## Table of Contents

Preface vii

1 Natural numbers 1

1.1 Counting 1

1.2 Properties of numbers 3

1.3 Magic squares 7

1.4 Prime numbers 8

1.5 From Pythagoras to Fermat 13

1.6 What are natural numbers? 16

1.7 Applications: Cryptography 19

2 Representations of numbers 23

2.1 How were numbers written in former times? 23

2.2 The abacus and counting board 26

2.3 The decimal system 31

2.4 Divisibility rules 33

2.5 Binary numbers 36

2.6 Applications: Barcodes 38

3 Rational and irrational 41

3.1 Fractions 41

3.2 Ratios 44

3.3 Rational numbers 48

3.4 Irrational numbers: The first crisis 52

3.5 Decimal numbers 57

4 Transcendental numbers 61

4.1 The most mysterious number 61

4.2 Limits 64

4.3 How many transcendental numbers are there? 70

5 Imaginary and complex 75

5.1 Linear equations 76

5.2 Quadratic equations 77

5.3 The drama of the cubic equation 80

5.4 The tragedy of the quintic equation 82

5.5 Every equation can be solved! 84

Additional Notes 89

Bibliography 99

Index 101