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6

Complex Numbers

6.1 Introduction
2
2
Since x > 0 for all real numbers x, the equation x = –1 admits no real number
as a solution. To deal with this problem, mathematicians in the 18th century
introduced the imaginary number i =− =1 j. (So as not to confuse the usual
symbol for a current with this quantity, electrical engineers prefer the use of
the j symbol. MATLAB accepts either symbol, but always gives the answer
with the symbol i).
Expressions of the form:

z = a + jb (6.1)

where a and b are real numbers called complex numbers. As illustrated in
Section 6.2, this representation has properties similar to that of an ordered
pair (a, b), which is represented by a point in the 2-D plane.
The real number a is called the real part of z, and the real number b is called
the imaginary part of z. These numbers are referred to by the symbols a =
Re(z) and b = Im(z).
When complex numbers are represented geometrically in the x-y coordi-
nate system, the x-axis is called the real axis, the y-axis is called the imaginary
axis, and the plane is called the complex plane.

6.2 The Basics

In this section, you will learn how, using MATLAB, you can represent a com-
plex number in the complex plane. It also shows how the addition (or sub-
traction) of two complex numbers, or the multiplication of a complex number
by a real number or by j, can be interpreted geometrically.

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© 2000 by CRC Press LLC
© 2001 by CRC Press LLC