How to get to the first step of solving $z^2=i$?

Question

How do I get from $z^2=i$ to $z=x+iy$? Is it a rule you use to solve the equation in general or specifically for this equation? (I don not understand the step from $z^2=i$ to $z=x+iy$) Thank you!

Answer 1

As $z$ is complex, we can set $z=x+iy$ where $x,y$ are real and $i=\sqrt{-1}$

So, $z^2=(x+iy)^2=x^2-y^2+2xyi$

Equate the real & the imaginary part of $x^2-y^2+2xyi=i$

$2xy=1\implies x=\frac1{2y}$

and $x^2-y^2=0\implies y^2=x^2=\left(\frac1{2y}\right)^2\implies y^4=4\implies y^2=2$ as $y$ is real

$\implies y=\pm\frac1{\sqrt2}$

$x=\frac1{2y}=?$

Answer 2

Another way is to write the equation in polar form $z=re^{i\theta}$, so that $$i=e^{i \frac{\pi}2}=z^2=r^2e^{2i\theta}$$

Whence $r=1$ and $\theta = \cfrac {\pi}4$ so that $$z=e^\frac{\pi}4=\cos \frac{\pi}4+i\sin \frac{\pi}4$$

There is another solution with $2\theta=2\pi+\frac{\pi}2$

This can be particularly convenient for taking roots or calculating powers.