" y="0" xlink:href="#MJMAIN-29"></use><use x="15590" y="0" xlink:href="#MJMAIN-28"></use><use x="15984" y="0" xlink:href="#MJMAIN-2212"></use><use x="16767" y="0" xlink:href="#MJMAIN-32"></use><use x="17272" y="0" xlink:href="#MJMAIN-29"></use><use x="17666" y="0" xlink:href="#MJMAIN-28"></use><use x="18060" y="0" xlink:href="#MJMAIN-2212"></use><use x="18843" y="0" xlink:href="#MJMAIN-32"></use><use x="19348" y="0" xlink:href="#MJMAIN-29"></use><use x="20019" y="0" xlink:href="#MJMAIN-3D"></use><use x="21080" y="0" xlink:href="#MJMAIN-28"></use><use x="21474" y="0" xlink:href="#MJMAIN-2212"></use><use x="22257" y="0" xlink:href="#MJMAIN-38"></use><use x="22762" y="0" xlink:href="#MJMAIN-29"></use><use x="23156" y="0" xlink:href="#MJMAIN-28"></use><use x="23550" y="0" xlink:href="#MJMAIN-2212"></use><use x="24333" y="0" xlink:href="#MJMAIN-32"></use><use x="24838" y="0" xlink:href="#MJMAIN-29"></use><use x="25510" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(26571,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-2B"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(783,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-36"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(2070,0)"><use xlink:href="#MJMAIN-3D"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(3131,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-36"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mn>4</mn></msup><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><mn>8</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mstyle color="#00aef0"><mo>+</mo><mn>16</mn><mo>=</mo><mn>16</mn></mstyle></mrow></math></script></p>
<p class="hangg"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="29ex" height="2.833ex" viewBox="0 -978.9 12493.8 1252.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-33"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-32"></use><use x="2810" y="0" xlink:href="#MJMAIN-3D"></use><use x="3871" y="0" xlink:href="#MJMAIN-28"></use><use x="4265" y="0" xlink:href="#MJMAIN-2212"></use><use x="5048" y="0" xlink:href="#MJMAIN-33"></use><use x="5553" y="0" xlink:href="#MJMAIN-29"></use><use x="5947" y="0" xlink:href="#MJMAIN-28"></use><use x="6341" y="0" xlink:href="#MJMAIN-2212"></use><use x="7124" y="0" xlink:href="#MJMAIN-33"></use><use x="7629" y="0" xlink:href="#MJMAIN-29"></use><use x="8301" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(9362,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-2B"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(783,0)"><use xlink:href="#MJMAIN-39"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1565,0)"><use xlink:href="#MJMAIN-3D"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(2626,0)"><use xlink:href="#MJMAIN-39"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo><mo>=</mo><mstyle color="#00aef0"><mo>+</mo><mn>9</mn><mo>=</mo><mn>9</mn></mstyle></mrow></math></script></p>
</div>
<p class="noindent">In base ai risultati nei precedenti esempi è giustificata la regola seguente.</p>
<div class="rule" title_dea="Potenza di un numero intero relativo" key_dea="potenza, potenze, elevamento a potenza, numero intero relativo, numeri interi relativi, interi relativi, intero relativo, potenza di un numero intero relativo">
<h4 class="noindent"><span class="red"><b>REGOLA</b></span></h4>
<p class="noindent">Per calcolare la potenza di un numero intero relativo, si calcola la potenza del valore assoluto della base; essa sarà il valore assoluto della potenza. Se la base ha segno +, anche la potenza ha segno +; se invece la base ha segno −, la potenza ha segno + se l’esponente è pari e segno − se l’esponente è dispari.</p>
</div>
<div class="sidetable">
<table width="30%" cellspacing="0" cellpadding="0">
<colgroup><col width="10%">
<col width="10%">
<col width="10%">
</colgroup><tbody>
<tr>
<td class="tabv">&nbsp;</td>
<td class="tabvab"><b><i>p</i> pari</b></td>
<td class="tabvab"><b><i>d</i> dispari</b></td>
</tr>
<tr>
<td class="tabvbb"><b><i>a</i> &gt; 0</b></td>
<td class="tabv">
<i>a<sup>p</sup></i> &gt; 0</td>
<td class="tabv">
<i>a<sup>d</sup></i> &gt; 0</td>
</tr>
<tr>
<td class="tabvbb"><b><i>a</i> &lt; 0</b></td>
<td class="tabv"><span class="red"><i>a<sup>p</sup></i> &gt; 0</span></td>
<td class="tabv">
<i>a<sup>d</sup></i> &lt; 0</td>
</tr>
</tbody>
</table>
</div>
<div class="box">
<p class="noindent"><span class="red"><b>IMPORTANTE</b></span></p>
<p class="noindent">Le potenze hanno la priorità sulle altre operazioni; in un’espressione, se non vi sono parentesi, le potenze vanno calcolate per prime. In particolare, richiamiamo l’attenzione su espressioni tipo</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="12.833ex" height="3.333ex" viewBox="0 -978.9 5498.1 1423.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2212"></use><g transform="translate(783,0)"><use xlink:href="#MJMAIN-35"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-32"></use></g><g transform="translate(1745,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(2355,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(2964,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-35"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-32"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>−</mo><msup><mn>5</mn><mn>2</mn></msup><mtext> </mtext><mtext> </mtext><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></math></script></p>
<p class="noindent">Nella prima si deve calcolare innanzitutto la potenza 5<sup>2</sup> = 25 e poi, come indicato dal segno meno, considerare l’opposto del risultato, ossia −25. Nella seconda, invece, le parentesi indicano che si deve elevare alla seconda potenza il numero negativo −5, ottenendo (−5) · (−5) = +25. Perciò si ha</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="28.667ex" height="3.333ex" viewBox="0 -978.9 12371.1 1423.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2212"></use><g transform="translate(783,0)"><use xlink:href="#MJMAIN-35"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-32"></use></g><use x="2022" y="0" xlink:href="#MJMAIN-3D"></use><use x="3083" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(3866,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-35"></use></g><g transform="translate(4876,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(5486,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(6096,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(6706,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-35"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-32"></use></g><use x="9517" y="0" xlink:href="#MJMAIN-3D"></use><use x="10578" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(11361,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-35"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>−</mo><msup><mn>5</mn><mn>2</mn></msup><mo>=</mo><mo>−</mo><mn>25</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>=</mo><mo>+</mo><mn>25</mn></mrow></math></script></p>
</div>
<p class="noindent">Quindi:</p>
<ul class="blist">
<li>
<p class="noindent">la potenza di un numero positivo è sempre positiva:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="27.5ex" height="3.333ex" viewBox="0 -986.7 11843.4 1431.2"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2B"></use><use x="1177" y="0" xlink:href="#MJMAIN-36"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-33"></use><use x="2810" y="0" xlink:href="#MJMAIN-3D"></use><use x="3871" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(4654,0)"><use xlink:href="#MJMAIN-36"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-33"></use></g><g transform="translate(5616,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(6226,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2B"></use><use x="1177" y="0" xlink:href="#MJMAIN-33"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-34"></use></g><use x="9037" y="0" xlink:href="#MJMAIN-3D"></use><use x="10098" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(10881,0)"><use xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="714" y="568" xlink:href="#MJMAIN-34"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mo stretchy="false">(</mo><mo>+</mo><mn>6</mn><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><mo>=</mo><mo>+</mo><msup><mn>6</mn><mn>3</mn></msup><mtext> </mtext><msup><mrow><mo stretchy="false">(</mo><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><mn>4</mn></msup><mo>=</mo><mo>+</mo><msup><mn>3</mn><mn>4</mn></msup></mrow></math></script></p>
</li>
<li>
<p class="noindent">la potenza di un numero negativo con esponente pari è positiva:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="13ex" height="2.833ex" viewBox="0 -978.9 5616.7 1252.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-34"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-36"></use><use x="2810" y="0" xlink:href="#MJMAIN-3D"></use><use x="3871" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(4654,0)"><use xlink:href="#MJMAIN-34"></use><use transform="scale(0.707)" x="714" y="585" xlink:href="#MJMAIN-36"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><mn>6</mn></msup><mo>=</mo><mo>+</mo><msup><mn>4</mn><mn>6</mn></msup></mrow></math></script></p>
</li>
</ul>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"></div>
</div></div>
<div data-page-container="32" id="page-32" class="row chapters-content"><div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">32</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<ul class="blist">
<li>
<p class="noindent"><span class="pagebreak" epub:type="pagebreak" title="32" id="page32"></span>la potenza di un numero negativo con esponente dispari è negativa:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="13ex" height="2.833ex" viewBox="0 -978.2 5616.7 1252.1"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-37"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-33"></use><use x="2810" y="0" xlink:href="#MJMAIN-3D"></use><use x="3871" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(4654,0)"><use xlink:href="#MJMAIN-37"></use><use transform="scale(0.707)" x="714" y="584" xlink:href="#MJMAIN-33"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>7</mn><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><mo>=</mo><mo>−</mo><msup><mn>7</mn><mn>3</mn></msup></mrow></math></script></p>
</li>
<li>
<p class="noindent"><b>una potenza con esponente pari non cambia se si cambia il segno della base:</b></p>
<p class="math"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="15ex" height="2.5ex" viewBox="0 -821.2 6467 1095.2"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMATHI-61"></use><use x="1711" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2976" y="688" xlink:href="#MJMATHI-70"></use><use x="2841" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(3902,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2B"></use><use x="1177" y="0" xlink:href="#MJMATHI-61"></use><use x="1711" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2976" y="688" xlink:href="#MJMATHI-70"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mi>p</mi></msup><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mo>+</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mi>p</mi></msup></mrow></math></script></span></p>
<p class="noindent">Ad esempio:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="53.167ex" height="3.333ex" viewBox="0 -986.7 22904.8 1431.2"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-33"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-34"></use><use x="2810" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(3871,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2B"></use><use x="1177" y="0" xlink:href="#MJMAIN-33"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-34"></use></g><g transform="translate(6404,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(7014,0)"><use xlink:href="#MJMAIN-69"></use><use x="283" y="0" xlink:href="#MJMAIN-6E"></use><use x="844" y="0" xlink:href="#MJMAIN-66"></use><use x="1155" y="0" xlink:href="#MJMAIN-61"></use><use x="1660" y="0" xlink:href="#MJMAIN-74"></use><use x="2054" y="0" xlink:href="#MJMAIN-74"></use><use x="2448" y="0" xlink:href="#MJMAIN-69"></use></g><g transform="translate(9745,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(10355,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-33"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-34"></use></g><use x="13166" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(14227,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-2B"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(783,0)"><use xlink:href="#MJMAIN-38"></use><use x="505" y="0" xlink:href="#MJMAIN-31"></use></g></g></g><g transform="translate(16020,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(17240,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2B"></use><use x="1177" y="0" xlink:href="#MJMAIN-33"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-34"></use></g><use x="20051" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(21111,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-2B"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(783,0)"><use xlink:href="#MJMAIN-38"></use><use x="505" y="0" xlink:href="#MJMAIN-31"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><mn>4</mn></msup><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><mn>4</mn></msup><mtext> </mtext><mtext>infatti</mtext><mtext> </mtext><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><mn>4</mn></msup><mo>=</mo><mstyle color="#00aef0"><mo>+</mo><mn>81</mn></mstyle><mtext>  </mtext><msup><mrow><mo stretchy="false">(</mo><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><mn>4</mn></msup><mo>=</mo><mstyle color="#00aef0"><mo>+</mo><mn>81</mn></mstyle></mrow></math></script></p>
</li>
</ul>
<p class="noindent1">Vediamo ora alcune applicazioni delle proprietà delle potenze.</p>
