Complex Numbers
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
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LIBRARY OF MATHEMATICS edited by
WALTER LEDERMANN D.Sc., Ph.D., F.R.S.Ed., Professor of Mathematics, University of Sussex
Linear Equations Sequences and Series Differential Calculus Elementary Differential Equations and Operators Partial Derivatives Complex Numbers Principles of Dynamics Electrical and Mechanical Oscillations Vibrating Strings Vibrating Systems Fourier Series Solutions of Laplace's Equation Solid Geometry Numerical Approximation Integral Calculus Sets and Groups Differential Geometry Probability Theory Multiple Integrals
P. M. Cohn J. A. Green P. J. Hilton G. E. H. Reuter P. J. Hilton w. Ledermann M. B. Glauert D. S. Jones D. R. Bland R. F. Chisnell I. N. Sneddon D. R. Bland P. M. Cohn B. R. Morton W. Ledermann J. A. Green K. L. Wardle A. M. Arthurs w. 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 essentially 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 page v
Preface
1. Algebraic Theory of Complex Numbers 1. Number Systems 2. T
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