# The History of Mathematics: An Introduction

7^{th}Edition

ISBN10: 0073383155

ISBN13: 9780073383156

Copyright: 2011

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# The History of Mathematics: An Introduction, 7e by David M. Burton

## Preface

## 1 Early Number Systems and Symbols

#### 1.1

#### Primitive Counting

#### A Sense of Number

#### Notches as Tally Marks

#### The Peruvian Quipus: Knots as Numbers

#### 1.2

#### Number Recording of the Egyptians and Greeks

#### The History of Herodotus

#### Hieroglyphic Representation of Numbers

#### Egyptian Hieratic Numeration

#### The Greek Alphabetic Numeral System

#### 1.3

#### Number Recording of the Babylonians

#### Babylonian Cuneiform Script

#### Deciphering Cuneiform: Grotefend and Rawlinson

#### The Babylonian Positional Number System

#### Writing in Ancient China

## 2 Mathematics in Early Civilizations

#### 2.1

#### The Rhind Papyrus

#### Egyptian Mathematical Papyri

#### A Key to Deciphering: The Rosetta Stone

#### 2.2

#### Egyptian Arithmetic

#### Early Egyptian Multiplication

#### The Unit Fraction Table

#### Representing Rational Numbers

#### 2.3

#### Four Problems from the Rhind Papyrus

#### The Method of False Position

#### A Curious Problem

#### Egyptian Mathematics as Applied Arithmetic

#### 2.4

#### Egyptian Geometry

#### Approximating the Area of a Circle

#### The Volume of a Truncated Pyramid

#### Speculations About the Great Pyramid

#### 2.5

#### Babylonian Mathematics

#### A Tablet of Reciprocals

#### The Babylonian Treatment of Quadratic Equations

#### Two Characteristic Babylonian Problems

#### 2.6

#### Plimpton

#### A Tablet Concerning Number Triples

#### Babylonian Use of the Pythagorean Theorem

#### The Cairo Mathematical Papyrus

## 3 The Beginnings of Greek Mathematics

#### 3.1

#### The Geometric Discoveries of Thales

#### Greece and the Aegean Area

#### The Dawn of Demonstrative Geometry: Thales of Miletos

#### Measurements Using Geometry

#### 3.2

#### Pythagorean Mathematics

#### Pythagoras and His Followers

#### Nichomachus' Introductio Arithmeticae

#### The Theory of Figurative Numbers

#### Zeno's Paradox

#### 3.3

#### The Pythagorean Problem

#### Geometric Proofs of the Pythagorean Theorem

#### Early Solutions of the Pythagorean Equation

#### The Crisis of Incommensurable Quantities

#### Theon's Side and Diagonal Numbers

#### Eudoxus of Cnidos

#### 3.4

#### Three Construction Problems of Antiquity

#### Hippocrates and the Quadrature of the Circle

#### The Duplication of the Cube

#### The Trisection of an Angle

#### 3.5

#### The Quadratrix of Hippias

#### Rise of the Sophists

#### Hippias of Elis

#### The Grove of Academia: Plato's Academy

## 4 The Alexandrian School: Euclid

#### 4.1

#### Euclid and the Elements

#### A Center of Learning: The Museum

#### Euclid's Life and Writings

#### 4.2

#### Euclidean Geometry

#### Euclid's Foundation for Geometry

#### Book I of the Elements

#### Euclid's Proof of the Pythagorean Theorem

#### Book II on Geometric Algebra

#### Construction of the Regular Pentagon

#### 4.3

#### Euclid's Number Theory

#### Euclidean Divisibility Properties

#### The Algorithm of Euclid

#### The Fundamental Theorem of Arithmetic

#### An Infinity of Primes

#### 4.4

#### Eratosthenes, the Wise Man of Alexandria

#### The Sieve of Eratosthenes