<div class="example">
<h4 class="noindent"><span class="gep"><span class="sgr">ESEMPI</span></span></h4>
<p class="hang"><b>3</b>&nbsp;&nbsp;<span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="66ex" height="2.833ex" viewBox="0 -978.9 28433.1 1252.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-32"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-33"></use><use x="2755" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(3260,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-32"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-32"></use></g><use x="6015" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(6521,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-32"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-35"></use></g><use x="9331" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(10392,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-32"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><g transform="translate(2076,486)"><use transform="scale(0.707)" xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-2B"></use><use transform="scale(0.707)" x="1288" y="0" xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="1793" y="0" xlink:href="#MJMAIN-2B"></use><use transform="scale(0.707)" x="2576" y="0" xlink:href="#MJMAIN-35"></use></g></g><use x="15025" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(16085,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-32"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><g transform="translate(2076,486)"><use transform="scale(0.707)" xlink:href="#MJMAIN-31"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-30"></use></g></g><use x="19253" y="0" xlink:href="#MJMAIN-3D"></use><use x="20314" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(21097,0)"><use xlink:href="#MJMAIN-32"></use><g transform="translate(505,402)"><use transform="scale(0.707)" xlink:href="#MJMAIN-31"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-30"></use></g></g><use x="22694" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(23755,0)"><use xlink:href="#MJMAIN-32"></use><g transform="translate(505,402)"><use transform="scale(0.707)" xlink:href="#MJMAIN-31"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-30"></use></g></g><use x="25352" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(26413,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use><use x="1010" y="0" xlink:href="#MJMAIN-32"></use><use x="1515" y="0" xlink:href="#MJMAIN-34"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><mo>⋅</mo><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>⋅</mo><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mn>5</mn></msup><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mrow><mn>3</mn><mo>+</mo><mn>2</mn><mo>+</mo><mn>5</mn></mrow></msup><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mrow><mn>10</mn></mrow></msup><mo>=</mo><mo>+</mo><msup><mn>2</mn><mrow><mn>10</mn></mrow></msup><mo>=</mo><msup><mn>2</mn><mrow><mn>10</mn></mrow></msup><mo>=</mo><mn color="#00aef0">1024</mn></mrow></math></script></p>
<p class="hang"><b>4</b>&nbsp;&nbsp;<span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="49.5ex" height="2.833ex" viewBox="0 -978.9 21279 1252.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-33"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-38"></use><use x="2810" y="0" xlink:href="#MJMAIN-3A"></use><g transform="translate(3371,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-33"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-33"></use></g><use x="6182" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(7243,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-33"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><g transform="translate(2076,486)"><use transform="scale(0.707)" xlink:href="#MJMAIN-38"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-2212"></use><use transform="scale(0.707)" x="1288" y="0" xlink:href="#MJMAIN-33"></use></g></g><use x="10964" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(12025,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-33"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-35"></use></g><use x="14836" y="0" xlink:href="#MJMAIN-3D"></use><use x="15897" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(16680,0)"><use xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="714" y="568" xlink:href="#MJMAIN-35"></use></g><use x="17920" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(18980,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-2212"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(783,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-34"></use><use x="1010" y="0" xlink:href="#MJMAIN-33"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><mn>8</mn></msup><mo>:</mo><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><mrow><mn>8</mn><mo>−</mo><mn>3</mn></mrow></msup><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><mn>5</mn></msup><mo>=</mo><mo>−</mo><msup><mn>3</mn><mn>5</mn></msup><mo>=</mo><mstyle color="#00aef0"><mo>−</mo><mn>243</mn></mstyle></mrow></math></script></p>
<p class="hang"><b>5</b>&nbsp;&nbsp;<span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="43.167ex" height="3.167ex" viewBox="0 -1109.8 18608.8 1383.7"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(1177,0)"><use xlink:href="#MJMAIN-35"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-34"></use></g><use x="2139" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="3582" y="870" xlink:href="#MJMAIN-36"></use><use x="3267" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(4328,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-35"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><g transform="translate(2076,486)"><use transform="scale(0.707)" xlink:href="#MJMAIN-34"></use><g transform="translate(357,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><use transform="scale(0.707)" x="1114" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(988,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(50.74127551116547) matrix(1 0 0 -1 0 0)"> </text></g><use transform="scale(0.707)" x="2007" y="0" xlink:href="#MJMAIN-36"></use></g></g><use x="8559" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(9620,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-35"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><g transform="translate(2076,486)"><use transform="scale(0.707)" xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-34"></use></g></g><use x="12788" y="0" xlink:href="#MJMAIN-3D"></use><use x="13848" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(14631,0)"><use xlink:href="#MJMAIN-35"></use><g transform="translate(505,402)"><use transform="scale(0.707)" xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-34"></use></g></g><use x="16228" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(17289,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-35"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(505,402)"><g fill="#00aef0" stroke="#00aef0"><use transform="scale(0.707)" xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-34"></use></g></g></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><msup><mn>5</mn><mn>4</mn></msup><mo stretchy="false">)</mo></mrow><mn>6</mn></msup><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><mrow><mn>4</mn><mtext> </mtext><mo>⋅</mo><mtext> </mtext><mn>6</mn></mrow></msup><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><mrow><mn>24</mn></mrow></msup><mo>=</mo><mo>+</mo><msup><mn>5</mn><mrow><mn>24</mn></mrow></msup><mo>=</mo><mstyle color="#00aef0"><msup><mn>5</mn><mrow><mn>24</mn></mrow></msup></mstyle></mrow></math></script></p>
<p class="hang"><b>6</b>&nbsp;&nbsp;<span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="65.667ex" height="2.833ex" viewBox="0 -978.9 28298.6 1252.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-33"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-35"></use><use x="2755" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(3260,0)"><use xlink:href="#MJMAIN-37"></use><use transform="scale(0.707)" x="714" y="584" xlink:href="#MJMAIN-35"></use></g><use x="4444" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(4950,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-32"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-35"></use></g><use x="7760" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(8821,0)"><use xlink:href="#MJMAIN-5B"></use><use x="283" y="0" xlink:href="#MJMAIN-28"></use><use x="677" y="0" xlink:href="#MJMAIN-2212"></use><use x="1460" y="0" xlink:href="#MJMAIN-33"></use><use x="1965" y="0" xlink:href="#MJMAIN-29"></use><use x="2581" y="0" xlink:href="#MJMAIN-22C5"></use><use x="3086" y="0" xlink:href="#MJMAIN-28"></use><use x="3480" y="0" xlink:href="#MJMAIN-2B"></use><use x="4263" y="0" xlink:href="#MJMAIN-37"></use><use x="4768" y="0" xlink:href="#MJMAIN-29"></use><use x="5384" y="0" xlink:href="#MJMAIN-22C5"></use><use x="5889" y="0" xlink:href="#MJMAIN-28"></use><use x="6283" y="0" xlink:href="#MJMAIN-2212"></use><use x="7066" y="0" xlink:href="#MJMAIN-32"></use><use x="7571" y="0" xlink:href="#MJMAIN-29"></use><use x="7965" y="0" xlink:href="#MJMAIN-5D"></use><use transform="scale(0.707)" x="11665" y="688" xlink:href="#MJMAIN-35"></use></g><use x="17805" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(18866,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(1177,0)"><use xlink:href="#MJMAIN-34"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g><use x="2187" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="3650" y="688" xlink:href="#MJMAIN-35"></use></g><use x="22182" y="0" xlink:href="#MJMAIN-3D"></use><use x="23242" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(24025,0)"><use xlink:href="#MJMAIN-34"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use><use transform="scale(0.707)" x="1428" y="585" xlink:href="#MJMAIN-35"></use></g><use x="25770" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(26831,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-34"></use><use x="505" y="0" xlink:href="#MJMAIN-32"></use></g></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1010,413)"><use transform="scale(0.707)" xlink:href="#MJMAIN-35"></use></g></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><mn>5</mn></msup><mo>⋅</mo><msup><mn>7</mn><mn>5</mn></msup><mo>⋅</mo><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mn>5</mn></msup><mo>=</mo><msup><mrow><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo>+</mo><mn>7</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mn>5</mn></msup><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mo>+</mo><mn>42</mn><mo stretchy="false">)</mo></mrow><mn>5</mn></msup><mo>=</mo><mo>+</mo><msup><mrow><mn>42</mn></mrow><mn>5</mn></msup><mo>=</mo><mstyle color="#00aef0"><msup><mrow><mn>42</mn></mrow><mn>5</mn></msup></mstyle></mrow></math></script></p>
<p class="hang"><b>7</b>&nbsp;&nbsp;<span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="55.333ex" height="2.833ex" viewBox="0 -978.9 23792.8 1252.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-34"></use><use transform="scale(0.707)" x="1428" y="585" xlink:href="#MJMAIN-35"></use><use x="1744" y="0" xlink:href="#MJMAIN-3A"></use><g transform="translate(2305,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-36"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-35"></use></g><use x="5116" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(6177,0)"><use xlink:href="#MJMAIN-5B"></use><use x="283" y="0" xlink:href="#MJMAIN-28"></use><use x="677" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(1460,0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-34"></use></g><use x="2470" y="0" xlink:href="#MJMAIN-29"></use><use x="3141" y="0" xlink:href="#MJMAIN-3A"></use><use x="3702" y="0" xlink:href="#MJMAIN-28"></use><use x="4096" y="0" xlink:href="#MJMAIN-2212"></use><use x="4879" y="0" xlink:href="#MJMAIN-36"></use><use x="5384" y="0" xlink:href="#MJMAIN-29"></use><use x="5778" y="0" xlink:href="#MJMAIN-5D"></use><use transform="scale(0.707)" x="8572" y="688" xlink:href="#MJMAIN-35"></use></g><use x="12973" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(14034,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-34"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-35"></use></g><use x="16845" y="0" xlink:href="#MJMAIN-3D"></use><use x="17906" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(18689,0)"><use xlink:href="#MJMAIN-34"></use><use transform="scale(0.707)" x="714" y="585" xlink:href="#MJMAIN-35"></use></g><use x="19929" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(20989,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-2212"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(783,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use><use x="1010" y="0" xlink:href="#MJMAIN-32"></use><use x="1515" y="0" xlink:href="#MJMAIN-34"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mn>24</mn></mrow><mn>5</mn></msup><mo>:</mo><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>6</mn><mo stretchy="false">)</mo></mrow><mn>5</mn></msup><mo>=</mo><msup><mrow><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mo>+</mo><mn>24</mn><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><mo>−</mo><mn>6</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mn>5</mn></msup><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><mn>5</mn></msup><mo>=</mo><mo>−</mo><msup><mn>4</mn><mn>5</mn></msup><mo>=</mo><mstyle color="#00aef0"><mo>−</mo><mn>1024</mn></mstyle></mrow></math></script></p>
