Easy as π?: An Introduction to Higher Mathematics PDF
Preview Easy as π?: An Introduction to Higher Mathematics
Easy as 1[? Springer-Science+Business Media, LLC O.A.lvanov Easy as n? An Introduction to Higher Mathematics Translated by Robert G. Bums With 60 Illustrations , Springer O.A. Ivanov Translator: Department of Mathematics and Robert G. Bums Mechanies Department of Mathematics St. Petersburg State University and Statistics Bibliotechnaya pI. 2, Stary Petergof York University St. Petersburg, 198904 4700 Keele Street Russia Toronto, Ontario MB IP3 Canada Library of Congress CataIoging-in-Publication Data Ivanov, O.A. (Oleg A.) Easy as pi? : an introduction to higher mathematics I O.A. Ivanov. p. cm. Inc1udes bibliographical references and index. ISBN 978-0-387-98521-3 ISBN 978-1-4612-0553-1 (eBook) DOI: 10.1007/978-1-4612-0553-1 I. Mathematics. I. Title. QA37.2.187 1998 51O-dc21 98-16710 Printed on acid-free paper. First Russian Edition: 1136paHHble rAaBbI 3AeMeHTapHoß MaTeManIKH, St. Petersburg, 1995. © 1999 Springer Science+Business Media New York Originally published by Springer-Verlag New Y ork Berlin Heidelberg in 1999 Softcover reprint ofthe hardcover I st edition 1999 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC, except for brief excerpts in cOmIection with reviews or scholarly analysis. Use in cOmIection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, Irade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Timothy Taylor; manufacturing supervised by Jeffrey Taub. Photocomposed copy prepared from the translator's ~TEJX file. 9 8 7 6 5 4 3 2 ISBN 978-0-387-98521-3 SPIN 10773312 Foreword The present book is rare, even unique of its kind, at least among mathematics texts published in Russian. You have before you neither a textbook nor a monograph, although these selected chapters from elementary mathematics certainly constitute a fine educational tool. It is my opinion that this is more than just another book about mathematics and the art of teaching that subject. Without considering the actual topics treated (the author himself has described these in sufficient detail in the Introduction), I shall attempt to convey a general idea of the book as a whole, and describe the impressions it makes on the reader. Almost every chapter begins by considering well-known problems ofe lementary mathematics. Now, every worthwhile elementary problem has hidden behind its diverting formulation what might be called "higher mathematics," or, more simply, mathematics, and it is this that the author demonstrates to the reader in this book. It is thus to be expected that every chapter should contain subject matter that is far from elementary. The end result of reading the book is that the material treated has become for the reader "three-dimensional" as it were, as in a hologram, capable of being viewed from all sides. It is difficult to say exactly how this effect is achieved, whether through the apt choice of problems, or their well-planned arrangement, or the judicious placement of an exercise (whose formulation is by itself often important), or even literary technique-all are significant: the subject matter, style, language, and also a sense of the author himself behind the text. As with every serious book, far from everyone will find "Easy as n?" so easy: in order for the reader to understand it in all of its ramifications, he or she will need to be possessed already of a rather sophisticated culture--in the present case mathematical. It is even possible that the greatest pleasure from reading this book will be derived by the professional mathematician. Be that as it may, one hopes that some of its readers will at least trace through the developments out of the vi Foreword elementary topics, as expounded here, and then relay these to his or her students. In any case it is certainly a good thing that such a book has been written, and I recommend it with pleasure to a wide readership. A.S. Merkurjev University of California at Los Angeles (Formerly at St. Petersburg State University, Russia) Thus ev 'ry kind their pleasure find. Robert Burns (1759-1796) Preface This book represents an expanded version oflecture notes of a course given by the author over a period of several years to fourth-and fifth-year students specializing in education in the Mathematics--Mechanics Faculty of St. Petersburg (formerly Leningrad) University. This course (consisting of 60 hours' worth of lectures) was conceived as being of a summarizing, general-mathematical nature, intended to make extensive reference to the concepts and propositions of the more basic mathematics courses. It had seemed clear that mathematics teachers in particular would like the idea of such a course, with "mathematics as a whole" as its subject matter, i.e., without the traditional subdivision into algebra, analysis, geometry, etc. It goes without saying that this course was not in the least intended to replace the more standard basic courses, but rather to supplement and clarify them. Thus Easy as 7r? should be suitable as a text for a wide readership interested in being introduced to modem higher mathematics, i.e. in "enriching" their knowledge of elementary mathematics. However, since it was originally conceived as a textbook for teachers at (special ized) academic middle schools and highschools, and for those training to become such teachers, a few words relevant to this aspect of the book may be in order. (Note in this connection that almost every chapter concludes with pedagogical remarks concerning the material of that chapter.) A teacher at such an institution should have sufficient depth and breadth of knowledge and understanding of math ematics to be capable of designing a mathematical curriculum appropriate to the mandate of his or her institution. For this reason in particular, the education of a teacher must not be restricted to the immediate material being taught. To give an example: while it is not appropriate for a teacher even to mention Peano's axioms for arithmetic in class (except perhaps where the class consists of an "elite," in the best sense of that word), he or she should have an understanding of these axioms so as to be capable at least of explaining to an intelligent eighth-grader where the viii Preface multiplication table comes from, or helping a clever eleventh--grader for whom the method of mathematical induction is not obvious. I am grateful to the translator, Robert Bums, not only for his expert translation of this book, but also for his editorial work. I am sure that the book has been considerably improved as a result of the continual communication between author and translator throughout the course of the translation. O.A. Ivanov St. Petersburg, Russia Translator's Acknowledgments I am grateful to Lydia Bums, Xiao-Min Dong, Anna and Wolfgang Herfort, Siu Man Kam, Yuri Medvedev, Abe Shenitzer, and Pavel Zalesskii variously for help with Russian, mathematics, and Jb.TEX, and for many other kindnesses, while this translation was being carried out at York University, Toronto, and the Technical University, Vienna. Robert G. Bums Toronto, Canada Contents Foreword v Preface vii Introduction xiii 1 Induction 1 1.1 Principle or method? . 1.2 The set of integers . . 3 1.3 Peano's axioms ... 5 1.4 Addition, order, and multiplication 6 1.5 The method of mathematical induction 8 2 Combinatorics 13 2.1 Elementary problems ........ . 13 2.2 Combinations and recurrence relations 16 2.3 Recurrence relations and power series. 21 2.4 Generating functions. . . . . . . . 23 2.5 The numbers 11:, e, and n-factorial . 28 3 Geometric Transformations 32 3.1 Translations, rotations, and other symmetries, in the context of problem-solving . . . . . . . . . . . . . . . . . . . 32 3.2 Problems involving composition of transformations 35 3.3 The group of Euclidean motions of the plane 37 3.4 Ornaments................ 39 3.5 Mosaics and discrete groups of motions ... 42
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