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Which Numbers Are Real? Are Numbers Which / MAA AMS AMS / MAA CLASSROOM RESOURCE MATERIALS VOL AMS / MAA CLASSROOM RESOURCE MATERIALS VOL 42 42 Which Numbers Are Real? Which Numbers are Real? MICHAEL HENLE Everyone knows the real numbers, those fundamental quantities that make possible all of mathematics from high school algebra and Euclidean geometry THE SURREAL NUMBERS through the Calculus and beyond; and also serve as the basis for measurement in science, industry, and ordinary life. This book surveys alternative real number systems: systems that generalize and extend the real numbers yet stay close to these properties that make the reals central to mathematics. Alternative real numbers include many different kinds of numbers, for example multidimen- THE HYPERREAL NUMBERS sional numbers (the complex numbers, the quaternions and others), infinitely small and infinitely large numbers (the hyperreal numbers and the surreal numbers), and numbers that represent positions in games (the surreal numbers). Each system has a well-developed theory, including applications to other areas of mathematics and science, such as physics, the theory of games, multi-dimen- sional geometry, and formal logic. They are all active areas of current mathemat- THE CONSTRUCTIVE REALS ical research and each has unique features, in particular, characteristic methods of proof and implications for the philosophy of mathematics, both highlighted in this book. Alternative real number systems illuminate the central, unifying Michael Henle role of the real numbers and include some exciting and eccentric parts of math- ematics. Which Numbers Are Real? Will be of interest to anyone with an interest in THE COMPLEX NUMBERS numbers, but specifically to upper-level undergraduates, graduate students, and professional mathematicians, particularly college mathematics teachers. THE QUATERNIONS AMS/ MAA PRESS 4-Color Process 230 pages on 50lb stock • trim size: 6"x9" • Spine 7/16" i i “master” — 2012/6/18 — 10:53 — page i — #1 i i 10.1090/clrm/042 Which Numbers are Real? i i i i i i “master” — 2012/6/18 — 10:53 — page ii — #2 i i c 2012 by the Mathematical Association of America, Inc. Library of Congress Catalog Card Number 2012937493 Print edition ISBN 978-0-88385-777-9 Electronic edition ISBN 978-1-61444-107-6 Printed in the United States of America Current Printing (last digit): 10987654321 i i i i i i “master” — 2012/6/18 — 10:53 — page iii — #3 i i Which Numbers are Real? Michael Henle Oberlin College Published and Distributed by The Mathematical Association of America i i i i i i “master” — 2012/6/18 — 10:53 — page iv — #4 i i Council on Publications and Communications Frank Farris, Chair Committee on Books Gerald M. Bryce, Chair Classroom Resource Materials Editorial Board Gerald M. Bryce, Editor Michael Bardzell Jennifer Bergner Diane L. Herrmann Philip P. Mummert Barbara E. Reynolds Susan G. Staples Philip D. Straffin Cynthia J Woodburn Holly S. Zullo i i i i i i “master” — 2012/6/18 — 10:53 — page v — #5 i i CLASSROOM RESOURCE MATERIALS Classroom Resource Materials is intended to provide supplementary class- room material for students—laboratory exercises, projects, historical in- formation, textbooks with unusual approaches for presenting mathematical ideas, career information, etc. 101 Careers in Mathematics, 2nd edition edited by Andrew Sterrett Archimedes: What Did He Do Besides Cry Eureka?, Sherman Stein Calculus: An Active Approach with Projects, Stephen Hilbert, Diane Driscoll Schwartz, Stan Seltzer, John Maceli, and Eric Robinson The Calculus Collection: A Resource for AP and Beyond, edited by Caren L. Diefenderfer and Roger B. Nelsen Calculus Mysteries and Thrillers, R. Grant Woods Conjecture and Proof, Mikl´os Laczkovich Counterexamples in Calculus, Sergiy Klymchuk Creative Mathematics, H. S. Wall Environmental Mathematics in the Classroom, edited by B. A. Fusaro and P. C. Kenschaft Excursions in Classical Analysis: Pathways to Advanced Problem Solving and Undergraduate Research, by Hongwei Chen Exploratory Examples for Real Analysis, Joanne E. Snow and Kirk E. Weller Geometry From Africa: Mathematical and EducationalExplorations, Paulus Gerdes Historical Modules for the Teaching and Learning of Mathematics (CD), edited by Victor Katz and Karen Dee Michalowicz Identification Numbers and Check Digit Schemes, Joseph Kirtland Interdisciplinary Lively Application Projects, edited by Chris Arney Inverse Problems: Activities for Undergraduates, Charles W. Groetsch Laboratory Experiences in Group Theory, Ellen Maycock Parker Learn from the Masters, Frank Swetz, John Fauvel, Otto Bekken, Bengt Johansson, and Victor Katz Math Made Visual: Creating Images for UnderstandingMathematics, Claudi Alsina and Roger B. Nelsen Mathematics Galore!: The First Five Years of the