Since $\sqrt{-1}=i$ is wrong, what about $\sqrt{-4}$? Is it $2i$ or $\pm 2i$ or something else?

Question

I recently herd that saying $i= \sqrt{-1}$ is wrong (did not understated very well).

So my problem is that:

The simplification of $\sqrt{-4}=\sqrt{4}\times\sqrt{-1}=2i$, is this correct?

Or $\sqrt{-4}=\pm2i$ is this correct?

Or both wrong?

Answer 1

$x^2=4$ is an equation with two solutions, $2$ and $-2$. We can write $\sqrt{4}=2$ because mathematicians have decided (as a convention) that the square root symbol will always mean the positive solution.

$x^2=-4$ is also an equation with two solutions, in this case, $2i$ and $-2i$. However, for complex numbers no convention similar to the one above is in place and so you should not use the square root symbol.

Answer 2

You may factor the given equation in order to obtain \begin{align*} x = \sqrt{-4} \Longleftrightarrow x^{2} = -4 \Longleftrightarrow x^{2} + 4 = 0 \Longleftrightarrow (x+2i)(x-2i) = 0 \Longleftrightarrow x = \pm 2i \end{align*}

Here, it is explicitly assumed we are dealing with complex numbers.