why is $\sqrt{-1} = i$ and not $\pm i$? this is something that came up when working with one of my students today and it has been bothering me since.  It is more of a maths question than a pedagogical question so i figured i would ask here instead of MESE.
Why is $\sqrt{-1} = i$ and not $\sqrt{-1}=\pm i$?
With positive numbers the square root function always returns both a positive and negative number, is it different for negative numbers? 

To be honest, I would refrain from saying that $$\sqrt{-1} = i$$
It's not even like the real numbers, where one can agree that the square root function takes on the positive values, because what is positive, in the complex plane? Even if you did establish the identity as convention, how would you extend the convention when dealing with: $$\sqrt[\Large 3]{1}$$
for example?
If you need to define the imaginary unit, the best choice is to say that $i$ is such that $$i^2 = -1
$$
There are even "better" choices, but it doesn't look like you'll be needing them.
