TITLE: $1/i=i$. I must be wrong but why?
QUESTION [28 upvotes]: $$\frac{1}{i} = \frac{1}{\sqrt{-1}} = \frac{\sqrt{1}}{\sqrt{-1}} = \sqrt{\frac{1}{-1}} = \sqrt{-1} = i$$
I know this is wrong, but why? I often see people making simplifications such as $\frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}}$, and I would calculate such a simplification in the manner shown above, namely
$$\frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{\sqrt{4}} = \sqrt{\frac{2}{4}} = \frac{1}{\sqrt{2}}$$

REPLY [3 votes]: The inverse of i is -i
$$1/i = i/(i*i) = i/-1 = -i$$
so you take the negative result from \sqrt