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Numbers and Geometry John Stillwell

Numbers and Geometry By John Stillwell

Numbers and Geometry by John Stillwell


$83.99
Condition - Very Good
Out of stock

Summary

NUMBERS AND GEOMETRY is a beautiful and relatively elementary account of a part of mathematics where three main fields--algebra, analysis and geometry--meet. Stillwell has chosen an array of exciting and worthwhile topics and elegantly combines mathematical history with mathematics.

Numbers and Geometry Summary

Numbers and Geometry by John Stillwell

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. 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. The key is algebra, which brings arithmetic and geometry together, and allows them to flourish and branch out in new directions. Stillwell has chosen an array of exciting and worthwhile topics and elegantly combines mathematical history with mathematics. He believes that most of mathematics is about numbers, curves and functions, and the links between these concepts can be suggested by a thorough study of simple examples, such as the circle and the square. This book covers the main ideas of Euclid--geometry, arithmetic and the theory of real numbers, but with 2000 years of extra insights attached. NUMBERS AND GEOMETRY presupposes only high school algebra and therefore can be read by any well prepared student entering university. Moreover, this book will be popular with graduate students and researchers in mathematics because it is such an attractive and unusual treatment of fundamental topics. Also, it will serve admirably in courses aimed at giving students from other areas a view of some of the basic ideas in mathematics. There is a set of well-written exercises at the end of each section, so new ideas can be instantly tested and reinforced.

Table of Contents

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* Kleins 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* Huberts 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 Fermats 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* Pells 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.

Additional information

GOR011490349
9780387982892
0387982892
Numbers and Geometry by John Stillwell
Used - Very Good
Hardback
Springer-Verlag New York Inc.
1997-10-30
343
N/A
Book picture is for illustrative purposes only, actual binding, cover or edition may vary.
This is a used book - there is no escaping the fact it has been read by someone else and it will show signs of wear and previous use. Overall we expect it to be in very good condition, but if you are not entirely satisfied please get in touch with us

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