An Introduction to Mathematics

360 Pages·1974·50.752 MB·English

by  Bevan K. Youse

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An Introduction to Mathematics
Second Edition
BEVAN K YOUSE
Emory University
ALLYN AND BACON, INC.
Boston
© COPYRIGHT 1974 BY ALLYN AND BACON, INC.
© COPYRIGHT 1970 BY ALLYN AND BACON, INC.
470 ATLANTIC AVENUE, BOSTON.
ALL RIGHTS RESERVED.
No part of the material protected by this copyright notice may be
reproduced or utilized in any form or by any means, electronic or
mechanical, including photocopying, recording, or by any informa-
tional storage and retrieval system, without written permission
from the copyright owner.
LIBRARY OF CONGRESS CATALOG CARD NUMBER: 73-88136
\
PRINTED IN THE UNITED STATES OF AMERICA
Contents
Preface vii
The Nature of Mathematics 1 CHAPTER
1
1. What is Mathematics? 2
Numbers and Numerals 7 CHAPTER
2
1. Sets and Cardinal Numbers 8
2. Rational Operations and Basic Properties 12
3. Algorithms for a Positional Notation 18
4. Other Notational Systems 27
5. Computing in Other Bases {Optional) 35
Checks and Pseudo-checks 41 CHAPTER
3
1. The Missing Digit 42
2. Divisibility Tests 45
3. Halve-double-sum Technique 54
4. Casting Out Nines 56
The Expanding World of Numbers 63 CHAPTER
4
1. Rational Numbers 64
2. Negative Numbers 70
3. Real Numbers 76
4. Decimal Notation 80
5. Complex Numbers {Optional) 86
Number Patterns 93 CHAPTER
5
1. Arithmetic Sequences 94
2. Geometric Sequences 100
3. A Look at the Primes 107
4. Making Conjectures 111
5. Perfect Numbers 115
6. Other Types of Sequences {Optional) 122
v
Contents
CHAPTER A Look at Topology 129
6
1. Houses, Maps, and Vests 130
2. Networks 134
3. The Regular Polyhedra 145
CHAPTER Probability and Statistics 157
7
1. Introduction 158
2. Counting Collections 159
3. Basic Ideas of Probability 168
4. Introduction to Statistics 178
5. Measures of Central Tendency and Dispersion 183
CHAPTER Logic and the Axiomatic Approach 189
8
1. Introduction to Logic 190
2. Testing the Validity of Arguments 200
3. Introduction to Axiomatics 209
4. Venn Diagrams and Lewis Carroll Puzzles 217
CHAPTER The Principles of Computer Programming 227
9
1. Simple Computer Programs 228
2. Programming Repetitive Procedures 235
3. Applications 241
CHAPTER Number Theory 247
10
1. Nurn,ber Congruence 248
2. Pythagorean Numbers 253
3. Mathematical Induction 259
4. Fundamental Theorem of Arithmetic 264
5. Euclidean Algorithm 269
Appendix A Suggested Reading A-1
Appendix B Lame’s Theorem A-3
Appendix C Tables A-7
Appendix D Answers to Odd-numbered Exercises A-20
Index A-59
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