Complex numbers can be represented in either the cartesian coordinate system ($ z = a + bi $) or the polar coordinate system ($ z = re^{\theta i} $). Here, operations in the polar system will be explained.
Polar Form
One can convert from cartesian to polar with the following formulas:
- $ r = \sqrt{a^2 + b^2} $
- $ \theta = \arctan(\frac{b}{a}) $
In the polar system, multiplication and division are easier than in the cartesian system.
Multiplication of two polar coordinates:
- $ r_1e^{\theta_1i} * r_2e^{\theta_2i} = r_1r_2e^{(\theta_1+\theta_2)i} $
Division of two polar coordinates:
- $ \frac{r_1e^{\theta_1i}}{r_2e^{\theta_2i}} = \frac{r_1}{r_2}e^{(\theta_1-\theta_2)i} $
