A complex number can be described by its horizontal and vertical components or by its magnitude and angle in the complex plane.

If

$$z=a+bi,$$

write

$$r=|z|$$

and let $\theta$ be its argument. Then

$$a=r\cos\theta,\qquad b=r\sin\theta.$$

So

$$z=r(\cos\theta+i\sin\theta).$$

Euler's formula

Euler's formula connects complex exponentials with trigonometry:

$$e^{i\theta}=\cos\theta+i\sin\theta.$$

Therefore the polar form becomes

$$z=re^{i\theta}.$$

Multiplication as scaling and rotation

If

$$z_1=r_1e^{i\theta_1},\qquad z_2=r_2e^{i\theta_2},$$

then

$$z_1z_2=r_1r_2e^{i(\theta_1+\theta_2)}.$$

Magnitudes multiply and angles add. Complex multiplication therefore combines scaling with rotation.

Powers

From the same representation,

$$z^n=r^ne^{in\theta}.$$

This geometric behavior is the reason complex exponentials are so effective for describing sinusoidal oscillations and phase shifts.