Complex Multiplication as Geometrical Transformation

Theorem
Let $z_1 = \left\langle{r_1, \theta_1}\right\rangle$ and $z_2 = \left\langle{r_2, \theta_2}\right\rangle$ be complex numbers expressed in polar form.

Let $z_1$ and $z_2$ be represented on the complex plane $\C$ in vector form.

Let $z = z_1 z_2$ be the product of $z_1$ and $z_2$.

Then $z$ can be interpreted as the result of:
 * rotating $z_1$ about the origin of $\C$ by $\theta_2$ in the positive direction
 * multiplying the modulus of $z_1$ by $r_2$.

Proof

 * Complex-Multiplication-as-Rotation.png

Let $z = r e^{i \alpha}$.

By Product of Complex Numbers in Exponential Form:
 * $z = r_1 r_2 e^{i \left({\theta_1 + \theta_2}\right)}$

Adding $\theta_2$ to $\theta_1$ is equivalent to rotation about the origin of $\C$ by $\theta_2$ in the positive direction.

Similarly, the modulus of $z$ is obtained by multiplying the modulus of $z_1$ by $r_2$.