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458 lines
17 KiB
HTML
458 lines
17 KiB
HTML
<!DOCTYPE html>
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<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
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<!-- Mirrored from www.mathsisfun.com/square-root.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:35:41 GMT -->
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<head>
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<meta http-equiv="content-type" content="text/html; charset=UTF-8">
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<!-- #BeginEditable "doctitle" -->
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<title>Squares and Square Roots</title>
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<script src="numbers/images/square-root.js" type="text/javascript"></script>
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<link rel="stylesheet" type="text/css" href="stylejs.css">
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<!-- #EndEditable -->
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<meta name="keywords" content="math, maths, mathematics, school, homework, education">
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<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
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<meta name="HandheldFriendly" content="true">
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gtag('js', new Date());
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gtag('config', 'UA-29771508-1');
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</script>
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</head>
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<body id="bodybg">
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<div id="stt"></div>
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<div id="adTop"></div>
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<header>
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<div id="hdr"></div>
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<div id="tran"></div>
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<div id="adHide"></div>
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<div id="cookOK"></div>
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</header>
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<div class="mid">
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<nav>
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<div id="menuWide" class="menu"></div>
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<div id="logo"><a href="index.html"><img src="images/style/logo.svg" alt="Math is Fun"></a></div>
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<div id="search" role="search"></div>
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<div id="linkto"></div>
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<div id="menuSlim" class="menu"></div>
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<div id="menuTiny" class="menu"></div>
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</nav>
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<div id="extra"></div>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Squares and Square Roots</h1>
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<p class="center">First learn about Squares, then Square Roots are easy.</p>
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<h2>How to Square A Number</h2>
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<div class="center">
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<p><span class="larger">To square a number: <b>multiply it by itself</b>.</span></p>
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</div>
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<div class="example">
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<h3>Example: What is 3 squared?</h3>
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<table align="center" cellpadding="6" border="0">
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<tbody>
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<tr>
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<td class="larger">3 Squared</td>
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<td class="larger">=</td>
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<td height="180" valign="top"><img src="algebra/images/3-squared-box.svg" alt="3 by 3 squares = 9 squares"></td>
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<td class="larger">= 3 × 3 = <b>9</b></td>
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</tr>
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</tbody></table>
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</div>
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<p> </p>
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<p>"Squared" is often written as a little 2 like this:</p>
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<p class="center"><img src="algebra/images/4-squared.svg" alt="4 squared is 16"><br>
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This says <i><b>"4 Squared equals 16"</b></i><span class="larger"><br>
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</span>(the little 2 says
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the number appears twice in multiplying)</p>
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<h2 align="left">Squares From <span class="larger">0<sup>2</sup></span> to <span class="larger">6<sup>2</sup></span></h2>
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<table class="larger" align="center" cellspacing="4" cellpadding="4" border="0">
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<tbody>
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<tr>
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<td class="larger">0 Squared</td>
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<td class="larger">=</td>
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<td class="larger">0<sup>2</sup></td>
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<td class="larger">=</td>
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<td class="larger">0 × 0</td>
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<td class="larger">=</td>
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<td class="larger" align="center"><b>0</b></td>
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</tr>
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<tr>
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<td class="larger">1 Squared</td>
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<td class="larger">=</td>
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<td class="larger">1<sup>2</sup></td>
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<td class="larger">=</td>
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<td class="larger">1 × 1</td>
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<td class="larger">=</td>
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<td class="larger" align="center"><b>1</b></td>
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</tr>
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<tr>
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<td class="larger">2 Squared</td>
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<td class="larger">=</td>
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<td class="larger">2<sup>2</sup></td>
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<td class="larger">=</td>
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<td class="larger">2 × 2</td>
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<td class="larger">=</td>
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<td class="larger" align="center"><b>4</b></td>
