convert complex number to polar form

Question

i have this complex number

$\sqrt x/2 + \sqrt x/2 i$

i am trying to convert it to polar form. I know that $r = \sqrt (x^2 + y^2)$ but what are the x and y, $1/2$ ?

Answer 1

First, $$\frac{\sqrt x}{2} + \frac{i\sqrt x}{2} = \frac{\sqrt x}{\sqrt{2}}\left(\frac{\sqrt{2}}{2}+\frac{i\sqrt{2}}{2} \right)$$Now recall that the cartesian coordinate $\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$ is the polar coordinate $(1,\pi/4)$ and hence $$\frac{\sqrt{2}}{2}+\frac{i\sqrt{2}}{2}=e^{i\pi/4} \\ \implies \frac{\sqrt x}{\sqrt{2}}\left(\frac{\sqrt{2}}{2}+\frac{i\sqrt{2}}{2} \right) = \frac{\sqrt x}{\sqrt{2}}e^{i\pi/4}$$

Answer 2

The $x$ is $\sqrt{x}/2$, and the $y$ is $\sqrt{x}/2$ as well.