## 3.3. Complex numbers

Complex numbers are used in engineering, science, and other technology applications, especially, in control engineering.

Definition of complex number

A complex number is a number that can be written in standard form as

a +

bi, where

a and

b are real numbers and

i is an imaginary unit

$\left(i=\sqrt{-1}\right)$.

We call a the real part of the complex number and bi the imaginary part of the complex number. The imaginary unit (number) i is sometimes denoted by j.

For a complex number, z has the rectangular form as z = x + yi, where x and y are real numbers,

$\begin{array}{c}x=\mathrm{Re}z\left(\text{real}\text{part}\text{of}z\right)\text{,}\\ y=\mathrm{Im}z\left(\text{imaginary}\text{part}\text{of}z\right)\end{array}$

For example, the following are complex numbers:

$\begin{array}{c}0+0i=\end{array}$