Complex Numbers

94 Pages·2017·2.1 MB·English

by  Demetrios P. Kanoussis, Kalamos Attikis

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COMPLEX NUMBERS

1) A concise and complete introduction to Complex Numbers Theory
2) An excellent supplementary text for all Mathematics, Physics and Engineering students
3) 90 solved illustrative examples and 180 characteristic problems to be solved
4) Odd-numbered problems are provided with answers

Demetrios P. Kanoussis
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About the Author
Demetrios P. Kanoussis, Ph.D
Kalamos Attikis, Greece
[email protected]

Dr. Kanoussis is a professional Electrical Engineer and Mathematician. He received his Ph.D
degree in Engineering and his Master degree in Mathematics from Tennessee Technological
University, U.S.A, and his Bachelor degree in Electrical Engineering from the National
Technical University of Athens (N.T.U.A), Greece.
As a professional Electrical Engineer, Dr. Kanoussis has been actively involved in the design
and in the implementation of various projects, mainly in the area of the Integrated Control
Systems.
Regarding his teaching experience, Dr. Kanoussis has long teaching experience in the field of
Applied Mathematics and Electrical Engineering.
His original scientific research and contribution, in Mathematics and Electrical Engineering,
is published in various, high impact international journals.
Additionally to his professional activities, teaching and research, Dr. Kanoussis is the author
of several textbooks in Electrical Engineering and Applied Mathematics. A list of his
publications is shown below:
Mathematics Textbooks
1) Complex Numbers, an Approach of Understanding, e-book, June 2017.
2) Infinite Series and Products, e-book, April 2017.
3) Sequences of Real and Complex Numbers, e-book, March 2017.
4) Algebraic Equations, e-book, February 2015.
5) Topics in Applied Mathematics, paperback, November 2011, (Greek Edition).
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Electrical Engineering Textbooks
1) Direct Current Circuits Analysis, Vol.2, e-book, May 2017.
2) Introduction to Electric Circuits Theory, Vol. 1, e-book, May 2017.
3) Introduction to Electric Circuits Theory, paperback, August 2013, (Greek Edition).
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Complex Numbers
Copyright 2017, Author: Demetrios P. Kanoussis.
All rights reserved. No part of this publication may be reproduced, distributed or
transmitted in any form or by any means, electronic or mechanical, without the prior
written permission of the author, except in the case of brief quotations and certain other
noncommercial uses permitted by copyright law.
Inquires should be addressed directly to the author,
Demetrios P. Kanoussis
[email protected]
This e book is licensed for your personal use only. This e book may not be resold or given
away to other people. If you would like to share this book with another person, please
purchase an additional copy for each recipient.
Thank you for respecting the work of this author.
First edition, June 2017
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Preface
Solving polynomial equations has been one of the most traditional Algebra topics during the
past few centuries. However, there are equations which do not have any real solution.
For instance the equation has no real roots, since if is any real number,
.
Mathematicians efforts to solve such types of equations, led gradually to the development
of complex numbers. In the set of complex numbers, the symbol does make sense and
has a certain meaning, while in the set of real numbers the symbol does not make any
sense at all.
The introduction of the symbol (the imaginary unit) to stand for , by the great
Mathematician L. Euler in 1777, has contributed immensely to the development of complex
numbers and complex functions in general.
In contemporary terms, a complex number is any number in the form , where and
are real numbers and .
Complex numbers have amazing applications, not only in Mathematics but in many areas of
Physics and Engineering as well. In Mathematics complex numbers are used to evaluate
certain types of infinite series and real valued improper integrals, while in Physics and
Engineering in general, complex numbers help to study the flow of fluids around objects, to
analyze Alternating Current Circuits, (a problem of great practical importance), to
investigate the propagation of radio waves through various media, etc.
Also several features of complex numbers make them extremely useful in proving various
geometrical propositions, in simple and elegant ways. As a striking example, we quote the
Cote’s Theorem, (proved in Example 14-8), the proof of which without complex numbers
would be extremely difficult if not impossible.
The current text is a complete and self contained presentation of fundamental concepts,
definitions, theorems and techniques on complex numbers and has been designed to be an
excellent supplementary textbook for all Mathematics, Physics and Engineering students.
In Chapter 1 the complex numbers are introduced as ordered pairs of real numbers and the
elementary operations with them (equality, addition, subtraction, multiplication and
division) are defined.
In Chapter 2 we define the fundamental laws in the Algebra of complex numbers, treated as
ordered pairs.
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In Chapter 3 we introduce the imaginary unit and show the fundamental
property that every complex number can be expressed in the equivalent form
, where and are real numbers.
In Chapter 4 the concept of the conjugate complex number of is introduced and various
identities involving complex conjugate numbers are proved. Two important Theorems
concerning the roots of polynomial equations with real coefficients are also presented and
proved, with the aid of the properties of complex conjugate numbers.
In Chapter 5 we define the absolute value or the modulus of a complex number , and
derive a number of important properties related to .
In Chapter 6 we derive the so called Trigonometric or Polar form of a complex number, and
prove two important Theorems about the product and the division of two complex numbers
expressed in Trigonometric form.
In Chapter 7 we state and prove the De Moivre’s Theorem and show how this Theorem is
applied in order to express and in terms of and , .
In Chapter 8 we show that every complex number has exactly distinct roots.
In Chapter 9 we derive the roots of unity and discuss some fundamental properties of
the cyclotomic equation .
In Chapter 10 we introduce the famous Euler’s formulas and by means of these formulas
we extent the trigonometric functions from real values of the argument to complex values
of the argument.
In Chapter 11 we introduce the hyperbolic functions of real and complex arguments and
derive the relationship between trigonometric and hyperbolic functions.
In Chapter 12 the exponential form of a complex number is derived, by means of Euler’s
formulas.
In Chapter 13 we define the logarithm of complex numbers, and as a consequence the
logarithm of negative numbers are obtained, (note that within the set of real numbers the
logarithm of a negative number simply does not exist)
In Chapter 14 we discuss the geometrical representation of complex numbers on the
complex plane (Argand’s diagram), and show that geometrically a complex number can be
equivalently represented by a vector, and therefore various properties of complex numbers
can be proved geometrically, and conversely, geometrical propositions can be proved with
the aid of complex numbers.
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The 90 illustrative solved Examples and the 180 characteristic Problems to be solved have
been chosen to help students develop a solid theoretical background, broaden their
knowledge and sharpen their analytical skills on the subject.
A brief Hint or a detailed outline in solving more complicated Problems is often given.
Finally answers are provided to the odd-numbered Problems.

Demetrios P. Kanoussis
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Table of Contents
1. Introduction………………………………………………………………………………………………………… 01
2. The Fundamental Laws in the Algebra of Complex Numbers……………………............. 13
3. The Imaginary Unit …………………………………………………………………. 16
4. Conjugate Complex Numbers……………………………………………………………………………… 23
5. The Absolute Value or Modulus of a Complex Number………………………………………. 28
6. The Trigonometric or Polar Form of a Complex Number……………………………………… 34
7. De Moivre’s Theorem………………………………………………………………………………………….. 41
8. Roots of Complex Numbers…………………………………………………………………………………. 47
9. The Roots of Unity…………………………………………………………………………………………. 53
10. The Euler’s Formulas………………………………………………………………………………………….. 59
11. The Hyperbolic Functions…………………………………………………………………………………… 65
12. The Exponential Form of a Complex Number……………………………………………………… 71
13. The Logarithm of a Complex Number…………………………………………………………………. 74
14. The Complex Plane (or The Argand’s Diagram)…………………………………………………… 82