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</tr>
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<tr>
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<td class="larger">3 Squared</td>
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<td class="larger">=</td>
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<td class="larger">3<sup>2</sup></td>
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<td class="larger">=</td>
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<td class="larger">3 × 3</td>
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<td class="larger">=</td>
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<td class="larger" align="center"><b>9</b></td>
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</tr>
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<tr>
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<td class="larger">4 Squared</td>
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<td class="larger">=</td>
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<td class="larger">4<sup>2</sup></td>
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<td class="larger">=</td>
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<td class="larger">4 × 4</td>
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<td class="larger">=</td>
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<td class="larger" align="center"><b>16</b></td>
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</tr>
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<tr>
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<td class="larger">5 Squared</td>
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<td class="larger">=</td>
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<td class="larger">5<sup>2</sup></td>
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<td class="larger">=</td>
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<td class="larger">5 × 5</td>
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<td class="larger">=</td>
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<td class="larger" align="center"><b>25</b></td>
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</tr>
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<tr>
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<td class="larger">6 Squared</td>
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<td class="larger">=</td>
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<td class="larger">6<sup>2</sup></td>
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<td class="larger">=</td>
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<td class="larger">6 × 6</td>
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<td class="larger">=</td>
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<td class="larger" align="center"><b>36</b></td>
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</tr>
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</tbody></table>
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<p> </p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td style="text-align:right;">The squares are also<br>
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on the <a href="tables.html">Multiplication Table</a>:</td>
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<td style="text-align:right;"> </td>
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<td><img src="images/squares-times-table.gif" alt="squares in times table" height="122" width="151"></td>
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</tr>
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</tbody></table>
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<h2>Negative Numbers</h2>
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<p>We can also square <b>negative numbers</b>.</p>
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<div class="example">
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<h3>Example: What happens when we square (−5) ?</h3>
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<p>Answer:</p>
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<p class="center larger">(−5) × (−5) = <b>25</b></p>
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<p class="center">(because a <a href="multiplying-negatives.html">negative times a negative gives a positive</a>)</p>
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</div>
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<p>That was interesting!</p>
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<p class="center larger">When we square a <b>negative</b> number we get a <b>positive</b> result.</p>
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<p>Just the same as squaring a positive number:</p>
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<p class="center"><img src="algebra/images/square-root-pos-neg.svg" alt="5x5 = -5x-5"></p>
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<p class="center">(For more detail read <a href="algebra/square-root.html">Squares and Square Roots in Algebra</a>)</p>
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<h2>Square Roots</h2>
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<p class="larger">A <b>square root</b> goes the other way:</p>
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<p class="center"><img src="algebra/images/square-vs-square-root.svg" alt="square root of 9 is 3"></p>
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<p class="center"><span class="larger">3 squared is 9, so a <b>square root
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of 9 is 3</b></span></p>
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<p class="center larger"> </p>
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<p>A square root of a number is ...</p>
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<div align="right"><span class="larger">... a value that can be<b> multiplied by itself</b> to give the original number.</span> </div>
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<p class="larger">A square root of <b>9</b> is ...</p>
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<div class="center"><span class="larger">... <b>3</b>, because <b>when 3 is multiplied by itself</b> we get <b>9</b>.</span> </div>
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<p>It is like asking:</p>
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<div class="center80">
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<p class="center large">What can we multiply by itself to get this?</p>
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</div>
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<table align="center" width="80%" border="0">
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<tbody>
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<tr>
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<td><img src="algebra/images/tree-root.jpg" alt="tree root" height="118" width="67"></td>
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<td>
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<p><b>To help you remember</b> think of the root of a tree:</p>
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<p class="center"><i>"I know the tree</i><i>, but what root made it?</i>"</p>
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<p>In this case the tree is "9", and the root is "3".</p></td>
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</tr>
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</tbody></table>
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<p>Here are some more squares and square roots:</p>
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<table class="larger" style="width:340px;" align="center" border="1">