St. Marks Institute of Mathematics, James Tanton i i i i i i “master” — 2012/6/18 — 10:53 — page vi — #6 i i Ordinary Differential Equations: A Brief Eclectic Tour, David A. S´anchez Oval Track and Other Permutation Puzzles, John O. Kiltinen A Primer of Abstract Mathematics, Robert B. Ash Proofs Without Words, Roger B. Nelsen Proofs Without Words II, Roger B. Nelsen Rediscovering Mathematics: You Do the Math, Shai Simonson She Does Math!, edited by Marla Parker Solve This: Math Activities for Students and Clubs, James S. Tanton Student Manual for Mathematics for Business Decisions Part 1: Probabil- ity and Simulation, David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic Student Manual for Mathematics for Business Decisions Part 2: Calculus and Optimization, David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic Teaching Statistics Using Baseball, Jim Albert Visual Group Theory, Nathan C. Carter Writing Projects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go, Annalisa Crannell, Gavin LaRose, Thomas Ratliff, Elyn Rykken MAA Service Center P.O. Box 91112 Washington, DC 20090-1112 1-800-331-1MAA FAX: 1-301-206-9789 i i i i i i “master” — 2012/6/18 — 10:53 — page vii — #7 i i Introduction Alternative reality The real numbers are fundamental. Although mostly taken for granted, they are what make possible all of mathematics from high school algebra and Euclidean geometry through the calculus and beyond, and also serve as the basis for measurement in science, industry, and ordinary life. In this book we study alternative systems of numbers: systems that generalize and ex- tend the reals yet stay close to the fundamental properties that make the reals central to so much mathematics. By an alternative number system we mean a set of objects that can be combined using two operations, addition and multiplication, and that share some significant algebraic and geometric properties with the real numbers. Exactly what these properties are is made clear in Chapter One. We are not concerned with numeration, however. A numeration system is a means of giving names to numbers, for example, the decimal system for writing real numbers. We go beyond numeration to describe number systems that in- clude numbers differentfrom ordinary numbers includingmulti-dimensional numbers, infinitely small and infinitely large numbers, and numbers that represent positions in games. Although we present some eccentric and relatively unexplored parts of mathematics, each system that we study has a well-developed theory. Each system has applications to other areas of mathematics and science, in par- ticular to physics, the theory of games, multi-dimensional geometry, formal logic, and the philosophy of mathematics. Most of these number systems are active areas of current mathematical research and several were discov- vii i i i i i i “master” — 2012/6/18 — 10:53 — page viii — #8 i i viii Introduction ered relatively recently. As a group, they illuminate the central, unifying role of the reals in mathematics. Design of this book This book is designed to encourage readers to participate in the mathemat- ical development themselves. The proofs of many results are either con- tained in problems or depend on results proved in problems. The problems should be read at least, if not worked out. With twoexceptions the chapters are independent and can be read in any order. The exceptions are that the first two chapters contain essential back- ground for the rest of the book and that Chapter Four depends somewhat on the proceeding chapter. Use of this book This book presents material that, in addition to being of general interest to mathematics students, is appropriate for an upper level course for under- graduates that can serve as introductionto or sequel to a course in advanced calculus. Alternatively, it can take the place of a course in the foundations of the real number system, or be given as an upper level seminar emphasiz- ing different methods of proof. Prerequisites are standard sophomore level courses: discrete mathematics, multivariable calculus, and linear algebra. A course in advanced calculus or foundationsof analysis would also be useful. The goal The goal is to present some interesting, even exotic, mathematics. I hope to convey a sense of the immense freedom available in mathematics, where even in a mundane and well-established area such as the real numbers, al- ternatives are always possible. Acknowledgements This book owes a tremendous debt to previous expositors in this area to whose work this book is closely tied, in particular, to the work of Elwyn Berlekamp, Errett Bishop, John H. Conway, Richard Guy, James Henle, and Eugene Kleinberg. i i i i i i “master” — 2012/6/18 — 10:53 — page ix — #9 i i Contents Introduction vii I THE REALS 1 1 Axiomsfor
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