1.3 Complex numbers and complex arithmetic

Before we begin studying signals, we need to review some basic aspects of complex numbers and complex arithmetic. The rectangular coordinate representation of a complex number $z$ is $z$ has the form:

where $a$ and $b$ are real numbers and $j=\sqrt{-1}$ . The real part of $z$ is the number $a$ , while the imaginary part of $z$ is the number $b$ . We also note that $jb\left(jb\right)=-b^2$ (a real number) since $j\left(j\right)=-1$ . Any number having the form

where $b$ is a real number is an imaginary number . A complex number can also be represented in polar coordinates

where

is the magnitude and

is the phase of the complex number $z$ . The notation for the magnitude and phase of a complex number is given by $\left|z\right|$ and $\angle z$ , respectively. Using Euler's Identity:

it follows that $a=rcos\left(\theta \right)$ and $b=rsin\left(\theta \right)$ . [link] illustrates how polar coordinates and rectangular coordinates are related.

Rectangular coordinates and polar coordinates are each useful depending on the type of mathematical operation performed on the complex numbers. Often, complex numbers are easier to add in rectangular coordinates, but multiplication and division is easier in polar coordinates. If $z=a+jb$ is a complex number then its complex conjugate is defined by

in polar coordinates we have

note that $z{z}^{*}={\left|z\right|}^2=r^2$ and $z+z^{*}=2a$ . Also, if $z_1,z_2,...,z_N$ are complex numbers it can be easily shown that

and

[link] indicates how two complex numbers combine in terms of addition, multiplication, and division when expressed in rectangular and in polar coordinates.