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<!doctype html>
<html lang="en">
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<meta name="author" content="Max Muster<sup>1</sup>,Lisa Master<sup>2</sup>">
<meta name="date" content="Sat May 01 2021 00:00:00 GMT+0200 (Mitteleuropäische Sommerzeit)">
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<title>Euler's Identity</title>
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<header>
<h1>Euler's Identity</h1>
<h3>How to combine 5 important math constants to a short formula</h3>
<h4>Max Muster<sup>1</sup>, Lisa Master<sup>2</sup></h4>
<h5><sup>1</sup>Hochschule Gartenstadt<br><sup>2</sup>Universität Übersee</h5>
<h5>May 2021</h5>
<h5><b>Keywords:</b> markdown+math, VSCode, static page, publication, LaTeX, math</h5>
</header>
<main>
<h3 id="abstract-1">Abstract</h3>
<p>Euler's identity makes a valid formula out of five mathematical constants.</p>
<h2 id="1-introduction-1">1. Introduction</h2>
<p>Euler's identity is often cited as an example of deep mathematical beauty.
Three basic arithmetic operations occur exactly once and combine five fundamental mathematical constants [<a href="#1">1</a>].</p>
<h2 id="2-the-identity-1">2. The Identity</h2>
<p>Starting from Euler's formula <eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>e</mi><mrow><mi>i</mi><mi>x</mi></mrow></msup><mo>=</mo><mi>cos</mi><mo></mo><mi>x</mi><mo>+</mo><mi>i</mi><mi>sin</mi><mo></mo><mi>x</mi></mrow><annotation encoding="application/x-tex">e^{ix}=\cos x + i\sin x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.824664em;vertical-align:0em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.824664em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66786em;vertical-align:0em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">x</span></span></span></span></eq> for any real number <eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">x</span></span></span></span></eq>, we get to Euler's identity with the special case of <eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>π</mi></mrow><annotation encoding="application/x-tex">x = \pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span></span></span></span></eq></p>
<section class="eqno"><eqn><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>e</mi><mrow><mi>i</mi><mi>π</mi></mrow></msup><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn><mtext> </mtext><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">e^{i\pi}+1=0\,.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9579939999999999em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8746639999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">iπ</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">.</span></span></span></span></span></eqn><span>(1)</span></section></math>
<p>The arithmetic operations <em>addition</em>, <em>multiplication</em> and <em>exponentiation</em> combine the fundamental constants</p>
<ul>
<li>the additive identity <eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></eq>.</li>
<li>the multiplicative identity <eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></eq>.</li>
<li>the circle constant <eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span></span></span></span></eq>.</li>
<li>Euler's number <eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">e</span></span></span></span></eq>.</li>
<li>the imaginary constant <eq><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathnormal">i</span></span></span></span></eq>.</li>
</ul>
<h2 id="3-conclusion-1">3. Conclusion</h2>
<p>It has been shown, how Euler's identity makes a valid formula from five mathematical constants.</p>
<h3 id="references-1">References</h3>
<p><span id='1'>[1] <a href="https://en.wikipedia.org/wiki/Euler%27s_identity">Euler's identity</a></p>
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