Numbers (Graduate Texts in Mathematics, 123)

✍ Scribed by Heinz-Dieter Ebbinghaus, John H. Ewing (editor)

• Publisher: Springer
• Year: 1995
• Tongue: English
• Leaves: 418
• Edition: Corrected
• Category: Library

✦ Synopsis

A book about numbers sounds rather dull. This one is not. Instead it is a lively story about one thread of mathematics-the concept of "number"­ told by eight authors and organized into a historical narrative that leads the reader from ancient Egypt to the late twentieth century. It is a story that begins with some of the simplest ideas of mathematics and ends with some of the most complex. It is a story that mathematicians, both amateur and professional, ought to know. Why write about numbers? Mathematicians have always found it diffi­ cult to develop broad perspective about their subject. While we each view our specialty as having roots in the past, and sometimes having connec­ tions to other specialties in the present, we seldom see the panorama of mathematical development over thousands of years. Numbers attempts to give that broad perspective, from hieroglyphs to K-theory, from Dedekind cuts to nonstandard analysis.

✦ Table of Contents

NUMBERS
Graduate Texts in Mathematics
Numbers
Copyright Page
Preface to the English Edition
Preface to the Second Edition
Preface to the First Edition
Contents
Introduction
Part A: From the Natural Numbers, to the Complex Numbers, to the p-adics
Chapter 1. Natural Numbers, Integers, and Rational Numbers
1. Historical
1.1 Egyptians and Babylonians
1.2 Greece
1.3. Indo-Arabic Arithmetical Practice
1.4. Modern Times
2. Natural Numbers
2.1 Definition of the Natural Numbers
2.2 The Recursion Theorem and the Uniqueness of N
2.3 Addition, Multiplication and Ordering of the Natural Numbers
2.4 Peano's Axioms
3. The Integers
3.1 The Additive Group Z
3.2 The Integral Domain Z
3.3 The Order Relation in Z
4. The Rational Numbers
4.1 Historical
4.2 The Field Q
4.3 The Ordering of Q
References
Further Readings
Chapter 2. Real Numbers
1. Historical
1.1 Hippasus and the Pentagon
1.2 Eudoxus and the Theory of Proportion
1.3 Irrational Numbers in Modern (that is, post-mediaeval) Mathematics
1.4 The Formulation of More Precise Definitions in the Nineteenth Century
2. Dedekind Cuts
2.1 The Set R of Cuts
2.2 The Order Relation in R
2.3 Addition in R
2.4 Multiplication in R
3. Fundamental Sequences
3.1 Historical Remarks
3.2 Cauchy's Criterion for Convergence
3.3 The Ring of Fundamental Sequences
3.4 The Residue Class Field F/N of Fundamental Sequences Modulo the Null Sequence
3.5 The Completely Ordered Residue Class Field F/N
4. Nesting of Intervals
4.1 Historical Remarks
4.2 Nested Intervals and Completeness
5. Axiomatic Definition of Real Numbers
5.1 The Natural Numbers, the Integers, and the Rational Numbers in the Real Number Field
5.2 Completeness Theorem
5.3 Existence and Uniqueness of the Real Numbers
References
Further Reading
Chapter 3. Complex Numbers
1. Genesis of the Complex Numbers
1.1 Cardano (1501-1576)
1.2 Bombelli (1526-1572)
1.3 Descartes (1596-1650), Newton (1642-1727) and Leibniz (1646-1716)
1.4 Euler (1707-1783)
1.5 Wallis (1616-1703), Wessel (1745-1818) and Argand (1768-1822)
1.6 Gauss (1777-1855)
1.7 Cauchy (1789-1857)
1.8 Hamilton (1805-1865)
1.9 Later Developments
2. The Field C
2.1 Definition by Pairs of Real Numbers
2.2 The Imaginary Unit i
2.3 Geometric Representation
2.4 Impossibility of Ordering the Field C
2.5 Representation by Means of 2? Real Matrices
3. Algebraic Properties of the Field C

3.2 The Field Automorphisms of C
3.3 The Natural Scalar Product Re(wz?) and Euclidean Length |z|
3.4 Product Rule and the "Two Squares" Theorem
3.5 Quadratic Roots and Quadratic Equations
3.6 Square Roots and nth Roots
4. Geometric Properties of the Field C

4.2 Cosine Theorem and the Triangle Inequality
4.3 Numbers on Straight Lines and Circles. Cross-Ratio
4.4 Cyclic Quadrilaterals and Cross-Ratio
4.5 Ptolemy's Theorem
4.6 Wallace's Line
5. The Groups O(C) and SO(2)
5.1 Distance Preserving Mappings of C
5.2 The Group O(C)