<p class="hang"><b>8</b>&nbsp;&nbsp;Esprimiamo come unica potenza il prodotto <span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="13.5ex" height="3ex" viewBox="0 -986.7 5793.6 1260.6"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-35"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-33"></use><use x="2755" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(3260,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2B"></use><use x="1177" y="0" xlink:href="#MJMAIN-35"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-34"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><mo>·</mo><msup><mrow><mo stretchy="false">(</mo><mo>+</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><mn>4</mn></msup></mrow></math></script>.</p>
<p class="hangg">Le basi delle due potenze sono opposte e quindi non possiamo sommare direttamente gli esponenti. Poiché la potenza (+5)<sup>4</sup> ha esponente pari, possiamo cambiare il segno della base e scrivere (+5)<sup>4</sup> = (−5)<sup>4</sup>; quindi</p>
<p class="math"><img src="images/pg50.jpg" alt="Image"></p>
<p class="hangg">Possiamo anche procedere così:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="50ex" height="3ex" viewBox="0 -986.7 21516.5 1260.6"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-35"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-33"></use><use x="2755" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(3260,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2B"></use><use x="1177" y="0" xlink:href="#MJMAIN-35"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-34"></use></g><use x="6071" y="0" xlink:href="#MJMAIN-3D"></use><use x="7132" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(7915,0)"><use xlink:href="#MJMAIN-35"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-33"></use></g><use x="9099" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(9604,0)"><use xlink:href="#MJMAIN-35"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-34"></use></g><use x="10844" y="0" xlink:href="#MJMAIN-3D"></use><use x="11905" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(12688,0)"><use xlink:href="#MJMAIN-35"></use><g transform="translate(505,402)"><use transform="scale(0.707)" xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-2B"></use><use transform="scale(0.707)" x="1288" y="0" xlink:href="#MJMAIN-34"></use></g></g><use x="14838" y="0" xlink:href="#MJMAIN-3D"></use><use x="15899" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(16682,0)"><use xlink:href="#MJMAIN-35"></use><use transform="scale(0.707)" x="714" y="569" xlink:href="#MJMAIN-37"></use></g><use x="17922" y="0" xlink:href="#MJMAIN-3D"></use><g fill="#00aef0" stroke="#00aef0" transform="translate(18983,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMAIN-28"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(394,0)"><use xlink:href="#MJMAIN-2212"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1177,0)"><use xlink:href="#MJMAIN-35"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1682,0)"><use xlink:href="#MJMAIN-29"></use></g></g><g fill="#00aef0" stroke="#00aef0" transform="translate(2076,486)"><use transform="scale(0.707)" xlink:href="#MJMAIN-37"></use></g></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><mo>⋅</mo><msup><mrow><mo stretchy="false">(</mo><mo>+</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><mn>4</mn></msup><mo>=</mo><mo>−</mo><msup><mn>5</mn><mn>3</mn></msup><mo>⋅</mo><msup><mn>5</mn><mn>4</mn></msup><mo>=</mo><mo>−</mo><msup><mn>5</mn><mrow><mn>3</mn><mo>+</mo><mn>4</mn></mrow></msup><mo>=</mo><mo>−</mo><msup><mn>5</mn><mn>7</mn></msup><mo>=</mo><mstyle color="#00aef0"><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><mn>7</mn></msup></mstyle></mrow></math></script></p>
</div>
</div></div>
</div></div>
<div class="small-12 medium-4 columns"></div>
</div></div>
<div data-page-container="33" id="page-33" class="row chapters-content"><div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">33</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<div class="prob_solution">
<h3 class="sec_title">
<span class="pagebreak" epub:type="pagebreak" title="33" id="page33"></span>Calciatori e automobili</h3>
<p class="noindent"><b>Numeri, continuamente numeri: dal numero di giri di una gara automobilistica al numero sulle auto, dal tempo trascorso in una partita di calcio al numero sulla maglia dei calciatori... Servono veramente tutti questi numeri? Sono essenziali o se ne potrebbe fare a meno?</b></p>
<p class="noindent1">Abbiamo sottolineato all’inizio del capitolo che i numeri si trovano in tantissimi contesti. Tuttavia non sempre il loro uso è essenziale, nel senso che, se non vengono utilizzati per le loro proprietà, si potrebbero adoperare altre entità.</p>
<p class="noindent">Consideriamo ad esempio i numeri sulle auto da corsa: il loro scopo è identificare in modo veloce e chiaro un pilota, ma il valore non ha nessun significato particolare. Se un pilota ha il numero 8 e un altro il numero 24, il fatto che tra di loro ci siano 16 numeri, o che uno sia il triplo dell’altro, non vuole dire nulla. O che la somma dia 32 non significa che insieme siano bravi come il pilota con il numero 32... Insomma le operazioni tra i numeri che rappresentano le auto non hanno senso, e del resto se si usassero le lettere o i disegni di animali per identificarli non cambierebbe molto.</p>
<div class="figure">
<p class="img" id="ch1.fg18"><img src="images/c01u01f01.jpg" alt="Image"></p>
<p class="figcap">FIGURA 18</p>
</div>
<div class="figure">
<p class="img" id="ch1.fg19"><img src="images/c01u01f02.jpg" alt="Image"></p>
<p class="figcap">FIGURA 19</p>
</div>
<p class="noindent1">Vale lo stesso per il numero del giro a cui è giunta la gara? Assolutamente no, qui l’uso dei numeri è importante. Se la gara dura 65 giri e siamo al 48°, posso sapere che mancano 17 giri (perché 65 − 48 = 17) mentre se ne ho percorsi 34 prima di cambiare le gomme e 20 dopo averle cambiate, posso dire di essere al 54° giro, perché 34 + 20 = 54. Quindi in questo caso è possibile attribuire un significato alle operazioni e alla distanza tra due diversi valori.</p>
<p class="noindent">Per quanto riguarda le maglie dei calciatori, ovviamente vale lo stesso discorso dei numeri sulle auto: due portieri non fanno un terzino...</p>
<p class="noindent">Se però vogliamo contare i minuti giocati in una partita, usare i numeri è utilissimo: possiamo sapere quanto manca per dosare accortamente le energie.</p>
<p class="noindent">Analogamente per contare i goal: se stiamo vincendo quattro a uno siamo tranquilli, se il risultato è due a uno è meglio fare attenzione, un solo goal di vantaggio può essere rimontato in un attimo. L’importanza dei numeri è evidenziata anche dalla cosiddetta «differenza reti» che in alcuni contesti viene applicata in caso di parità: se le due squadre hanno lo stesso punteggio in classifica, si considera davanti quella che, negli scontri diretti, ha il risultato più alto nella differenza tra le reti segnate e le reti subite.</p>
<p class="noindent">Perché usare allora i numeri per identificare giocatori, automobili o anche corridori di una maratona e non lettere, colori, nomi di piante...?</p>
<p class="noindent">È ovvio che in questi casi non si possono sfruttare le proprietà aritmetiche dei numeri, tuttavia i numeri hanno il grande vantaggio di essere... infiniti, quindi se ne possono utilizzare a piacimento, caratteristica particolarmente comoda per i grandi raduni di atletica o in generale per situazioni in cui è presente un numero elevato di «elementi».</p>
<p class="noindent">Detto ciò... in una partitella di calcio, sceglierò comunque la maglia numero 10 per distinguermi, proprio come Del Piero, Totti, Maradona.</p>
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<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">34</div></div>
<div class="small-12 medium-11 columns"><div class="ch1"><div>
<h2 class="para_title" id="teoria">
<span class="pagebreak" epub:type="pagebreak" title="34" id="page34"></span>Teoria.zip Numeri naturali e numeri interi relativi</h2>
<h3 class="sec_title">L’insieme dei numeri naturali</h3>
<ul class="blist">
<li><p class="noindent">Insieme ℕ dei numeri naturali: ℕ = {0 ; 1 ; 2 ; 3 ; ...}</p></li>
<li><p class="noindent">Insieme ℕ* dei numeri naturali, escluso lo zero: ℕ* = {1 ; 2 ; 3 ; ...}</p></li>
</ul>
<h3 class="sec_title">Le quattro operazioni aritmetiche con i numeri naturali</h3>
<div class="table">
<table width="100%" cellspacing="0" cellpadding="0">
<colgroup><col width="20%">
<col width="30%">
<col width="10%">
<col width="40%">
</colgroup><tbody>
<tr>
<td class="tabv1c"><b>operazione</b></td>
<td class="tabvab"><b>nome dei termini</b></td>
<td class="tabvab"><b>risultato</b></td>
<td class="tabvab"><b>proprietà</b></td>
</tr>
<tr>
<td class="tabvf" valign="middle">
<p class="center">addizione</p>
<p class="center"><i>a</i> + <i>b</i></p>
</td>
<td class="tabvm" valign="middle"><p class="center"><i>a</i> e <i>b</i> addendi</p></td>
<td valign="middle" class="tabvm"><p class="center">somma</p></td>
<td valign="middle" class="tabvm">
<p class="noindentf">commutativa: <span class="cyan"><i>a</i> + <i>b</i> = <i>b</i> + <i>a</i></span></p>
<p class="noindentf">associativa: <span class="cyan">(<i>a</i> + <i>b</i>) + <i>c</i> = <i>a</i> + (<i>b</i> + <i>c</i>)</span></p>
<p class="noindentf">elemento neutro: <span class="cyan"><i>a</i> + 0 = 0 + <i>a</i> = <i>a</i></span></p>
</td>
</tr>
<tr>
<td valign="middle" class="tabvf">
<p class="center">sottrazione</p>
<p class="center"><i>a</i> − <i>b</i> con <i>a</i> ≥ <i>b</i></p>
</td>
<td valign="middle" class="tabvm">
<p class="center"><i>a</i> minuendo</p>
<p class="center"><i>b</i> sottraendo</p>
</td>
<td valign="middle" class="tabvm"><p class="center">differenza</p></td>
<td valign="middle" class="tabvm">
<p class="noindentf">invariantiva: <span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -3ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="27.333ex" height="7ex" viewBox="0 -1772.9 11770.9 3045.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use x="0" y="-1" xlink:href="#MJSZ4-27E8"></use><g transform="translate(978,0)"><g transform="translate(-11,0)"><g transform="translate(0,725)"><use xlink:href="#MJMATHI-61"></use><use x="756" y="0" xlink:href="#MJMAIN-2212"></use><use x="1761" y="0" xlink:href="#MJMATHI-62"></use><use x="2473" y="0" xlink:href="#MJMAIN-3D"></use><use x="3534" y="0" xlink:href="#MJMAIN-28"></use><use x="3928" y="0" xlink:href="#MJMATHI-61"></use><use x="4684" y="0" xlink:href="#MJMAIN-2B"></use><use x="5689" y="0" xlink:href="#MJMATHI-63"></use><use x="6127" y="0" xlink:href="#MJMAIN-29"></use><use x="6743" y="0" xlink:href="#MJMAIN-2212"></use><use x="7748" y="0" xlink:href="#MJMAIN-28"></use><use x="8142" y="0" xlink:href="#MJMATHI-62"></use><use x="8799" y="0" xlink:href="#MJMAIN-2B"></use><use x="9804" y="0" xlink:href="#MJMATHI-63"></use><use x="10242" y="0" xlink:href="#MJMAIN-29"></use></g><g transform="translate(0,-766)"><use xlink:href="#MJMATHI-61"></use><use x="756" y="0" xlink:href="#MJMAIN-2212"></use><use x="1761" y="0" xlink:href="#MJMATHI-62"></use><use x="2473" y="0" xlink:href="#MJMAIN-3D"></use><use x="3534" y="0" xlink:href="#MJMAIN-28"></use><use x="3928" y="0" xlink:href="#MJMATHI-61"></use><use x="4684" y="0" xlink:href="#MJMAIN-2212"></use><use x="5689" y="0" xlink:href="#MJMATHI-63"></use><use x="6127" y="0" xlink:href="#MJMAIN-29"></use><use x="6743" y="0" xlink:href="#MJMAIN-2212"></use><use x="7748" y="0" xlink:href="#MJMAIN-28"></use><use x="8142" y="0" xlink:href="#MJMATHI-62"></use><use x="8799" y="0" xlink:href="#MJMAIN-2212"></use><use x="9804" y="0" xlink:href="#MJMATHI-63"></use><use x="10242" y="0" xlink:href="#MJMAIN-29"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo>〈</mo><mtable columnalign="left"><mtr><mtd><mi>a</mi><mo>−</mo><mi>b</mi><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>c</mi><mo stretchy="false">)</mo><mo>−</mo><mo stretchy="false">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>a</mi><mo>−</mo><mi>b</mi><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo>−</mo><mi>c</mi><mo stretchy="false">)</mo><mo>−</mo><mo stretchy="false">(</mo><mi>b</mi><mo>−</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow></math></script></span></p>
<p class="noindentf">due o più sottrazioni consecutive vanno eseguite nell’ordine in cui sono indicate</p>
</td>
</tr>
<tr>
<td valign="middle" class="tabvf">
<p class="center">moltiplicazione</p>
<p class="center"><i>a</i> · <i>b</i></p>
</td>
<td valign="middle" class="tabvm"><p class="center"><i>a</i> e <i>b</i> fattori</p></td>
<td valign="middle" class="tabvm"><p class="center">prodotto</p></td>
<td valign="middle" class="tabvm">
<p class="noindentf">commutativa: <span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="11ex" height="1.667ex" viewBox="0 -717.9 4729.4 752.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-61"></use><use x="756" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1261" y="0" xlink:href="#MJMATHI-62"></use><use x="1973" y="0" xlink:href="#MJMAIN-3D"></use><use x="3034" y="0" xlink:href="#MJMATHI-62"></use><use x="3690" y="0" xlink:href="#MJMAIN-22C5"></use><use x="4195" y="0" xlink:href="#MJMATHI-61"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>a</mi><mo>·</mo><mi>b</mi><mo>=</mo><mi>b</mi><mo>·</mo><mi>a</mi></mrow></math></script></span></p>