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<tbody>
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<tr style="text-align:center;">
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<td colspan="3"><img src="algebra/images/powers-square-arrows.svg" alt="square vs square root"></td>
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</tr>
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<tr style="text-align:center;">
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<td class="large" width="33%">4</td>
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<td class="larger" width="33%"> </td>
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<td class="larger" width="33%">16</td>
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</tr>
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<tr style="text-align:center;">
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<td class="large" width="33%">5</td>
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<td class="larger" width="33%"> </td>
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<td class="larger" width="33%">25</td>
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</tr>
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<tr style="text-align:center;">
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<td class="large" width="33%">
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<div class="center">6</div></td>
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<td class="larger" width="33%"> </td>
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<td class="larger" width="33%">36</td>
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</tr>
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<tr style="text-align:center;">
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<td class="large" width="33%">
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<div class="center">7</div></td>
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<td class="larger" width="33%"> </td>
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<td class="larger" width="33%">49</td>
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</tr>
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</tbody></table>
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<h2>Decimal Numbers</h2>
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<p class="larger">It also works for decimal numbers.</p>
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<p>Try the sliders below (note: '...' means the decimals continue on forever):</p>
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<script>squarerootMain();</script>
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<p>Using the sliders:</p>
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<ul>
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<li>What is the square root of <b>8</b>?</li>
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<li>What is the square root of <b>9</b>?</li>
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<li>What is the square root of <b>10</b>?</li>
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<li>What is <b>1</b> squared?</li>
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<li>What is <b>1.1</b> squared?</li>
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<li>What is <b>2.6</b> squared?</li>
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</ul>
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<h2>Negatives</h2>
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<p>We discovered earlier that we can square negative numbers:</p>
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<div class="example">
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<h3>Example: (−3) squared</h3>
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<p class="center larger">(−3) × (−3) = <b>9</b></p>
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</div>
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<p>And of course 3 × 3 = <b>9</b> also.</p>
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<p class="larger center">So the square root of 9 could be <b>−3</b> or <b>+3</b></p>
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<div class="example">
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<h3>Example: What are the square roots of 25?</h3>
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<p class="center larger">(−5) × (−5) = 25</p>
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<p class="center larger">5 × 5 = 25</p>
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<p>So the square roots of 25 are <b>−5</b> and <b>+5</b></p>
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</div>
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<h2>The Square Root Symbol</h2>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td style="text-align:center;"><img src="images/square-root-symbol.gif" alt="radical symbol" height="56" width="31"></td>
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<td> </td>
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<td>This is the special symbol that means "square root",
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it is sort of like a tick,<br>
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and actually started hundreds of years
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ago as a dot with a flick upwards.<br>
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<br>
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It is called the <i><b>radical</b></i>, and always makes mathematics look important!</td>
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</tr>
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</tbody></table>
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<p>We use it like this:</p>
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<p class="center"><img src="images/square-root-9.gif" alt="square root of 9 is 3" align="absmiddle" height="33" width="92"><br>
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and we say <i><b>"square root of 9 equals 3"</b></i></p>
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<div class="example">
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<h3>Example: What is <span class="larger">√</span>25?</h3>
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<p>25 = 5 × 5, in other words when we multiply
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5 by itself (5 × 5) we get 25</p>
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<p><b>So the answer is:</b></p>
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<p class="center larger">√25 = 5</p>
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</div>
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<p>But wait a minute! Can't the square root <b>also be −5</b>? Because (−5) × (−5) = <b>25</b> too.</p>
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<ul>
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<li>Well the <b>square root of 25</b> could be −5 or +5.</li>
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<li>But when we use the <b>radical symbol <span class="larger">√</span></b> we only give the <b>positive (or zero) result</b>.</li>
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</ul>
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<div class="example">
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<h3>Example: What is √36 ?</h3>
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<p>Answer: 6 × 6 = 36, so <b>√36 = 6</b></p>
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</div>
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<h2>Perfect Squares</h2>
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<p>The Perfect Squares (also called "Square Numbers") are the squares of the <a href="whole-numbers.html">integers</a>:</p>
