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326 lines
16 KiB
HTML
326 lines
16 KiB
HTML
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<title>e - Euler's number</title>
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<!-- #BeginEditable "Body" -->
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<h1 class="center"><i><b>e</b></i> (Euler's Number)</h1>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/e1.svg" alt="e (eulers number)" height="100" width="100"></p>
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<p>The number <i><b>e</b></i> is one of the most important numbers in mathematics.</p>
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<p>The first few digits are:</p>
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<p class="center"><b>2.7182818284590452353602874713527</b> (and more ...)</p>
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<div class="words">
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<p><i>It is often called <b>Euler's number</b> after Leonhard Euler (pronounced "Oiler").</i></p>
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</div>
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<p><b><i>e</i></b> is an <a href="../irrational-numbers.html">irrational number</a> (it cannot be written as a simple fraction).</p>
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<p><b><i>e</i></b> is the base of the Natural <a href="../sets/function-logarithmic.html">Logarithms</a> (invented by John Napier).</p>
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<p><b><i>e</i></b> is found in many interesting areas, so is worth learning about.</p>
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<h2>Calculating</h2>
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<p>There are many ways of calculating the value of <b><i>e</i></b>, but none of them ever give a totally exact answer, because <b><i>e</i></b> is <a href="../irrational-numbers.html">irrational</a> and its digits go on forever without repeating.</p>
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<p>But it <b>is</b> known to over 1 trillion digits of accuracy!</p>
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<p>For example, the value of <span class="large">(1 + 1/n)<sup>n</sup></span> approaches <i><b>e</b></i> as n gets bigger and bigger:</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="../calculus/images/graph-1-1-n-n.gif" alt="graph of (1+1/n)^n" height="218" width="357"></p>
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<table>
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<tbody>
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<tr style="text-align:right;">
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<td style="width:100px;"><span class="large">n</span></td>
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<td style="width:150px;"><span class="large">(1 + 1/n)<sup>n</sup></span></td>
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</tr>
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<tr style="text-align:right;">
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<td style="width:100px;">1</td>
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<td style="width:150px;">2.00000</td>
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</tr>
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<tr style="text-align:right;">
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<td style="width:100px;">2</td>
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<td style="width:150px;">2.25000</td>
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</tr>
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<tr style="text-align:right;">
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<td style="width:100px;">5</td>
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<td style="width:150px;">2.48832</td>
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</tr>
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<tr style="text-align:right;">
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<td style="width:100px;">10</td>
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<td style="width:150px;">2.59374</td>
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</tr>
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<tr style="text-align:right;">
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<td style="width:100px;">100</td>
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<td style="width:150px;">2.70481</td>
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</tr>
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<tr style="text-align:right;">
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<td style="width:100px;">1,000</td>
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<td style="width:150px;">2.71692</td>
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</tr>
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<tr style="text-align:right;">
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<td style="width:100px;">10,000</td>
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<td style="width:150px;">2.71815</td>
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</tr>
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<tr style="text-align:right;">
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<td style="width:100px;">100,000</td>
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<td style="width:150px;">2.71827</td>
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</tr>
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</tbody></table>
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<p> </p>
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<div class="fun">
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<p><b>Try it!</b> Put "(1 + 1/100000)^100000" into the <a href="calculator.html">calculator</a>:</p>
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<p class="center larger">(1 + 1/100000)<sup>100000</sup></p>
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<p>What do you get?</p>
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<!-- <p>And here is an interesting one:</p>
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<p class="center larger">(1 + 9<sup>-4<sup>6×7</sup></sup>)<sup>3<sup>2<sup>85</sup></sup></sup></p>
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<p>It is very close to <i><b>e</b></i>, uses all the digits 1 to 9, but the values are too big for normal calculators.</p>
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<p>But none of these are totally accurate, just approximations.</p>
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--> </div>
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<!--<div class="fun">
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<p><b>Fun Fact</b>: We know that</p>
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<p class="center larger"><i>e</i> ≈ (1 + 1/100,000)<sup>100,000</sup></p>
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<p>So we can choose n = 9<sup>4<sup>6×7</sup></sup> which is <i>also</i> equal to 3<sup>2<sup>85</sup></sup> and get:</p>
