NUMBER SYSTEMS: AN INTRODUCTION TO ALGEBRA AND ANALYSIS. VOLUME: 23

This book offers a rigorous and coherent introduction to the five basic number systems of mathematics, namely natural numbers, integers, rational numbers, real numbers, and complex numbers. It is a subject that many mathematicians believe should be learned by any student of mathematics including future teachers.

The book starts with the development of Peano arithmetic in the first chapter which includes mathematical induction and elements of recursion theory. It proceeds to an examination of integers that also covers rings and ordered integral domains. The presentation of rational numbers includes material on ordered fields and convergence of sequences in these fields. Cauchy and Dedekind completeness properties of the field of real numbers are established, together with some properties of real continuous functions. An elementary proof of the Fundamental Theorem of Algebra is the highest point of the chapter on complex numbers. The great merit of the book lies in its extensive list of exercises following each chapter. These exercises are designed to assist the instructor and to enhance the learning experience of the students.

Readership
Undergraduate students interested in foundations of algebra and analysis.

Author
Sergei Ovchinnikov, San Francisco State University, CA.

Contents
Preface vii
Chapter 1. Natural Numbers 1
1.1. Peano Systems 2
1.2. Addition 4
1.3. Multiplication 9
1.4. Order 12
1.5. Isomorphism of Peano Systems 13
1.6. A Set-Theoretic Model 17
1.7. Recursion 19
1.8. Mathematical Induction 24
1.9. Algebraic Structures 27
Notes 28
Exercises 29
Chapter 2. Integers 37
2.1. Definition of the Integers 37
2.2. Addition of Integers 40
2.3. Multiplication of Integers 44
2.4. Order 47
2.5. Rings and Integral Domains 49
Notes 54
Exercises 55
Chapter 3. Rational Numbers 59
3.1. Definition of Rational Numbers 60
3.2. Addition of Rational Numbers 61
3.3. Multiplication of Rational Numbers 62
3.4. Order 64
3.5. Algebraic Structures on Q 68
3.6. Convergence in an Ordered Field 71
3.7. Limitations of Q 77
Notes 80
Exercises 80
Chapter 4. Real Numbers 85
4.1. Definition of Real Numbers 85
4.2. Operations on R 87
4.3. R as a Field 88
4.4. R as an Ordered Field 90
4.5. Cauchy Completeness of R 93
4.6. Dedekind Completeness of R 96
4.7. Continuous Functions on R 101
Notes 103
Exercises 104
Chapter 5. Complex Numbers 107
5.1. Definition of Complex Numbers 108
5.2. The Field C of Complex Numbers 108
5.3. C as a Vector Space 111
5.4. C as a Normed Algebra 112
5.5. Convergence in C 113
5.6. Roots of Complex Numbers 117
5.7. Continuous functions 118
5.8. The Fundamental Theorem of Algebra 120
Notes 122
Exercises 123
Appendix A. Sets, Relations, Functions 127
A.1. Sets 127
A.2. Operations on Sets 129
A.3. Relations 132
A.4. Functions and Operations 133
Notes 134
Exercises 135
Bibliography 139
Index 141