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 $r_2$.

Corollary

Let $z = \polar {r, \theta}$ be a complex number expressed in polar form.

Let $z$ be represented on the complex plane $\C$ in vector form.

The effect of multiplying $z$ by $e^{i \alpha}$ is to rotate it about the origin of $\C$ by $\alpha$ in the positive direction

Proof

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 $r_2$.

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations