Log In
Or create an account ->
Imperial Library
Home
About
News
Upload
Forum
Help
Login/SignUp
Index
Cover
About the Author
Title Page
Copyright Page
Dedication
CONTENTS
PREFACE
IMAGINING NUMBERS
PART I
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?
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
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
4: PERMISSION AND LAWS
20. Permission
21. Forced conventions, or definitions?
22. What kind of “law” is the distributive law?
5: ECONOMY OF EXPRESSION
23. Charting the plane
24. The geometry of qualities
25. The spareness of the inventory of the imagination
6: JUSTIFYING LAWS
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
7: BOMBELLI'S PUZZLE
31. The argument between Cardano and Tartaglia
32. Bombelli's L'Algebra
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
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
9: PUTTING GEOMETRY INTO NUMBERS
45. Many hands
46. Imagining the dynamics of multiplication by …
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
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 …
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
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
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
ACKNOWLEDGMENTS
INDEX
PERMISSIONS ACKNOWLEDGMENTS
Footnotes
PART I
1: THE IMAGINATION AND SQUARE ROOTS
1. Picture this
Page 6
Page 10
Page 11
Page 12
2. Imagination
Page 18
Page 23
2: SQUARE ROOTS AND THE IMAGINATION
8. The quadratic formula
Page 32
Page 33
3: LOOKING AT NUMBERS
12. The problem of describing how we imagine
Page 44
17. Words, things, pictures
Page 54
Page 56
4: PERMISSION AND LAWS
22. What kind of “law” is the distributive law?
Page 74
6: JUSTIFYING LAWS
27. Defining the operation of multiplication
Page 95
28. The distributive law and its momentum
Page 100
PART II
7: BOMBELLI'S PUZZLE
31. The argument between Cardano and Tartaglia
Page 107
Page 108
32. Bombelli's L'Algebra
Page 115
8: STRETCHING THE IMAGE
39. The inventors of writing
Page 143
40. Arithmetic in the realm of imaginary numbers
Page 145
41. The absence of time in mathematics
Page 149
9: PUTTING GEOMETRY INTO NUMBERS
48. The power of notation
Page 166
49. A plane of numbers
Page 167
Page 167
Page 170
PART III
12: UNDERSTANDING ALGEBRA VIA GEOMETRY
65. But…
Page 227
← Prev
Back
Next →
← Prev
Back
Next →