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.

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