The Enjoyment of Math The Enjoyment of Math BY HANS RADEMACHER AND OTTO TOEPLITZ TRANSLATED BY HERBERT ZUCKERMAN PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY Published 1957 by Princeton University Press, Princeton, New Jersey In the United Kingdom: Princeton University Press, Chichester, West Sussex All Rights Reserved Library of Congress Card No.: 57-627 ISBN 0-691-07958-7 (hardcover edition) ISBN 0-691-02351-4 (paperback edition) This is a translation from Von Zahlen und Figuren: Proben Mathematischen Denkens fUr Liebhaber der Mathematik, by Hans Rademacher and Otto Toeplitz, second edition originally published by Julius Springer, Berlin, 1933. Chapters 15 and 28 by Herbert Zuckerman have been added to the English language edition. Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources First Princeton Paperback edition, 1966 Second paperback and eighth hardcover printing, 1970 Seventh printing, for the Princeton Science Library, 1994 Printed in the United States of America 15 14 13 12 II 10 9 8 7 Preface Otto Toeplitz, co-author of this book, died in Jerusalem on Fe­ bruary 19, 1940, after having left Germany in the Spring of 1939. Toeplitz began his academic career in Gottingen as a disciple of David Hilbert, was then professor in Kiel and later in Bonn. His scientific work is centered around the theory of integral equations and the theory of functions of infinitely many variables, fields to which he has made lasting contributions. The plan for this book arose at frequent meetings which the authors had, while Toeplitz was in Kiel and I was at the University of Hamburg. Both of us had repeatedly lectured about the subject matter of this book to a wider public. The manuscript was rewritten several times under mutual criticism. Toeplitz's great interest in the history of mathematics is still visible in the present edition. I re­ member with warm feelings the summer days in 1929 at Bonn, when together we put the last touches to the German manuscript. I am sure that Toeplitz would have been pleased and proud of the present English edition, a project of which he had often thought. I wish to thank the translator Professor Herbert S. Zuckerman for his painstaking and understanding work. Not only do I admire his apt translation of the German text, but I also think that in his pre­ sentation of the content he has brought many of its arguments closer to the English speaking reader. He has added two chapters (15 and 28) to the book, which faithfully reflect the spirit in which this book was written. My thanks go to my friends Emil Grosswald, D. H. Lehmer, and Herbert Robbins for help and valuable advice, and also to the publisher for his sympathetic cooperation. Philadelphia, 1956 HANS RADEMAGHER CONTENTS Preface ν Introduction 5 1. The Sequence of Prime Numbers 9 2. Traversing Nets of Curves 13 3. Some Maximum Problems 17 4. Incommensurable Segments and Irrational Numbers 22 5. A Minimum Property of the Pedal Triangle 27 6. A Second Proof of the Same Minimum Property 30 7. The Theory of Sets 34 8. Some Combinatorial Problems 43 9. On Waring's Problem 52 10. On Closed Self-Intersecting Curves 61 11. Is the Factorization of a Number into Prime Factors Unique? 66 12. The Four-Color Problem 73 13. The Regular Polyhedrons 82 14. Pythagorean Numbers and Fermat's Theorem 88 15. The Theorem of the Arithmetic and Geometric Means 95 16. The Spanning Circle of a Finite Set of Points 103 17. Approximating Irrational Numbers by Means of Rational Numbers 111 18. Producing Rectilinear Motion by Means of Linkages 119 19. Perfect Numbers 129 20. Euler's Proof of the Infinitude of the Prime Numbers 135 21. Fundamental Principles of Maximum Problems 139 22. The Figure of Greatest Area with a Given Perimeter 142 23. Periodic Decimal Fractions 147 24. A Characteristic Property of the Circle 160 25. Ciuves of Constant Breadth 163 26. The Indispensability of the Compass for the Constructions of Elementary Geometry 177 27. A Property of the Number 30 187 28. An Improved Inequality 192 Notes and Remarks 197 The Enjoyment of Math Introduction Mathematics, because of its language and notation and its odd- looking special symbols, is closed off from the surrounding world as by a high wall. What goes on behind that wall is, for the most part, a secret to the layman. He thinks of dull uninspiring numbers, of a lifeless mechanism which functions according to laws of inescapable necessity. On the other hand, that wall very often limits the view of him who stays within. He is prone to measure all mathematical things with a special yardstick and he prides himself that nothing profane shall enter his realm. Is it possible to breach this wall, to present mathematics in such a way that the spectator may enjoy it? Cannot the enjoyment of mathematics be extended beyond the small circle of those who are "mathematically gifted"? Indeed, only a few are mathematically gifted in the sense that they are endowed with the talent to discover new mathematical facts. But by the same token, only very few are musically gifted in that they are able to compose music. Never­ theless there are many who can understand and perhaps reproduce music, or who at least enjoy it. We believe that the number of people who can understand simple mathematical ideas is not relatively smaller than the number of those who are commonly called musical, and that their interest will be stimulated if only we can eliminate the aversion toward mathematics that so many have acquired from childhood experiences. It is the aim of these pages to show that the aversion toward mathematics vanishes if only truly mathematical, essential ideas are presented. This book is intended to give samples of the diversified phenomena which comprise mathematics, of mathematics for its own sake, and of the intrinsic values which it possesses. The attempt to present mathematics to nonmathematicians has often been made, but this has usually been done by emphasizing the usefulness of mathematics in other fields of human endeavor in an effort to secure the comprehension and interest of the reader. Frequently the advantages which it offers in technological and other applications have been described and these advantages have been illustrated by numerous examples. On the other hand, many books INTRODUCTION have been written on mathematical games and pastimes. Although these books contain much interesting material, they give at best a very distorted picture of what mathematics really is. Finally, other books have discussed the foundations of mathematics with regard to their general philosophical validity. A reader of the following pages who is primarily interested in the pure, the absolute mathema­ tics will naturally direct his attention toward just such an epistemolo- gical evaluation of mathematics. But this seems to us to be attaching an extraneous value to mathematics, to be judging its value according to measures outside itself. In the following pages we will not be able to demonstrate the effects of the ideas to be presented on the domain of mathematics itself. We cannot consider the interior applications, so to speak, of mathematics, the use of the ideas and results of one field in other fields of mathematics. This means that we must omit something that is quite essential in the nature of the mathematical edifice, the great and surprising cross-connections that permeate this edifice in all directions. This omission is quite involuntary on our part, for the greatest mathematical discoveries are those which have revealed just such far-reaching interrelations. In order to present these inter­ relations, however, we would need long and comprehensive prepara­ tions and would have to assume a thorough training on the part of the reader. This is not our intention here. In other words, our presentation will emphasize not the facts as other sciences can disclose them to the outsider but the types of phenomena, the method of proposing problems, and the method of solving problems. Indeed, in order to understand the great mathematical events, the comprehensive theories, long schooling and persistent application would be required. But this is also true with music. On going to a concert for the first time one is not able to appreciate fully Bach's "The Art of Fugue," nor can one immediately visualize the structure of a symphony. But besides the great works of music there are the smaller pieces which have something of true sublimity and whose spirit reveals itself to everyone. We plan to select such "smaller pieces" from the huge realm of mathematics: a sequence of subjects each one complete in itself, none requiring more than an hour to read and understand. The subjects are independent, so that one need not remember what has gone before when reading any chapter. Also, the reader is not required to remember what he may have been compelled to learn in his younger years. No use is made of logarithms or trigonometry. No mention is made of β .