5.4 Rational Parametrization of Properly Orthogonal 2? Matrices
6. Polar Coordinates and nth Roots
6.1 Polar Coordinates
6.2 Multiplication of Complex Numbers in Polar Coordinates
6.3 De Moivre's Formula
6.4 Roots of Unity
Chapter 4. The Fundamental Theorem of Algebra
1. On the History of the Fundamental Theorem
1.1 Girard (1595-1632) and Descartes (1596-1650)
1.2 Leibniz (1646-1716)
1.3 Euler (1707-1783)
1.4 d'Alembert (1717-1783)
1.5 Lagrange (1736-1813) and Laplace (1749-1827)
1.6 Gauss's Critique
1.7 Gauss's Four Proofs
1.8 Argand (1768-1822) and Cauchy (1798-1857)
1.9 The Fundamental Theorem of Algebra: Then and Now
1.10 Brief Biographical Notes on Carl Friedrich Gauss
2. Proof of the Fundamental Theorem Based on Argand
2.1 Cauchy's Minimum Theorem
2.2 Proof of the Fundamental Theorem
2.3 Proof of Argand's Inequality
2.4 Variant of the Proof
2.5 Constructive Proofs of the Fundamental Theorem
3. Application of the Fundamental Theorem
3.1 Factorization Lemma
3.2 Factorization of Complex Polynomials
3.3 Factorization of Real Polynomials
3.4 Existence of Eigenvalues
3.5 Prime Polynomials in C[Z] and R[X]
3.6 Uniqueness of C
3.7 The Prospects for "Hypercomplex Numbers"
Appendix: Proof of the Fundamental Theorem, after Laplace
A.1 Results Used
A.2 Proof
A.3 Historical Note


1.1 Definition by Measuring a Circle
1.2 Practical Approximations
1.3 Systematic Approximation
1.4 Analytical Formulae
1.5 Baltzer's Definition
1.6 Landau and His Contemporary Critics

2.1 The Addition Theorem
2.2 Elementary Consequences
2.3 Epimorphism Theorem

Appendix: Elementary Proof of Lemma 3

3.1 Definitions of cos z and sin z
3.2 Addition Theorem













Further Reading
Chapter 6. The p-Adic Numbers
1. Numbers as Functions
2. The Arithmetic Significance of the p-Adic Numbers
3. The Analytical Nature of p-Adic Numbers
4. The p-Adic Numbers
References
Part B: Real Division Algebras
Introduction
Repertory. Basic Concepts from the Theory of Algebras
1. Real Algebras
2. Examples of Real Algebras
3. Subalgebras and Algebra Homomorphisms
4. Determination of All One-Dimensional Algebras
5. Division Algebras
6. Construction of Algebras by Means of Bases
Additional Reading
Chapter 7. Hamilton's Quaternions
Introduction
1. The Quaternion Algebra H
1.1 The Algebra H of the Quaternions

1.3 The Imaginary Space of H
1.4 Quaternion Product, Vector Product and Scalar Product
1.5 Noncommutativity of H. The Center
1.6 The Endomorphisms of the R-Vector Space H
1.7 Quaternion Multiplication and Vector Analysis
1.8 The Fundamental Theorem of Algebra for Quaternions
2. The Algebra H as a Euclidean Vector Space
2.1 Conjugation and the Linear Form Re
2.2 Properties of the Scalar Product
2.3 The "Four Squares Theorem"
2.4 Preservation of Length, and of the Conjugacy Relation Under Automorphisms
2.5 The Group S?of Quaternions of Length 1

3. The Orthogonal Groups O(3), O(4) and Quaternions
3.1 Orthogonal Groups
3.2 The Group O(H). Cayley's Theorem
3.3 The Group O(Im H). Hamilton's Theorem

3.5 Axis of Rotation and Angle of Rotation
3.6 Euler's Parametric Representation of SO(3)
Reference
Chapter 8. The Isomorphism Theorems of Frobenius, Hopf and Gelfand-Mazur
Introduction
1. Hamiltonian Triples in Alternative Algebras
1.1 The Purely Imaginary Elements of an Algebra
1.2 Hamiltonian Triple
1.3 Existence of Hamiltonian Triples in Alternative Algebras
1.4 Alternative Algebras
2. Frobenius's Theorem
2.1 Frobenius's Lemma
2.2 Examples of Quadratic Algebras
2.3 Quaternions Lemma
2.4 Theorem of Frobenius (1877)
3. Hopf's Theorem
3.1 Topologization of Real Algebras

3.3 Hopf's Theorem
3.4 The Original Proof by Hopf
3.5 Description of All 2-Dimensional Algebras with Unit Element
4. The Gelfand-Mazur Theorem
4.1 Banach Algebras

