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<main class="main"><div id="content" class="pretext-content"><section xmlns:svg="http://www.w3.org/2000/svg" class="section" id="section-44"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">7.1</span> <span class="title">Named properties and special subsets of real numbers</span>
</h2>
<section class="subsection" id="subsection-96"><h3 class="heading hide-type">
<span class="type">Subsection</span> <span class="codenumber">7.1.1</span> <span class="title">Special subsets of the real numbers</span>
</h3>
<p id="p-1285">For our purposes, the <dfn class="terminology">real numbers</dfn> consists of all possible decimals. These include whole numbers, both positive and negative, fractions, simple decimals like \(1.25\text{,}\) repeating decimals like \(1.333\ldots=\frac43\) and irrational numbers like \(\sqrt2\) and \(\pi\text{.}\)</p>
<p id="p-1286">We have some special notation that is used with the real numbers:</p>
<ul class="disc">
<li id="li-421"><p id="p-1287">\(\N\text{,}\) the <dfn class="terminology">natural numbers</dfn>, is the set of all nonnegative integers: \(\{0, 1, 2, 3,\ldots\}\text{.}\)</p></li>
<li id="li-422"><p id="p-1288">\(\Z\text{,}\) the <dfn class="terminology">integers</dfn>, consists of \(\{0, \pm1, \pm2, \pm3, \ldots\}\text{.}\)</p></li>
<li id="li-423"><p id="p-1289">\(\Q\text{,}\) the <dfn class="terminology">rationals</dfn>, is the set of all fractions: \(\frac mn\) where \(m\) and \(n\) are integers with \(n\not=0\text{.}\)</p></li>
<li id="li-424"><p id="p-1290">\(\R\) is the set of <em class="emphasis">all</em> real numbers, that is, ordinary decimals.</p></li>
</ul></section><section class="subsection" id="RealNumberProperties"><h3 class="heading hide-type">
<span class="type">Subsection</span> <span class="codenumber">7.1.2</span> <span class="title">Named properties of real numbers</span>
</h3>
<figure class="table table-like" id="table-10"><figcaption><span class="type">Table</span><span class="space"> </span><span class="codenumber">7.1.1<span class="period">.</span></span><span class="space"> </span>Properties of real numbers</figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr>
<td class="l m b0 r0 l0 t0 lines"></td>
<td class="l m b0 r0 l0 t0 lines"><em class="emphasis">Additive properties</em></td>
<td class="l m b0 r0 l0 t0 lines"><em class="emphasis">Multiplicative properties</em></td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines">Closure:</td>
<td class="l m b0 r0 l0 t0 lines">If \(x\) and \(y\) are real then so is \(x+y\)</td>
<td class="l m b0 r0 l0 t0 lines">If \(x\) and \(y\) are real then so is \(xy\)</td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines">Associativity:</td>
<td class="l m b0 r0 l0 t0 lines">\(x+(y+z)=(x+y)+z\)</td>
<td class="l m b0 r0 l0 t0 lines">\((xy)z=x(yz)\)</td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines">Identity:</td>
<td class="l m b0 r0 l0 t0 lines">\(x+0=x\)</td>
<td class="l m b0 r0 l0 t0 lines">\(1x=x\)</td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines">Inverse:</td>
<td class="l m b0 r0 l0 t0 lines">For any \(x\) there is a \(-x\) so that \(x+(-x)=0\)</td>
<td class="l m b0 r0 l0 t0 lines">For any \(x\not=0\) there is a number \(x^{-1}\) so that \(xx^{-1}=1\)</td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines">Commutativity:</td>
<td class="l m b0 r0 l0 t0 lines">\(x+y=y+x\)</td>
<td class="l m b0 r0 l0 t0 lines">\(xy=yx\)</td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines">\(\strut\)</td>
<td class="l m b0 r0 l0 t0 lines"></td>
<td class="l m b0 r0 l0 t0 lines"></td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines">Distributive:</td>
<td class="c m b0 r0 l0 t0 lines" colspan="2">\(x(y+z)=xy+xz\) and \((x+y)z=xz+yz\)</td>
</tr>
</table></div></figure></section></section></div></main>
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