Table Of ContentJourney
Mathematics
into
An Introduction to Proofs
Joseph J. Rotman
University of Illinois, Urbana
Dover Publications, Inc.
Mineola, New York
Copyright
Copyright CO 1998~ 2007 by Joseph J. Rotnlan
All rights reserved.
Bibliographical Note
This Dover edition~ first published in 2007, is a corrected republication
of the first edition of the work originally published by Prentice Hall~ Inc.,
Upper Saddle River, New Jersey~in 1998. Readers of this book who would
like to receive the solutions to the exercises l11ay request thel11 fr0l11 the
publishcr at the following e-111ail address: editors(!iJdoverpublications.com *
International Standard Book Nll/nhe/:' 0-486-45306-5
Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street Mineola, N.Y. 1150I
* The 92-page answer booklet froin Dover follows the index.
To Marganit.
Contents
Preface vii
To the Reader xi
To the Instructor xiii
1. Setting Out 1
Induction . 1
Binomial Coefficients . 22
2. Things Pythagorean 31
Area . 31
The Pythagorean Theorem . . . . . 45
Pythagorean Triples . . . . . 56
The Method ofDiophantus 64
Trigonometry . . . . . 72
Integration ... 86
3. Circles and 98
Jr
Approximations . . . . . . . . 98
The Area of a Disk . . . . . 107
The Circumference of a Disk . 117
Sequences . 134
4. Polynomials 149
Quadratics . . . . . . . . 149
Complex Numbers ... 160
De Moivre's Theorem .. 171
Cubics and Quartics . 183
Irrationalities . . . . . 204
Epilogue 213
Bibliography 215
v
vi CONTENTS
Glossary ofLogic 217
Propositional Calculus 217
Proofs ... 222
Set Theory . 224
Quantifiers . 228
Index 233
Preface
Instructors have observed, when teaching junior level courses in ab
stract algebra, number theory, or real variables, that many students have diffi
culty out ofproportion to the level ofdifficulty of the material. In an abstract
algebra course introducing groups and rings, students' struggles are not af
fected by the changing oftexts, instructors, or the order ofpresentation. Sim
ilarly, experimentingwithcourses in real variables (say, by treating only func
tions ofa single variable instead offunctions on euclidean n-space) offers lit
tle relief. The cause of this problem is plain when one considers the previous
mathematics courses. Thestandardcalculussequenceispresented, nowadays,
to students from various disciplines who have different backgrounds, abili
ties, and goals, with the aim of teaching them how to differentiate, how to
integrate, and how to use these techniques to solve problems. Theorems are
stated but usually not proved; hypotheses of theorems are often not verified
before applying the theorems (e.g., does one always check whether a given
function is continuous?); definitions are given (e.g., limit and convergence)
but not taken seriously. After two years ofsuch "mathematics," is it any won
derthatajunior-levelstudentis woefully unprepared to read and do real math
ematics?
There are two possible solutions to this problem. The obvious solu
tion, revise the calculus sequence, is impractical. Many have tried; many are
trying. I wishsuccess to those still fighting the goodfight, butI am pessimistic
about there being a revolution in undergraduate mathematics, and I am even
more pessimistic aboutthere being such a revolutiontomorrow. After all, sci
entistsand engineers cannotafford the extratimebeforeusing calculusin their
vii
viii PREFACE
own domains, and so calculus courses are necessarily compromises between
teaching the techniques ofcalculus and teaching an understanding ofits prin
ciples.
