What did I do here? This can't be right... ($i = -1$)?

Question

I was messing around in Geometry class today and found a very odd 'proof'.

It relies on only two facts, $1^2=1$ and $i=\sqrt{-1}$

From here I did this:

$$i = \sqrt{-1}$$ $$i^2 = -1$$ $$-i^2 = 1$$

therefore, since $-i^2 = 1$ and $1^2 = 1$:

$$-i^2=1^2$$ $$-i=1$$ $$i=-1$$

but this is obviously inconsistent, since $-1^2 \neq i^2$

what in the world (of math) did I do wrong, since this result is clearly impossible??

Also, any tags I should add to this question?

Answer 1

Not quite; at the line where you have $-i^2=1^2$ and you take a square root, you should have $$\sqrt{-i^2} = \sqrt{-1}\sqrt{i^2} = i \cdot i = -1 \neq 1$$

Answer 2

The flaw is that you can't conclude $-i = 1$ from $-i^2 = 1^2$.