<p class="noindentf">associativa: <span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="29.833ex" height="2.5ex" viewBox="0 -773.9 12835.8 1047.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-61"></use><use x="756" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1261" y="0" xlink:href="#MJMATHI-62"></use><use x="1917" y="0" xlink:href="#MJMAIN-22C5"></use><use x="2422" y="0" xlink:href="#MJMATHI-63"></use><use x="3138" y="0" xlink:href="#MJMAIN-3D"></use><use x="4199" y="0" xlink:href="#MJMAIN-28"></use><use x="4593" y="0" xlink:href="#MJMATHI-61"></use><use x="5349" y="0" xlink:href="#MJMAIN-22C5"></use><use x="5854" y="0" xlink:href="#MJMATHI-62"></use><use x="6288" y="0" xlink:href="#MJMAIN-29"></use><use x="6905" y="0" xlink:href="#MJMAIN-22C5"></use><use x="7410" y="0" xlink:href="#MJMATHI-63"></use><use x="8126" y="0" xlink:href="#MJMAIN-3D"></use><use x="9186" y="0" xlink:href="#MJMATHI-61"></use><use x="9943" y="0" xlink:href="#MJMAIN-22C5"></use><use x="10448" y="0" xlink:href="#MJMAIN-28"></use><use x="10842" y="0" xlink:href="#MJMATHI-62"></use><use x="11498" y="0" xlink:href="#MJMAIN-22C5"></use><use x="12003" y="0" xlink:href="#MJMATHI-63"></use><use x="12441" y="0" xlink:href="#MJMAIN-29"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>a</mi><mo>·</mo><mi>b</mi><mo>·</mo><mi>c</mi><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo>·</mo><mi>b</mi><mo stretchy="false">)</mo><mo>·</mo><mi>c</mi><mo>=</mo><mi>a</mi><mo>·</mo><mo stretchy="false">(</mo><mi>b</mi><mo>·</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></math></script></span></p>
<p class="noindentf">distributiva: <span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -3ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="37.167ex" height="7ex" viewBox="0 -1772.9 16016.7 3045.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use x="0" y="-1" xlink:href="#MJSZ4-27E8"></use><g transform="translate(978,0)"><g transform="translate(-11,0)"><g transform="translate(0,725)"><use xlink:href="#MJMATHI-61"></use><use x="756" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1261" y="0" xlink:href="#MJMAIN-28"></use><use x="1655" y="0" xlink:href="#MJMATHI-62"></use><use x="2311" y="0" xlink:href="#MJMAIN-2B"></use><use x="3316" y="0" xlink:href="#MJMATHI-63"></use><use x="3977" y="0" xlink:href="#MJMAIN-2B"></use><use x="4982" y="0" xlink:href="#MJMATHI-64"></use><use x="5510" y="0" xlink:href="#MJMAIN-29"></use><use x="6182" y="0" xlink:href="#MJMAIN-3D"></use><use x="7242" y="0" xlink:href="#MJMATHI-61"></use><use x="7999" y="0" xlink:href="#MJMAIN-22C5"></use><use x="8504" y="0" xlink:href="#MJMATHI-62"></use><use x="9160" y="0" xlink:href="#MJMAIN-2B"></use><use x="10165" y="0" xlink:href="#MJMATHI-61"></use><use x="10922" y="0" xlink:href="#MJMAIN-22C5"></use><use x="11427" y="0" xlink:href="#MJMATHI-63"></use><use x="12087" y="0" xlink:href="#MJMAIN-2B"></use><use x="13092" y="0" xlink:href="#MJMATHI-61"></use><use x="13848" y="0" xlink:href="#MJMAIN-22C5"></use><use x="14354" y="0" xlink:href="#MJMATHI-64"></use></g><g transform="translate(0,-766)"><use xlink:href="#MJMATHI-61"></use><use x="756" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1261" y="0" xlink:href="#MJMAIN-28"></use><use x="1655" y="0" xlink:href="#MJMATHI-62"></use><use x="2311" y="0" xlink:href="#MJMAIN-2212"></use><use x="3316" y="0" xlink:href="#MJMATHI-63"></use><use x="3754" y="0" xlink:href="#MJMAIN-29"></use><use x="4426" y="0" xlink:href="#MJMAIN-3D"></use><use x="5487" y="0" xlink:href="#MJMATHI-61"></use><use x="6243" y="0" xlink:href="#MJMAIN-22C5"></use><use x="6748" y="0" xlink:href="#MJMATHI-62"></use><use x="7405" y="0" xlink:href="#MJMAIN-2212"></use><use x="8410" y="0" xlink:href="#MJMATHI-61"></use><use x="9166" y="0" xlink:href="#MJMAIN-22C5"></use><use x="9671" y="0" xlink:href="#MJMATHI-63"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo>〈</mo><mtable columnalign="left"><mtr><mtd><mi>a</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mo>⋅</mo><mi>b</mi><mo>+</mo><mi>a</mi><mo>⋅</mo><mi>c</mi><mo>+</mo><mi>a</mi><mo>⋅</mo><mi>d</mi></mtd></mtr><mtr><mtd><mi>a</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>b</mi><mo>−</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mo>⋅</mo><mi>b</mi><mo>−</mo><mi>a</mi><mo>⋅</mo><mi>c</mi></mtd></mtr></mtable></mrow></mrow></math></script></span></p>
<p class="noindentf">raccoglimento: <span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -3ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="37.167ex" height="7ex" viewBox="0 -1772.9 16016.7 3045.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use x="0" y="-1" xlink:href="#MJSZ4-27E8"></use><g transform="translate(978,0)"><g transform="translate(-11,0)"><g transform="translate(0,725)"><use xlink:href="#MJMATHI-61"></use><use x="756" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1261" y="0" xlink:href="#MJMATHI-62"></use><use x="1917" y="0" xlink:href="#MJMAIN-2B"></use><use x="2922" y="0" xlink:href="#MJMATHI-61"></use><use x="3679" y="0" xlink:href="#MJMAIN-22C5"></use><use x="4184" y="0" xlink:href="#MJMATHI-63"></use><use x="4844" y="0" xlink:href="#MJMAIN-2B"></use><use x="5849" y="0" xlink:href="#MJMATHI-61"></use><use x="6606" y="0" xlink:href="#MJMAIN-22C5"></use><use x="7111" y="0" xlink:href="#MJMATHI-64"></use><use x="7917" y="0" xlink:href="#MJMAIN-3D"></use><use x="8977" y="0" xlink:href="#MJMATHI-61"></use><use x="9734" y="0" xlink:href="#MJMAIN-22C5"></use><use x="10239" y="0" xlink:href="#MJMAIN-28"></use><use x="10633" y="0" xlink:href="#MJMATHI-62"></use><use x="11289" y="0" xlink:href="#MJMAIN-2B"></use><use x="12294" y="0" xlink:href="#MJMATHI-63"></use><use x="12954" y="0" xlink:href="#MJMAIN-2B"></use><use x="13960" y="0" xlink:href="#MJMATHI-64"></use><use x="14488" y="0" xlink:href="#MJMAIN-29"></use></g><g transform="translate(0,-766)"><use xlink:href="#MJMATHI-61"></use><use x="756" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1261" y="0" xlink:href="#MJMATHI-62"></use><use x="1917" y="0" xlink:href="#MJMAIN-2212"></use><use x="2922" y="0" xlink:href="#MJMATHI-61"></use><use x="3679" y="0" xlink:href="#MJMAIN-22C5"></use><use x="4184" y="0" xlink:href="#MJMATHI-63"></use><use x="4900" y="0" xlink:href="#MJMAIN-3D"></use><use x="5960" y="0" xlink:href="#MJMATHI-61"></use><use x="6717" y="0" xlink:href="#MJMAIN-22C5"></use><use x="7222" y="0" xlink:href="#MJMAIN-28"></use><use x="7616" y="0" xlink:href="#MJMATHI-62"></use><use x="8272" y="0" xlink:href="#MJMAIN-2212"></use><use x="9277" y="0" xlink:href="#MJMATHI-63"></use><use x="9715" y="0" xlink:href="#MJMAIN-29"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo>〈</mo><mtable columnalign="left"><mtr><mtd><mi>a</mi><mo>⋅</mo><mi>b</mi><mo>+</mo><mi>a</mi><mo>⋅</mo><mi>c</mi><mo>+</mo><mi>a</mi><mo>⋅</mo><mi>d</mi><mo>=</mo><mi>a</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>a</mi><mo>⋅</mo><mi>b</mi><mo>−</mo><mi>a</mi><mo>⋅</mo><mi>c</mi><mo>=</mo><mi>a</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>b</mi><mo>−</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow></math></script></span></p>
<p class="noindentf">elemento neutro: <span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="15.667ex" height="1.667ex" viewBox="0 -689.9 6744 723.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-61"></use><use x="756" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1261" y="0" xlink:href="#MJMAIN-31"></use><use x="2044" y="0" xlink:href="#MJMAIN-3D"></use><use x="3105" y="0" xlink:href="#MJMAIN-31"></use><use x="3832" y="0" xlink:href="#MJMAIN-22C5"></use><use x="4337" y="0" xlink:href="#MJMATHI-61"></use><use x="5149" y="0" xlink:href="#MJMAIN-3D"></use><use x="6210" y="0" xlink:href="#MJMATHI-61"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>a</mi><mo>·</mo><mn>1</mn><mo>=</mo><mn>1</mn><mo>·</mo><mi>a</mi><mo>=</mo><mi>a</mi></mrow></math></script></span></p>
<p class="noindentf">elemento annullatore: <span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="15.667ex" height="1.667ex" viewBox="0 -689.9 6715 735.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-61"></use><use x="756" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1261" y="0" xlink:href="#MJMAIN-30"></use><use x="2044" y="0" xlink:href="#MJMAIN-3D"></use><use x="3105" y="0" xlink:href="#MJMAIN-30"></use><use x="3832" y="0" xlink:href="#MJMAIN-22C5"></use><use x="4337" y="0" xlink:href="#MJMATHI-61"></use><use x="5149" y="0" xlink:href="#MJMAIN-3D"></use><use x="6210" y="0" xlink:href="#MJMAIN-30"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>a</mi><mo>·</mo><mn>0</mn><mo>=</mo><mn>0</mn><mo>·</mo><mi>a</mi><mo>=</mo><mn>0</mn></mrow></math></script></span></p>
<p class="noindentf">legge di annullamento del prodotto:<a id="ind140"></a><!--<?"prodotto|legge di annullamento del",4,0,2>--> <span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="63.167ex" height="3ex" viewBox="0 -875 27174.5 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-73"></use><use x="399" y="0" xlink:href="#MJMAIN-65"></use><g transform="translate(1014,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="1624" y="0" xlink:href="#MJMATHI-61"></use><use x="2380" y="0" xlink:href="#MJMAIN-22C5"></use><use x="2886" y="0" xlink:href="#MJMATHI-62"></use><use x="3597" y="0" xlink:href="#MJMAIN-3D"></use><use x="4658" y="0" xlink:href="#MJMAIN-30"></use><g transform="translate(5163,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(5940,0)"><use xlink:href="#MJMAIN-61"></use><use x="505" y="0" xlink:href="#MJMAIN-6C"></use><use x="788" y="0" xlink:href="#MJMAIN-6C"></use><use x="1071" y="0" xlink:href="#MJMAIN-6F"></use><use x="1576" y="0" xlink:href="#MJMAIN-72"></use><use x="1973" y="0" xlink:href="#MJMAIN-61"></use></g><g transform="translate(8584,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="9194" y="0" xlink:href="#MJMATHI-61"></use><use x="10006" y="0" xlink:href="#MJMAIN-3D"></use><use x="11067" y="0" xlink:href="#MJMAIN-30"></use><g transform="translate(11572,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(12349,0)"><use xlink:href="#MJMAIN-6F"></use><use x="505" y="0" xlink:href="#MJMAIN-70"></use><use x="1066" y="0" xlink:href="#MJMAIN-70"></use><use x="1627" y="0" xlink:href="#MJMAIN-75"></use><use x="2188" y="0" xlink:href="#MJMAIN-72"></use><use x="2585" y="0" xlink:href="#MJMAIN-65"></use></g><g transform="translate(15549,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="16159" y="0" xlink:href="#MJMATHI-62"></use><use x="16871" y="0" xlink:href="#MJMAIN-3D"></use><use x="17932" y="0" xlink:href="#MJMAIN-30"></use><g transform="translate(18437,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(19213,0)"><use xlink:href="#MJMAIN-6F"></use><use x="505" y="0" xlink:href="#MJMAIN-70"></use><use x="1066" y="0" xlink:href="#MJMAIN-70"></use><use x="1627" y="0" xlink:href="#MJMAIN-75"></use><use x="2188" y="0" xlink:href="#MJMAIN-72"></use><use x="2585" y="0" xlink:href="#MJMAIN-65"></use></g><g transform="translate(22414,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="italic" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="23024" y="0" xlink:href="#MJMATHI-61"></use><use x="23836" y="0" xlink:href="#MJMAIN-3D"></use><use x="24896" y="0" xlink:href="#MJMATHI-62"></use><use x="25608" y="0" xlink:href="#MJMAIN-3D"></use><use x="26669" y="0" xlink:href="#MJMAIN-30"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>se</mi><mtext> </mtext><mi>a</mi><mo>·</mo><mi>b</mi><mo>=</mo><mn>0</mn><mi> </mi><mi>allora</mi><mi> </mi><mi>a</mi><mo>=</mo><mn>0</mn><mi> </mi><mi>oppure</mi><mi> </mi><mi>b</mi><mo>=</mo><mn>0</mn><mi> </mi><mi>oppure</mi><mi> </mi><mi>a</mi><mo>=</mo><mi>b</mi><mo>=</mo><mn>0</mn></mrow></math></script></span></p>
</td>
</tr>
<tr>
<td valign="middle" class="tabvf">
<p class="center">divisione esatta</p>
<p class="center"><i>a</i> : <i>b</i> con <i>b</i> ≠ 0</p>
<p class="center"><i>a</i> multiplo di <i>b</i></p>
<p class="center"><i>b</i> divisore di <i>a</i></p>
</td>
<td valign="middle" class="tabvm">
<p class="center"><i>a</i> dividendo</p>
<p class="center"><i>b</i> divisore</p>
</td>
<td valign="middle" class="tabvm"><p class="center">quoziente</p></td>
<td valign="middle" class="tabvm">
<p class="noindentf">invariantiva: <span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -3ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="23.667ex" height="7ex" viewBox="0 -1772.9 10215.3 3045.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use x="0" y="-1" xlink:href="#MJSZ4-27E8"></use><g transform="translate(978,0)"><g transform="translate(-11,0)"><g transform="translate(0,725)"><use xlink:href="#MJMATHI-61"></use><use x="811" y="0" xlink:href="#MJMAIN-3A"></use><use x="1372" y="0" xlink:href="#MJMATHI-62"></use><use x="2084" y="0" xlink:href="#MJMAIN-3D"></use><use x="3145" y="0" xlink:href="#MJMAIN-28"></use><use x="3539" y="0" xlink:href="#MJMATHI-61"></use><use x="4295" y="0" xlink:href="#MJMAIN-22C5"></use><use x="4800" y="0" xlink:href="#MJMATHI-63"></use><use x="5238" y="0" xlink:href="#MJMAIN-29"></use><use x="5910" y="0" xlink:href="#MJMAIN-3A"></use><use x="6471" y="0" xlink:href="#MJMAIN-28"></use><use x="6865" y="0" xlink:href="#MJMATHI-62"></use><use x="7521" y="0" xlink:href="#MJMAIN-22C5"></use><use x="8026" y="0" xlink:href="#MJMATHI-63"></use><use x="8464" y="0" xlink:href="#MJMAIN-29"></use></g><g transform="translate(0,-766)"><use xlink:href="#MJMATHI-61"></use><use x="811" y="0" xlink:href="#MJMAIN-3A"></use><use x="1372" y="0" xlink:href="#MJMATHI-62"></use><use x="2084" y="0" xlink:href="#MJMAIN-3D"></use><use x="3145" y="0" xlink:href="#MJMAIN-28"></use><use x="3539" y="0" xlink:href="#MJMATHI-61"></use><use x="4350" y="0" xlink:href="#MJMAIN-3A"></use><use x="4911" y="0" xlink:href="#MJMATHI-63"></use><use x="5349" y="0" xlink:href="#MJMAIN-29"></use><use x="6021" y="0" xlink:href="#MJMAIN-3A"></use><use x="6582" y="0" xlink:href="#MJMAIN-28"></use><use x="6976" y="0" xlink:href="#MJMATHI-62"></use><use x="7687" y="0" xlink:href="#MJMAIN-3A"></use><use x="8248" y="0" xlink:href="#MJMATHI-63"></use><use x="8686" y="0" xlink:href="#MJMAIN-29"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo stretchy="true">〈</mo><mtable columnalign="left"><mtr><mtd><mi>a</mi><mo>:</mo><mi>b</mi><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo>⋅</mo><mi>c</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><mi>b</mi><mo>⋅</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>a</mi><mo>:</mo><mi>b</mi><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo>:</mo><mi>c</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><mi>b</mi><mo>:</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow></math></script></span></p>