#### Measurement of the Earth

#### The Almagest of Claudius Ptolemy

#### Ptolemy's Geographical Dictionary

#### 4.5

#### Archimedes

#### The Ancient World's Genius

#### Estimating the Value of π

#### The Sand-Reckoner

#### Quadrature of a Parabolic Segment

#### Apollonius of Perga: The Conics

## 5 The Twilight of Greek Mathematics: Diophantus

#### 5.1

#### The Decline of Alexandrian Mathematics

#### The Waning of the Golden Age

#### The Spread of Christianity

#### Constantinople, A Refuge for Greek Learning

#### 5.2

#### The Arithmetica

#### Diophantus's Number Theory

#### Problems from the Arithmetica

#### 5.3

#### Diophantine Equations in Greece, India, and China

#### The Cattle Problem of Archimedes

#### Early Mathematics in India

#### The Chinese Hundred Fowls Problem

#### 5.4

#### The Later Commentators

#### The Mathematical Collection of Pappus

#### Hypatia, the First Woman Mathematician

#### Roman Mathematics: Boethius and Cassiodorus

#### 5.5

#### Mathematics in the Near and Far East

#### The Algebra of al-Khowârizmî

#### Abû Kamil and Thâbit ibn Qurra

#### Omar Khayyam

#### The Astronomers al-Tusi and al-Karashi

#### The Ancient Chinese Nine Chapters

#### Later Chinese Mathematical Works

## 6 The First Awakening: Fibonacci

#### 6.1

#### The Decline and Revival of Learning

#### The Carolingian Pre-Renaissance

#### Transmission of Arabic Learning to the West

#### The Pioneer Translators: Gerard and Adelard

#### 6.2

#### The Liber Abaci and Liber Quadratorum

#### The Hindu-Arabic Numerals

#### Fibonacci's Liver Quadratorum

#### The Works of Jordanus de Nemore

#### 6.3

#### The Fibonacci Sequence

#### The Liber Abaci's Rabbit Problem

#### Some Properties of Fibonacci Numbers

#### 6.4

#### Fibonacci and the Pythagorean Problem

#### Pythagorean Number Triples

#### Fibonacci's Tournament Problem

## 7 The Renaissance of Mathematics: Cardan and Tartaglia

#### 7.1

#### Europe in the Fourteenth and Fifteenth Centuries

#### The Italian Renaissance

#### Artificial Writing: The Invention of Printing

#### Founding of the Great Universities

#### A Thirst for Classical Learning

#### 7.2

#### The Battle of the Scholars

#### Restoring the Algebraic Tradition: Robert Recorde

#### The Italian Algebraists: Pacioli, del Ferro and Tartaglia

#### Cardan, A Scoundrel Mathematician

#### 7.3

#### Cardan's Ars Magna

#### Cardan's Solution of the Cubic Equation

#### Bombelli and Imaginary Roots of the Cubic

#### 7.4

#### Ferrari's Solution of the Quartic Equation

#### The Resolvant Cubic

#### The Story of the Quintic Equation: Ruffini, Abel and Galois

## 8 The Mechanical World: Descartes and Newton

#### 8.1

#### The Dawn of Modern Mathematics

#### The Seventeenth Century Spread of Knowledge

#### Galileo's Telescopic Observations

#### The Beginning of Modern Notation: Francois Vièta

#### The Decimal Fractions of Simon Steven

#### Napier's Invention of Logarithms

#### The Astronomical Discoveries of Brahe and Kepler

#### 8.2

#### Descartes: The Discours de la Méthod

#### The Writings of Descartes

#### Inventing Cartesian Geometry

#### The Algebraic Aspect of La Géometrie

#### Descartes' Principia Philosophia

#### Perspective Geometry: Desargues and Poncelet

#### 8.3

#### Newton: The Principia Mathematica

#### The Textbooks of Oughtred and Harriot

#### Wallis' Arithmetica Infinitorum

