https://www.explainxkcd.com/wiki/api.php?action=feedcontributions&feedformat=atom&user=66.214.89.97 2026-07-10T19:25:46Z User contributions MediaWiki 1.30.0 https://www.explainxkcd.com/wiki/index.php?title=179:_e_to_the_pi_times_i&diff=10612 2012-08-26T05:06:21Z

<p>66.214.89.97: </p> <hr /> <div>{{comic<br /> | horizontal = yes<br /> | number = 179<br /> | date = November 3, 2006<br /> | title = e to the pi times i<br /> | image = e_to_the_pi_times_i.png<br /> | titletext = I have never been totally satisfied by the explanations for why e to the ix gives a sinusoidal wave.<br /> | imagesize = <br /> }}<br /> <br /> ==Explanation==<br /> The comic largely references {{w|Euler's identify}}. This identity states that e^(i*π) + 1 = 0. Therefore, e^(i*π) = -1. <br /> <br /> The humour from this comic is because of the seemingly arbitraty relationship between e, π, and the identity of i (the square root of -1). e is the mathematical identity of which the derivative of e^x with respect to x is still e^x, while π is the relationship between the circumfrance of a circle divided by its diameter. Taking these two values and applying them to the value of i in such a manner seems counterintuitive to getting i^2 (-1) from basic analysis. The above linked Wikipedia page goes into good detail of how to derive this identity. <br /> <br /> The title text refers to how Euler's identity is called upon in complex form (separating real and imaginary numbers): e^(i*π) = cos(x) + i*sin(x).<br /> <br /> {{Comic discussion}}</div>

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