The Enjoyment of Math PDF
Preview The Enjoyment of Math
ThEen joyomfMe antth ThEen joyomfe nt Math BY HANS RADEMACHER AND OTTO TOEPLITZ TRANSLATED BY HERBERT ZUCKERMAN With a new foreword by Alex Kontorovich PRINCETON UNIVERSITY PRESS PRINCETON OXFORD Published 1957 by Princeton University Press, 41 William Street, Princeton, New Jersey In the United Kingdom: Princeton University Press, 99 Banbury Road, Oxford OX2 6JX Foreword to theA Plrl iRncigehttosn R Secsieernvecde Library edition copyright © 2023, by Princeton University Press This is a translation from Von Zahlen undFiguren: Proben Mathematischen Denkens fiir 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. First published 1957 New Princeton Science Library paperback edition, with a new foreword by Alex Kontorovich, 2023 Paperback ISBN 9780691241548 E-book ISBN 9780691241531 LCCN: 2022942877 Cover design by Michael Boland for TheBolandDesignCo.com Cover image: iStock press.princeton.edu Printed in the United States of America CONTENTS Foreword by Alex Kontorovich vii P reface 1 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. Curves of Constant Breadth 163 v contents 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 vi Foreword The art form of mathem atics is well-k nown to have a marketing prob- lem. The great German mathematician Carl Friedrich Gauss famously said, “Mathem atics is the queen of science.”1 And yet this queen, direct- ly responsible for much of our modern existence and all its extranatural comforts, is greatly underappreciated, if not outright despised, by most of the educated public. I c an’t tell you how many people I’ve met who have told me, “I hate math.” Upon further prodding, they’ll admit that in fact they loved math up to Subject X (where X might be calculus, or trigonometry, or algebra, or what ever), and then Subject X + 1 was the worst and they’ve hated math ever since. (Upon even further prodding, it turns out they really hated their teacher in Subject X + 1, hence never learning it properly, and everyt hing thereafter was gobbledygook. Many of them also admit that as adults, they love struggling to solve difficult puzzles like Sudoku, KenKen, and so forth). You don’t hear such statements nearly as often about a closely related subject, physics. I’ve always wondered why. For one explanation, physics is less often a requirement in school, so the only people subjected to its rigors are there voluntarily. Another, perhaps, and much closer to the discussion at hand, is that a g reat number of Nobel laureate physicists write public accounts of their discoveries, wowing us with faraway galax- ies, quantum field theories, Earth-l ike Goldilocks planets, black holes, quantum entanglement, and so on. The rec ord for mathematicians of the highest caliber “wasting” their im mense talents and invaluable time by trying to reason with the general public is far, far sparser. This is what makes the pres ent book all the more remarkable. H ere are two world-r enowned mathematicians making a valiant attempt at showing the public some true gems of our beloved subject. Hans Rademacher and Otto Toeplitz were preeminent researchers who made fundamental contributions to twentieth- century mathe matics. Toeplitz, born in 1881 in Breslau (modern- day Poland) to a Jewish family, cut his teeth at the University of Gottingen, which was the mathematical home of every one from Gauss to Riemann to his own teachers Hilbert, Klein, and Minkowski. When the Nazis came to power, Toeplitz was dismissed from his professorship at Bonn, and he moved to the Hebrew Universi- 1 He continued, “and Number Theory”— which features heavi ly in the book before you— “is the queen of Mathe matics.” vii foreword ty in Jerusalem, where he died in 1940 from tuberculosis. Rademacher, about a de cade Toeplitz’s ju nior, hailed from the Hamburg area and overlapped with his coauthor in Gottingen, where he earned his PhD in 1916. While Rademacher was racially acceptable to the Nazis, his pacifist views were not, and he was likewise dismissed from his position at the University of Breslau. Rademacher moved to the United States to take an assistant professorship at the University of Pennsylvania (despite already having served a dec ade as a full professor in Germany), from which he retired in 1962, passing away not long a fter in 1969. Their combined interest in and passion for popularizing mathe matics, crys- talized over many lectures to the public, came to fruition as Von Zahlen und Figuren, released in German in 1930 and translated into the text before you in 1957. The topics presented within are not the recreational material one sometimes encounters; these are the “real deal.” The first chapter, for example, begins with Euclid’s argument that the prime numbers are plentiful: for any finite collection of them, more exist; this is followed immediately with a contradistinction: one may chance upon an arbi- trarily long distance between one prime to the next. Such issues are quite deep, forming the basis of ongoing research today, and yet their pre sen ta tion here is completely “elementary,” in the sense of not re- quiring any technical knowledge beyond the fifth grade. Instead, what this book asks of its audience is a penchant for finding t hese questions of interest, as well as the determination and stamina to chew over the arguments presented until they coalesce. This duality of render- ing advanced disciplines in an elementary style applies to most of the topics presented, ranging from geometry and arithmetic to topology and graph theory. Some chapters, like that describing the Four-C olor Prob lem (now a theorem, resolved only in 1976), describe the state of things in the 1930s but of course not our present- day understanding. Other chapters pre sent topics that have since been showcased in new ways to the public. For example, chapter 25 discusses curves of constant breadth; today, someone wanting to learn about such curves can visit the National Museum of Mathe matics in New York City and take a spin on some non- spherical coasters which nonetheless ride smoothly. Overall, the topics seem to me on the advanced side for t oday’s general audience. Maybe today we know more about what the general public can and cannot typically follow. Or maybe previous generations experienced so much more daily hardship that a few hours of strugg le with a mathematical chapter (or just a paragraph, or even a sentence) viii foreword would, by comparison, not seem so insurmountable. On the other hand, I’m reminded of a story that Keith Devlin, the author of Introduc- tion to Mathematical Thinking and other popu lar math books, told me some time ago about the first time he wrote something and submitted it to a local newspaper. The editor told him that he’d be delighted to have him contribute something, but what Keith had written was way too com plicated for the audience. He’d need to craft something that people could read for five minutes over a cup of coffee (and when Keith’s me dium moved to radio, he was similarly told that his topic needed to be digestible while the average Joe was pumping his gas). Now, by keeping his material accessible, Keith has been tremendously successful with communicating with the public. But the modern experience of mass media, with significantly less gate keeping and editorializing, has shown several t hings about the public audience. In defiance of conventional media wisdom, audiences have an appetite for longf orm content, whether it’s a fourteenh our Netflix show or an extended series of podcasts, and this has proven true across virtually all va ri e ties of content form and subject matter. People will gladly watch, by the millions and tens of millions, videos on very ad vanced graduatel evel mathe matics, when presented by masters of expo sition like the YouTube channels 3Blue1Brown, Veritasium, and many others. All this is to say that, perhaps if this book were to be proposed today, its authors might be told by editors to water down the subjects and discussion; but since it has already achieved a significant amount of success, it has earned its right to be as it is, and it remains an essential entry point for those audiences who hunger for a deeper exploration of mathe matics. I sincerely hope that you, dear reader, will give its message the time required to strugg le with the difficult, import ant, and timeless material; you will be greatly rewarded. We seem to be in a bit of a re nais sance for math exposition, both in terms of mathematicians showing greater re spect for expository writing and in the appreciation of the general public for books like these. Let us hope that another generation may be inspired by the writings within. New Brunswick, New Jersey, 2022 ALEX KONTOROVICH ix
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