#### The Lucasian Professorship: Barrow and Newton

#### Newton's Golden Years

#### The Laws of Motion

#### Later Years: Appointment to the Mint

#### 8.4

#### Gottfried Leibniz: The Calculus Controversy

#### The Early Work of Leibniz

#### Leibniz's Creation of the Calculus

#### Newton's Fluxional Calculus

#### The Dispute over Priority

#### Maria Agnesi and Emilie du Châtelet

## 9 The Development of Probability Theory: Pascal, Bernoulli, and Laplace

#### 9.1

#### The Origins of Probability Theory

#### Graunt's Bills of Mortality

#### Games of Chance: Dice and Cards

#### The Precocity of the Young Pascal

#### Pascal and the Cycloid

#### De Méré's Problem of Points

#### 9.2

#### Pascal's Arithmetic Triangle

#### The Traité du Triangle Arithmétique

#### Mathematical Induction

#### Francesco Maurolico's Use of Induction

#### 9.3

#### The Bernoullis and Laplace

#### Christiaan Huygens's Pamphlet on Probability

#### The Bernoulli Brothers: John and James

#### De Moivre's Doctrine of Chances

#### The Mathematics of Celestial Phenomena: Laplace

#### Mary Fairfax Somerville

#### Laplace's Research on Probability Theory

#### Daniel Bernoulli, Poisson, and Chebyshev

## 10 The Revival of Number Theory: Fermat, Euler, and Gauss

#### 10.1

#### Martin Mersenne and the Search for Perfect Numbers

#### Scientific Societies

#### Marin Mersenne's Mathematical Gathering

#### Numbers, Perfect and Not So Perfect

#### 10.2

#### From Fermat to Euler

#### Fermat's Arithmetica

#### The Famous Last Theorem of Fermat

#### The Eighteenth-Century Enlightenment

#### Maclaurin's Treatise on Fluxions

#### Euler's Life and Contributions

#### 10.3

#### The Prince of Mathematicians: Carl Friedrich Gauss

#### The Period of the French Revolution: Lagrange, Monge, and Carnot

#### Gauss's Disquisitiones Arithmeticae

#### The Legacy of Gauss: Congruence Theory

#### Dirichlet and Jacobi

## 11 Nineteenth-Century Contributions: Lobachevsky to Hilbert

#### 11.1

#### Attempts to Prove the Parallel Postulate

#### The Efforts of Proclus, Playfair, and Wallis

#### Saccheri Quadrilaterals

#### The Accomplishments of Legendre

#### Legendre's Eléments de géometrie

#### 11.2

#### The Founders of Non-Euclidean Geometry

#### Gauss's Attempt at a New Geometry

#### The Struggle of John Bolyai

#### Creation of Non-Euclidean Geometry: Lobachevsky

#### Models of the New Geometry: Riemann, Beltrami, and Klein

#### Grace Chisholm Young

#### 11.3

#### The Age of Rigor

#### D'Alembert and Cauchy on Limits

#### Fourier's Series

#### The Father of Modern Analysis, Weierstrass

#### Sonya Kovalevsky

#### The Axiomatic Movement: Pasch and Hilbert

#### 11.4

#### Arithmetic Generalized

#### Babbage and the Analytical Engine

#### Peacock's Treatise on Algebra

#### The Representations of Complex Numbers

#### Hamilton's Discovery of Quaternions

#### Matrix Algebra: Cayley and Sylvester

#### Boole's Algebra of Logic

## 12 Transition to the Twenthieth Century: Cantor and Kronecker

#### 12.1

#### The Emergence of American Mathematics

#### Ascendency of the German Universities

#### American Mathematics Takes Root: 1800-1900

#### The Twentieth Century Consolidation

#### 12.2

#### Counting the Infinite

#### The Last Universalist: Poincaré

#### Cantor's Theory of Infinite Sets

#### Kronecker's View of Set Theory

#### Countable and