Hjem

1 Arithmetic.-
1.1 The Natural Numbers.-
1.2 Division, Divisors, and Primes.-
1.3 The Mysterious Sequence of Primes.-
1.4 Integers and Rationals.-
1.5 Linear Equations.-
1.6 Unique Prime Factorization.-
1.7 Prime Factorization and Divisors.-
1.8 Induction.-
1.9* Foundations.-
1.10 Discussion.- 2 Geometry.-
2.1 Geometric Intuition.-
2.2 Constructions.-
2.3 Parallels and Angles.-
2.4 Angles and Circles.-
2.5 Length and Area.-
2.6 The Pythagorean Theorem.-
2.7 Volume.-
2.8* The Whole and the
Part.-
2.9 Discussion.- 3 Coordinates.-
3.1 Lines and Circles.-
3.2 Intersections.-
3.3 The Real Numbers.-
3.4 The Line.-
3.5 The Euclidean Plane.-
3.6 Isometries of the Euclidean Plane.-
3.7 The Triangle Inequality.-
3.8* Klein's Definition of Geometry.-
3.9* The Non-Euclidean Plane.-
3.10 Discussion.- 4 Rational Points.-
4.1 Pythagorean Triples.-
4.2 Pythagorean Triples in Euclid.-
4.3 Pythagorean Triples in Diophantus.-
4.4 Rational Triangles.-
4.5 Rational Points on Quadratic Curves.-
4.6* Rational Points on the Sphere.-
4.7* The Area of Rational Right Triangles.-
4.8 Discussion.- 5 Trigonometry.-
5.1 Angle Measure.-
5.2 Circular Functions.-
5.3 Addition Formulas.-
5.4 A Rational Addition Formula.-
5.5* Hubert's Third Problem.-
5.6* The Dehn Invariant.-
5.7* Additive Functions.-
5.8* The Tetrahedron and the Cube.-
5.9 Discussion.- 6 Finite Arithmetic.-
6.1 Three Examples.-
6.2 Arithmetic mod n.-
6.3 The Ring ?/n?.-
6.4 Inverses mod n.-
6.5 The Theorems of Fermat and Wilson.-
6.6 The Chinese Remainder Theorem.-
6.7 Squares mod p.-
6.8* The Quadratic Character of-1 and.-
6.9* Quadratic Reciprocity.-
6.10 Discussion.- 7 Complex Numbers.-
7.1 Addition, Multiplication, and Absolute Value.-
7.2 Argument and the Square Root of -1.-
7.3 Isometries of the Plane.-
7.4 The GaussianIntegers.-
7.5 Unique Gaussian Prime Factorization.-
7.6 Fermat's TWo Squares Theorem.-
7.7* Factorizing a Sum of Two Squares.-
7.8 Discussion.- 8 Conic Sections.-
8.1 Too Much, Too Little, and Just Right.-
8.2 Properties of Conic Sections.-
8.3 Quadratic Curves.-
8.4* Intersections.-
8.5 Integer Points on Conics.-
8.6* Square Roots and the Euclidean Algorithm.-
8.7* Pell's Equation.-
8.8 Discussion.- 9 Elementary Functions.-
9.1 Algebraic and Transcendental Functions.-
9.2 The Area Bounded by a Curve.-
9.3 The Natural Logarithm and the Exponential.-
9.4 The Exponential Function.-
9.5 The Hyperbolic Functions.-
9.6 The Pell Equation Revisited.-
9.7 Discussion.