## Square-roots of Complex Numbers

If $z=re^{i\theta}$ is a complex number in polar coordinates, with say $0<\theta<\pi$, then one of its square-roots is $w=\sqrt{r}e^{i\theta/2}$. But what if one wants to avoid using the exponential function?

This trick was related to me by my colleague Bob Burckel.

Draw the parallelogram generated by $0,z,|z|$ and draw its diagonal through $0$. In complex notation that’s simply $z+|z|$. By simple geometry, the angle that $z+|z|$ forms with the positive $x$-axis is half the angle that $z$ forms. So it will be enough to renormalize by dividing by $|z+|z||$ and multiplying by $\sqrt{|z|}$ and get

$\displaystyle w=\sqrt{|z|}\frac{z+|z|}{|z+|z||}.$

One can then check computationally that $w^2=z$. Try it, it’s actually not entirely straight-forward.