How do I find the square roots of i?

Oct 26, 2014

Let $z = r {e}^{i \theta}$ be the square-roots of $i$.

${z}^{2} = i R i g h t a r r o w {r}^{2} {e}^{i \left(2 \theta\right)} = {e}^{i \left(\frac{\pi}{2} + 2 n \pi\right)}$

$R i g h t a r r o w \left\{\begin{matrix}{r}^{2} = 1 R i g h t a r r o w r = 1 \\ 2 \theta = \frac{\pi}{2} + 2 n \pi R i g h t a r r o w \theta = \frac{\pi}{4} + n \pi\end{matrix}\right.$

$z = \left\{{e}^{i \frac{\pi}{4}} , {e}^{i \frac{5 \pi}{4}}\right\}$

$= \left\{\cos \left(\frac{\pi}{4}\right) + i \sin \left(\frac{\pi}{4}\right) , \cos \left(\frac{5 \pi}{4}\right) + i \sin \left(\frac{5 \pi}{4}\right)\right\}$

$= \left\{\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} i , - \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} i\right\}$

I hope that this was helpful.