Preface ... 8,
Contents ... 12,
Chapter 1 Arithtmetic ... 16,
1.1 The Natural Numbers ... 16,
1.2 Division,Divisors, and Primes ... 19,
1.3 The Mysterious Sequence of Primes ... 22,
1.4 Integers and Rationals ... 24,
1.5 Linear Equations ... 28,
1.6 Unique Prime Factorization ... 32,
1.7 Prime Factorization and Divisors ... 35,
1.8 Induction ... 38,
1.9* Foundations ... 41,
I. I 0 Discussion ... 45,
Chapter 2 Geometry ... 52,
2.1 Geometric Intuition ... 52,
2.2 Constructions ... 55,
2.3 Parallels and Angles ... 59,
2.4 Angles and Circles ... 62,
2.5 Length and Area ... 65,
2.6 The Pythagorean Theorem ... 68,
2.7 Volume ... 71,
2.8* The Whole and the Part ... 74,
2.9 Discussion ... 79,
Chapter 3 Coordinates ... 84,
3.1 Lines and Circles ... 84,
3.2 Intersections ... 87,
3.3 The Real Numbers ... 92,
3.4 The Line ... 97,
3.5 The Euclidean Plane ... 100,
3.6 Isometries of the Euclidean Plane ... 104,
3.7 The Triangle Inequality ... 108,
3.8* Klein's Definition of Geometry ... 110,
3.9* The Non-Euclidean Plane ... 115,
3.10 Discussion ... 120,
Algebra and Geometry ... 120,
The Jump from Q to R ... 121,
A Different Definition of Euclidean Geometry ... 124,
Chapter 4 Rational Points ... 126,
4.1 Pythagorean Triples ... 126,
4.2 Pythagorean Triples in Euclid ... 128,
4.3 Pythagorean Triples in Diophantus ... 131,
4.4 Rational Triangles ... 135,
4.5 Rational Points on Quadratic Curves ... 139,
4.6* Rational Points on the Sphere ... 142,
4.7 The Area of Rational Right Triangles ... 146,
4.8 Discussion ... 151,
Diophantus and His Legacy ... 151,
Fermat's Last Theorem ... 154,
Elliptic Curves ... 156,
Chapter 5 Trigonometry ... 158,
5.1 Angle Measure ... 158,
5.2 Circular Functions ... 162,
5.3 Addition Formulas ... 167,
5.4 A Rational Addition Formula ... 171,
5.5 Hilbert's Third Problem ... 174,
5.6* The Dehn Invariant ... 176,
5.7* Additive Functions ... 180,
5.8* The Tetrahedron and the Cube ... 183,
5.9 Discussion ... 186,
Chapter 6 Finite Arithtmetic ... 192,
6.1 Three Examples ... 192,
6.2 Arithmetic mod n ... 194,
6.3 The Ring Z/nZ ... 198,
6.4 Inverses mod n ... 201,
6.5 The Theorems of Fermat and Wilson ... 205,
6.6 The Chinese Remainder Theorem ... 209,
6.7 Square s mod p ... 212,
6.8* The Quadratic Character of -1 and 2 ... 215,
6.9 Quadratic Reciprocity ... 218,
6.10 Discussion ... 223,
Congruences and Congruence Classes ... 223,
Rings, Fields, and Abelian Groups ... 224,
Applied Number Theory ... 227,
Chapter 7 Complex Numbers ... 230,
7. 1 Addition, Multiplication, and Absolute Value ... 230,
7.2 Argument and the Square Root of-I ... 234,
7.3 Isometries of the Plane ... 238,
7.4 The Gaussian Integers ... 242,
7.5 Unique Gaussian Prime Factorization ... 245,
7.6 Fermat's Two Squares Theorem ... 248,
7.7* Factorizing a Sum of Two Square s ... 252,
7.8 Discussion ... 253,
Complex Numbers and Geometry ... 253,
Quadratic Forms ... 257,
Quadratic Integers and Lattices ... 258,
Chapter 8 Conic Sections ... 262,
8.1 Too Much, Too Little and Just Right ... 262,
8.2 Properties of Conic Sections ... 266,
8.3 Quadratic Curves ... 270,
8.4* Intersections ... 275,
8.5 Integer Points on Conics ... 278,
8.6* Square Roots and the Euclidean Algorithm ... 282,
8.7* Pell's Equation ... 286,
8.8 Discussion ... 290,
The Projective View of Conic Sections ... 290,
Chapter 9 Elementary Functions ... 296,
9.1 Algebraic and Transcendental Functions ... 296,
9.2 The Area Bounded by a Curve ... 299,
9.3 The Natural Logarithm and the Exponential ... 302,
9.4 The Exponential Function ... 307,
9.5 The Hyperbolic Functions ... 310,
9.6 The Pell Equation Revisited ... 313,
9.7 Discussion ... 318,
From Natural Numbers to Complex Numbers ... 318,
The Exponential Function ... 320,
From Pythagoras to Pell ... 322,
Bibliography ... 326,
Index ... 332,