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<p class="center larger"><i>e</i> ≈ (1 + 9<sup>-4<sup>6×7</sup></sup>)<sup>3<sup>2<sup>85</sup></sup></sup></p><p>Which is very close to <i><b>e</b></i>, and uses all the digits 1 to 9.</p>
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</div>-->
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<h2>Another Calculation</h2>
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<p>The value of <i><b>e</b></i> is also equal to <span class="intbl"><em>1</em><strong>0!</strong></span> + <span class="intbl"><em>1</em><strong>1!</strong></span> + <span class="intbl"><em>1</em><strong>2!</strong></span> + <span class="intbl"><em>1</em><strong>3!</strong></span> + <span class="intbl"><em>1</em><strong>4!</strong></span> + <span class="intbl"><em>1</em><strong>5!</strong></span> + <span class="intbl"><em>1</em><strong>6!</strong></span> + <span class="intbl"><em>1</em><strong>7!</strong></span> + ... (etc)</p>
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<p class="center"><i>(Note: "!" means <a href="factorial.html">factorial</a>)</i></p>
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<p>The first few terms add up to: 1 + 1 + <span class="intbl"><em>1</em><strong>2</strong></span> + <span class="intbl"><em>1</em><strong>6</strong></span> + <span class="intbl"><em>1</em><strong>24</strong></span> + <span class="intbl"><em>1</em><strong>120</strong></span> = 2.71666...</p>
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<p>In fact Euler himself used this method to calculate <i><b>e</b></i> to 18 decimal places.</p>
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<p>You can try it yourself at the <a href="sigma-calculator.html">Sigma Calculator</a>.</p>
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<h2>Remembering</h2>
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<p>To remember the value of <i><b>e</b></i> (to 10 places) just remember this saying (count the letters!):</p>
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<ul>
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<li>To</li>
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<li>express</li>
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<li><b><i>e</i></b></li>
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<li>remember</li>
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<li>to</li>
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<li>memorize</li>
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<li>a</li>
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<li>sentence</li>
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<li>to</li>
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<li>memorize</li>
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<li>this</li>
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</ul>
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<p>Or you can remember the curious pattern that after the "2.7" the number "1828" appears TWICE:</p>
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<p class="center"><b>2.7 1828 1828</b></p>
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<p>And following THAT are the digits of the angles 45°, 90°, 45° in a <a href="../triangle.html">Right-Angled Isosceles Triangle</a> (no real reason, just how it is):</p>
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<p class="center"><b>2.7 1828 1828 45 90 45</b></p>
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<p><i>(An instant way to seem really smart!)</i></p>
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<h2>Growth</h2>
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<p><b><i>e</i></b> is used in the <b>"Natural</b>" Exponential Function:</p>
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<p class="center"><img src="../sets/images/function-exponential-e.svg" alt="natural exponential function" height="161" width="263"><br>
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<span class="larger">Graph of <b>f(x) = e<sup>x</sup></b></span></p>
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<p>It has this wonderful property: <span class="larger">"its slope is its value"</span></p>
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<div class="def">
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<p>At any point the slope of <i><b>e</b></i><sup>x</sup> equals the value of <i><b>e</b></i><sup>x</sup> :</p>
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<p class="center"><img src="../sets/images/function-exponential-slopes.svg" alt="natural exponential function" height="161" width="263"><br>
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when x=0, the value <i><b>e</b></i><sup>x</sup> = <i><b>1</b></i>, and the slope = <i><b>1</b></i><br>
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when x=1, the value <i><b>e</b></i><sup>x</sup> = <i><b>e</b></i>, and the slope = <i><b>e</b></i><br>
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etc...</p>
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</div>
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<p>This is true anywhere for <i><b>e</b></i><sup>x</sup>, and helps us a lot in <a href="../calculus/introduction.html">Calculus</a> when we need to find slopes etc.</p>
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<p>So <i><b>e</b></i> is perfect for <b>natural growth</b>, see <a href="../algebra/exponential-growth.html">exponential growth</a> to learn more.</p>
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<h2>Area</h2>
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<p>The area <b>up to</b> any x-value is also equal to <b><i>e</i></b><sup>x</sup> :</p>
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<p class="center"><img src="../sets/images/function-exponential-area.svg" alt="natural exponential function" height="161" width="263"></p>
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<h2>An Interesting Property</h2>
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<h3>Just for fun, try "Cut Up Then Multiply"</h3>
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<p>Let us say that we cut a number into equal parts and then multiply those parts together.</p>
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<div class="example">
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<h3>Example: Cut 10 into 2 pieces and multiply them:</h3>
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<p>Each "piece" is 10/2 = <b>5</b> in size</p>
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<p class="center">5×5 = <b>25</b></p>
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</div>
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<p>Now, ... how could we get the answer to be <b>as big as possible</b>, what size should each piece be?</p>
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<p class="center larger">The answer: make the parts as close as possible to "<i><b>e</b></i>" in size.</p>
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<div class="example">
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<h3>Example: <b>10</b></h3>
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<div class="tbl">
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<div class="row"><span class="left">10 cut into 2 equal parts is 5:</span><span class="right">5×5 = 5<sup>2</sup> = 25</span></div>
