e39465ad2f
new file: Files/flashplayer_32_sa.exe
new file: favicon.ico
new file: globe.gif
new file: imgs/download.png
new file: imgs/zuck.jpg
new file: index.html
new file: other.ico
new file: script.js
new file: site.webmanifest
new file: sitemap.html
new file: styles/backround.css
new file: styles/border.css
new file: styles/fonts/Titillium_Web/OFL.txt
new file: styles/fonts/Titillium_Web/TitilliumWeb-Black.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Bold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-BoldItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLight.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Italic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Light.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-LightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Regular.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBoldItalic.ttf
new file: styles/fonts/webfontkit-20221027-163353/generator_config.txt
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-demo.html
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff2
new file: styles/fonts/webfontkit-20221027-165950/generator_config.txt
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-demo.html
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff2
new file: styles/style.css
new file: tools/2048/.gitignore
new file: tools/2048/.jshintrc
new file: tools/2048/CONTRIBUTING.md
new file: tools/2048/LICENSE.txt
new file: tools/2048/README.md
new file: tools/2048/Rakefile
new file: tools/2048/favicon.ico
new file: tools/2048/index.html
new file: tools/2048/js/animframe_polyfill.js
new file: tools/2048/js/application.js
new file: tools/2048/js/bind_polyfill.js
new file: tools/2048/js/classlist_polyfill.js
new file: tools/2048/js/game_manager.js
new file: tools/2048/js/grid.js
new file: tools/2048/js/html_actuator.js
new file: tools/2048/js/keyboard_input_manager.js
new file: tools/2048/js/local_storage_manager.js
new file: tools/2048/js/tile.js
new file: tools/2048/meta/apple-touch-icon.png
new file: tools/webretro/cores/neocd_libretro.js
new file: tools/webretro/cores/neocd_libretro.wasm
new file: tools/webretro/cores/nestopia_libretro.js
new file: tools/webretro/cores/nestopia_libretro.wasm
new file: tools/webretro/cores/o2em_libretro.js
new file: tools/webretro/cores/o2em_libretro.wasm
new file: tools/webretro/cores/opera_libretro.js
new file: tools/webretro/cores/opera_libretro.wasm
366 lines
13 KiB
HTML
366 lines
13 KiB
HTML
<!DOCTYPE html>
|
||
<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
|
||
|
||
<!-- Mirrored from www.mathsisfun.com/numbers/digital-root.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 01:03:58 GMT -->
|
||
<head>
|
||
<meta http-equiv="content-type" content="text/html; charset=UTF-8">
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<!-- #BeginEditable "doctitle" -->
|
||
<title>Digital Root</title>
|
||
|
||
|
||
<meta name="Description" content="Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.">
|
||
<!-- #EndEditable -->
|
||
<meta name="keywords" content="math, maths, mathematics, school, homework, education">
|
||
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
|
||
<meta name="HandheldFriendly" content="true">
|
||
<meta name="referrer" content="always">
|
||
<link rel="preload" href="../images/style/font-champ-bold.ttf" as="font" type="font/ttf" crossorigin="">
|
||
<link rel="preload" href="../style4.css" as="style">
|
||
<link rel="preload" href="../main4.js" as="script">
|
||
<link rel="stylesheet" href="../style4.css">
|
||
<script src="../main4.js" defer="defer"></script>
|
||
<!-- Global site tag (gtag.js) - Google Analytics -->
|
||
<script async="" src="https://www.googletagmanager.com/gtag/js?id=UA-29771508-1"></script>
|
||
<script>
|
||
window.dataLayer = window.dataLayer || [];
|
||
function gtag(){dataLayer.push(arguments);}
|
||
gtag('js', new Date());
|
||
gtag('config', 'UA-29771508-1');
|
||
</script>
|
||
</head>
|
||
|
||
<body id="bodybg">
|
||
|
||
<div id="stt"></div>
|
||
<div id="adTop"></div>
|
||
<header>
|
||
<div id="hdr"></div>
|
||
<div id="tran"></div>
|
||
<div id="adHide"></div>
|
||
<div id="cookOK"></div>
|
||
</header>
|
||
|
||
<div class="mid">
|
||
|
||
<nav>
|
||
<div id="menuWide" class="menu"></div>
|
||
<div id="logo"><a href="../index.html"><img src="../images/style/logo.svg" alt="Math is Fun"></a></div>
|
||
|
||
<div id="search" role="search"></div>
|
||
<div id="linkto"></div>
|
||
|
||
<div id="menuSlim" class="menu"></div>
|
||
<div id="menuTiny" class="menu"></div>
|
||
</nav>
|
||
|
||
<div id="extra"></div>
|
||
|
||
<article id="content" role="main">
|
||
|
||
<!-- #BeginEditable "Body" -->
|
||
|
||
|
||
<h1 class="center">Digital Root</h1>
|
||
<p class="center">
|
||
|
||
<img src="images/digital-root.svg" alt="digital root of 314 = 3+1+4=8">
|
||
</p><p>The Digital Root is the sum of the individual digits of a number, repeating this process until we get a one-digit result.
