Numbers and Geometry

A beautiful and relatively elementary account of a part of mathematics where three main fields - algebra, analysis and geometry - meet. The book provides a broad view of these subjects at the level of calculus, without being a calculus book. Its roots are in arithmetic and geometry, the two opposite poles of mathematics, and the source of historic conceptual conflict. The resol...

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A beautiful and relatively elementary account of a part of mathematics where three main fields - algebra, analysis and geometry - meet. The book provides a broad view of these subjects at the level of calculus, without being a calculus book. Its roots are in arithmetic and geometry, the two opposite poles of mathematics, and the source of historic conceptual conflict. The resolution of this conflict, and its role in the development of mathematics, is one of the main stories in the book. Stillwell has chosen an array of exciting and worthwhile topics and elegantly combines mathematical history with mathematics. He covers the main ideas of Euclid, but with 2000 years of extra insights attached. Presupposing only high school algebra, it can be read by any well prepared student entering university. Moreover, this book will be popular with graduate students and researchers in mathematics due to its attractive and unusual treatment of fundamental topics. A set of well-written exercises at the end of each section allows new ideas to be instantly tested and reinforced.

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 Gaussian Integers.- 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.

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 Gaussian Integers.- 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.
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