Contents

Preface vii

Chapter 1. From the Real Numbers to the Complex Numbers 1

1. Introduction 1

2. Number systems 1

3. Inequalities and ordered ﬁelds 7

4. The complex numbers 15

5. Alternative deﬁnitions of C 17

6. A glimpse at metric spaces 21

Chapter 2. Complex Numbers 25

1. Complex conjugation 25

2. Existence of square roots 27

3. Limits 30

4. Convergent inﬁnite series 32

5. Uniform convergence and consequences 35

6. The unit circle and trigonometry 40

7. The geometry of addition and multiplication 44

8. Logarithms 45

Chapter 3. Complex Numbers and Geometry 49

1. Lines, circles, and balls 49

2. Analytic geometry 52

3. Quadratic polynomials 53

4. Linear fractional transformations 60

5. The Riemann sphere 63

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