Uncountable Sets

#### Transcendental Numbers

#### The Continuum Hypothesis

#### 12.3

#### The Paradoxes of Set Theory

#### The Early Paradoxes

#### Zermelo and the Axiom of Choice

#### The Logistic School: Frege, Peano and Russell

#### Hilbert's Formalistic Approach

#### Brouwer's Intuitionism

## 13 Extensions and Generalizations: Hardy, Hausdorff, and Noether

#### 13.1

#### Hardy and Ramanujan

#### The Tripos Examination

#### The Rejuvenation of English Mathematics

#### A Unique Collaboration: Hardy and Littlewood

#### India's Prodigy, Ramanujan

#### 13.2

#### The Beginnings of Point-Set Topology

#### Frechet's Metric Spaces

#### The Neighborhood Spaces of Hausdorff

#### Banach and Normed Linear Spaces

#### 13.3

#### Some Twentieth-Century Developments

#### Emmy Noether's Theory of Rings

#### Von Neumann and the Computer

#### Women in Modern Mathematics

#### A Few Recent Advances

## General Bibliography

## Additional Reading

## The Greek Alphabet

## Solutions to Selected Problems

## Index

# The History of Mathematics: An Introduction, 7e by David M. Burton

## Preface

## 1 Early Number Systems and Symbols

#### 1.1

#### Primitive Counting

#### A Sense of Number

#### Notches as Tally Marks

#### The Peruvian Quipus: Knots as Numbers

#### 1.2

#### Number Recording of the Egyptians and Greeks

#### The History of Herodotus

#### Hieroglyphic Representation of Numbers

#### Egyptian Hieratic Numeration

#### The Greek Alphabetic Numeral System

#### 1.3

#### Number Recording of the Babylonians

#### Babylonian Cuneiform Script

#### Deciphering Cuneiform: Grotefend and Rawlinson

#### The Babylonian Positional Number System

#### Writing in Ancient China

## 2 Mathematics in Early Civilizations

#### 2.1

#### The Rhind Papyrus

#### Egyptian Mathematical Papyri

#### A Key to Deciphering: The Rosetta Stone

#### 2.2

#### Egyptian Arithmetic

#### Early Egyptian Multiplication

#### The Unit Fraction Table

#### Representing Rational Numbers

#### 2.3

#### Four Problems from the Rhind Papyrus

#### The Method of False Position

#### A Curious Problem

#### Egyptian Mathematics as Applied Arithmetic

#### 2.4

#### Egyptian Geometry

#### Approximating the Area of a Circle

#### The Volume of a Truncated Pyramid

#### Speculations About the Great Pyramid

#### 2.5

#### Babylonian Mathematics

#### A Tablet of Reciprocals

#### The Babylonian Treatment of Quadratic Equations

#### Two Characteristic Babylonian Problems

#### 2.6

#### Plimpton

#### A Tablet Concerning Number Triples

#### Babylonian Use of the Pythagorean Theorem

#### The Cairo Mathematical Papyrus

## 3 The Beginnings of Greek Mathematics

#### 3.1

#### The Geometric Discoveries of Thales

#### Greece and the Aegean Area

#### The Dawn of Demonstrative Geometry: Thales of Miletos

#### Measurements Using Geometry

#### 3.2

#### Pythagorean Mathematics

#### Pythagoras and His Followers

#### Nichomachus' Introductio Arithmeticae

#### The Theory of Figurative Numbers

#### Zeno's Paradox

#### 3.3

#### The Pythagorean Problem

#### Geometric Proofs of the Pythagorean Theorem

#### Early Solutions of the Pythagorean Equation

#### The Crisis of Incommensurable Quantities

#### Theon's Side and Diagonal Numbers

#### Eudoxus of Cnidos

#### 3.4

#### Three Construction Problems of Antiquity

#### Hippocrates and the Quadrature of the