Cover 1
Title 3
Copyright 4
Table of Contents 5
Opening thoughts: Welcome to the jungle 7
Modules 13
Module 1. A bit of foreshadowing and some rational rationale 13
Module 2. Building the rationals via Farey Sequences 17
Module 3. Discoveries of Dirichlet and Hurwitz 23
Module 4. The theory of continued fractions 29
Module 5. Enforcing the law of best approximates 35
Module 6. Markoff's spectrum and numbers 41
Module 7. Badly approximabie numbers and quadratics 49
Module 8. Solving the alleged "Pell" equation 55
Module 9. Liouville's work on numbers algebraic and not 61
Module 10. Roth's stunning result and its consequences 67
Module 11. Pythagorean triples through diophantine geometry 73
Module 12. A quick tour through elliptic curves 81
Module 13. The geometry of numbers 87
Module 14. Simultaneous diophantine approximation 95
Module 15. Using geometry to sum some squares 101
Module 16. Spinning around irrationally and uniformly 105
Module 17. A whole new world of p-adic numbers 111
Module 18. A glimpse into p-adic analysis 119
Module 19. A new twist on Newton's method 123
Module 20. The power of acting locally while thinking globally 129
Appendices 135
1. Selected Big Picture Question commentaries 135
2. Hints and remarks 141
3. Further reading 151
Acknowledgments 153
Index 155
A 155
B 155
C 155
D 155
E 155
F 155
G 155
H 156
I 156
J 156
K 156
L 156
M 156
N 156
O 156
P 156
Q 156
R 156
S 156
T 156
U 157
Back Cover 160
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