دانلود کتاب A readable introduction to real mathematics

فهرست مطالب :

Preface to the Second Edition......Page 7
Preface for Readers......Page 8
Preface for Instructors......Page 11
Contents......Page 14
1 Introduction to the Natural Numbers......Page 16
1.1 Prime Numbers......Page 17
1.2 Unanswered Questions......Page 20
Basic Exercises......Page 21
Challenging Problems......Page 22
2.1 The Principle of Mathematical Induction......Page 23
2.2 The Principle of Complete Mathematical Induction......Page 30
Interesting Problems......Page 35
Challenging Problems......Page 36
3.1 The Basics......Page 37
3.2 Some Applications......Page 39
Interesting Problems......Page 41
Challenging Problems......Page 42
4.1 Proof of the Fundamental Theorem of Arithmetic......Page 44
Challenging Problems......Page 47
5.1 Fermat\'s Little Theorem......Page 49
5.2 Wilson\'s Theorem......Page 51
Interesting Problems......Page 53
Challenging Problems......Page 54
6 Sending and Receiving Secret Messages......Page 55
6.1 The RSA Method......Page 56
Basic Exercises......Page 60
7 The Euclidean Algorithm and Applications......Page 61
7.1 The Euclidean Algorithm......Page 62
7.2 Applications......Page 63
Basic Exercises......Page 71
Interesting Problems......Page 72
Challenging Problems......Page 73
8.1 Rational Numbers......Page 74
8.2 Irrational Numbers......Page 76
Interesting Problems......Page 81
Challenging Problems......Page 82
9.1 What is a Complex Number?......Page 84
9.2 The Complex Plane......Page 87
9.3 The Fundamental Theorem of Algebra......Page 94
Interesting Problems......Page 98
Challenging Problems......Page 99
10.1 Cardinality......Page 100
10.2 Countable Sets and Uncountable Sets......Page 104
10.3 Comparing Cardinalities......Page 108
Interesting Problems......Page 122
Challenging Problems......Page 123
11.1 Triangles......Page 125
11.2 The Parallel Postulate......Page 130
11.3 Areas and Similarity......Page 133
Basic Exercises......Page 138
Interesting Problems......Page 139
Challenging Problems......Page 140
12 Constructibility......Page 143
12.1 Constructions with Straightedge and Compass......Page 144
12.2 Constructible Numbers......Page 148
12.3 Surds......Page 155
12.4 Constructions of Geometric Figures......Page 163
Basic Exercises......Page 170
Interesting Problems......Page 171
Challenging Problems......Page 172
13.1 Convergence......Page 175
13.2 Geometric Series......Page 181
13.3 Convergence of Related Series......Page 183
13.4 Least Upper Bounds......Page 185
13.5 The Comparison Test......Page 187
13.6 Representing Real Numbers by Infinite Decimals......Page 189
13.7 Further Examples of Infinite Series......Page 191
Basic Exercises......Page 195
Interesting Problems......Page 196
Challenging Problems......Page 198
14.1 Two-Dimensional Space......Page 203
14.2 Three-Dimensional Space......Page 206
14.3 Spaces of Dimension Four and Higher......Page 208
14.4 Norms and Inner Products......Page 209
14.5 Infinite-Dimensional Spaces......Page 213
14.6 A Difference Between Finite and Infinite-Dimensional Spaces......Page 218
Basic Exercises......Page 220
Challenging Problems......Page 221
Index......Page 224