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? 

Well, if you are considering that $y=\sqrt{x}$ is the relation $y^2=x$, then, yes, $\pm i$ are both solutions to $\sqrt{-1}$. However, this is not usually how square roots are defined. Typically we say:
$$\sqrt{1}=1$$
Not plus or minus $1$ - just $1$. This means that $\sqrt{x}$ is a "right inverse" of $x^2$ - that is, we have that
$$\left(\sqrt{x}\right)^2=x$$
but not necessarily that
$$\sqrt{(x^2)}=x.$$
The difference here is that, to make $\sqrt{x}$ a function, we need it to yield unique output values - so we choose a principal root. We generally define that whenever we take the square root of a positive number, we get a positive number. And the definition $\sqrt{-1}=i$ is equally innocuous, since if we defined it as $\sqrt{-1}=-i$, we could just relabel every number by its complex conjugate, and get back to our typical definition (that is to say, $\sqrt{-1}=i$ more defines $i$ than it defines $\sqrt{-1}$). So, though the equation $x^2+1$ has two solutions (as does any equation $x^2-c$ for complex $c\neq 0$), for the sake of making square roots act like a function, we must choose one of them to be the square root of $-1$, and we call this number $i$.
