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Numbers and Geometry

NUMBERS AND GEOMETRY is a beautiful and relatively elementary account of a part of mathematics where three main fields--algebra, analysis and geometry--meet. The aim of this book is to give a broad view of these subjects at the level of calculus, without being a calculus (or a pre-calculus) book. It...

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Bibliographic Details
Main Author: Stillwell, John
Format: eBook
Language:English
Published: New York, NY Springer New York 1998, 1998
Edition:1st ed. 1998
Series:Undergraduate Texts in Mathematics
Subjects:
Number Theory
Geometry
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 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
  • 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

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