The power series definition and deMoivre formula is cool and it makes sense, but it frankly unsatisfying as an intuitive reason. I like to think of it in terms

The power series definition and deMoivre formula is cool and it makes sense, but it frankly unsatisfying as an intuitive reason. I like to think of it in terms of rotations and velocity. This stack overflow answer gives a great intuitive reason for why e^(pi I) = -1. https://math.stackexchange.com/a/27051 https://math.stackexchange.com/a/27051