<p class="noindentf">distributiva: <span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -3ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="38.167ex" height="7ex" viewBox="0 -1772.9 16449.1 3045.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use x="0" y="-1" xlink:href="#MJSZ4-27E8"></use><g transform="translate(978,0)"><g transform="translate(-11,0)"><g transform="translate(0,725)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMATHI-61"></use><use x="1150" y="0" xlink:href="#MJMAIN-2B"></use><use x="2155" y="0" xlink:href="#MJMATHI-62"></use><use x="2811" y="0" xlink:href="#MJMAIN-2B"></use><use x="3816" y="0" xlink:href="#MJMATHI-63"></use><use x="4254" y="0" xlink:href="#MJMAIN-29"></use><use x="4926" y="0" xlink:href="#MJMAIN-3A"></use><use x="5487" y="0" xlink:href="#MJMATHI-64"></use><use x="6293" y="0" xlink:href="#MJMAIN-3D"></use><use x="7353" y="0" xlink:href="#MJMATHI-61"></use><use x="8165" y="0" xlink:href="#MJMAIN-3A"></use><use x="8726" y="0" xlink:href="#MJMATHI-64"></use><use x="9476" y="0" xlink:href="#MJMAIN-2B"></use><use x="10481" y="0" xlink:href="#MJMATHI-62"></use><use x="11193" y="0" xlink:href="#MJMAIN-3A"></use><use x="11754" y="0" xlink:href="#MJMATHI-64"></use><use x="12504" y="0" xlink:href="#MJMAIN-2B"></use><use x="13509" y="0" xlink:href="#MJMATHI-63"></use><use x="14225" y="0" xlink:href="#MJMAIN-3A"></use><use x="14786" y="0" xlink:href="#MJMATHI-64"></use></g><g transform="translate(0,-766)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMATHI-61"></use><use x="1150" y="0" xlink:href="#MJMAIN-2212"></use><use x="2155" y="0" xlink:href="#MJMATHI-62"></use><use x="2589" y="0" xlink:href="#MJMAIN-29"></use><use x="3261" y="0" xlink:href="#MJMAIN-3A"></use><use x="3822" y="0" xlink:href="#MJMATHI-63"></use><use x="4537" y="0" xlink:href="#MJMAIN-3D"></use><use x="5598" y="0" xlink:href="#MJMATHI-61"></use><use x="6410" y="0" xlink:href="#MJMAIN-3A"></use><use x="6971" y="0" xlink:href="#MJMATHI-63"></use><use x="7631" y="0" xlink:href="#MJMAIN-2212"></use><use x="8636" y="0" xlink:href="#MJMATHI-62"></use><use x="9348" y="0" xlink:href="#MJMAIN-3A"></use><use x="9909" y="0" xlink:href="#MJMATHI-63"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo stretchy="true">〈</mo><mtable columnalign="left"><mtr><mtd><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo stretchy="false">)</mo><mo>:</mo><mi>d</mi><mo>=</mo><mi>a</mi><mo>:</mo><mi>d</mi><mo>+</mo><mi>b</mi><mo>:</mo><mi>d</mi><mo>+</mo><mi>c</mi><mo>:</mo><mi>d</mi></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><mi>a</mi><mo>−</mo><mi>b</mi><mo stretchy="false">)</mo><mo>:</mo><mi>c</mi><mo>=</mo><mi>a</mi><mo>:</mo><mi>c</mi><mo>−</mo><mi>b</mi><mo>:</mo><mi>c</mi></mtd></mtr></mtable></mrow></mrow></math></script></span></p>
<p class="noindentf">la proprietà distributiva vale solo se il simbolo di divisione è scritto <b>a destra</b> dell’addizione o della sottrazione</p>
<p class="noindentf">due o più divisioni consecutive vanno eseguite nell’ordine con cui sono indicate</p>
</td>
</tr>
</tbody>
</table>
</div>
<ul class="blist">
<li>
<p class="noindent1"><b>Casi particolari di divisione</b></p>
<ul class="blist">
<li><p class="noindent"><span class="cyan"><i>a</i> : 1 = <i>a</i></span></p></li>
<li><p class="noindent"><span class="cyan"><i>a</i> : <i>a</i> = 1 <b>se</b> <i>a</i> ≠ 0</span></p></li>
<li><p class="noindent"><span class="cyan">0 : <i>a</i> = 0 <b>se</b> <i>a</i> ≠ 0</span></p></li>
</ul>
</li>
<li><p class="noindent1"><b>Non è possibile dividere per zero</b></p></li>
<li>
<p class="noindent1"><b>Divisione approssimata</b></p>
<p class="math"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="54.167ex" height="3ex" viewBox="0 -875 23289.3 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-61"></use><use x="811" y="0" xlink:href="#MJMAIN-3A"></use><use x="1372" y="0" xlink:href="#MJMATHI-62"></use><use x="2084" y="0" xlink:href="#MJMAIN-3D"></use><use x="3145" y="0" xlink:href="#MJMATHI-71"></use><g transform="translate(3610,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(4220,0)"><use xlink:href="#MJMAINB-63"></use><use x="516" y="0" xlink:href="#MJMAINB-6F"></use><use x="1096" y="0" xlink:href="#MJMAINB-6E"></use><g transform="translate(1740,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="bold" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="2349" y="0" xlink:href="#MJMAINB-72"></use><use x="2828" y="0" xlink:href="#MJMAINB-65"></use><use x="3360" y="0" xlink:href="#MJMAINB-73"></use><use x="3819" y="0" xlink:href="#MJMAINB-74"></use><use x="4271" y="0" xlink:href="#MJMAINB-6F"></use></g><g transform="translate(9072,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="9681" y="0" xlink:href="#MJMATHI-72"></use><g transform="translate(10137,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="11025" y="0" xlink:href="#MJMAIN-2194"></use><g transform="translate(12308,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="12918" y="0" xlink:href="#MJMATHI-61"></use><use x="13730" y="0" xlink:href="#MJMAIN-3D"></use><use x="14790" y="0" xlink:href="#MJMATHI-62"></use><use x="15447" y="0" xlink:href="#MJMAIN-22C5"></use><use x="15952" y="0" xlink:href="#MJMATHI-71"></use><use x="16639" y="0" xlink:href="#MJMAIN-2B"></use><use x="17644" y="0" xlink:href="#MJMATHI-72"></use><g transform="translate(18100,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(18710,0)"><use xlink:href="#MJMAINB-63"></use><use x="516" y="0" xlink:href="#MJMAINB-6F"></use><use x="1096" y="0" xlink:href="#MJMAINB-6E"></use></g><g transform="translate(20450,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="21060" y="0" xlink:href="#MJMATHI-72"></use><use x="21794" y="0" xlink:href="#MJMAIN-3C"></use><use x="22855" y="0" xlink:href="#MJMATHI-62"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>a</mi><mo>:</mo><mi>b</mi><mo>=</mo><mi>q</mi><mtext> </mtext><mstyle mathvariant="bold" mathsize="normal"><mi>c</mi><mi>o</mi><mi>n</mi><mi> </mi><mi>r</mi><mi>e</mi><mi>s</mi><mi>t</mi><mi>o</mi></mstyle><mtext> </mtext><mi>r</mi><mtext> </mtext><mo>↔</mo><mtext> </mtext><mi>a</mi><mo>=</mo><mi>b</mi><mo>⋅</mo><mi>q</mi><mo>+</mo><mi>r</mi><mtext> </mtext><mstyle mathvariant="bold" mathsize="normal"><mi>c</mi><mi>o</mi><mi>n</mi></mstyle><mtext> </mtext><mi>r</mi><mo>&lt;</mo><mi>b</mi></mrow></math></script></span></p>
<p class="noindent">Proprietà&nbsp;invariantiva: <span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -3ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="68.667ex" height="7ex" viewBox="0 -1772.9 29596.6 3045.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use x="0" y="-1" xlink:href="#MJSZ4-27E8"></use><g transform="translate(978,0)"><g transform="translate(-11,0)"><g transform="translate(0,895)"><use xlink:href="#MJMATHI-61"></use><use x="811" y="0" xlink:href="#MJMAIN-3A"></use><use x="1372" y="0" xlink:href="#MJMATHI-62"></use><use x="2084" y="0" xlink:href="#MJMAIN-3D"></use><use x="3145" y="0" xlink:href="#MJMATHI-71"></use><g transform="translate(3610,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(4220,0)"><use xlink:href="#MJMAINB-63"></use><use x="516" y="0" xlink:href="#MJMAINB-6F"></use><use x="1096" y="0" xlink:href="#MJMAINB-6E"></use></g><g transform="translate(5960,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(6570,0)"><use xlink:href="#MJMAINB-72"></use><use x="479" y="0" xlink:href="#MJMAINB-65"></use><use x="1011" y="0" xlink:href="#MJMAINB-73"></use><use x="1470" y="0" xlink:href="#MJMAINB-74"></use><use x="1922" y="0" xlink:href="#MJMAINB-6F"></use></g><g transform="translate(9072,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="9681" y="0" xlink:href="#MJMATHI-72"></use><g transform="translate(10137,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="11025" y="0" xlink:href="#MJMAIN-2192"></use><g transform="translate(12308,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="12918" y="0" xlink:href="#MJMAIN-28"></use><use x="13312" y="0" xlink:href="#MJMATHI-61"></use><use x="14068" y="0" xlink:href="#MJMAIN-22C5"></use><use x="14573" y="0" xlink:href="#MJMATHI-63"></use><use x="15011" y="0" xlink:href="#MJMAIN-29"></use><use x="15683" y="0" xlink:href="#MJMAIN-3A"></use><use x="16244" y="0" xlink:href="#MJMAIN-28"></use><use x="16638" y="0" xlink:href="#MJMATHI-62"></use><use x="17294" y="0" 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xlink:href="#MJMAINB-73"></use><use x="1470" y="0" xlink:href="#MJMAINB-74"></use><use x="1922" y="0" xlink:href="#MJMAINB-6F"></use></g><g transform="translate(25897,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="26507" y="0" xlink:href="#MJMATHI-72"></use><use x="27185" y="0" xlink:href="#MJMAIN-22C5"></use><use x="27690" y="0" xlink:href="#MJMATHI-63"></use></g><g transform="translate(0,-827)"><use xlink:href="#MJMATHI-61"></use><use x="811" y="0" xlink:href="#MJMAIN-3A"></use><use x="1372" y="0" xlink:href="#MJMATHI-62"></use><use x="2084" y="0" xlink:href="#MJMAIN-3D"></use><use x="3145" y="0" xlink:href="#MJMATHI-71"></use><g transform="translate(3610,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(4220,0)"><use xlink:href="#MJMAINB-63"></use><use x="516" y="0" xlink:href="#MJMAINB-6F"></use><use x="1096" y="0" xlink:href="#MJMAINB-6E"></use></g><g transform="translate(5960,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(6570,0)"><use xlink:href="#MJMAINB-72"></use><use x="479" y="0" xlink:href="#MJMAINB-65"></use><use x="1011" y="0" xlink:href="#MJMAINB-73"></use><use x="1470" y="0" xlink:href="#MJMAINB-74"></use><use x="1922" y="0" xlink:href="#MJMAINB-6F"></use></g><g transform="translate(9072,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="9681" y="0" xlink:href="#MJMATHI-72"></use><g 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</li>
</ul>
</div></div></div>
</div></div>
<div class="small-12 medium-4 columns"></div>
</div></div>
<div data-page-container="35" id="page-35" class="row chapters-content"><div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">35</div></div>
<div class="small-12 medium-11 columns"><div class="ch1"><div>
<h3 class="sec_title">
<span class="pagebreak" epub:type="pagebreak" title="35" id="page35"></span>Potenze in ℕ e loro proprietà</h3>
<ul class="blist">
<li>
<p class="noindent"><b>Elevamento a potenza</b> <span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><span style="display: inline-block; white-space: nowrap; padding: 1px 0px;"><span style="display: inline-block; position: relative; vertical-align: -5ex; width: 57ex; height: 11ex;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="margin-left: 0ex; margin-right: 0ex; position: absolute; left: 0px;" width="57ex" height="11ex" viewBox="0 -2632 24571 4764"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMATHI-61"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(534,362)"><use transform="scale(0.707)" xlink:href="#MJMATHI-6E"></use></g></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1339,0)"><use xlink:href="#MJMAIN-3D"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(2400,0)"><g fill="#00aef0" stroke="#00aef0" transform="translate(1265,0)"><g fill="#00aef0" stroke="#00aef0"><g fill="#00aef0" stroke="#00aef0"><use xlink:href="#MJMATHI-61"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(756,0)"><use xlink:href="#MJMAIN-22C5"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(1261,0)"><use xlink:href="#MJMATHI-61"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(2017,0)"><use xlink:href="#MJMAIN-22C5"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(2522,0)"><use xlink:href="#MJMAIN-2E"></use><use x="283" y="0" xlink:href="#MJMAIN-2E"></use><use x="566" y="0" xlink:href="#MJMAIN-2E"></use></g><g fill="#00aef0" stroke="#00aef0" transform="translate(3594,0)"><use xlink:href="#MJMAIN-22C5"></use></g><g fill="#00aef0" stroke="#00aef0" 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xlink:href="#MJMAIN-61"></use></g></g></g></g></g></svg></span></span></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mstyle color="#00aef0"><msup><mi>a</mi><mi>n</mi></msup><mo>=</mo><munder><munder><mrow><mi>a</mi><mo>⋅</mo><mi>a</mi><mo>⋅</mo><mn>...</mn><mo>⋅</mo><mi>a</mi></mrow><mo stretchy="true">︸</mo></munder><mrow><mi>n</mi><mtext> </mtext><mtext>fattori</mtext><mtext> </mtext><mtext>uguali</mtext><mtext> </mtext><mtext>ad</mtext><mtext> </mtext><mi>a</mi></mrow></munder></mstyle><mtext> </mtext><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mo>↗</mo></mtd><mtd columnalign="left"><mrow><msup><mi>a</mi><mi>n</mi></msup><mtext> </mtext><mtext>potenza</mtext></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mo>→</mo></mtd><mtd columnalign="left"><mrow><mi>a</mi><mtext> </mtext><mtext>base</mtext><mtext> </mtext><mtext>della</mtext><mtext> </mtext><mtext>potenza</mtext></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mo>↘</mo></mtd><mtd columnalign="left"><mrow><mi>n</mi><mtext> </mtext><mtext>esponente</mtext><mtext> </mtext><mtext>della</mtext><mtext> </mtext><mtext>potenza</mtext></mrow></mtd></mtr></mtable></mrow></math></script></p>
<ul class="blist">
<li><p class="noindent"><span class="cyan"><i>a</i><sup>1</sup> = <i>a</i></span></p></li>
<li><p class="noindent"><span class="cyan"><i>a</i><sup>0</sup> = 1 <b>per</b> <i>a</i> ≠ 0</span></p></li>
</ul>
<p class="noindent">Per convenzione si stabilisce che 0<sup>0</sup> non ha significato.</p>
</li>
<li>
<p class="noindent1"><b>Proprietà delle potenze</b></p>
