Ok so I just proved 1 = -1. Someone help me find my error.

[gumpzilla]

You're picking the two different branches, since both i and 1 / i when squared yield -1, but themselves differ by a factor of -1, and this is where the problem comes in. So the answer to your question is that you are being inconsistent in how you define sqrt(-1), and if you're consistent in that regard then the problem goes away. Difficulties of this kind are extremely common when working with complex variables, which is part of what makes them confusing.