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803 lines
15 KiB
HTML
803 lines
15 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/numbers/pi-normal.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:59:23 GMT -->
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<title>Is Pi Normal?</title>
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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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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Is Pi Normal?</h1>
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<p class="center"><img src="images/pi-normal.svg" alt="pi circle"></p>
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<p>A <b>Simply Normal Number</b> has all its digits spread out as if each one is chosen by a throw of a dice.</p>
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<p class="center"><img src="../geometry/images/10-sided-dice.jpg" alt="10-sided dice"><br>A dice can have 10 sides</p>
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<p>
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And we get something like this:
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</p>
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<p class="center"><img src="images/normal-digits.svg" alt="normal digits"></p>
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<div class="def">
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<p>Over the infinite sequence of decimal digits:</p>
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<ul>
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<li><span class="intbl"><em>1</em><strong>10</strong></span> are 0, <span class="intbl"><em>1</em><strong>10</strong></span> are 1, <span class="intbl"><em>1</em><strong>10</strong></span> are 2, etc (this makes it "<i>simply </i>normal")</li></ul>
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<p>But to be "normal" (without the "simply" in front) we also need:</p>
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<ul>
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<li><span class="intbl"><em>1</em><strong>100</strong></span> are 00, <span class="intbl"><em>1</em><strong>100</strong></span> are 01, <span class="intbl"><em>1</em><strong>100</strong></span> are 02, etc
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</li>
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<li><span class="intbl"><em>1</em><strong>1000</strong></span> are 000, <span class="intbl"><em>1</em><strong>1000</strong></span> are 001, <span class="intbl"><em>1</em><strong>1000</strong></span> are 002, etc
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</li>
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<li><b>and so on</b>, for every group of digits</li>
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</ul>
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</div>
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<p><br></p>
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<p class="center larger">Is this the case for <a href="pi.html">Pi</a> ?</p>
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<p>To investigate let us <b>count</b> how many of each single digit there are, and see if they appear more or less often than expected.</p>
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<p>We also need to do this for 2-digit pairs, 3-digit groups, etc.</p>
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<p>Let us have a go!</p>
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<h2>A Hundred Digits</h2>
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<p>Here are the first 100 digits of Pi:</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 class="large" align="center">314159265358979323846264338327950288<br>
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4197169399375105820974944592307816<br>
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406286208998628034825342117067</td>
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</tr>
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</tbody></table>
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<p>With 100 digits and ten possibilities (0,1,2,3,4,5,6,7,8,9) we should get 10 of each digit, right? </p>
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<p>Well I count 8 "0"s, does that seem right? </p>
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<p>Expecting a perfect 10 of each digit is unrealistic of course, so maybe just 8 is fine.</p>
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<p>Here is the count for each digit:</p>
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<div class="simple">
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<table style=" text-align: right; margin:auto;">
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<tbody>
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<tr>
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<th style="width: 70px;">Digit</th>
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<th style="width: 70px;">Count</th></tr>
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<tr>
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<td>0:</td>
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<td>8</td></tr>
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<tr>
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<td>1:</td>
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<td>8</td></tr>
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<tr>
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<td>2:</td>
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<td>12</td></tr>
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<tr>
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<td>3:</td>
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<td>12</td></tr>
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<tr>
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<td>4:</td>
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<td>10</td></tr>
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<tr>
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<td>5:</td>
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<td>8</td></tr>
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<tr>
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<td>6:</td>
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<td>9</td></tr>
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<tr>
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<td>7:</td>
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<td>8</td></tr>
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<tr>
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<td>8:</td>
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<td>12</td></tr>
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<tr>
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<td>9:</td>
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<td>13</td></tr>
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</tbody></table>
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</div>
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<p>They are mostly around 10, but wow: "9" occurred 13 times!</p>
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<p>Should that be expected? Let us count the counts:</p>
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<ul>
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<li>4 digits (0,1,5 and 7) occur <b>8</b> times, </li>
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<li>1 digit (6) occurs <b>9 </b>times, </li>
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<li>1 digit (4) occurs <b>10</b> times, </li>
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<li>3 digits (2,3 and 8) occur <b>12</b> times and </li>
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<li>1 digits (9) occur <b>13</b> times</li></ul>
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<p>As a graph:</p>
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<p class="center"><img src="images/pi-normal-ten-100.svg"></p>
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<p>They are kind of near 10, but also a little spread out ... is this "normal"?</p>
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<p>Let us try two-digit pairs:</p>
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<div class="simple">
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<table style=" text-align: right; margin:auto;">
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<tbody>
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<tr>
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<th style="width: 70px;">Digits</th>
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<th style="width: 70px;">Count</th></tr>
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<tr>
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<td>00:</td>
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<td>0</td></tr>
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<tr>
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<td>01:</td>
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<td>0</td></tr>
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<tr>