Circle

#### The Duplication of the Cube

#### The Trisection of an Angle

#### 3.5

#### The Quadratrix of Hippias

#### Rise of the Sophists

#### Hippias of Elis

#### The Grove of Academia: Plato's Academy

## 4 The Alexandrian School: Euclid

#### 4.1

#### Euclid and the Elements

#### A Center of Learning: The Museum

#### Euclid's Life and Writings

#### 4.2

#### Euclidean Geometry

#### Euclid's Foundation for Geometry

#### Book I of the Elements

#### Euclid's Proof of the Pythagorean Theorem

#### Book II on Geometric Algebra

#### Construction of the Regular Pentagon

#### 4.3

#### Euclid's Number Theory

#### Euclidean Divisibility Properties

#### The Algorithm of Euclid

#### The Fundamental Theorem of Arithmetic

#### An Infinity of Primes

#### 4.4

#### Eratosthenes, the Wise Man of Alexandria

#### The Sieve of Eratosthenes

#### Measurement of the Earth

#### The Almagest of Claudius Ptolemy

#### Ptolemy's Geographical Dictionary

#### 4.5

#### Archimedes

#### The Ancient World's Genius

#### Estimating the Value of π

#### The Sand-Reckoner

#### Quadrature of a Parabolic Segment

#### Apollonius of Perga: The Conics

## 5 The Twilight of Greek Mathematics: Diophantus

#### 5.1

#### The Decline of Alexandrian Mathematics

#### The Waning of the Golden Age

#### The Spread of Christianity

#### Constantinople, A Refuge for Greek Learning

#### 5.2

#### The Arithmetica

#### Diophantus's Number Theory

#### Problems from the Arithmetica

#### 5.3

#### Diophantine Equations in Greece, India, and China

#### The Cattle Problem of Archimedes

#### Early Mathematics in India

#### The Chinese Hundred Fowls Problem

#### 5.4

#### The Later Commentators

#### The Mathematical Collection of Pappus

#### Hypatia, the First Woman Mathematician

#### Roman Mathematics: Boethius and Cassiodorus

#### 5.5

#### Mathematics in the Near and Far East

#### The Algebra of al-Khowârizmî

#### Abû Kamil and Thâbit ibn Qurra

#### Omar Khayyam

#### The Astronomers al-Tusi and al-Karashi

#### The Ancient Chinese Nine Chapters

#### Later Chinese Mathematical Works

## 6 The First Awakening: Fibonacci

#### 6.1

#### The Decline and Revival of Learning

#### The Carolingian Pre-Renaissance

#### Transmission of Arabic Learning to the West

#### The Pioneer Translators: Gerard and Adelard

#### 6.2

#### The Liber Abaci and Liber Quadratorum

#### The Hindu-Arabic Numerals

#### Fibonacci's Liver Quadratorum

#### The Works of Jordanus de Nemore

#### 6.3

#### The Fibonacci Sequence

#### The Liber Abaci's Rabbit Problem

#### Some Properties of Fibonacci Numbers

#### 6.4

#### Fibonacci and the Pythagorean Problem

#### Pythagorean Number Triples

#### Fibonacci's Tournament Problem

## 7 The Renaissance of Mathematics: Cardan and Tartaglia

#### 7.1

#### Europe in the Fourteenth and Fifteenth Centuries

#### The Italian Renaissance

#### Artificial Writing: The Invention of Printing

#### Founding of the Great Universities

#### A Thirst for Classical Learning

#### 7.2

#### The Battle of the Scholars

#### Restoring the Algebraic Tradition: Robert Recorde

#### The Italian Algebraists: Pacioli, del Ferro and Tartaglia

#### Cardan, A Scoundrel Mathematician

#### 7.3

#### Cardan's Ars Magna

#### Cardan's Solution of the Cubic Equation

#### Bombelli and Imaginary Roots of the Cubic

#### 7.4

#### Ferrari's Solution of the Quartic