Multiplication in polar format

Complex numbers: The complex plane

Multiplication in polar format

The modulus and argument of a complex number are central in working in polar form. If you multiply two complex numbers, then the modulus and argument of the product is easyly expressed in these two factors of the product.

Let $z_1=r_1\,e^{\,\mathrm{i}\,\varphi_1}$ and $z_2=r_2\,e^{\,\mathrm{i}\,\varphi_2}$ , then $\begin{aligned}z_1\cdot z_2&= r_1r_2\, e^{\,\mathrm{i}\,(\varphi_1+\varphi_2)}\\ \\ \frac{z_1}{z_2}&= \frac{r_1}{r_2}\, e^{\,\mathrm{i}\,(\varphi_1-\varphi_2)}\end{aligned}$

There is also a geometrical structure that corresponds with the multiplication and can be visualized by the following dynamic figure: the draggable red points represent complex numbers and for the green dot represents the product of these numbers. The drawn triangles are similar.