<p class="center"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -3ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="74ex" height="7ex" viewBox="0 -1770.9 31843.8 3041.7"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(167,0)"><g transform="translate(-11,0)"><g transform="translate(0,895)"><use xlink:href="#MJMATHI-61"></use><use transform="scale(0.707)" x="755" y="513" xlink:href="#MJMATHI-6D"></use><use x="1480" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(1985,0)"><use xlink:href="#MJMATHI-61"></use><use transform="scale(0.707)" x="755" y="513" xlink:href="#MJMATHI-6E"></use></g><use x="3325" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(4386,0)"><use xlink:href="#MJMATHI-61"></use><g 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</li>
</ul>
<h3 class="sec_title">Espressioni con i numeri naturali</h3>
<ul class="blist">
<li>
<p class="noindent"><b>Grado di priorità delle operazioni</b>: è l’ordine di precedenza tra le operazioni.</p>
<p class="noindent">Occorre eseguire:</p>
<ol class="olist">
<li><p class="noindent">prima gli elevamenti a potenza;</p></li>
<li><p class="noindent">poi le moltiplicazioni e le divisioni, nell’ordine in cui sono scritte;</p></li>
<li><p class="noindent">infine le addizioni e le sottrazioni, nell’ordine in cui sono scritte.</p></li>
</ol>
</li>
<li><p class="noindent"><b>Parentesi</b>: alterano il grado di priorità delle operazioni.</p></li>
<li>
<p class="noindent"><b>Altre proprietà</b></p>
<ul class="blist">
<li>
<p class="noindent">Dividere un prodotto per un numero: <span class="cyan">(<i>a</i> · <i>b</i> · <i>c</i>) : <i>d</i> = <i>a</i> · (<i>b</i> : <i>d</i>) · <i>c</i></span></p>
<p class="noindent">In particolare: <span class="cyan">(<i>a</i> · <i>b</i> · <i>c</i>) : <i>b</i> = <i>a</i> · <i>c</i></span></p>
</li>
<li><p class="noindent">Dividere un numero per un prodotto: <span class="cyan"><i>a</i> : (<i>b</i> · <i>c</i>) = (<i>a</i> : <i>b</i>) : <i>c</i></span></p></li>
<li><p class="noindent">Moltiplicare un numero per un quoziente: <span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -2.5ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="27ex" height="6ex" viewBox="0 -1539.2 11597.1 2578.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-61"></use><use x="756" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1261" y="0" xlink:href="#MJMAIN-28"></use><use x="1655" y="0" xlink:href="#MJMATHI-62"></use><use x="2367" y="0" xlink:href="#MJMAIN-3A"></use><use x="2928" y="0" xlink:href="#MJMATHI-63"></use><use x="3366" y="0" xlink:href="#MJMAIN-29"></use><use x="4037" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(5098,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(5875,0)"><g transform="translate(-11,0)"><use x="0" y="725" xlink:href="#MJMAIN-2197"></use><use x="0" y="-766" xlink:href="#MJMAIN-2198"></use></g><g transform="translate(1794,0)"><g transform="translate(0,725)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMATHI-61"></use><use x="1150" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1655" y="0" xlink:href="#MJMATHI-62"></use><use x="2089" y="0" xlink:href="#MJMAIN-29"></use><use x="2761" y="0" xlink:href="#MJMAIN-3A"></use><use x="3322" y="0" xlink:href="#MJMATHI-63"></use></g><g transform="translate(0,-766)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMATHI-61"></use><use x="1205" y="0" xlink:href="#MJMAIN-3A"></use><use x="1766" y="0" xlink:href="#MJMATHI-63"></use><use x="2204" y="0" xlink:href="#MJMAIN-29"></use><use x="2820" y="0" xlink:href="#MJMAIN-22C5"></use><use x="3326" y="0" xlink:href="#MJMATHI-62"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>a</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>b</mi><mo>:</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mtext> </mtext><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mo>↗</mo></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>⋅</mo><mi>b</mi><mo stretchy="false">)</mo><mo>:</mo><mi>c</mi></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mo>↘</mo></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>:</mo><mi>c</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>b</mi></mrow></mtd></mtr></mtable></mrow></math></script></span></p></li>
<li><p class="noindent">Dividere un numero per un quoziente: <span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -2.5ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="27.167ex" height="6ex" viewBox="0 -1539.2 11708.2 2578.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-61"></use><use x="811" y="0" xlink:href="#MJMAIN-3A"></use><use x="1372" y="0" xlink:href="#MJMAIN-28"></use><use x="1766" y="0" xlink:href="#MJMATHI-62"></use><use x="2478" y="0" xlink:href="#MJMAIN-3A"></use><use x="3039" y="0" xlink:href="#MJMATHI-63"></use><use x="3477" y="0" xlink:href="#MJMAIN-29"></use><use x="4148" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(5209,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(5986,0)"><g transform="translate(-11,0)"><use x="0" y="725" xlink:href="#MJMAIN-2197"></use><use x="0" y="-766" xlink:href="#MJMAIN-2198"></use></g><g transform="translate(1794,0)"><g transform="translate(0,725)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMATHI-61"></use><use x="1205" y="0" xlink:href="#MJMAIN-3A"></use><use x="1766" y="0" xlink:href="#MJMATHI-62"></use><use x="2200" y="0" xlink:href="#MJMAIN-29"></use><use x="2816" y="0" xlink:href="#MJMAIN-22C5"></use><use x="3322" y="0" xlink:href="#MJMATHI-63"></use></g><g transform="translate(0,-766)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMATHI-61"></use><use x="1150" y="0" xlink:href="#MJMAIN-22C5"></use><use x="1655" y="0" xlink:href="#MJMATHI-63"></use><use x="2093" y="0" xlink:href="#MJMAIN-29"></use><use x="2765" y="0" xlink:href="#MJMAIN-3A"></use><use x="3326" y="0" xlink:href="#MJMATHI-62"></use></g></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>a</mi><mo>:</mo><mo stretchy="false">(</mo><mi>b</mi><mo>:</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mtext> </mtext><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mo>↗</mo></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>:</mo><mi>b</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>c</mi></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mo>↘</mo></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>⋅</mo><mi>c</mi><mo stretchy="false">)</mo><mo>:</mo><mi>b</mi></mrow></mtd></mtr></mtable></mrow></math></script></span></p></li>
</ul>
</li>
</ul>
<h3 class="sec_title">Divisibilità e numeri primi</h3>
<ul class="blist">
<li>
<p class="noindent"><b>Multiplo, divisore</b></p>
<table width="100%" cellspacing="0" cellpadding="0">
<colgroup><col width="30%">
<col width="5%">
<col width="65%">
</colgroup><tbody>
<tr>
<td valign="top"><p class="noindent"><i>a</i> = <i>b</i> · <i>n</i></p></td>
<td valign="top"><p class="noindent">→</p></td>
<td valign="top"><p class="noindent"><i>a</i> è multiplo di <i>b</i>; se <i>b</i> ≠ 0, <i>a</i> è divisibile per <i>b</i> e <i>b</i> è divisore di <i>a</i></p></td>
</tr>
<tr>
<td valign="top"><p class="noindent">0 : <i>b</i> = 0 con <i>b</i> ≠ 0</p></td>
<td valign="top"><p class="noindent">→</p></td>
<td valign="top"><p class="noindent">0 è divisibile per qualsiasi numero diverso da 0; qualsiasi numero diverso da 0 è divisore di 0</p></td>
</tr>
<tr>
<td valign="top"><p class="noindent"><i>a</i> : 0 non si può eseguire</p></td>
<td valign="top"><p class="noindent">→</p></td>
<td valign="top"><p class="noindent">nessun numero è divisibile per 0</p></td>
</tr>
<tr>
<td valign="top"><p class="noindent"><i>a</i> : 1 = <i>a</i></p></td>
<td valign="top"><p class="noindent">→</p></td>
<td valign="top"><p class="noindent">qualsiasi numero è divisibile per 1; 1 è divisore di qualsiasi numero</p></td>
</tr>
<tr>
<td valign="top"><p class="noindent"><i>a</i> : <i>a</i> = 1 con <i>a</i> ≠ 0</p></td>
<td valign="top"><p class="noindent">→</p></td>
<td valign="top"><p class="noindent">qualsiasi numero diverso da 0 è divisibile per se stesso.</p></td>
</tr>
</tbody>
</table>
</li>
<li>
<p class="noindent"><b>Criteri di divisibilità</b></p>
<p class="noindent">Un numero è divisibile per</p>
<table width="100%" cellspacing="0" cellpadding="0">
<colgroup><col width="10%">
<col width="90%">
</colgroup><tbody>
<tr>
<td valign="top"><p class="noindent">• 2</p></td>
<td valign="top"><p class="noindent">se termina per cifra pari;</p></td>
</tr>
<tr>
<td valign="top"><p class="noindent">• 3 o 9</p></td>
<td valign="top"><p class="noindent">se lo è la somma delle sue cifre;</p></td>
</tr>
<tr>
<td valign="top"><p class="noindent">• 5</p></td>
<td valign="top"><p class="noindent">se termina per 0 o per 5;</p></td>
</tr>
<tr>
<td valign="top"><p class="noindent">• 10</p></td>
<td valign="top"><p class="noindent">se l’ultima cifra è 0;</p></td>
</tr>
<tr>
<td valign="top"><p class="noindent">• 4 o 25</p></td>
<td valign="top"><p class="noindent">se lo è il numero formato dalle sue ultime due cifre o se termina con due zeri;</p></td>
</tr>
<tr>
<td valign="top"><p class="noindent">• 11</p></td>
<td valign="top"><p class="noindent">se la differenza tra la somma delle cifre di posto dispari (contandole ad esempio da destra a sinistra), eventualmente aumentata di un multiplo di 11, e la somma di quelle di posto pari è divisibile per 11.</p></td>
</tr>
</tbody>
</table>
</li>
</ul>
</div></div></div>
</div></div>
<div class="small-12 medium-4 columns"></div>
</div></div>
<div data-page-container="36" id="page-36" class="row chapters-content"><div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">36</div></div>
<div class="small-12 medium-11 columns"><div class="ch1"><div>
<ul class="blist">
<li>
<p class="noindent"><span class="pagebreak" epub:type="pagebreak" title="36" id="page36"></span><b>Numero primo</b></p>
<p class="noindent">Un numero naturale diverso da 1 è primo se è divisibile solo per se stesso e per 1. Ogni numero maggiore di 0 che non sia primo si può scomporre, in un unico modo, nel prodotto di fattori primi.</p>
</li>
</ul>
<h3 class="sec_title">Massimo comune divisore e minimo comune multiplo</h3>
<ul class="blist">
<li>
<p class="noindent1"><b><i>MCD</i> (massimo comune divisore)</b></p>
<p class="noindent">Il <i>MCD</i> di due o più numeri naturali diversi da 0 è il maggiore tra i loro divisori comuni.</p>
<p class="noindent">Il <i>MCD</i> di due o più numeri si ottiene calcolando il prodotto dei fattori primi comuni a tutti i numeri dati, presi una volta sola, ciascuno con il minimo esponente con cui figura.</p>
</li>
<li>
<p class="noindent1"><b><i>mcm</i> (minimo comune multiplo)</b></p>
<p class="noindent">Il <i>mcm</i> di due o più numeri naturali diversi da 0 è il minore tra i loro multipli comuni diversi da 0.</p>
<p class="noindent">Il <i>mcm</i> di due o più numeri naturali si ottiene calcolando il prodotto dei fattori primi, comuni e non comuni a tutti i numeri dati, presi una volta sola, ciascuno con il massimo esponente con cui figura.</p>
</li>
<li>
<p class="noindent1"><b>Numeri primi tra loro</b></p>
<p class="noindent">Due numeri naturali sono primi tra loro (o coprimi) se il loro <i>MCD</i> è 1.</p>
</li>
</ul>
<h3 class="sec_title">Sistemi di numerazione</h3>
<ul class="blist">
<li>
<p class="noindent1"><b>Sistema di numerazione</b></p>
<p class="noindent">Insieme di simboli (cifre) e regole con cui è possibile rappresentare tutti i numeri.</p>
</li>
<li>
<p class="noindent1"><b>Sistema posizionale</b></p>
<p class="noindent">A ogni simbolo viene associato un valore diverso che dipende, oltre che dal simbolo stesso, dalla posizione che esso occupa nella scrittura del numero.</p>
</li>
<li>
<p class="noindent1"><b>Sistemi di numerazione posizionali<a id="ind141"></a><!--<?"sistemi di numerazione posizionali",4,0,2>--></b></p>
<ul class="blist">
<li><p class="noindent"><i>Base</i>: è il numero di cifre utilizzato in un sistema posizionale.<a id="ind142"></a><!--<?"base|in un sistema posizionale",4,0,2>--></p></li>
<li><p class="noindent"><i>Ordine di una cifra</i>: è il posto che essa occupa contando, a partire da zero, dall’ultima cifra a destra verso sinistra; nel numero 57 032 la cifra 7 è di ordine 3.</p></li>
<li>
<p class="noindent"><i>Valore associato a una cifra</i>:<a id="ind143"></a><!--<?"valore|associato a una cifra",4,0,2>--> in un sistema di numerazione in base <i>b</i> a ogni cifra è associato il valore che si ottiene moltiplicando tale cifra per <i>b</i> elevato all’esponente dato dall’ordine della cifra stessa.</p>
<p class="noindent">Nel numero che, in base 8, si scrive 57032, alla cifra 7 è associato il valore 7 · 8<sup>3</sup> (valore espresso in base 10).</p>
</li>
<li><p class="noindent"><i>Sistema decimale</i>: è il sistema posizionale a base 10, che usiamo quotidianamente.</p></li>
<li><p class="noindent"><i>Sistema binario</i>: è il sistema posizionale a base 2. Nel sistema binario si usano solo due cifre: 0 e 1.</p></li>
</ul>
</li>
<li>
<p class="noindent"><b>Cambiamenti di base</b></p>
<ul class="blist">
<li>
<p class="noindent"><b>Dalla base <i>b</i> alla base 10</b></p>