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<td>02:</td>
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<td>1</td></tr>
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<tr>
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<td>03:</td>
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<td>1</td></tr>
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<tr>
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<td>04:</td>
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<td>0</td></tr>
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<tr>
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<td>05:</td>
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<td>1</td></tr>
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<tr>
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<td>06:</td>
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<td>2</td></tr>
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<tr>
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<td>...</td>
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<td>...</td></tr>
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<tr>
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<td>98:</td>
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<td>1</td></tr>
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<tr>
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<td>99:</td>
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<td>2</td></tr>
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</tbody></table>
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</div>
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<p>Mostly 0's and 1's. We need more raw data!</p>
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<h2>A Thousand Digits</h2>
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<p>Using the first thousand digits of Pi gets us this:</p>
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<div class="simple">
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<table style=" text-align: right; margin:auto;">
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<tbody>
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<tr>
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<th style="width: 70px;">Digit</th>
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<th style="width: 70px;">Count</th></tr>
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<tr>
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<td>0:</td>
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<td>93</td></tr>
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<tr>
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<td>1:</td>
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<td>116</td></tr>
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<tr>
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<td>2:</td>
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<td>103</td></tr>
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<tr>
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<td>3:</td>
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<td>103</td></tr>
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<tr>
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<td>4:</td>
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<td>93</td></tr>
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<tr>
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<td>5:</td>
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<td>97</td></tr>
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<tr>
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<td>6:</td>
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<td>94</td></tr>
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<tr>
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<td>7:</td>
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<td>95</td></tr>
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<tr>
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<td>8:</td>
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<td>101</td></tr>
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<tr>
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<td>9:</td>
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<td>105</td></tr>
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</tbody></table>
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</div>
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<p>
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Counting the counts:
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</p>
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<p class="center"><img src="images/pi-normal-ten-1000.svg"></p>
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<p>Zoomed out:</p>
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<p class="center"><img src="images/pi-normal-ten-1000b.svg"></p>
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<p>It now looks more grouped around 100, our first hint that Pi might be "normal".</p>
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<p>Let's look at two-digit pairs:</p>
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<div class="simple">
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<table style=" text-align: right; margin:auto;">
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<tbody>
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<tr>
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<th style="width: 70px;">Digits</th>
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<th style="width: 70px;">Count</th></tr>
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<tr>
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<td>00:</td>
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<td>7</td></tr>
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<tr>
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<td>01:</td>
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<td>10</td></tr>
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<tr>
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<td>02:</td>
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<td>10</td></tr>
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<tr>
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<td>03:</td>
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<td>9</td></tr>
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<tr>
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<td>04:</td>
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<td>4</td></tr>
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<tr>
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<td>05:</td>
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<td>14</td></tr>
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<tr>
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<td>06:</td>
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<td>8</td></tr>
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<tr>
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<td>07:</td>
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<td>6</td></tr>
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<tr>
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<td>...</td>
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<td>...</td></tr>
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<tr>
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<td>93:</td>
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<td>14</td></tr>
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<tr>
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<td>94:</td>
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<td>11</td></tr>
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<tr>
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<td>95:</td>
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<td>16</td></tr>
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<tr>
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<td>96:</td>
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<td>9</td></tr>
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<tr>
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<td>97:</td>
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<td>7</td></tr>
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<tr>
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<td>98:</td>
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<td>8</td></tr>
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<tr>
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<td>99:</td>
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<td>11</td></tr>
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</tbody></table>
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</div>
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<p>
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Good, we are getting values around 10, but let us see the count of counts:</p>
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<p class="center"><img src="images/pi-normal-hun-1000.svg"></p>Wow, it looks nicely clustered around 10, maybe a little skewed.
