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6.4 Polar Form of Complex Numbers
If we use polar coordinates, we can write the real and imaginary parts of a
complex number z = a + jb in terms of the modulus of z and the polar angle θ:

a = cos( )θ = z cos( )θ (6.13)
r

b = sin( )θ = z sin( )θ (6.14)
r

and the complex number z can then be written in polar form as:

z = z cos( )θ + j z sin( )θ = z(cos( )θ + j sin( ))θ (6.15)

The angle θ is called the argument of z and is usually evaluated in the interval
–π≤θ≤π. However, we still have the same complex number if we added to
the value of θ an integer multiple of 2π.

θ = arg( )z
(6.16)
b
tan( θ =)
a
From the above results, it is obvious that the argument of the complex con-
jugate of a complex number is equal to minus the argument of this complex
number.
In MATLAB, the convention for arg(z) is angle(z).

In-Class Exercise
Pb. 6.10 Find the modulus and argument for each of the following complex
numbers:

z =+ j; z = + j; z =− j 2 ; z = − + j 2 ; z = − − j 2
1
1
1
12
2
1 2 3 4 5
Plot these points. Can you detect any geometrical pattern? Generalize.
The main advantage of writing complex numbers in polar form is that it
makes the multiplication and division operations more transparent, and pro-
vides a simple geometric interpretation to these operations, as shown below.

© 2001 by CRC Press LLC