# What is the geometric interpretation of multiplying two complex numbers?

Dec 12, 2014

Let ${z}_{1}$ and ${z}_{2}$ be two complex numbers.

By rewriting in exponential form,

$\left\{\begin{matrix}{z}_{1} = {r}_{1} {e}^{i {\theta}_{1}} \\ {z}_{2} = {r}_{2} {e}^{i {\theta}_{2}}\end{matrix}\right.$

So,

z_1 cdot z_2 =r_1e^{i theta_1}cdot r_2 e^{i theta_2} =(r_1 cdot r_2)e^{i(theta_1+theta_2)}

Hence, the product of two complex numbers can be geometrically interpreted as the combination of the product of their absolute values (${r}_{1} \cdot {r}_{2}$) and the sum of their angles (${\theta}_{1} + {\theta}_{2}$) as shown below.

I hope that this was clear.