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<p>Let's go for more data!</p>
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<h2>Ten Thousand Digits</h2>
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<p>Using the first ten thousand digits of Pi gets us this:</p>
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<div class="simple">
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<table style=" text-align: right; margin:auto;">
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<tbody>
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<tr>
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<th style="width: 70px;">Digit</th>
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<th style="width: 70px;">Count</th></tr>
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<tr>
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<td>0:</td>
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<td>968</td></tr>
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<tr>
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<td>1:</td>
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<td>1026</td></tr>
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<tr>
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<td>2:</td>
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<td>1021</td></tr>
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<tr>
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<td>3:</td>
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<td>975</td></tr>
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<tr>
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<td>4:</td>
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<td>1012</td></tr>
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<tr>
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<td>5:</td>
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<td>1046</td></tr>
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<tr>
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<td>6:</td>
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<td>1021</td></tr>
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<tr>
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<td>7:</td>
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<td>970</td></tr>
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<tr>
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<td>8:</td>
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<td>947</td></tr>
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<tr>
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<td>9:</td>
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<td>1014</td></tr>
|
|
</tbody></table>
|
|
</div>
|
|
<p>
|
|
And the count of counts:
|
|
</p>
|
|
|
|
<p class="center"><img src="images/pi-normal-ten-10000.svg"></p><br><br>And for two-digit pairs:<br>
|
|
<div class="simple">
|
|
<table style=" text-align: right; margin:auto;">
|
|
<tbody>
|
|
<tr>
|
|
<th style="width: 70px;">Digits</th>
|
|
<th style="width: 70px;">Count</th></tr>
|
|
<tr>
|
|
<td>00:</td>
|
|
<td>85</td></tr>
|
|
<tr>
|
|
<td>01:</td>
|
|
<td>103</td></tr>
|
|
<tr>
|
|
<td>02:</td>
|
|
<td>98</td></tr>
|
|
<tr>
|
|
<td>03:</td>
|
|
<td>103</td></tr>
|
|
<tr>
|
|
<td>04:</td>
|
|
<td>98</td></tr>
|
|
<tr>
|
|
<td>05:</td>
|
|
<td>89</td></tr>
|
|
<tr>
|
|
<td>06:</td>
|
|
<td>101</td></tr>
|
|
<tr>
|
|
<td>07:</td>
|
|
<td>93</td></tr>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<tr>
