Examples from Polar form

Convert between the two forms

For $z=1+i$,

The point lies in the first quadrant and $\tan\theta=1$, so $\theta=\pi/4$ is a principal argument. Hence

Conversely,

Perform product and quotient calculations

Let

Both factors point down the imaginary axis. Their arguments add to $3\pi$, so the product points along the negative real axis:

If instead $z_2=\pi e^{3\pi i/4}$, then

Calculate a power

For $z=\sqrt2\,e^{3\pi i/4}$,

Polar form handles the large exponent by treating the modulus and the argument separately.

Find the eighth roots of unity

Since $1=e^{2\pi i k}$ for every integer $k$, the eighth roots of unity are

They are the eight equally spaced points on the unit circle. For example,

and the remaining roots continue around the circle.