COMPLEX NUMBERS LIBRARY OF MATHEMATICS edited by WA LTER LEDERMANN D.Sc., Ph.D., F.R.S.Ed., Professor of Mathematics, University of Sussex Linear Equations P. M. Cohn Sequences and Series J. A. Green Differential Calculus P. J. Hilton Elementary Differential Equations and Operators G. E. H. Reuter Partial Derivatives P. J. Hilton w. Complex Numbers Ledermann Principles of Dynamics M. B. Glauert Electrical and Mechanical Oscillations D. S. Jones Vibrating Strings D. R. Bland Vibrating Systems R. F. Chisnell Fourier Series I. N. Sneddon Solutions of Laplace's Equation D. R. Bland Solid Geometry P. M. Cohn Numerical Approximation B. R. Morton Integral Calculus W. Ledermann Sets and Groups J. A. Green Differential Geometry K. L. Wardle Probability Theory A. M. Arthurs w. Multiple Integrals Ledermann COMPLEX NUMBERS BY WALTER LEDERMANN LONDON: Routledge & Kegan Paul Ltd NEW YORK: Dover Publications Inc First published I960 in Great Britain by Routledge & Kegan Paul Limited Broadway House, 68-74 Carter Lane London, E.C.4 and in the U.S.A. by Dover Publications Inc. I80 Varick Street New York, IOOI4 © Walter Ledermann I960, I96z Second impression (with some corrections) I96z Reprinted I964, I96S, I967 No part of this book may be reproduced in any form without permission from the publisher, except for the quotation of brief passages in criticism Library of Congress Catalogue Card Number: 66-ZIZ4Z ISBN-13: 978-0-7100-8634-1 e-ISBN-13: 978-1-4684-7730-6 DOl: 10.1 007/978-1-4684-7730-6 by Latimer Trend & Co Ltd Plymouth Preface THE purpose of this book is to present a straightforward introduction to complex numbers and their properties. Complex numbers, like other kinds of numbers, are essen tially objects with which to perform calculations according to certain rules, and when this principle is borne in mind, the nature of complex numbers is no more mysterious than that of the more familiar types of numbers. This formal approach has recently been recommended in a Reportt prepared for the Mathematical Association. We believe that it has distinct advantages in teaching and that it is more in line with modern algebraical ideas than the alternative geometrical or kinematical definitions of ..; - 1 that used to be proposed. On the other hand, an elementary textbook is clearly not the place to enter into a full discussion of such questions as logical consistency, which would have to be included in a rigorous axiomatic treatment. However, the steps that had to be omitted (with due warning) can easily be filled in by the methods of abstract algebra, which do not conflict with the 'naive' attitude adopted here. I should like to thank my friend and colleague Dr. J. A. Green for a number of valuable suggestions, especially in connection with the chapter on convergence, which is a sequel to his volume Sequences and Series in this Library. WAL TER LEDERMANN t The Teaching of Algebra in Sixth Forms, Chapter 3. (G. Bell & Sona, Ltd., London, 1957.) v Contents Preface page v 1. Algebraic Theory of Complex Numbers 1. Number Systems 1 2. The Algebraic Theory 5 2. Geometrical Representations 14 3. Roots of Unity 33 4. Elementary Functions of a Complex Variable 1. Introduction 43 2. Sequences 44 3. Series 47 4. Power Series 49 5. The Functions eZ, cos z, sin z 52 6. The Logarithm 56 Answers to Exercises 60 Index 62 VI CHAPTER ONE Algebraic Theory of Complex Numbers 1. NUMBER SYSTEMS Before defining complex numbers let us briefly review the more familiar types of numbers and let us examine why there are different kinds of numbers. The most primitive type of number is the set of natural numbers 1, 2, 3, ..., which the child learns for counting objects. Arithmetic, the science of numbers, is based on the fact that numbers can be added and multiplied, subject to certain rules, to which we shall presently return in more detail. It is the existence of these two laws of composition and their mutual relation that we shall regard as the typical feature of all numbers and that will serve us as a guide for introducing new systems of numbers for various pur poses. Let us recall how in the school curriculum we proceed from the natural numbers to more elaborate systems. The attempt to make subtraction always possible, that is to solve the equation a+x=b for x when a and b are given, leads to the introduction of zero (one of the great achievements of the human mind \) and of the negative numbers. We now have the set of all integers (whole numbers) •.• - 3, - 2, - 1,0, 1, 2, 3, .