|
|
<td>...</td>
|
|
<td>...</td></tr>
|
|
<tr>
|
|
<td>94:</td>
|
|
<td>98</td></tr>
|
|
<tr>
|
|
<td>95:</td>
|
|
<td>107</td></tr>
|
|
<tr>
|
|
<td>96:</td>
|
|
<td>106</td></tr>
|
|
<tr>
|
|
<td>97:</td>
|
|
<td>88</td></tr>
|
|
<tr>
|
|
<td>98:</td>
|
|
<td>100</td></tr>
|
|
<tr>
|
|
<td>99:</td>
|
|
<td>91</td></tr>
|
|
</tbody></table>
|
|
</div>
|
|
<p>And as a count of counts:</p>
|
|
<p class="center"><img src="images/pi-normal-hun-10000.svg"></p>
|
|
<p>It is starting to look like a <a href="../data/standard-normal-distribution.html">normal distribution</a> isn't it?</p>
|
|
<p class="center"><img src="../data/images/normal-distribution-1.svg" alt="bell curve" height="174" width="384"></p>
|
|
<h2>Hundred Thousand Digits</h2>
|
|
<p>Using the first hundred thousand digits of Pi gets us this:</p>
|
|
<div class="simple">
|
|
<table style=" text-align: right; margin:auto;">
|
|
<tbody>
|
|
<tr>
|
|
<th style="width: 70px;">Digit</th>
|
|
<th style="width: 70px;">Count</th></tr>
|
|
<tr>
|
|
<td>0:</td>
|
|
<td>9999</td></tr>
|
|
<tr>
|
|
<td>1:</td>
|
|
<td>10137</td></tr>
|
|
<tr>
|
|
<td>2:</td>
|
|
<td>9908</td></tr>
|
|
<tr>
|
|
<td>3:</td>
|
|
<td>10026</td></tr>
|
|
<tr>
|
|
<td>4:</td>
|
|
<td>9971</td></tr>
|
|
<tr>
|
|
<td>5:</td>
|
|
<td>10026</td></tr>
|
|
<tr>
|
|
<td>6:</td>
|
|
<td>10028</td></tr>
|
|
<tr>
|
|
<td>7:</td>
|
|
<td>10025</td></tr>
|
|
<tr>
|
|
<td>8:</td>
|
|
<td>9978</td></tr>
|
|
<tr>
|
|
<td>9:</td>
|
|
<td>9902</td></tr>
|
|
</tbody></table>
|
|
</div>
|
|
|
|
<p>The count of counts is:</p>
|
|
<p class="center"><img src="images/pi-normal-ten-100000.svg"></p><br><br>And for two-digit pairs:
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<div class="simple">
|
|
<table style=" text-align: right; margin:auto;">
|
|
<tbody>
|
|
<tr>
|
|
<th style="width: 70px;">Digits</th>
|
|
<th style="width: 70px;">Count</th></tr>
|
|
<tr>
|
|
<td>00:</td>
|
|
<td>998</td></tr>
|
|
<tr>
|
|
<td>01:</td>
|
|
<td>1027</td></tr>
|
|
<tr>
|
|
<td>02:</td>
|
|
<td>962</td></tr>
|
|
<tr>
|
|
<td>03:</td>
|
|
<td>993</td></tr>
|
|
<tr>
|
|
<td>04:</td>
|
|
<td>968</td></tr>
|
|
<tr>
|
|
<td>05:</td>
|
|
<td>1007</td></tr>
|
|
<tr>
|
|
<td>06:</td>
|
|
<td>1009</td></tr>
|
|
<tr>
|
|
<td>07:</td>
|
|
<td>1017</td></tr>
|
|
<tr>
|
|
<td>08:</td>
|
|
<td>1001</td></tr>
|
|
<tr>
|
|
<td>...</td>
|
|
<td>...</td></tr>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<tr>
|
|
<td>94:</td>
|
|
<td>1033</td></tr>
|
|
<tr>
|
|
<td>95:</td>
|
|
<td>1014</td></tr>
|
|
<tr>
|
|
<td>96:</td>
|
|
<td>952</td></tr>
|
|
<tr>
|
|
<td>97:</td>
|
|
<td>934</td></tr>
|
|
<tr>
|
|
<td>98:</td>
|
|
<td>997</td></tr>
|
|
<tr>
|
|
<td>99:</td>
|
|
<td>968</td></tr>
|
|
</tbody></table>
|
|
</div>
|
|
|
|
<p>And the count of counts:</p>
|
|
<p class="center"><img src="images/pi-normal-hun-100000.svg"></p>
|
|
<p>The numbers are all around 1000, and, using the <a href="../data/standard-deviation-calculator.html">standard deviation calculator</a> gets a standard deviation of only <b>32.3</b>, which is a reasonably tight grouping, don't you think?</p>
|
|
<h2>Summary</h2>
|
|
<p>Just looking at single digits:</p>
|
|
<div class="simple">
|
|
<table style=" text-align: right; margin:auto;">
|
|
<tbody>
|
|
<tr>
|
|
<th style="width: 70px;">Digit</th>
|
|
<th style="width: 70px;">100</th>
|
|
<th style="width: 70px;">1000</th>
|
|
<th style="width: 70px;">10,000</th>
|
|
<th style="width: 70px;">100,000</th>
|
|
</tr>
