Chapters in this book

1. Frontmatter i
2. Contents vii
3. Preface xi
4. Part I: Real Infinity
5. 1 Real Numbers, Extended
6. 1.1 Real numbers: Operations and properties 1
7. 1.2 Two new numbers: ∞ and −∞ 7
8. 1.3 Computing with ∞ and −∞ 11
9. For your consideration 13
10. 2 Infinite Decimals
11. 2.1 Place value notation 15
12. 2.2 What is 0.3333 . . . ? 16
13. 2.3 Infinite sums as sequences 17
14. 2.4 Smaller terms but infinite sum 19
15. For your consideration 21
16. 3 Tropical Arithmetic
17. 3.1 New operations: ⊕ and ⊙ 23
18. 3.2 Polynomials 26
19. 3.3 Tropical algebraic curves 28
20. For your consideration 33
21. 4 Hyperreals
22. 4.1 Starting small 35
23. 4.2 The transfer principle 37
24. 4.3 Derivatives made easy 41
25. For your consideration 43
26. Part II: Plane Infinity
27. 5 Complex Numbers, Extended
28. 5.1 Complex numbers 47
29. 5.2 One infinity for all 49
30. 5.3 Stereographic projection and the Riemann sphere 50
31. 5.4 Linear fractional transformations 52
32. For your consideration 56
33. 6 Line at Infinity I: Projective Plane
34. 6.1 Parallel lines meet at infinity 59
35. 6.2 The projective plane 60
36. 6.3 Points and lines, algebraically 63
37. 6.4 Desargues’s theorem 67
38. 6.5 Topology of the projective plane 70
39. 6.6 Bézout’s theorem 74
40. For your consideration 80
41. 7 Line at Infinity II: Hyperbolic Plane
42. 7.1 The parallel postulate 83
43. 7.2 The hyperbolic plane 85
44. 7.3 Tessellations 92
45. 7.4 Distance and area in the hyperbolic plane 97
46. For your consideration 100
47. Part III: Infinite Things
48. 8 Counting Infinity
49. 8.1 Counting things 103
50. 8.2 Sets, functions, bijections 104
51. 8.3 Finite and infinite sets 110
52. 8.4 Comparing infinite sets 113
53. 8.5 Transfinite cardinal numbers 118
54. 8.6 Transcendental numbers 123
55. For your consideration 125
56. 9 Order Infinity
57. 9.1 Ordered sets 127
58. 9.2 Well-ordered sets 129
59. 9.3 Ordinal numbers 133
60. For your consideration 139
61. 10 Infinite Shapes
62. 10.1 Circles 141
63. 10.2 How long is the diagonal of a square? 143
64. 10.3 Sierpiński’s carpet 145
65. 10.4 Hilbert’s space-filling curve 151
66. 10.5 Mandelbrot set 156
67. For your consideration 162
68. Bibliography 167
69. Index 171