•. Next, when we wish to carry out division, we have to solve equations of the form ax=b, where a and b are given integers and a is non-zero. In order to make the solution possible in all cases it is necessary to introduce the rational numbers (fractions). These numbers are denoted by symbols bfa, where a and b are integers and a is non-zero. When this stage has been reached, the four rules of arith- 1 ALGEBRAIC THEORY OF COMPLEX NUMBERS metic, that is addition, subtraction, multiplication and division apply without restriction, always excepting division by zero. These basic operations are governed by the follow ing general laws, which are of fundamental importance in mathematics. I. a+b=b+a (commutative law of addition). II. (a+b)+c=a+(b+c) (associative law of addi tion). III. a+x=b has a unique solution, written x=b-a (law of subtraction). IV. ab=ba (commutative law of multiplication). V. (ab)c=a(bc) (associative law of multiplica tion). VI. ax=b (a:;&O) has a unique solution x=bja (law of division). VII. (a+b)c=ac+bc (distributive law). Most of these laws, perhaps in a different guise, are so familiar to the reader that he might be unaware of their existence. Thus the associative law of addition implies that a column of figures can be added by starting either from the top or from the bottom. Again, the distributive law is more popularly known as the principle of multiplying out brackets. The rational numbers are adequate for dealing with the more elementary questions of arithmetic, but their defici ency becomes apparent when we consider such problems as extracting square roots. For example, it can be shown that y2 cannot be expressed in the form min, where m and n are integers, i.e. there are no integers m, n ( :F 0) such that ml=2nl. Again, when we pass from algebra to analysis, where limits of sequences playa fundamental part, we find that the limit of a sequence of rational numbers is not necessarily a rational number.t The situation may be described by using a single co-ordinate axis t See J. A. Green, S.qu.nces nnd Series, in this series, p. 7. 2 NUMBER SYSTEMS -~a1 ------~I---4I--+I-----~tI -+I--~I~·--+I~----+I------+I~-+~ -2.,.f -I 0 1:1 2 3 Figure I on which in the first place we mark all the integers in a certain scale. Then we imagine all the rational numbers inserted, e.g. - 7/5, -1/4, 1/2, ... But even when this has been done, there will be many points on the line against which no number has been entered. For instance when we lay down a segment of length .y2 (the diagonal of a square of unit sides) by placing one end at 0, the other end-point falls on a point of the scale which has as yet no number attached to it. On the other hand, we intuitively accept the fact that every segment ought to have a length which is measured by some 'number'. In other words, we postulate that every point on the axis possesses a co-ordinate which is a definite number, positive if the point is on the right of 0 and negative if it is on the left of O. This number need not be a rational number. The set of numbers which in this way fill the whole line, is called the set of real numbers; they comprise the familiar rational numbers, the remaining real numbers being called irrational, such as .y2, e, log 2, 'IJ', etc. (Of course, the word irrational means that the number is not the ratio of two integers and has nothing to do with the idea that something irrational is beyond the realm of reason.) Alternatively, the real numbers may be described as the set of all decimal fractions. A terminating or a recurrent decimal fraction corresponds to a rational number, whilst the other fractions represent irrational numbers. From the way in which real numbers are depicted on a line it is clear that there exists an order relation among them, that is any two real numbers a and b satisfy either a=b or a <b or a >b. This is indeed an important property when we wish to use numbers for measuring. But in the present algebraical context we are much more concerned with the fact that real numbers, like rational numbers, can be added 3 ALGEBRAIC THEORY OF COMPLEX NUMBERS and multiplied and that they obey the laws I to VII listed on p. 2. We take the view that the existence of the two modes of composition with their laws makes numbers de serve their name. Numbers are essentially things to be com puted, and any other properties, however useful for certain purposes, are not part of the definition of number. One of these secondary properties is the fact that real numbers can be classified into positive and negative numbers together with the usual deductions from it, such as 'the product of two negative numbers is positive'. For a long time it was held that arithmetic had reached saturation with the introduction of the complete set of real numbers. Indeed, there was no obvious geometrical or technical problem that called for the creation of new num bers. Yet, one of the simplest algebraical questions remains in an unsatisfactory state when only real numbers are avail able. For we should then be forced to admit that some quadratic equations have solutions whilst others have none. On the other hand, it is easy to see that all quadratic equa tions would have solutions if only we could solve the special equation (1.1) for this would assign a meaning to 1/ - 1 and hence to 1/ - a, where a is any positive number. Indeed, we could simply put 1/-a=1/-11/a. Now it is obvious that (1.1) cannot have a real solution, since if x is real, x2 is never negative and cannot therefore be equal to - 1. So in order to make (1.1) soluble we have to introduce a new type of number, for which the rule 'the square of any number is positive' certainly does not hold. But this rule, or indeed anything else concerning positiveness and negativeness is not a consequence of the seven fundamental laws listed on p. 2, and it is therefore quite conceivable that these laws can be satisfied by symbols or numbers to which the terms positive and negative do not apply. We now formally introduce a symbol i which wo treat in 4