|
|
|
|
|
|
<tr>
|
|
<td><i>expect</i>:</td>
|
|
<td><i>10</i></td>
|
|
<td><i>100</i></td>
|
|
<td><i>1000</i></td>
|
|
<td><i>10000</i></td></tr>
|
|
<tr>
|
|
<td>0:</td>
|
|
<td>8</td>
|
|
<td>93</td>
|
|
<td>968</td>
|
|
<td>9999</td></tr>
|
|
<tr>
|
|
<td>1:</td>
|
|
<td>8</td>
|
|
<td>116</td>
|
|
<td>1026</td>
|
|
<td>10137</td></tr>
|
|
<tr>
|
|
<td>2:</td>
|
|
<td>12</td>
|
|
<td>103</td>
|
|
<td>1021</td>
|
|
<td>9908</td></tr>
|
|
<tr>
|
|
<td>3:</td>
|
|
<td>12</td>
|
|
<td>103</td>
|
|
<td>975</td>
|
|
<td>10026</td></tr>
|
|
<tr>
|
|
<td>4:</td>
|
|
<td>10</td>
|
|
<td>93</td>
|
|
<td>1012</td>
|
|
<td>9971</td></tr>
|
|
<tr>
|
|
<td>5:</td>
|
|
<td>8</td>
|
|
<td>97</td>
|
|
<td>1046</td>
|
|
<td>10026</td></tr>
|
|
<tr>
|
|
<td>6:</td>
|
|
<td>9</td>
|
|
<td>94</td>
|
|
<td>1021</td>
|
|
<td>10028</td></tr>
|
|
<tr>
|
|
<td>7:</td>
|
|
<td>8</td>
|
|
<td>95</td>
|
|
<td>970</td>
|
|
<td>10025</td></tr>
|
|
<tr>
|
|
<td>8:</td>
|
|
<td>12</td>
|
|
<td>101</td>
|
|
<td>947</td>
|
|
<td>9978</td></tr>
|
|
<tr>
|
|
<td>9:</td>
|
|
<td>13</td>
|
|
<td>105</td>
|
|
<td>1014</td>
|
|
<td>9902</td></tr>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</tbody></table> </div>
|
|
<p>Just by eye we can see that the more digits we use the more the counts tend to group around the expected value. </p>
|
|
<p>This is called the "Law of Large Numbers": <span itemprop="articleBody">in the long run random events tend to average out at the expected value.</span></p>
|
|
<p>We see the same thing with two-digit pairs.</p>
|
|
<p>So this increases our confidence that Pi is a normal number.</p>
|
|
<h2>Conclusion</h2>
|
|
<p>
|
|
|
|
</p><p>So it <b>seems</b> that the single digits, and double digit pairs, occur no more frequently than random chance. At least within the first 100,000 digits. </p>
|
|
<p>But we need to check more digits and also 3-digit, 4-digit etc groups. A very big job. </p>
|
|
<h2>Research</h2>
|
|
<p>People with access to powerful computers have checked trillions of digits of Pi:</p>
|
|
<p><a href="https://arxiv.org/abs/1612.00489v1">(external link) Digit Statistics of the First 22.4 Trillion Decimal Digits of Pi</a></p>
|
|
<p> and have also found no reason to doubt that Pi is a Normal Number. <b>Not a proof</b>, just a very strong suspicion that Pi is normal.</p>
|
|
<h2>Why is this Important?</h2>
|
|
<p>Firstly, isn't it curious that the digits of Pi (which form an exact sequence) nonetheless seem to be randomly distributed?</p>
|
|
<p>But the other mind-boggling implication is that an infinite number of randomly occuring digits will contain every finite sequence of digits, including:</p>
|
|
<ul>
|
|
<li>your pin number</li>
|
|
<li>your phone number</li>
|
|
<li>your birth date</li>
|
|
<li>the winning numbers in next weeks lottery</li>
|
|
</ul>
|
|
<p>And by simply converting text to digits (such as hello = 8 5 12 12 15) then Pi's digits will also include:</p>
|
|
<ul>
|
|
<li>your name</li>
|
|
<li>your address</li>
|
|
<li>all of your text messages</li>
|
|
<li>any or every book you could imagine</li>
|
|
<li>and perhaps the secret to life?</li></ul>
|
|
<p>Now if we could also get an index to where those things are ...</p><br>
|
|
<div class="related">
|
|
<a href="pi.html">Pi</a>
|
|
<a href="pi-digits.html">1 Hundred to 1 Million Digits of Pi</a>
|
|
<a href="../data/index.html">Data Index</a>
|
|
<a href="index.html">Numbers Index</a>
|
|
</div>
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