|
||
|
||
</p>
|
||
<div class="example">
|
||
|
||
<h3>Example: 25</h3>
|
||
<p>Sum the digits of 25: <b>2 + 5 = 7</b></p>
|
||
<p>Answer: the digital root of 25 is <b>7</b></p>
|
||
</div>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: 3456</h3>
|
||
<p>Sum the digits of 3456: <b>3 + 4 + 5 + 6 = 18</b></p>
|
||
<p>But 18 has two digits, not one! So let's repeat the process.</p>
|
||
<p>Sum the digits of 18: <b>1 + 8 = 9</b></p>
|
||
<p>Answer: the Digital Root of 3456 is <b>9</b></p> </div>
|
||
|
||
|
||
<p>We always end up with a number between 0 and 9.</p>
|
||
<h2>Useful</h2>
|
||
|
||
<p>The digital root is useful as it helps us check the accuracy of
|
||
our arithmetic, and can also check divisibility by 3 and 9.</p>
|
||
<h3>Check Addition</h3>
|
||
<p>Try adding the digital roots, their sum should also be correct:</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: 1234 plus 411</h3>
|
||
<table style="text-align:right; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td style="width:80px;">Sum</td>
|
||
<td style="width:240px;">Digital Root</td>
|
||
<td style="width:20px;"> </td>
|
||
<td style="width:50px;"><br>
|
||
</td> </tr>
|
||
<tr>
|
||
<td class="large">1234</td>
|
||
<td>1+2+3+4=10 → 1+0 = 1</td>
|
||
<td><br></td>
|
||
<td class="large">1</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="large">+ 411</td>
|
||
<td>4+1+1 = 6</td>
|
||
<td><br></td>
|
||
<td class="large">+ 6</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="border-top: solid 2px;" class="large">1645</td>
|
||
<td>1+6+4+5=16 → 1+6 = 7</td>
|
||
<td><br></td>
|
||
<td class="large" style="border-top: solid 2px;">7</td>
|
||
</tr>
|
||
</tbody>
|
||
</table><p>After doing the sum 1234+411=1645, check to see that the sum of the digital roots works out fine too. If it doesn't you must have made a mistake somewhere!</p></div>
|
||
<p>This also works when adding long lists of numbers.</p>
|
||
|
||
<h3>Check Subtraction</h3>
|
||
<p>Same idea but using subtraction:</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: 1234 minus 411</h3>
|
||
<table style="text-align:right; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td style="width:80px;">Sum</td>
|
||
<td style="width:240px;">Digital Root</td>
|
||
<td style="width:20px;"><br></td>
|
||
<td style="width:50px;"><br>
|
||
</td> </tr>
|
||
<tr>
|
||
<td class="large">1234</td>
|
||
<td>1+2+3+4=10, 1+0 = 1</td>
|
||
<td><br></td>
|
||
<td class="large">1</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="large">− 411</td>
|
||
<td>4+1+1 = 6</td>
|
||
<td><br></td>
|
||
<td class="large">− 6</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="border-top: solid 2px;" class="large">823</td>
|
||
<td>8+2+3=13, 1+3 = 4</td>
|
||
<td><br></td>
|
||
<td style="border-top: solid 2px;" class="large">−5</td>
|
||
</tr>
|
||
</tbody>
|
||
</table><p>Hang on ... minus 5? OK, when we get a negative digital root we can <b>add 9</b></p>
|
||
<p>So −5 + 9 = 4, and it checks nicely.</p></div>
|
||
<p>Why did we add 9? Well the reason all this works is that <b>9 is the key number</b> because it is one less than 10 (the basis of our decimal number system).</p>
|
||
|
||
<h3>Casting Out 9s</h3>
|
||
<p>In fact the method we are using is sometimes called "casting out 9s": we are, in effect, casting out (getting rid of) all <b>whole multiples of 9</b>.</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: The digital root of 123</h3>
|
||
<p>123 is also 1×100 + 2×10 + 3</p>
|
||
<p>Which is also 1×(99+1) + 2×(9+1) + 3×1</p>
|
||
<p>Which can be rearranged into <b>1×99 + 2×9</b> + 1×1 + 2×1 + 3×1</p>
