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Complex Numbers and Complex Arithmetic

Module by: Carlos E. Davila. E-mail the author

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 zz is zz has the form:

z = a + j b z = a + j b (1)

where aa and bb are real numbers and j=-1j=-1. The real part of zz is the number aa, while the imaginary part of zz is the number bb. We also note that jb(jb)=-b2jb(jb)=-b2 (a real number) since j(j)=-1j(j)=-1. Any number having the form

z = j b z = j b (2)

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

z = r e j θ z = r e j θ (3)

where

r = a 2 + b 2 r = a 2 + b 2 (4)

is the magnitude and

θ = arctan b a θ = arctan b a (5)

is the phase of the complex number zz. The notation for the magnitude and phase of a complex number is given by zz and ∠z∠z, respectively. Using Euler's Identity:

e ± j θ = cos ( θ ) ± j sin ( θ ) e ± j θ = cos ( θ ) ± j sin ( θ ) (6)

it follows that a=rcos(θ)a=rcos(θ) and b=rsin(θ)b=rsin(θ). Figure 1 illustrates how polar coordinates and rectangular coordinates are related.

Figure 1: Relationship between rectangular and polar coordinates.
Figure 1 (pol_plot.png)

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+jbz=a+jb is a complex number then its complex conjugate is defined by

z * = a - j b z * = a - j b (7)

in polar coordinates we have

z * = r e - j θ z * = r e - j θ (8)

note that zz*=|z|2=r2zz*=|z|2=r2 and z+z*=2az+z*=2a. Also, if z1,z2,...,zNz1,z2,...,zN are complex numbers it can be easily shown that

z 1 + z 2 + ⋯ + z N * = z 1 * + z 2 * + ⋯ + z N * z 1 + z 2 + ⋯ + z N * = z 1 * + z 2 * + ⋯ + z N * (9)

and

z 1 z 2 ⋯ z N * = z 1 * z 2 * ⋯ z N * z 1 z 2 ⋯ z N * = z 1 * z 2 * ⋯ z N * (10)

Table 1 indicates how two complex numbers combine in terms of addition, multiplication, and division when expressed in rectangular and in polar coordinates.

Table 1: Operations on two complex numbers, z1=a1+jb1=r1ejθ1z1=a1+jb1=r1ejθ1 and z2=a2+jb2=r2ejθ2z2=a2+jb2=r2ejθ2. The sum of two complex numbers is cumbersome to express in polar coordinates, and is not shown.
operation rectangular polar
z 1 + z 2 z 1 + z 2 ( a 1 + a 2 ) + j ( b 1 + b 2 ) ( a 1 + a 2 ) + j ( b 1 + b 2 )  
z 1 z 2 z 1 z 2 a 1 a 2 - b 1 b 2 + j ( a 1 b 2 + a 2 b 1 ) a 1 a 2 - b 1 b 2 + j ( a 1 b 2 + a 2 b 1 ) r 1 r 2 e j ( θ 1 + θ 2 ) r 1 r 2 e j ( θ 1 + θ 2 )
z 1 / z 2 z 1 / z 2 ( a 1 a 2 + b 1 b 2 ) + j ( b 1 a 2 - a 1 b 2 ) a 2 2 + b 2 2 ( a 1 a 2 + b 1 b 2 ) + j ( b 1 a 2 - a 1 b 2 ) a 2 2 + b 2 2 r 1 r 2 e j ( θ 1 - θ 2 ) r 1 r 2 e j ( θ 1 - θ 2 )

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