<p class="noindent">Occorre utilizzare la <i>forma polinomiale</i>. Ad esempio:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.5ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="48.333ex" height="2.5ex" viewBox="0 -905.9 20825.3 1092.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-32"></use><use x="505" y="0" xlink:href="#MJMAIN-30"></use><use x="1010" y="0" xlink:href="#MJMAIN-33"></use><use transform="scale(0.707)" x="2142" y="-213" xlink:href="#MJMAIN-34"></use><use x="2249" y="0" xlink:href="#MJMAIN-3D"></use><use x="3310" y="0" xlink:href="#MJMAIN-32"></use><use x="4037" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(4543,0)"><use xlink:href="#MJMAIN-34"></use><use transform="scale(0.707)" x="714" y="585" xlink:href="#MJMAIN-32"></use></g><use x="5727" y="0" xlink:href="#MJMAIN-2B"></use><use x="6732" y="0" xlink:href="#MJMAIN-30"></use><use x="7459" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(7965,0)"><use xlink:href="#MJMAIN-34"></use><use transform="scale(0.707)" x="714" y="585" xlink:href="#MJMAIN-31"></use></g><use x="9149" y="0" xlink:href="#MJMAIN-2B"></use><use x="10154" y="0" xlink:href="#MJMAIN-33"></use><use x="10881" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(11387,0)"><use xlink:href="#MJMAIN-34"></use><use transform="scale(0.707)" x="714" y="585" xlink:href="#MJMAIN-30"></use></g><use x="12626" y="0" xlink:href="#MJMAIN-3D"></use><use x="13687" y="0" xlink:href="#MJMAIN-32"></use><use x="14414" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(14920,0)"><use xlink:href="#MJMAIN-31"></use><use x="505" y="0" xlink:href="#MJMAIN-36"></use></g><use x="16152" y="0" xlink:href="#MJMAIN-2B"></use><use x="17157" y="0" xlink:href="#MJMAIN-33"></use><use x="17940" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(19001,0)"><use xlink:href="#MJMAIN-33"></use><use x="505" y="0" xlink:href="#MJMAIN-35"></use><g transform="translate(1010,-150)"><use transform="scale(0.707)" xlink:href="#MJMAIN-31"></use><use transform="scale(0.707)" x="505" y="0" xlink:href="#MJMAIN-30"></use></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mn>203</mn></mrow><mrow><mn>4</mn></mrow></msub><mo>=</mo><mn>2</mn><mo>·</mo><msup><mrow><mn>4</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>0</mn><mo>·</mo><msup><mrow><mn>4</mn></mrow><mrow><mn>1</mn></mrow></msup><mo>+</mo><mn>3</mn><mo>·</mo><msup><mrow><mn>4</mn></mrow><mrow><mn>0</mn></mrow></msup><mo>=</mo><mn>2</mn><mo>·</mo><mn>16</mn><mo>+</mo><mn>3</mn><mo>=</mo><msub><mrow><mn>35</mn></mrow><mrow><mn>10</mn></mrow></msub></mrow></math></script></p>
</li>
<li>
<p class="noindent"><b>Dalla base 10 alla base <i>b</i></b></p>
<p class="noindent">Per determinare la rappresentazione, in base <i>b</i>, di un numero di cui conosciamo la rappresentazione nel sistema decimale, si utilizza il seguente algoritmo.</p>
<p class="noindent"><b>Algoritmo delle divisioni successive</b></p>
<ol class="olist">
<li><p class="noindent">Si divide il numero, rappresentato in base 10, per <i>b</i>. Il resto ottenuto è l’ultima cifra a destra del numero scritto in base <i>b</i>.</p></li>
</ol>
</li>
</ul>
</li>
</ul>
</div></div></div>
</div></div>
<div class="small-12 medium-4 columns"></div>
</div></div>
<div data-page-container="37" id="page-37" class="row chapters-content"><div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">37</div></div>
<div class="small-12 medium-11 columns"><div class="ch1"><div>
<ul class="blist">
<li>
<ul class="blist">
<li>
<ol class="olist">
<li><p class="noindent"><span class="pagebreak" epub:type="pagebreak" title="37" id="page37"></span>Si divide nuovamente per <i>b</i> il quoziente ottenuto. Il nuovo resto è la cifra, in base <i>b</i>, immediatamente a sinistra di quella ottenuta precedentemente.</p></li>
<li><p class="noindent">Si ripete l’operazione <span class="orangeb">b.</span> fino a ottenere un quoziente nullo.</p></li>
</ol>
<p class="noindent">L’ultimo resto è la prima cifra a sinistra del numero espresso nella base <i>b</i>.</p>
<p class="noindent">Il numero 35<sub>10</sub> in base 4 diventa</p>
<p class="img"><img src="images/pg55.jpg" alt="Image"></p>
</li>
</ul>
</li>
</ul>
<h3 class="sec_title">L’insieme dei numeri interi relativi</h3>
<ul class="blist">
<li>
<p class="noindent1"><b>Numero intero relativo</b>: è un numero naturale preceduto dal segno + o dal segno −.</p>
<p class="noindent">I numeri preceduti dal segno + sono <i>positivi</i>; quelli preceduti dal segno meno − sono <i>negativi</i>. Il segno + può essere omesso.</p>
</li>
<li><p class="noindent1"><b>Numeri concordi</b>: sono numeri che hanno lo stesso segno.</p></li>
<li><p class="noindent1"><b>Numeri discordi</b>: sono numeri che hanno segni diversi.</p></li>
<li>
<p class="noindent1"><b>Valore assoluto</b></p>
<p class="noindent">Il valore assoluto di <i>a</i> si indica con |<i>a</i>|.</p>
<p class="noindent"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><span style="display: inline-block; white-space: nowrap; padding: 1px 0px;"><span style="display: inline-block; position: relative; vertical-align: -5ex; width: 76.167ex; height: 11.167ex;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="margin-left: 0ex; margin-right: 0ex; position: absolute; left: 0px;" width="76.167ex" height="11.167ex" viewBox="0 -2666.7 32785.5 4833.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(167,0)"><g transform="translate(-11,0)"><g transform="translate(0,1791)"><use xlink:href="#MJMAIN-2219"></use><g transform="translate(505,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g 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</li>
<li>
<p class="noindent1"><b>Numeri opposti</b>: sono due numeri interi con lo stesso valore assoluto e segni diversi.</p>
<p class="noindent">L’opposto di un numero <i>a</i> si indica con −<i>a</i>.</p>
</li>
<li>
<p class="noindent1"><b>Ordinamento dei numeri interi</b></p>
<ul class="blist">
<li><p class="noindent1">Tra due numeri discordi, il numero negativo è minore del numero positivo.</p></li>
<li><p class="noindent1">Lo zero è maggiore di qualsiasi numero negativo e minore di qualsiasi numero positivo.</p></li>
<li><p class="noindent1">Tra due numeri positivi il minore è quello che ha il minore valore assoluto.</p></li>
<li><p class="noindent1">Tra due numeri negativi il minore è quello che ha il maggiore valore assoluto.</p></li>
</ul>
</li>
</ul>
<h3 class="sec_title">Le quattro operazioni aritmetiche con i numeri interi relativi</h3>
<ul class="blist">
<li>
<p class="noindent1"><b>Addizione</b></p>
<ul class="blist">
<li><p class="noindent1">La somma di due <i>numeri</i> interi relativi <i>concordi</i> è il numero che ha per segno lo stesso segno degli addendi e per valore assoluto la somma dei valori assoluti degli addendi.</p></li>
<li><p class="noindent1">La somma di due <i>numeri</i> interi relativi <i>discordi</i> è il numero che ha per segno il segno dell’addendo maggiore in valore assoluto e per valore assoluto la differenza tra il maggiore e il minore dei due valori assoluti.</p></li>
<li><p class="noindent1">La somma di due <i>numeri opposti</i> è zero.</p></li>
</ul>
</li>
<li>
<p class="noindent1"><b>Proprietà dell’addizione</b></p>
<ul class="blist">
<li><p class="noindent1">Proprietà commutativa:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="cyan"><i>a</i> + <i>b</i> = <i>b</i> + <i>a</i></span></p></li>
<li><p class="noindent1">Proprietà associativa:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="cyan"><i>a</i> + <i>b</i> + <i>c</i> = (<i>a</i> + <i>b</i>) + <i>c</i> = <i>a</i> + (<i>b</i> + <i>c</i>)</span></p></li>
<li><p class="noindent1">Lo zero è elemento neutro:<a id="ind144"></a><!--<?"elemento neutro",4,0,2>--> <span class="cyan"><i>a</i> + 0 = 0 + <i>a</i> = <i>a</i></span></p></li>
</ul>
</li>
</ul>
</div></div></div>
</div></div>
<div class="small-12 medium-4 columns"></div>
</div></div>
<div data-page-container="38" id="page-38" class="row chapters-content"><div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">38</div></div>
<div class="small-12 medium-11 columns"><div class="ch1"><div><ul class="blist">
<li>
<p class="noindent1"><span class="pagebreak" epub:type="pagebreak" title="38" id="page38"></span><b>Sottrazione</b></p>
<p class="noindent">La differenza di due numeri interi relativi è la somma del minuendo con l’opposto del sottraendo.</p>
<p class="noindent">Casi particolari: <span class="cyan"><i>a</i> − <i>a</i> = 0&nbsp;&nbsp;&nbsp;&nbsp;<i>a</i> − 0 = <i>a</i></span></p>
</li>
<li>
<p class="noindent1"><b>Proprietà invariantiva della sottrazione</b></p>
<p class="noindent">Sommando o sottraendo uno stesso numero sia al minuendo sia al sottraendo, la differenza non cambia:</p>
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</li>
<li>
<p class="noindent1"><b>Addizione algebrica</b></p>
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<p class="noindent">Davanti al primo termine il segno + può essere omesso, il segno − dev’essere sempre scritto.</p>
</li>
<li>
<p class="noindent1"><b>Addizione algebrica e proprietà commutativa</b></p>
<p class="noindent">In una somma algebrica si può cambiare l’ordine dei termini, seguendo queste regole:</p>
<ul class="blist">
<li><p class="noindent">ogni termine deve conservare il proprio segno;</p></li>
<li><p class="noindent">quando un termine viene portato al primo posto, se è positivo si può scrivere senza segno, se è negativo occorre scrivere davanti a esso il segno −;</p></li>
<li><p class="noindent">quando si sposta il termine che si trova al primo posto, se non è preceduto da un segno, esso dovrà essere scritto con il segno +.</p></li>
</ul>
</li>
<li>
<p class="noindent1"><b>Addizione algebrica ed eliminazione delle parentesi</b></p>
<p class="noindent">Per liberare una somma algebrica da una coppia di parentesi, se la prima parentesi è preceduta dal segno + si riscrivono i termini contenuti nella coppia di parentesi con il loro segno, se invece è preceduta dal segno − si riscrivono i termini contenuti nella coppia di parentesi con il segno cambiato:</p>
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</li>
<li>
<p class="noindent1"><b>Moltiplicazione</b></p>
<p class="noindent">Il prodotto di due numeri interi relativi è il numero che ha per valore assoluto il prodotto dei valori assoluti dei due fattori ed è positivo se i due fattori sono concordi, negativo se sono discordi.</p>
<p class="noindent">Il prodotto di tre o più numeri interi relativi è il numero che ha per valore assoluto il prodotto dei valori assoluti dei fattori ed è positivo se il numero dei fattori negativi è pari, negativo se il numero dei fattori negativi è dispari.</p>
</li>
<li>
<p class="noindent1"><b>Proprietà della moltiplicazione</b></p>
<ul class="blist">
<li><p class="noindent1">Proprietà commutativa: <span class="cyan"><i>a</i> · <i>b</i> = <i>b</i> · <i>a</i></span></p></li>
<li><p class="noindent1">Proprietà associativa: <span class="cyan"><i>a</i> · <i>b</i> · <i>c</i> = (<i>a</i> · <i>b</i>) · <i>c</i> = <i>a</i> · (<i>b</i> · <i>c</i>)</span></p></li>
<li><p class="noindent1">Proprietà distributiva rispetto all’addizione: <span class="cyan"><i>a</i> · (<i>b</i> + <i>c</i>) = <i>a</i> · <i>b</i> + <i>a</i> · <i>c</i></span></p></li>
<li><p class="noindent1">Proprietà distributiva rispetto alla sottrazione: <span class="cyan"><i>a</i> · (<i>b</i> − <i>c</i>) = <i>a</i> · <i>b</i> − <i>a</i> · <i>c</i></span></p></li>
<li><p class="noindent1">Raccoglimento: <span class="cyan"><i>a</i> · <i>b</i> + <i>a</i> · <i>c</i> + <i>a</i> · <i>d</i> = <i>a</i> · (<i>b</i> + <i>c</i> + <i>d</i>)  <i>a</i> · <i>b</i> − <i>a</i> · <i>c</i> = <i>a</i> · (<i>b</i> − <i>c</i>)</span></p></li>
<li><p class="noindent1">Elemento neutro: <span class="cyan"><i>a</i> · 1 = 1 · <i>a</i> = <i>a</i></span></p></li>
<li><p class="noindent1">Elemento annullatore: <span class="cyan"><i>a</i> · 0 = 0 · <i>a</i> = 0</span></p></li>
<li><p class="noindent1">Legge di annullamento del prodotto: se <i>a</i> · <i>b</i> = 0, allora <i>a</i> = 0 oppure <i>b</i> = 0 oppure <i>a</i> = <i>b</i> = 0.</p></li>
</ul>
</li>
<li>
<p class="noindent1"><b>Divisione</b></p>
<p class="noindent">Il quoziente di due numeri interi relativi, di cui il secondo diverso da 0, è il numero intero relativo, se esiste, che ha per valore assoluto il quoziente della divisione tra i valori assoluti del dividendo e del divisore, e che è positivo se i due numeri sono concordi, negativo se sono discordi. <i>Non è possibile la divisione per</i> 0, <i>a</i> : 0 non ha significato.</p>
<p class="noindent">Casi particolari: <span class="cyan"><i>a</i> : 1 = <i>a</i>  <i>a</i> : <i>a</i> = 1 <b>per</b> <i>a</i> ≠ 0  0 : <i>a</i> = 0 <b>per</b> <i>a</i> ≠ 0</span></p>
</li>
</ul></div></div></div>
</div></div>
<div class="small-12 medium-4 columns"></div>
</div></div>
<div data-page-container="39" id="page-39" class="row chapters-content"><div class="row">
<div class="small-12 medium-8 columns"><div class="row">
<div class="small-12 medium-1 columns"><div class="pagenumber">39</div></div>
<div class="small-12 medium-11 columns"><div class="ch1">
<div>
<ul class="blist">
<li>
<p class="noindent1"><span class="pagebreak" epub:type="pagebreak" title="39" id="page39"></span><b>Proprietà della divisione<a id="ind145"></a><!--<?"divisione|propriet&#x00E0; della",4,0,2>--></b></p>
<ul class="blist">
<li><p class="noindent1">Proprietà invariantiva: <span class="cyan"><i>a</i> : <i>b</i> = (<i>a</i> · <i>c</i>): (<i>b</i> · <i>c</i>)&nbsp;&nbsp;&nbsp;&nbsp;<i>a</i> : <i>b</i> = (<i>a</i> : <i>c</i>) : (<i>b</i> : <i>c</i>)</span></p></li>
<li><p class="noindent1">Proprietà distributiva rispetto all’addizione:&nbsp;&nbsp;<span class="cyan">(<i>a</i> + <i>b</i>) : <i>c</i> = <i>a</i> : <i>c</i> + <i>b</i> : <i>c</i></span></p></li>
<li><p class="noindent1">Proprietà distributiva rispetto alla sottrazione: <span class="cyan">(<i>a</i> − <i>b</i>) : <i>c</i> = <i>a</i> : <i>c</i> − <i>b</i> : <i>c</i></span></p></li>
</ul>
<p class="noindent">La proprietà distributiva vale solo se il simbolo di divisione è scritto <i>a destra</i> dell’addizione o della sottrazione.</p>
</li>
</ul>
<h3 class="sec_title">Potenza di un numero intero relativo</h3>
<ul class="blist">
<li>
<p class="noindent1"><b>Definizione di potenza</b></p>
<ul class="blist">
<li><p class="noindent1">La potenza di base <i>a</i> ed esponente <i>n</i>, che indichiamo con il simbolo <i>a<sup>n</sup></i>, è il prodotto di <i>n</i> fattori uguali ad <i>a</i>. Nell’insieme ℤ dei numeri interi relativi si considerano solo le potenze con esponente naturale.</p></li>
<li><p class="noindent1"><span class="cyan">1<sup><i>n</i></sup> = 1&nbsp;&nbsp;&nbsp;&nbsp;0<sup><i>n</i></sup> = 0 (<i>n</i> ≠ 0)&nbsp;&nbsp;&nbsp;&nbsp;<i>a</i><sup>1</sup> = <i>a</i>&nbsp;&nbsp;&nbsp;&nbsp;<i>a</i><sup>0</sup> = 1 (<i>a</i> ≠ 0)&nbsp;&nbsp;&nbsp;&nbsp;0<sup>0</sup> <b>non ha significato</b></span></p></li>
