# Square root of 1 is (not) -1 [duplicate]

Possible Duplicate:
$i^2$ why is it $-1$ when you can show it is $1$?

I was thinking on the following line of thoughts: $1 = \sqrt{1} = \sqrt{-1 \cdot -1} = \sqrt{-1} \cdot \sqrt{-1} = i^2 = -1$

Of course this is not true, but I was wondering which step in this 'line of thoughts' is forbidden to make?

Thanks for the explanation.

• The incorrect step is the assumption that $\sqrt{ab} = \sqrt{a} \sqrt{b}$ While this holds for nonnegative numbers, it does not hold for negative numbers. This has to do with the convention that $\sqrt 4 = 2$ instead of $-2$. I should also note that this is a duplicate-post-in-idea, so it will probably be closed. Commented May 15, 2012 at 8:07
• This question is almost an exact duplicate; others whose answers you may find helpful include this one and this one. Commented May 15, 2012 at 8:10

$\sqrt{-1 \cdot -1}$ is not equal to $\sqrt{-1} \cdot \sqrt{-1}$. The formula $\sqrt{ab} = \sqrt{a}\sqrt{b}$ is only valid when both $a,b$ are nonnegative real numbers.