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<div class="simple">
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<table align="center">
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<tbody>
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<tr>
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<td> </td>
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<td style="text-align:right;"><b>Perfect<br>
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Squares</b></td>
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</tr>
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<tr>
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<td>0</td>
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<td style="text-align:right;"><b>0</b></td></tr>
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<tr>
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<td>1</td>
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<td style="text-align:right;"><b>1</b></td></tr>
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<tr>
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<td>2</td>
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<td style="text-align:right;"><b>4</b></td></tr>
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<tr>
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<td>3</td>
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<td style="text-align:right;"><b>9</b></td></tr>
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<tr>
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<td>4</td>
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<td style="text-align:right;"><b>16</b></td></tr>
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<tr>
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<td>5</td>
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<td style="text-align:right;"><b>25</b></td></tr>
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<tr>
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<td>6</td>
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<td style="text-align:right;"><b>36</b></td></tr>
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<tr>
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<td>7</td>
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<td style="text-align:right;"><b>49</b></td></tr>
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<tr>
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<td>8</td>
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<td style="text-align:right;"><b>64</b></td></tr>
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<tr>
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<td>9</td>
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||
<td style="text-align:right;"><b>81</b></td></tr>
|
||
<tr>
|
||
<td>10</td>
|
||
<td style="text-align:right;"><b>100</b></td></tr>
|
||
<tr>
|
||
<td>11</td>
|
||
<td style="text-align:right;"><b>121</b></td></tr>
|
||
<tr>
|
||
<td>12</td>
|
||
<td style="text-align:right;"><b>144</b></td></tr>
|
||
<tr>
|
||
<td>13</td>
|
||
<td style="text-align:right;"><b>169</b></td></tr>
|
||
<tr>
|
||
<td>14</td>
|
||
<td style="text-align:right;"><b>196</b></td></tr>
|
||
<tr>
|
||
<td>15</td>
|
||
<td style="text-align:right;"><b>225</b></td></tr>
|
||
<tr>
|
||
<td> </td>
|
||
<td style="text-align:right;">etc...</td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p>Try to remember them up to 12.</p>
|
||
<h2>Calculating Square Roots</h2>
|
||
<p>It is easy to work out the square root of a perfect square, but it
|
||
is <b>really hard</b> to work out other square roots.</p>
|
||
<div class="example">
|
||
<h3>Example: what is √10?</h3>
|
||
<p>Well, 3 × 3 = 9 and 4 × 4 = 16, so we can guess the answer is between 3 and 4.</p>
|
||
<ul>
|
||
<li>Let's try 3.5: <i>3.5 × 3.5 = 12.25</i></li>
|
||
<li>Let's try 3.2: <i>3.2 × 3.2 = 10.24</i></li>
|
||
<li>Let's try 3.1: <i>3.1 × 3.1 = 9.61</i></li>
|
||
<li>...</li>
|
||
</ul>
|
||
<p>Getting closer to 10, but it will take a long time to get a good answer!</p>
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td>
|
||
<p style="float:left; margin: 0 10px 5px 0;"><img src="images/style/calc.gif" alt="calculator" height="48" width="48"></p>
|
||
<p>At this point, I get out my calculator and it says:</p>
|
||
<div class="center">
|
||
<p><i>3.1622776601683793319988935444327</i></p>
|
||
</div>
|
||
<p>But the digits just go on and on, without any pattern.</p>
|
||
<p>So even
|
||
the calculator's answer is <b>only an <i>approximation</i> !</b></p></td>
|
||
</tr>
|
||
</tbody></table>
|
||
|
||
<p><i>Note: numbers like that are called <a href="irrational-numbers.html">Irrational Numbers</a>, if you want to know more.</i></p>
|
||
</div>
|
||
<h2>The Easiest Way to Calculate a Square Root</h2>
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td><img src="images/calculator-square-root.gif" alt="square root button" height="58" width="57"></td>
|
||
<td> </td>
|
||
<td class="larger">Use your calculator's square root button!</td>
|
||
<td class="larger"> </td>
|
||
<td class="larger"><span style="float:left; margin: 0 10px 5px 0;"><img src="numbers/images/square-root-button.jpg" alt="square root button" height="69" width="124"></span></td>
|
||
</tr>
|
||
</tbody></table>
|
||
|
||
<p>And also use your common sense to make sure you have the right answer.</p>
|
||
<h2>A Fun Way to Calculate a Square Root</h2>
|
||
<p>There is a fun method for calculating a square root that gets more and more accurate each time around:</p>
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td> </td>
|
||
<td>a) start with a <font color="blue">guess</font> (let's guess 4 is the square root of 10)</td>
|
||
</tr>
|
||
<tr>
|
||
<td><img src="images/style/rotate.gif" alt="around" height="94" width="86"></td>
|
||
<td>b) divide by the <font color="blue">guess</font> (10/4 = 2.5)<br>
|
||
c) add that to the <font color="blue">guess</font> (4 + 2.5 = 6.5)<br>
|
||
d) then divide <i>that</i> result by 2, in other words halve it. (6.5/2 = 3.25)<br>
|
||
e) now, set that as the <b><font color="blue">new guess</font></b>, and start at b) again</td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p> </p>
|
||
<ul>
|
||
<li>Our first attempt got us from 4 to <b>3.25</b></li>
|
||
<li>Going again (<i>b to e</i>) gets us: <b>3.163</b></li>
|
||
<li>Going again (<i>b to e</i>) gets us: <b>3.1623</b></li>
|
||
</ul>
|
||
<p>And so, after 3 times around the answer is 3.1623, which is pretty good, because:</p>
|
||
<p class="center"><b> 3.1623 x 3.1623 = 10.00014</b></p>
|
||
<p>Now ... why don't <b>you</b> try calculating the square root of 2 this way?</p>
|
||
<h3>How to Guess</h3>
|
||
<p>What if we have to guess the square root for a difficult number such as "82,163" ... ?</p>
|
||
<p>In that case we could think "82,163" has 5 digits, so the square root might have 3 digits (100x100=10,000), and the square root of 8 (the first digit) is about 3 (3x3=9), so 300 is a good start.</p>
|
||
<div class="fun">
|
||
<h3>Square Root Day</h3>
|
||
<p>The 4th of April 2016 is a Square Root Day, because the date looks like <b>4/4/16</b></p>
|
||
<p>The next after that is the 5th of May 2025 (5/5/25)</p>
|
||
</div>
|
||
<p> </p>
|
||
<div class="questions">309,310,315, 1082, 1083, 2040, 3156, 2041, 2042, 3154</div>
|
||
|
||
<div class="related">
|
||
<a href="algebra/square-root.html">Squares and Square Roots in Algebra</a>
|
||
<a href="irrational-numbers.html">Irrational Numbers</a>
|
||
<a href="surds.html">Surds</a>
|
||
<a href="scientific-calculator.html">Scientific Calculator</a>
|
||
<a href="algebra/index.html">Algebra Index</a>
|
||
</div>
|
||
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