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THIS DOCUMENT WILL PROVIDE YOU WITH THE KNOWLEDGE YOU NEED FOR COMPLEX NUMBERS AT UNDERGRADUATE LEVEL
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4. COMPLEX NUMBERSINTRODUCTION
Complex numbers often seem strange when first encountered but it is worth persevering
with them because they provide a powerful mathematical tool for solving several engineering
problems. One of the main applications is to the analysis of Alternating Current (AC) circuits.
Engineers are very interested in these because the mains supply is itself AC and electricity
generation and transportation are dominated by AC voltages and currents. A great deal of
signal analysis and processing uses mathematical models based on complex numbers
because they allow the manipulation of sinusoidal quantities to be undertaken more easily.
Furthermore, the design of filters to be used in communications equipment relies heavily on
their use. One area of particular relevance is control engineering – so much so that control
engineers often prefer to think of control system in terms of a ‘complex plane’ representation
rather than ‘time domain’ representation.
LEARNING OUTCOMES
On completion of this chapter, you will be able to:
Introduce and explain the complex number system.
Represent a complex number in various forms.
Perform algebraic operations with complex numbers in rectangular, polar or
exponential form.
Solve problems involving solutions to complex numbers.
COMPILED BY T. PAEPAE
,TABLE OF CONTENTS
4. COMPLEX NUMBERS .................................................................................................. 0
4.1 THE MEANING OF COMPLEX NUMBERS............................................................ 2
4.1.1 The Extended Number System ........................................................................ 3
4.1.2 Graphical Representation of Complex Numbers .............................................. 5
4.1.3 The Rectangular Form of a Complex Number .................................................. 6
4.1.4 The Polar Form of a Complex Number............................................................. 7
4.1.5 The Exponential Form of a Complex Number ................................................ 12
4.2 ALGEBRAIC OPERATIONS WITH COMPLEX NUMBERS ................................. 15
4.2.1 Powers of j ..................................................................................................... 16
4.2.2 Equality of Complex Numbers ........................................................................ 17
4.2.3 Algebraic Operations with Complex Numbers in Rectangular Form ............... 18
4.2.4 Arithmetic with Complex Numbers in Polar Form ........................................... 23
4.2.5 Arithmetic with Complex Numbers in Exponential Form ................................. 26
4.3 DE MOIVRE’S THEOREM.................................................................................... 30
4.3.1 Introduction .................................................................................................... 31
4.3.2 Powers of Complex Numbers ........................................................................ 32
4.3.3 Roots of Complex Numbers ........................................................................... 33
4.4 APPLICATIONS OF COMPLEX NUMBERS ........................................................ 36
4.4.1 Theory of Determinants Applied to Solve Complex Numbers ......................... 37
4.4.2 Application of Complex Numbers in Alternating Current Theory ..................... 39
1
,4.1 THE MEANING OF COMPLEX NUMBERS
Why it is important to understand: Complex Numbers
“Complex numbers are used in many scientific fields, including engineering,
electromagnetism, quantum physics, and applied mathematics, such as chaos theory. Any
physical motion which is periodic, such as an oscillating beam, string, wire, pendulum,
electronic signal, or electromagnetic wave can be represented by a complex number
function. This can make calculations with the various components simpler than with real
numbers and sines and cosines. In control theory, systems are often transformed from the
time domain to the frequency domain using the Laplace transform. In fluid dynamics,
complex functions are used to describe potential flow in two dimensions. In electrical
engineering, the Fourier transform is used to analyse varying voltages and currents.
Complex numbers are used in signal analysis and other fields for a convenient description
for periodically varying signals. This use is also extended into digital signal processing and
digital image processing, which utilise digital versions of Fourier analysis (and wavelet
analysis) to transmit, compress, restore, and otherwise process digital audio signals, still
images, and video signals. Knowledge of complex numbers is clearly essential for further
studies in so many engineering disciplines”. Bird, J., 2017. Higher engineering mathematics.
Routledge.
SPECIFIC OUTCOMES
On completion of this study unit, you will be able to:
Explain why the number system was expanded to include complex numbers.
Define a complex number.
Use an Argand diagram to represent a complex number pictorially.
Convert between different forms of complex numbers.
2
, 4.1.1 The Extended Number System
You have already learned that some quadratic equations have no real solutions. For
instance, the equation
x2 1 0
has no real solution. If we try to solve this equation, we get
x 2 1 so x 1
A problem now arises in that we need to find the square root of a negative number. We know
from experience that squaring both positive and negative numbers yield a positive result
hence, there is no number x that can be squared to produce 1 . To make it possible to
solve all quadratic equations, mathematicians invented an expanded number system, called
the complex number system using an imaginary unit j , defined as
j 1 a pure imaginary unit
where j 1 . By adding real numbers to real multiples of this imaginary unit, the set of
2
complex numbers is obtained. Each complex number can be written in the standard form
a bj . The real number a is called the real part of the complex number a bj , and the
number bj (where b is a real number) is called the imaginary part of the complex number.
The set of real numbers is a subset of the set of complex numbers, as shown in the figure
below. This is true because every real number a can be written as a complex number using
b 0 . That is, for every real number a , you can write a a 0 j .
Real Numbers
Complex Numbers
Imaginary Numbers
3
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