4.3 Local Inversion Theorem
4.4 The Multiplicative Group A ?240
4.5 The Gelfand-Mazur Theorem
4.6 Structure of Normed Associative Division Algebras
4.7 The Spectrum
4.8 Historical Remarks on the Gelfand-Mazur Theorem
References
4.9 Further Developments
References
Chapter 9. Cayley Numbers or Alternative Division Algebras
1. Alternative Quadratic Algebras
1.1 Quadratic Algebras
1.2 Theorem on the Bilinear Form
1.3 Theorem on the Conjugation Mapping
1.4 The Triple Product Identity
1.5 The Euclidean Vector Space A and the Orthogonal Group O(A)
2. Existence and Properties of Octonions
2.1 Construction of the Quadratic Algebra O of Octonions
2.2 The Imaginary Space, Linear Form, Bilinear Form, and Conjugation of O
2.3 O as an Alternative Division Algebra
2.4 The "Eight Squares" Theorem

2.6 Multiplication Table for O
3. Uniqueness of the Cayley Algebra
3.1 Duplication Theorem
3.2 Uniqueness of the Cayley Algebra (Zorn 1933)
3.3 Description of O by Zorn's Vector Matrices
Additional Reading
Chapter 10. Composition Algebras. Hurwitz's Theorem. Vector Product Algebras
1. Composition Algebras
1.1 Historical Remarks on the Theory of Composition
1.2 Examples
1.3 Composition Algebras with Unit Element
1.4 Structure Theorem for Composition Algebras with Unit Element
2. Mutation of Composition Algebras
2.1 Mutation of Algebras
2.2 Mutation Theorem for Finite-Dimensional Composition Algebras
2.3 Hurwitz's Theorem (1898)
3. Vector-Product Algebras
3.1 The Concept of a Vector-Product Algebra
3.2 Construction of Vector-Product Algebras
3.3 Specification of all Vector-Product Algebras
3.4 Malcev-Algebras
3.5 Historical Remarks
Chapter 11. Division Algebras and Topology
1. The Dimension of a Division Algebra Is a Power of 2
1.1 Odd Mappings and Hopf's Theorem
1.2 Homology and Cohomology with Coefficients in F 2
1.3 Proof of Hopf's Theorem
1.4 Historical Remarks on Homology and Cohomology Theory
1.5 Stiefel's Characteristic Homology Classes
2. The Dimension of a Division Algebra Is 1, 2, 4 or 8

2.2 Parallelizability of Spheres and Division Algebras
2.3 Vector Bundles
2.4 Whitney's Characteristic Cohomology Classes
2.5 The Ring of Vector Bundles
2.6 Bott Periodicity
2.7 Characteristic Classes of Direct Sums and Tensor Products
2.8 End of the Proof
2.9 Historical Remarks
3. Additional Remarks
3.1 Definition of the Hopf Invariant
3.2 The Hopf Construction
3.3 Adams's Theorem on the Hopf Invariants
3.4 Summary
3.5 Adams's Theorem about Vector Fields on Spheres
References
Part C: Infinitesimals, Games, and Sets
Chapter 12. Nonstandard Analysis
1. Introduction
2. The Nonstandard Number Domain R
2.1 Construction of R
2.2 Properties of R
3. Features Common to R and R
4. Differential and Integral Calculus
4.1 Differentiation
4.2 Integration
Epilogue
Uniqueness of *R
Extension of the Context
References
Chapter 13. Numbers and Games
1. Introduction
1.1 The Traditional Construction of the Real Numbers
1.2 The Conway Method
1.3 Synopsis
2. Conway Games
2.1 Discussion of the Dedekind Postulates
2.2 Conway's Modification of the Dedekind Postulates
2.3 Conway Games
3. Games
3.1 The Concept of a Game
3.2 Examples of Games
3.3 An Induction Principle for Games
4. On the Theory of Games
4.1 Winning Strategies
4.2 Positive and Negative Games
4.3 A Classification of Games
5. A Partially Ordered Group of Equivalent Games
5.1 The Negative of a Game
5.2 The Sum of Two Games
5.3 Isomorphic Games
5.4 A Partial Ordering of Games
5.5 Equality of Games
6. Games and Conway Games
6.1 The Fundamental Mappings
6.2 Extending to Conway Games the Definitions of the Relations and Operations Defined for Games
6.3 Examples
7. Conway Numbers
7.1 The Conway Postulates (C1) and (C2)
7.2 Elementary Properties of the Order Relation
8. The Field of Conway Numbers
8.1 The Arithmetic Operations for Numbers
8.2 Examples
8.3 Properties of the Field of Numbers
References
Chapter 14. Set Theory and Mathematics
Introduction
1. Sets and Mathematical Objects
1.1 Individuals and More Complex Objects
1.2 Set Theoretical Definitions of More Complex Objects
1.3 Urelements as Sets
2. Axiom Systems of Set Theory
2.1 The Russell Antinomy
2.2 Zermelo's and the Zermelo-Fraenkel Set Theory
2.3 Some Consequences
3. Some Metamathematical Aspects
3.1 The Von Neumann Hierarchy
3.2 The Axiom of Choice
3.3 Independence Proofs
Epilogue
References
Name Index
Subject Index
Portraits of Famous Mathematicians
Series Titles of GTM: Continued
Back Cover