|
||
<p>Ignoring all the multiples of 9 we get 1×1 + 2×1 + 3×1 = 1+2+3 = 6</p>
|
||
</div>
|
||
<h3>A Circle Explains It</h3>
|
||
<p>It is like we don't care how many times we go around, just <b>where we end up</b>.</p>
|
||
<div class="example">
|
||
|
||
<h3>Example: The digital root of 16</h3>
|
||
<p>Here is a circle with 9 points (0,1,2,3,4,5,6,7,8):</p>
|
||
<p class="center"><img src="images/mod9.svg" alt="mod 9" height="187" width="181"></p>
|
||
<p>16 goes once around, then ends up at 7:</p>
|
||
<p class="center"><img src="images/mod9-16.svg" alt="16 mod 9" height="187" width="181"></p>
|
||
<p>Or this way: 10 (which ends at 1) followed by 6, or just "1 + 6 = 7"</p>
|
||
<p class="center"><img src="images/mod9-10-6.svg" alt="16 mod 9" height="187" width="181"></p>
|
||
</div>
|
||
<p>Notice:</p>
|
||
<ul>
|
||
<li><b>10 </b>goes once around and ends at <b>1</b></li>
|
||
<li><b>20 </b>goes twice around and ends at <b>2</b></li>
|
||
<li><b>30 </b>goes three times around and ends at <b>3</b>, etc</li></ul>
|
||
<p>Likewise:</p>
|
||
<ul>
|
||
<li><b>100 </b>goes 11 times around and ends at <b>1</b></li>
|
||
<li><b>200 </b>ends at <b>2</b></li>
|
||
<li><b>300 </b>ends at <b>3</b></li></ul>
|
||
<p>So we can just add up the digits to find where we end up!</p>
|
||
<p>Play with it here:</p>
|
||
<div class="script" style="height: 360px;">
|
||
images/mod-anim.js
|
||
</div>
|
||
<p><br></p>
|
||
|
||
<h3>Going Backwards</h3>
|
||
<p>Remember earlier when we had "1−6 = −5" and we added 9 to get 4 ?</p>
|
||
<p>Well the circle explain how "−5" is the same as "4" :</p>
|
||
<p class="center"><img src="images/mod9-1-6.svg" alt="-5 mod 9 = 4" height="187" width="181"></p>
|
||
|
||
<h3>
|
||
|
||
</h3>
|
||
<p>So the digital root is all about "where do we end up on the circle" and we ignore how many times we go around.</p>
|
||
<p>Read about the <a href="modulo.html">Modulus Operation</a> to find out more.</p>
|
||
|
||
<h3>Shortcuts</h3>
|
||
<p>In fact there is a simple shortcut: find any <b>digits that add up to 9</b> (one full rotation), and just <b>ignore </b>them!</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: The digital root of 453</h3>
|
||
<p>4+5 is 9, so ignore them. So the digital root of 453 is simply 3</p>
|
||
</div>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: The digital root of 13331</h3>
|
||
<p>Three 3's make 9, so ignore them. So the digital root of 13331 is 1+1=2</p></div>
|
||
|
||
<h3>Multiplication</h3>
|
||
<p>We can also use digital roots to check Multiplication!</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: 345 times 107</h3>
|
||
<table style="text-align:right; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td style="width:80px;">Sum</td>
|
||
<td style="width:240px;">Digital Root</td>
|
||
<td style="width:20px;"><br></td>
|
||
<td style="width:50px;"><br>
|
||
</td> </tr>
|
||
<tr>
|
||
<td class="large">345</td>
|
||
<td>Ignore 4+5:</td>
|
||
<td><br></td>
|
||
<td class="large">3</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="large">× 107</td>
|
||
<td>1+7=8:</td>
|
||
<td><br></td>
|
||
<td class="large">× 8</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="border-top: solid 2px;" class="large">36915</td>
|
||
<td><i>(Ignore 3+6 and 9)</i> 1+5=6:</td>
|
||
<td><br></td>
|
||
<td style="border-top: solid 2px;" class="large">24</td>
|
||
</tr>
|
||
</tbody>
|
||
</table><p>The digital root of 24 is 2+4=6, which matches the digital root of 36915. Yay!</p></div>
|
||
|
||
<h3>Division</h3>
|
||
<p>Oh this is way too hard for me, you work it out!</p>
