179: e to the pi times i

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e to the pi times i
I have never been totally satisfied by the explanations for why e to the ix gives a sinusoidal wave.
Title text: I have never been totally satisfied by the explanations for why e to the ix gives a sinusoidal wave.

[edit] Explanation

The comic largely references Euler's identity. This identity states that e^(i*π) + 1 = 0. Therefore, e^(i*π) = -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 -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 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 -1 from basic analysis. The above linked Wikipedia page goes into good detail of how to derive this identity, as does this YouTube video.

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

[edit] Transcript

[Two people standing at a board with writing on. One person is pointing a stick at the board.]
Cueball: Numbers of the form n√-1 are "imaginary," but can still be used in equations.
Friend: Okay.
Cueball: And e^(π√-1)=-1.
Friend: Now you're just fucking with me.

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This is one of the few comics that were changed after release, as stated by Randall in his XKCD book. It first claimed e^(i*Pi) = 1, which lead to huge protest from the community and a correction from Randall. --Gefrierbrand (talk) 09:47, 3 September 2012 (UTC)

He must have been pie-eyed when he wrote that; he's usually pretty good about his math... -- IronyChef (talk) 05:09, 7 November 2012 (UTC)
I see what you did there. Daddy (talk) 15:18, 28 April 2013 (UTC)

Randall says in the title text that he's never been satisfied with explanations of the sinusoidal nature of the function of e^ix. http://www.math.toronto.edu/mathnet/questionCorner/epii.html really helps, at least for those who are obsessed with taylor series yet tragically horrible at math. --Jolbucley (talk) 03:39, 29 January 2014 (UTC)

Why e to the ix gives a sinusoidal wave? This is because neutrinos keep oscillating. They wouldn't without the e^ix support... 04:01, 24 May 2016 (UTC)

Technically i isn't "imaginary" at all, but is incorporated into equations to represent rotations perpendicular to the x-y plane. 15:51, 9 September 2016 (UTC)

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