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229 lines
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HTML
229 lines
8.6 KiB
HTML
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<div id="adTop"></div>
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<div id="hdr"></div>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Sigma Notation</h1>
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<div class="center def"><span class="times" style="font-size:400%; vertical-align:middle;">Σ</span> This symbol (called <b>Sigma</b>) means "sum up"</div>
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<p>I love Sigma, it is fun to use, and can do many clever things.</p><br>
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<p class="center">So <span style="font-size:150%; font-family: 'Times New Roman', Times, serif; ">Σ</span> means to sum things up ...</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 colspan="3">
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<h3>Sum What?</h3></td>
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</tr>
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<tr>
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<td>
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<p>Sum whatever is after the Sigma:</p></td>
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<td> </td>
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<td class="larger">
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<div style="display:inline-block; text-align:center;">
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<div class="intto"> </div>
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<div class="intsymb">Σ</div>
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<div class="intfrom"> </div>
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</div>
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<div class="inttext">n</div>
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</td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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<td>so we sum <span class="number"><i><b>n</b></i></span>
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</td></tr>
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<tr>
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<td colspan="3">
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<h3>But What Values of <i><b>n</b></i> ?</h3></td>
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</tr>
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<tr>
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<td>
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<p align="right">The values are shown below<br>
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and above the Sigma:</p></td>
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<td> </td>
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<td class="larger">
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<div style="display:inline-block; text-align:center;">
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<div class="intto">4</div>
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<div class="intsymb">Σ</div>
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<div class="intfrom">n=1</div>
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</div>
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<div class="inttext">n</div>
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</td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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<td> it says <span class="number"><i>n</i></span> goes from 1 to 4,<br>
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which is <b>1</b>, <b>2</b>, <b>3</b> and <b>4</b>
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</td></tr>
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<tr>
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<td colspan="3">
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<h3>OK, Let's Go ...</h3></td>
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</tr>
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<tr>
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<td>
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<p align="right">So now we add up 1,2,3 and 4:</p></td>
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<td> </td>
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<td class="larger">
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<div style="display:inline-block; text-align:center;">
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<div class="intto">4</div>
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<div class="intsymb">Σ</div>
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<div class="intfrom">n=1</div>
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</div>
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<div class="inttext">n = 1 + 2 + 3 + 4 = <b>10</b></div>
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</td>
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</tr>
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</tbody></table>
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<p>Here it is in one diagram:</p>
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<p class="center"><img src="images/sigma-notation.svg" alt="Sigma Notation" height="118" width="451"></p>
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<h2>More Powerful</h2>
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<p>But <span style="font-size:150%; font-family: 'Times New Roman', Times, serif; ">Σ</span> can do more powerful things than that!</p>
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<p>We can square <span class="number"><i>n</i></span> each time and sum the result:</p>
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<div class="center">
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<div style="display:inline-block; text-align:center;">
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<div class="intto">4</div>
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<div class="intsymb">Σ</div>
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<div class="intfrom">n=1</div>
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</div>
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<div class="inttext" style="transform: translateY(-100%);">n<sup>2</sup> = 1<sup>2</sup> + 2<sup>2</sup> + 3<sup>2</sup> + 4<sup>2</sup> = 30</div>
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</div>
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<p class="center"> </p>
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<p>We can add up the first four terms in the <a href="sequences-series.html">sequence</a> <b>2n+1</b>:</p>
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<div class="center">
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<div style="display:inline-block; text-align:center;">
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<div class="intto">4</div>
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<div class="intsymb">Σ</div>
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<div class="intfrom">n=1</div>
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</div>
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<div class="inttext">(2n+1) = 3 + 5 + 7 + 9 = 24</div>
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</div>
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<p> </p>
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<p>We can use other letters, here we use <span class="larger">i</span> and sum up <span class="larger">i × (i+1)</span>, going from <b>1</b> to <b>3</b>:</p>
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<div class="center">
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<div style="display:inline-block; text-align:center;">
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<div class="intto">3</div>
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<div class="intsymb">Σ</div>
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<div class="intfrom">i=1</div>
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</div>
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<div class="inttext">i(i+1) = 1×2 + 2×3 + 3×4 = 20</div>
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</div>
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<p> </p>
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<p>And we can start and end with any number. Here we go from <b>3</b> to <b>5</b>:</p>
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<div class="center">
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<div style="display:inline-block; text-align:center;">
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<div class="intto">5</div>
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<div class="intsymb">Σ</div>
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<div class="intfrom">i=3</div>
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</div>
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<div class="inttext" style="transform: translateY(-70%);"><span class="intbl"><em>i</em><strong>i + 1</strong></span> = <span class="intbl"><em>3</em><strong>4</strong></span> + <span class="intbl"><em>4</em><strong>5</strong></span> + <span class="intbl"><em>5</em><strong>6</strong></span></div>
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</div>
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<p> </p>
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<p class="larger">There are lots more examples in the more advanced topic <a href="partial-sums.html">Partial Sums</a>.</p>
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<div class="fun center">
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<p>You can try some of your own with the <a href="../numbers/sigma-calculator.html">Sigma Calculator</a>.<br>
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<a href="../numbers/sigma-calculator.html"><img src="../numbers/images/sigma-calculator.gif" alt="sigma calculator" height="81" width="306"></a></p> </div>
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<p> </p>
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<h2>Why is it called "Sigma"</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><img src="images/sigma-greek.svg" alt="Sigma is Greek" height="87" width="72"></td>
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<td>
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<p>Sigma is the upper case letter <b>S</b> in Greek. And <b>S</b> stands for <b>S</b>um.</p></td>
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</tr>
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</tbody></table>
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<p> </p>
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<div class="questions">8322, 8323, 600, 601, 1247, 3015, 3016, 3017, 3018, 1248</div>
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<div class="related">
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<a href="../numbers/sigma-calculator.html">Sigma Calculator</a>
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<a href="partial-sums.html">Partial Sums</a>
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<a href="infinite-series.html">infinite-series</a>
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<a href="index.html">Algebra Index</a>
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<div id="copyrt">Copyright © 2022 Rod Pierce</div>
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