Wow, how was that visual proof? For those who aren't looking forward to a lame video, the basic idea is this: e^(pi)(i) = -1. Why? Because you can express e

Wow, how was that visual proof? For those who aren't looking forward to a lame video, the basic idea is this: e^(pi)(i) = -1. Why? Because you can express e^x as a taylor series: e^x = 1 + x + x^2/2! + x^3/3! + ... thus e^ix = 1 + ix + ix^2/2! + ix^3/3! + .. e^ix = (1 - x^2/2! + x^4/4! + ..) + i(x - x^3/3!+x^5/5!+...) implies e^ix = cosx + isinx because those parenthetical quantities are the taylor series for cosx and sinx respectively So, e^i(pi) = cos(pi) + isin(pi). = -1 + i(0) = -1