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