I am proud of me. I just decided that I felt like actually proving Euler's formula, to see if I could do it. After a couple hours, I figured it out. I did have to take out the old Calculus book to remind myself what a Taylor expansion was.
If you take the Taylor series of sinx, you get x - x^3/3! + x^5/5! - ...
If you take the Taylor series of cosx, you get 1 - x^2/2! + x^4/4! - ...
If you take the Taylor series of e^ix, you get 1 + ix - x^2/2! - ix^3/3! + x^4/4! + ix^5/5! - ...
Therefore, e^ix = cosx + isinx. If you substitute π for x, you get cosπ + isinπ. cosπ is -1, and sinπ (and therefore isinπ) is 0.
Therefore, e^iπ = -1 which can be changed to the more fun version:
e^iπ + 1 = 0
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