<li>
<p class="noindent1">Per calcolare la potenza di un numero intero relativo si calcola la potenza del valore assoluto della base e si determina il segno secondo il seguente schema:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><span style="display: inline-block; white-space: nowrap; padding: 1px 0px;"><span style="display: inline-block; position: relative; vertical-align: -5ex; width: 59.833ex; height: 11ex;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="margin-left: 0ex; margin-right: 0ex; position: absolute; left: 0px;" width="59.833ex" height="11ex" viewBox="0 -2632 25776.4 4764"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(167,0)"><g transform="translate(-11,0)"><g transform="translate(9091,1756)"><use xlink:href="#MJMAIN-62"></use><use x="561" y="0" xlink:href="#MJMAIN-61"></use><use x="1066" y="0" xlink:href="#MJMAIN-73"></use><use x="1465" y="0" xlink:href="#MJMAIN-65"></use><g transform="translate(1914,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(2523,0)"><use xlink:href="#MJMAIN-70"></use><use x="561" y="0" xlink:href="#MJMAIN-6F"></use><use x="1066" y="0" xlink:href="#MJMAIN-73"></use><use x="1465" y="0" xlink:href="#MJMAIN-69"></use><use x="1748" y="0" xlink:href="#MJMAIN-74"></use><use x="2142" y="0" xlink:href="#MJMAIN-69"></use><use x="2425" y="0" xlink:href="#MJMAIN-76"></use><use x="2958" y="0" xlink:href="#MJMAIN-61"></use></g></g><g transform="translate(1243,34)"><use xlink:href="#MJMAIN-62"></use><use x="561" y="0" xlink:href="#MJMAIN-61"></use><use x="1066" y="0" xlink:href="#MJMAIN-73"></use><use x="1465" y="0" xlink:href="#MJMAIN-65"></use><g transform="translate(1914,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g 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xlink:href="#MJMAIN-74"></use><use x="2414" y="0" xlink:href="#MJMAIN-69"></use><use x="2697" y="0" xlink:href="#MJMAIN-76"></use><use x="3230" y="0" xlink:href="#MJMAIN-61"></use></g></g></g></g></g></svg></span></span></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mtext>base</mtext><mtext> </mtext><mtext>positiva</mtext></mrow></mtd><mtd columnalign="left"><mo>→</mo></mtd><mtd columnalign="left"><mrow><mtext>potenza</mtext><mtext> </mtext><mtext>positiva</mtext></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mtext>base</mtext><mtext> </mtext><mtext>negativa,</mtext><mtext> </mtext><mtext>esponente</mtext><mtext> </mtext><mtext>pari</mtext></mrow></mtd><mtd columnalign="left"><mo>→</mo></mtd><mtd columnalign="left"><mrow><mtext>potenza</mtext><mtext> </mtext><mtext>positiva</mtext></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mtext>base</mtext><mtext> </mtext><mtext>negativa,</mtext><mtext> </mtext><mtext>esponente</mtext><mtext> </mtext><mtext>dispari</mtext></mrow></mtd><mtd columnalign="left"><mo>→</mo></mtd><mtd columnalign="left"><mrow><mtext>potenza</mtext><mtext> </mtext><mtext>negativa</mtext></mrow></mtd></mtr></mtable></mrow></math></script></p>
</li>
<li>
<p class="noindent1">Se non ci sono parentesi, la potenza ha la priorità sul segno − che la precede:</p>
<p class="math"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="56.833ex" height="3.333ex" viewBox="0 -978.9 24456 1423.4"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-2212"></use><g transform="translate(783,0)"><use xlink:href="#MJMAIN-37"></use><use transform="scale(0.707)" x="714" y="584" xlink:href="#MJMAIN-32"></use></g><use x="2022" y="0" xlink:href="#MJMAIN-3D"></use><use x="3083" y="0" xlink:href="#MJMAIN-2212"></use><use x="3866" y="0" xlink:href="#MJMAIN-28"></use><use x="4260" y="0" xlink:href="#MJMAIN-37"></use><g transform="translate(4765,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="5597" y="0" xlink:href="#MJMAIN-22C5"></use><use x="6103" y="0" xlink:href="#MJMAIN-37"></use><use x="6608" y="0" xlink:href="#MJMAIN-29"></use><use x="7279" y="0" xlink:href="#MJMAIN-3D"></use><use x="8340" y="0" xlink:href="#MJMAIN-2212"></use><g transform="translate(9123,0)"><use xlink:href="#MJMAIN-34"></use><use x="505" y="0" xlink:href="#MJMAIN-39"></use></g><g transform="translate(10133,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><g transform="translate(609,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g></g><g transform="translate(11353,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(11963,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMAIN-2212"></use><use x="1177" y="0" xlink:href="#MJMAIN-37"></use><use x="1682" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="2935" y="688" xlink:href="#MJMAIN-32"></use></g><use x="14774" y="0" xlink:href="#MJMAIN-3D"></use><use x="15835" y="0" xlink:href="#MJMAIN-28"></use><use x="16229" y="0" xlink:href="#MJMAIN-2212"></use><use x="17012" y="0" xlink:href="#MJMAIN-37"></use><use x="17517" y="0" xlink:href="#MJMAIN-29"></use><g transform="translate(17911,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="18743" y="0" xlink:href="#MJMAIN-22C5"></use><use x="19248" y="0" xlink:href="#MJMAIN-28"></use><use x="19642" y="0" xlink:href="#MJMAIN-2212"></use><use x="20425" y="0" xlink:href="#MJMAIN-37"></use><use x="20930" y="0" xlink:href="#MJMAIN-29"></use><use x="21602" y="0" xlink:href="#MJMAIN-3D"></use><use x="22663" y="0" xlink:href="#MJMAIN-2B"></use><g transform="translate(23446,0)"><use xlink:href="#MJMAIN-34"></use><use x="505" y="0" xlink:href="#MJMAIN-39"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>−</mo><msup><mn>7</mn><mn>2</mn></msup><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><mn>7</mn><mtext> </mtext><mo>⋅</mo><mn>7</mn><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>49</mn><mtext>  </mtext><mtext> </mtext><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>7</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><mn>7</mn><mo stretchy="false">)</mo><mtext> </mtext><mo>⋅</mo><mo stretchy="false">(</mo><mo>−</mo><mn>7</mn><mo stretchy="false">)</mo><mo>=</mo><mo>+</mo><mn>49</mn></mrow></math></script></p>
</li>
</ul>
</li>
<li>
<p class="noindent1"><b>Proprietà delle potenze<a id="ind146"></a><!--<?"potenze|propriet&#x00E0; delle",4,0,2>--></b></p>
<ul class="blist">
<li><p class="noindent1"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="15.333ex" height="2ex" viewBox="0 -796.1 6626 830.1"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-61"></use><use transform="scale(0.707)" x="755" y="513" xlink:href="#MJMATHI-6D"></use><use x="1480" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(1985,0)"><use xlink:href="#MJMATHI-61"></use><use transform="scale(0.707)" x="755" y="513" xlink:href="#MJMATHI-6E"></use></g><use x="3325" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(4386,0)"><use xlink:href="#MJMATHI-61"></use><g transform="translate(534,362)"><use transform="scale(0.707)" xlink:href="#MJMATHI-6D"></use><use transform="scale(0.707)" x="883" y="0" xlink:href="#MJMAIN-2B"></use><use transform="scale(0.707)" x="1666" y="0" xlink:href="#MJMATHI-6E"></use></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>·</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup></mrow></math></script></span></p></li>
<li><p class="noindent1"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="39.333ex" height="3ex" viewBox="0 -875 16958.4 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-61"></use><use transform="scale(0.707)" x="755" y="513" xlink:href="#MJMATHI-6D"></use><use x="1536" y="0" xlink:href="#MJMAIN-3A"></use><g transform="translate(2096,0)"><use xlink:href="#MJMATHI-61"></use><use transform="scale(0.707)" x="755" y="513" xlink:href="#MJMATHI-6E"></use></g><use x="3436" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(4497,0)"><use xlink:href="#MJMATHI-61"></use><g transform="translate(534,362)"><use transform="scale(0.707)" xlink:href="#MJMATHI-6D"></use><use transform="scale(0.707)" x="883" y="0" xlink:href="#MJMAIN-2212"></use><use transform="scale(0.707)" x="1666" y="0" xlink:href="#MJMATHI-6E"></use></g></g><g transform="translate(6737,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(7513,0)"><use xlink:href="#MJMAINB-63"></use><use x="516" y="0" xlink:href="#MJMAINB-6F"></use><use x="1096" y="0" xlink:href="#MJMAINB-6E"></use></g><g transform="translate(9420,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="10030" y="0" xlink:href="#MJMATHI-61"></use><use x="10842" y="0" xlink:href="#MJMAIN-2265"></use><use x="11902" y="0" xlink:href="#MJMAIN-30"></use><use x="12407" y="0" xlink:href="#MJMAIN-2C"></use><g transform="translate(12857,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="13467" y="0" xlink:href="#MJMATHI-6D"></use><g transform="translate(14350,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text><use x="609" y="0" xlink:href="#MJMAIN-3E"></use></g><g transform="translate(15743,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="16353" y="0" xlink:href="#MJMATHI-6E"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>a</mi><mi>m</mi></msup><mo>:</mo><msup><mi>a</mi><mi>n</mi></msup><mo>=</mo><msup><mi>a</mi><mrow><mi>m</mi><mo>−</mo><mi>n</mi></mrow></msup><mtext> </mtext><mi mathvariant="bold">con</mi><mtext> </mtext><mi>a</mi><mo>≥</mo><mn>0</mn><mo>,</mo><mtext> </mtext><mi>m</mi><mtext> &gt;</mtext><mtext> </mtext><mi>n</mi></mrow></math></script></span></p></li>
<li><p class="noindent1"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="13.5ex" height="2.5ex" viewBox="0 -773.9 5799 1047.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMAIN-28"></use><g transform="translate(394,0)"><use xlink:href="#MJMATHI-61"></use><use transform="scale(0.707)" x="755" y="513" xlink:href="#MJMATHI-6D"></use></g><use x="1652" y="0" xlink:href="#MJMAIN-29"></use><g transform="translate(2046,0)"><use transform="scale(0.707)" x="0" y="513" xlink:href="#MJMATHI-6E"></use></g><use x="2851" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(3912,0)"><use xlink:href="#MJMATHI-61"></use><g transform="translate(534,362)"><use transform="scale(0.707)" xlink:href="#MJMATHI-6D"></use><use transform="scale(0.707)" x="883" y="0" xlink:href="#MJMAIN-22C5"></use><use transform="scale(0.707)" x="1166" y="0" xlink:href="#MJMATHI-6E"></use></g></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msup><mo stretchy="false">)</mo><msup><mrow></mrow><mi>n</mi></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi><mo>·</mo><mi>n</mi></mrow></msup></mrow></math></script></span></p></li>
<li><p class="noindent1"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -0.667ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="16.5ex" height="2.5ex" viewBox="0 -773.9 7100.8 1047.9"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-61"></use><use transform="scale(0.707)" x="755" y="513" xlink:href="#MJMATHI-6E"></use><use x="1284" y="0" xlink:href="#MJMAIN-22C5"></use><g transform="translate(1789,0)"><use xlink:href="#MJMATHI-62"></use><use transform="scale(0.707)" x="613" y="609" xlink:href="#MJMATHI-6E"></use></g><use x="3028" y="0" xlink:href="#MJMAIN-3D"></use><use x="4089" y="0" xlink:href="#MJMAIN-28"></use><use x="4483" y="0" xlink:href="#MJMATHI-61"></use><use x="5239" y="0" xlink:href="#MJMAIN-22C5"></use><use x="5745" y="0" xlink:href="#MJMATHI-62"></use><use x="6179" y="0" xlink:href="#MJMAIN-29"></use><g transform="translate(6573,0)"><use transform="scale(0.707)" x="0" y="513" xlink:href="#MJMATHI-6E"></use></g></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>·</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo>·</mo><mi>b</mi><mo stretchy="false">)</mo><msup><mrow></mrow><mi>n</mi></msup></mrow></math></script></span></p></li>
<li><p class="noindent1"><span class="cyan"><span class="MathJax_Preview"><span style="font-size: 100%; display: inline-block;" class="MathJax_SVG" role="textbox" aria-readonly="true"><svg xmlns:xlink="http://www.w3.org/1999/xlink" style="vertical-align: -1.167ex; margin-left: 0ex; margin-right: 0ex; margin: 1px 0px;" width="29.167ex" height="3ex" viewBox="0 -875 12560.5 1319.5"><g stroke="black" fill="black" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#MJMATHI-61"></use><use transform="scale(0.707)" x="755" y="513" xlink:href="#MJMATHI-6E"></use><use x="1339" y="0" xlink:href="#MJMAIN-3A"></use><g transform="translate(1900,0)"><use xlink:href="#MJMATHI-62"></use><use transform="scale(0.707)" x="613" y="513" xlink:href="#MJMATHI-6E"></use></g><use x="3139" y="0" xlink:href="#MJMAIN-3D"></use><g transform="translate(4200,0)"><use xlink:href="#MJMAIN-28"></use><use x="394" y="0" xlink:href="#MJMATHI-61"></use><use x="1205" y="0" xlink:href="#MJMAIN-3A"></use><use x="1766" y="0" xlink:href="#MJMATHI-62"></use><use x="2200" y="0" xlink:href="#MJMAIN-29"></use><use transform="scale(0.707)" x="3669" y="688" xlink:href="#MJMATHI-6E"></use></g><g transform="translate(7323,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><g transform="translate(7933,0)"><use xlink:href="#MJMAINB-63"></use><use x="516" y="0" xlink:href="#MJMAINB-6F"></use><use x="1096" y="0" xlink:href="#MJMAINB-6E"></use></g><g transform="translate(9673,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" font-style="" font-weight="" stroke="none" style="font-family: monospace" transform="scale(71.759) matrix(1 0 0 -1 0 0)"> </text></g><use x="10282" y="0" xlink:href="#MJMATHI-62"></use><use x="10994" y="0" xlink:href="#MJMAIN-2260"></use><use x="12055" y="0" xlink:href="#MJMAIN-30"></use></g></svg></span></span><script type="math/mml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mi>a</mi><mi>n</mi></msup><mo>:</mo><msup><mi>b</mi><mi>n</mi></msup><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>:</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><mi>n</mi></msup><mtext> </mtext><mstyle mathvariant="bold" mathsize="normal"><mi>c</mi><mi>o</mi><mi>n</mi></mstyle><mtext> </mtext><mi>b</mi><mo>≠</mo><mn>0</mn></mrow></math></script></span></p></li>
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