My solution is a one semester intermediate course between calculus
and the first courses in abstract algebra and real variables. This is not a new
idea. There are many such "transition courses" designed to prepare students
forjunior-levelcourses, butthey emphasizetheelements oflogic (frommodus
ponens and truth tables through quantifiers) and set theory (from Boolean op
erations through relations and functions). I find this material rather dull and
uninspiring, and I imagine that this feeling is shared by most students. Of
course, these things should be learned eventually; as Hermann Weyl wrote,
"Logicis thehygienethatthemathematician practices to keep his ideashealthy
and strong." It is cruel, however, to inflict an entire course comprised of such
things on defenseless students. And it doesn't work; my unscientific obser
vations indicate that those students who survive such tedious material do not
fare any better in subsequent courses than do those who were spared. George
P6lya wrote, "When introduced at the wrong time orplace, good logic may be
the worst enemy ofgood teaching." I have attached an appendix, Glossary of
Logic, at the end of the book, covering much of this material. Although this
section is too briefan account to qualify as a text for a standard course, it is a
place where readers can look to resolve the usual questions that tend to arise.
An introductorycourse shouldcontain valuablematerial, it mustbein
teresting, and it must give a fairly accurate pictureofwhat mathematics really
is and what mathematicians do. One learns how to read and write proofs by
reading and writing them; merely reading about mathematics is not an ade
quate substitute for actually doing mathematics. This book begins with some
very elementary mathematics - induction, binomial coefficients, and polygo
nal areas - because, when starting out, readers must be allowedto focus onthe
writing and reading of proofs without the distraction of absorbing unfamiliar
ideas at the same time. From the outset, complete proofs are given to serve as
models for the reader. The presentationis a coherent story, with historical and
etymological asides, because itis more interestingand more natural to watch a
subject grow and develop. The journey continues with elementary area prob
J2,
lems, the irrationality of the Pythagorean theorem, Pythagorean triples
after Diophantus, and trigonometry. The Diophantinemethod offinding Pyth
agorean triples by parametrizing the circle with rational functions is extended
to finding such parametrizations of other conic sections, and this leads to a
glimpse of elliptic integrals. Next, one passes to disks, proving the area and
circumference formulas (due to Eudoxus and Archimedes, respectively) es-
ix
sentially in the classical way. This early notion of approximation is subtle,
but it is digestiblebecause one can see areas ofinscribed polygons approach
ing the area of the disk. One notes afterward, however, that this early notion
has defects. Seeing how convergence remedies defects ofthe classical notion
gives the reader a better understanding and appreciation of the modem defi
+
nition of limit. We then see why (-1) x (-1) = 1, discuss the quadratic
formula, complex numbers, De Moivre's theorem, the cubic formulas (Car
dano's version in terms of radicals as well as Viete's trigonometric version),
discriminants, and the quartic formula. The text ends with proofs of the irra
tionality of e, the irrationality of some specific values of sine and cosine, and
the irrationality of Thus, geometry, algebra, number theory, and analysis
Jr.
are all intertwined. The journey travels a road from humble beginnings to a
fairly sophisticateddestination. I hope that students and instructors will enjoy
this text, and that it will serve the several aims set forth for it.
Ithank Paul Bateman, RichardBishop, PeterBraunfeld, Everett Dade,
Heini Halberstam, Carl Jockusch, Daniel Saltz, Donald Sherbert, and Ken
neth Stolarskyfor theirexcellentsuggestions. I give specialthanks toPhilippe
Tondeur, whose notes on similar material were the starting point of this text,
and I also give special thanks to J.-P. Tignol, for permitting me to quote an
excerpt from his wonderful book about Galois's theory of equations. I give
extra special thanks to John Wetzel and Leon McCulloh who made many fine
suggestionsas they were teaching from a preliminary version ofthistext; they
generously allowed me to use the ones I liked, and they did not complain in
the rare cases when I did not use one. It also gives me great pleasure to thank
my daughter, EllaRose, for drawing and producing all the figures in the book.
IthankJoyceWoodworthfor an excellentjoboftypingmy manuscript
in LaTeX, and I thank Adam Lewenberg for his expert help in the final stages
ofLaTeX typesetting. Lastly, I thankthe reviewers for theirgoodsuggestions:
Linda A. Bolte, Eastern Washington University
Thomas G. Clarke, North Carolina A&T State University
Donald Nowlin, Eastern Washington University
Michael Stecher, Texas A&M University
David Walnut, George Mason University.
Joseph Rotman
