文档介绍：Journey
into Mathematics
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(! *
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. 1150 I
* 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 of Diophantus 64
Trigonometry . . . . . 72
Integration ... 86
3. Circles and Jr 98
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 of proportion to the level of difficulty of the material. In an abstract
algebra course introducing groups and rings, students' struggles are not af•
fecte