Author | Stillwell, John. author |
---|---|
Title | Numbers and Geometry [electronic resource] / by John Stillwell |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1998 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-0687-3 |
Descript | XIV, 343 p. online resource |
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* Kleinโs 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* Hubertโs 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 Gaussian Integers -- 7.5 Unique Gaussian Prime Factorization -- 7.6 Fermatโs 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* Pellโs 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