Editing 179: e to the pi times i

{{comic
| number    = 179
| date      = November 3, 2006
| title     = e to the pi times i
| image     = e_to_the_pi_times_i.png
| titletext = I have never been totally satisfied by the explanations for why e to the ix gives a sinusoidal wave.
}}

==Explanation==
The comic largely references {{w|Euler's identity}}. This identity states that e<sup>iπ</sup> + 1 = 0. Therefore, e<sup>iπ</sup> = &minus;1.

The humor from this comic is because of the seemingly arbitrary relationship between e, π, and the identity of i (the square root of &minus;1). e is the mathematical identity of which the derivative of e<sup>x</sup> with respect to x is still e<sup>x</sup>, while π is the relationship between the circumference of a circle divided by its diameter. Taking these two values and applying them to the value of i in such a manner makes it seem counter-intuitive that it would yield &minus;1 from basic analysis. The above linked Wikipedia page goes into good detail of how to derive this identity, as does [https://www.youtube.com/watch?v=-dhHrg-KbJ0 this YouTube video].

The title text refers to how Euler's identity is called upon in complex form (separating real and imaginary numbers): e<sup>ix</sup> = cos(x) + i sin(x).

From ''[[xkcd: volume 0]]'':
{{quote|I initially got the sign wrong here, and got 40 emails in 3 hours.}}

==Transcript==
:[Two people standing at a board with writing on. One person is pointing a stick at the board.]
:Cueball: Numbers of the form <math>n\sqrt{-1}</math> are "imaginary," but can still be used in equations.
:Friend: Okay.
:Cueball: And <math>e^{\pi\sqrt{-1}}=-1</math>.
:Friend: Now you're just fucking with me.

{{comic discussion}}
[[Category:Comics featuring Cueball]]
[[Category:Math]]