How the elusive imaginary number was first imagined, and how to imagine it yourself
Imagining Numbers (particularly the square root of minus fifteen) is Barry Mazur's invitation to those who take delight in the imaginative work of reading poetry, but may have no background in math, to make a leap of the imagination in mathematics. Imaginary numbers entered into mathematics in sixteenth-century Italy and were used with immediate success, but nevertheless presented an intriguing challenge to the imagination. It took more than two hundred years for mathematicians to discover a satisfactory way of "imagining" these numbers.
With discussions about how we comprehend ideas both in poetry and in mathematics, Mazur reviews some of the writings of the earliest explorers of these elusive figures, such as Rafael Bombelli, an engineer who spent most of his life draining the swamps of Tuscany and who in his spare moments composed his great treatise "L'Algebra". Mazur encourages his readers to share the early bafflement of these Renaissance thinkers. Then he shows us, step by step, how to begin imagining, ourselves, imaginary numbers.
Author(s): Barry Mazur
Edition: 1
Publisher: Farrar, Straus and Giroux
Year: 2004
Language: English
Commentary: Front and back covers, OCR, bookmarks, paginated.
Pages: 288
Preface
PART I
Ch 1 The Imagination and Square Roots
1. Picture this
2. Imagination
3. Imagining what we read
4. Mathematical problems and square roots
5. What is a mathematical problem?
Ch 2 Square Roots and the Imagination
6. What is a square root?
7. What is a square root?
8. The quadratic formula
9. What kind of thing is the square root of a negative number?
10. Girolamo Cardano
11. Mental tortures
Ch 3 Looking at Numbers
12. The problem of describing how we imagine
13. Noetic, imaginary, impossible
14. Seeing and squinting
15. Double negatives
16. Are tulips yellow?
17. Words, things, pictures
18. Picturing numbers on lines
19. Real numbers and sophists
Ch 4 Permission and Laws
20. Permission
21. Forced conventions, or definitions?
22. What kind of "law" is the distributive law?
Ch 5 Economy of Expression 77
23. Charting the plane
24. The geometry of qualities
25. The spareness of the inventory of the imagination
Ch 6 justifying Laws 91
26. "Laws" and why we believe them
27. Defining the operation of multiplication
28. The distributive law and its momentum
29. Virtuous circles versus vicious circles
30. So, why does minus times minus equal plus?
PART II
Ch 7 Bombelli's Puzzle
31. The argument between Cardano and Tartaglia
32. Bombelli's L'Aigebra
33. "I have found another kind of cubic radical which is very different from the others"
34. Numbers as algorithms
35. The name of the unknown
36. Species and numbers
Ch 8 Stretching the Image
37. The elasticity of the number line
38. "To imagine" versus "to picture"
39. The inventors of writing
40. Arithmetic in the realm of imaginary numbers
41. The absence of time in mathematics
42. Questioning answers
43. Back to Bombelli's puzzle
44. Interviewing Bombelli
Ch 9 Putting Geometry into Numbers
45. Many hands
46. Imagining the dynamics of multiplication by sqrt -1 : algebra and geometry mixed
47. Writing and singing
48. The power of notation
49. A plane of numbers
50. Thinking silently, out loud
51. The complex plane of numbers
52. Telling a straight story
Ch 10 Seeing the Geometry in the Numbers
53. Critical moments in the story of discovery
54. What are we doing when we identify one thing with another?
55. Song and story
56. Multiplying in the complex plane. The geometry behind multiplication by sqrt -1, by 1 + sqrt -1, and by (1 + sqrt 3) /2
57. How can I be sure my guesses are right?
58. What is a number?
59. So, how can we visualize multiplication in the complex plane?
PART III
Ch 11 The Literature of Discovery of Geometry in Numbers
60. "These equations are of the same form as the equations for cosines, though they are things of quite a different nature"
61. A few remarks on the literature of discovery and the literature of use
Ch 12 Understanding Algebra via Geometry
62. Twins
63. Bombelli's cubic radicals revisited: Dal Ferro's expression as algorithm
64. Form and content
65. But ...
Appendix: The Quadratic Formula
Notes
Bibliography
Acknowledgements
Index
