Here's a little bit of a strange question: In class, my math teacher decided to let us take a break from the usual class to solve a mathematical puzzle:
\begin{align} i & = i \\ i^2 &= i^2 \\ \sqrt{-1} &= \sqrt{-1} \\ \sqrt{\frac{-1}{1}} &= \sqrt{\frac{1}{-1}} \\ \frac{\sqrt{-1}}{\sqrt{1}} &= \frac{\sqrt{1}}{\sqrt{-1}} \\ i &= \frac{1}{i} \end{align}
I assume that the transition from line 4 to 5 is wrong, but I don't see how :p.
Does anyone care to explain?
The function $t \mapsto \sqrt{t}$ is defined on $\mathbb{R}_{\geq 0}$. You can extend this notion to the whole real line (actually, to $\mathbb{C}$), but you lose some properties. Particularly, you lose the product rule you're using to "prove" the statement (the fourth equality to be precise).