|
||
|
||
<h3>Decimals</h3>
|
||
<p>And we can find the digital root of decimals, too:</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: The digital root of 0.125</h3>
|
||
<p>1+2+5 = 8</p>
|
||
</div>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: 0.125 plus 0.125</h3>
|
||
<p> 0.125 + 0.125 = 0.25</p>
|
||
<p>Digital roots give us:</p>
|
||
<ul>
|
||
<li>8 + 8 = 16, and 1+6 = 7,</li>
|
||
<li>which matches the digital root of 0.25: 2+5 = 7</li></ul>
|
||
<p>So we probably got that sum correct!</p>
|
||
</div>
|
||
|
||
|
||
<h2>Divisibility</h2>
|
||
<p>Digital roots also help us check <a href="../divisibility-rules.html">Divisibility</a> (after dividing one number by another do we get a <a href="whole-numbers.html">whole number</a> answer) for both 3 and 9.</p>
|
||
<h3>Divisible by 9</h3>
|
||
<p>Can we divide a number by 9 and get a whole number result? Only when the digital root is 9.</p>
|
||
<div class="example">
|
||
<h3>Example: is 709 divisible by 9?</h3>
|
||
<p>Sum the digits: 7+0+9 = 16, repeat for 16: 1 + 6 = <b>7</b></p>
|
||
<p>Answer: NO, 709 is not divisible by 9</p>
|
||
<p>(Check: 709/9 = 78<span class="intbl"><em>7</em><strong>9</strong></span>, not a whole number)</p>
|
||
</div>
|
||
<div style="float:right; margin: 0 0 5px 10px;"><br>
|
||
</div>
|
||
<div class="example">
|
||
<h3>Example: is 5436 divisible by 9?</h3>
|
||
<p>Sum the digits: 5+4+3+6 = 18, repeat for 18: 1 + 8 = <b>9</b></p>
|
||
<p>Answer: YES, 5436 is divisible by 9</p>
|
||
<p>(Check: 5436/9 = 604 exactly)</p>
|
||
</div>
|
||
<h3>Divisible by 3</h3>
|
||
|
||
<p>Can we divide a number by 3 and get a whole number result? Only when the digital root is a <b>multiple of 3</b>.</p>
|
||
<div class="example">
|
||
<h3>Example: is 428 divisible by 3?</h3>
|
||
<p>Sum the digits: 4+2+8 = 14, repeat for 14: 1 + 4 = <b>5</b></p>
|
||
<p>Answer: NO, 428 is not divisible by 3</p>
|
||
<p>(Check: 428/3 = 142<span class="intbl"><em>2</em><strong>3</strong></span>,
|
||
not a whole number)</p>
|
||
</div>
|
||
|
||
<div style="float:right; margin: 0 0 5px 10px;"><br>
|
||
</div>
|
||
<div class="example">
|
||
<h3>Example: is 1725 divisible by 3?</h3>
|
||
<p>Sum the digits: 1+7+2+5 = 15, repeat for 15: 1 + 5 = <b>6</b></p>
|
||
<p>Answer: 6 is a multiple of 3, so YES, 1725 is divisible by 3</p>
|
||
<p>(Check: 1725/3 = 575 exactly)</p>
|
||
</div>
|
||
<h3><a href="../divisibility-rules.html"></a></h3>
|
||
|
||
<h2>Puzzle Your Friends</h2>
|
||
<p>
|
||
Ask them "what is going on with this?"</p>
|
||
<ul>
|
||
<li>3 × 3 = 0</li>
|
||
<li>3 + 3 = 6</li>
|
||
<li>3 + 6 = 0</li>
|
||
|
||
<li>5 + 5 = 1</li></ul>
|
||
|
||
<h2>Other Bases</h2>
|
||
|
||
<p style="float:left; margin: 0 10px 5px 0;"><img src="images/paws-8.svg" alt="2 paws 8" height="108" width="200"></p>
|
||
<p>If Dogs ruled the world they might use base-8 instead of decimal, so they would "cast out 7s" instead!</p>
|
||
<p><br></p>
|
||
<p><br></p>
|
||
|
||
<div class="related">
|
||
<a href="modulo.html">Modulo Operation</a>
|
||
<a href="../divisibility-rules.html">Divisibility Rules</a>
|
||
<a href="index.html">Numbers Index</a>
|
||
</div>
|
||
<!-- #EndEditable -->
|
||
|
||
</article>
|
||
|
||
<div id="adend" class="centerfull noprint"></div>
|
||
<footer id="footer" class="centerfull noprint"></footer>
|
||
<div id="copyrt">Copyright © 2022 Rod Pierce</div>
|
||
|
||
</div>
|
||
</body><!-- #EndTemplate -->
|
||
<!-- Mirrored from www.mathsisfun.com/numbers/digital-root.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 01:03:59 GMT -->
|
||
</html> |