Complex Numbers

Complex numbers are numbers that consist of a real and imaginary part. Usually it is of the form $a + jb$ where $a$ and $b$ are real values and $j$ represents $\sqrt{-1}$. Complex numbers can be visualized as a plane where the real part is the x-axis and the imaginary part is the y-axis.

Polar Form

Complex numbers can be thought of as Cartesian points on a plane, and subsequently, they can be expressed as polar coordinates.

$$ \begin{align} z &= x + jy \ &= r\cdot(\cos \phi + j\sin \phi) \ &= r\cdot e^{j\phi} \end{align} $$

where

$$ \begin{align} r &= |x+y| = \sqrt{x^2 + y^2} \ \phi &= atan \left( \frac{y}{x} \right) \ \end{align} $$

Complex Conjugate

The complex conjugate is the number with an equal real part and an imaginary part opposite in sign.