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<div class="row"><span class="left">10 cut into 3 equal parts is 3<span class="intbl"><em>1</em><strong>3</strong></span>:</span><span class="right">(3<span class="intbl"><em>1</em><strong>3</strong></span>)×(3<span class="intbl"><em>1</em><strong>3</strong></span>)×(3<span class="intbl"><em>1</em><strong>3</strong></span>) = (3<span class="intbl"><em>1</em><strong>3</strong></span>)<sup>3</sup> = 37.0...</span></div>
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<div class="row"><span class="left">10 cut into 4 equal parts is 2.5:</span><span class="right">2.5×2.5×2.5×2.5 = 2.5<sup>4</sup> = <b>39.0625</b></span></div>
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<div class="row"><span class="left">10 cut into 5 equal parts is 2:</span><span class="right">2×2×2×2×2 = 2<sup>5</sup> = 32</span></div>
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</div>
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</div>
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<p>The winner is the number closest to "<i><b>e</b></i>", in this case 2.5.</p>
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<p>Try it with another number yourself, say 100, ... what do you get?</p>
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<h2>100 Decimal Digits</h2>
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<p>Here is <i><b>e</b></i> to 100 decimal digits:</p>
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<p class="center"><b>2.71828182845904523536028747135266249775724709369995957<br>
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49669676277240766303535475945713821785251664274...</b></p>
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<h2>Advanced: Use of <i><b>e</b></i> in Compound Interest</h2>
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<p>Often the number <i><b>e</b></i> appears in unexpected places. Such as in <b>finance</b>.</p>
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<div class="example">
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<p>Imagine a wonderful bank that pays 100% interest.</p>
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<p class="center">In one year you could turn $1000 into $2000.</p>
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<p>Now imagine the bank pays twice a year, that is 50% and 50%</p>
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<p class="center">Half-way through the year you have $1500,<br>
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you reinvest for the rest of the year and your $1500 grows to $2250</p>
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<p>You got <b>more money</b>, because you reinvested half way through.</p>
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<p>That is called <a href="../money/compound-interest-periodic.html">compound interest</a>.</p></div>
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<p>Could we get even <i>more</i> if we broke the year up into months?</p>
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<p>We can use this formula:</p>
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<p class="center large">(1+r/n)<sup>n</sup></p>
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<p class="center"><b>r</b> = annual interest rate (as a decimal, so <b>1</b> not 100%)<br>
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<b>n</b> = number of periods within the year</p>
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<p>Our half yearly example is:</p>
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<p class="center large">(1+1/2)<sup>2</sup> = 2.25</p>
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<p>Let's try it monthly:</p>
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<p class="center large">(1+1/12)<sup>12</sup> = 2.613...</p>
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<p>Let's try it 10,000 times a year:</p>
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<p class="center large">(1+1/10,000)<sup>10,000</sup> = 2.718...</p>
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<p><span class="large">Yes, it is heading towards <i><b>e</b></i></span> (and is how Jacob Bernoulli first discovered it).</p>
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<p> </p>
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<h3>Why does that happen?</h3>
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<p>The answer lies in the similarity between:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;">
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<td style="text-align:right;">Compounding Formula:</td>
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<td style="width:20px;"> </td>
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<td><span class="large">(1 + r/n)<sup>n</sup></span></td>
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</tr>
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<tr style="text-align:center;">
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<td>and </td>
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<td> </td>
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<td> </td>
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</tr>
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<tr style="text-align:center;">
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<td style="text-align:right;"><i><b>e</b></i> (as n approaches infinity):</td>
|
||
<td> </td>
|
||
<td><span class="large">(1 + 1/n)<sup>n</sup></span></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>The Compounding Formula is <b>very like</b> the formula for <i><b>e</b></i><i> (as n approaches infinity)</i>, just with an extra <b>r</b> (the interest rate).</p>
|
||
<p>When we chose an interest rate of 100% (= 1 as a decimal), the formulas became the same.</p>
|
||
<p>Read <a href="../money/compound-interest-periodic.html">Continuous Compounding</a> for more.</p>
|
||
|
||
|
||
<h2>Euler's Formula for Complex Numbers</h2>
|
||
|
||
<p><b><i>e</i></b> also appears in this most amazing equation: </p>
|
||
<p class="large center"><i>e</i><sup><i><b>i</b></i><span class="times">π</span></sup> + 1 = 0</p>
|
||
<p><a href="../algebra/eulers-formula.html">Read more here</a></p>
|
||
|
||
|
||
<h2>Transcendental</h2>
|
||
|
||
<p><b><i>e</i></b> is also a <a href="transcendental-numbers.html">transcendental</a> number.</p>
|
||
<h2>e-Day</h2>
|
||
<p style="float:right; margin: 0 0 5px 10px;">
|
||
<img src="../images/style/balloons.svg" alt="balloons">
|
||
</p>
|
||
<p>Celebrate this amazing number on </p>
|
||
<ul>
|
||
<li>27th January: <b>27/1 at 8:28</b> if you like writing your days first, or</li>
|
||
<li>February 7th: <b>2/7 at 18:28</b> if you like writing your months first, or</li>
|
||
|
||
<li>On both days!</li></ul>
|
||
|
||
<p><br></p>
|
||
<div class="questions">2011, 2012, 2013</div>
|
||
|
||
<div class="related">
|
||
<a href="../algebra/eulers-formula.html">Euler's Formula for Complex Numbers</a>
|
||
<a href="../irrational-numbers.html">Irrational Number</a>
|
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<a href="index.html">Numbers Index</a>
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