What is wrong with my proof: $-1 = 1$?

Question

I have some theories about why this could by wrong but I still haven't something that convinces me. What is wrong with this proof:

$ -1 = i^2 = i.i = \sqrt{-1}.\sqrt{-1} = \sqrt{(-1).(-1)}= \sqrt1 = 1 $

This would imply that: $1 = -1$

Which is obviously false.

So my theory is that it's not a great idea to write $i = \sqrt{-1}$, but I'm not sure why...

Answer 1

The flaw is in assuming that the rule $\sqrt x\sqrt y=\sqrt{xy}$ holds with imaginary numbers. You just show us a counter-example.

Answer 2

The step $\sqrt{-1}\times \sqrt{-1}=\sqrt{(-1)\times(-1)}$ is incorrect. Such a law is only valid for nonnegative arguments.