Equation

#### The Resolvant Cubic

#### The Story of the Quintic Equation: Ruffini, Abel and Galois

## 8 The Mechanical World: Descartes and Newton

#### 8.1

#### The Dawn of Modern Mathematics

#### The Seventeenth Century Spread of Knowledge

#### Galileo's Telescopic Observations

#### The Beginning of Modern Notation: Francois Vièta

#### The Decimal Fractions of Simon Steven

#### Napier's Invention of Logarithms

#### The Astronomical Discoveries of Brahe and Kepler

#### 8.2

#### Descartes: The Discours de la Méthod

#### The Writings of Descartes

#### Inventing Cartesian Geometry

#### The Algebraic Aspect of La Géometrie

#### Descartes' Principia Philosophia

#### Perspective Geometry: Desargues and Poncelet

#### 8.3

#### Newton: The Principia Mathematica

#### The Textbooks of Oughtred and Harriot

#### Wallis' Arithmetica Infinitorum

#### The Lucasian Professorship: Barrow and Newton

#### Newton's Golden Years

#### The Laws of Motion

#### Later Years: Appointment to the Mint

#### 8.4

#### Gottfried Leibniz: The Calculus Controversy

#### The Early Work of Leibniz

#### Leibniz's Creation of the Calculus

#### Newton's Fluxional Calculus

#### The Dispute over Priority

#### Maria Agnesi and Emilie du Châtelet

## 9 The Development of Probability Theory: Pascal, Bernoulli, and Laplace

#### 9.1

#### The Origins of Probability Theory

#### Graunt's Bills of Mortality

#### Games of Chance: Dice and Cards

#### The Precocity of the Young Pascal

#### Pascal and the Cycloid

#### De Méré's Problem of Points

#### 9.2

#### Pascal's Arithmetic Triangle

#### The Traité du Triangle Arithmétique

#### Mathematical Induction

#### Francesco Maurolico's Use of Induction

#### 9.3

#### The Bernoullis and Laplace

#### Christiaan Huygens's Pamphlet on Probability

#### The Bernoulli Brothers: John and James

#### De Moivre's Doctrine of Chances

#### The Mathematics of Celestial Phenomena: Laplace

#### Mary Fairfax Somerville

#### Laplace's Research on Probability Theory

#### Daniel Bernoulli, Poisson, and Chebyshev

## 10 The Revival of Number Theory: Fermat, Euler, and Gauss

#### 10.1

#### Martin Mersenne and the Search for Perfect Numbers

#### Scientific Societies

#### Marin Mersenne's Mathematical Gathering

#### Numbers, Perfect and Not So Perfect

#### 10.2

#### From Fermat to Euler

#### Fermat's Arithmetica

#### The Famous Last Theorem of Fermat

#### The Eighteenth-Century Enlightenment

#### Maclaurin's Treatise on Fluxions

#### Euler's Life and Contributions

#### 10.3

#### The Prince of Mathematicians: Carl Friedrich Gauss

#### The Period of the French Revolution: Lagrange, Monge, and Carnot

#### Gauss's Disquisitiones Arithmeticae

#### The Legacy of Gauss: Congruence Theory

#### Dirichlet and Jacobi

## 11 Nineteenth-Century Contributions: Lobachevsky to Hilbert

#### 11.1

#### Attempts to Prove the Parallel Postulate

#### The Efforts of Proclus, Playfair, and Wallis

#### Saccheri Quadrilaterals

#### The Accomplishments of Legendre

#### Legendre's Eléments de géometrie

#### 11.2

#### The Founders of Non-Euclidean Geometry

#### Gauss's Attempt at a New Geometry

#### The Struggle of John Bolyai

#### Creation of Non-Euclidean Geometry: Lobachevsky

#### Models of the New Geometry: Riemann, Beltrami, and Klein

#### Grace Chisholm Young

#### 11.3

#### The Age of Rigor

#### D'Alembert and Cauchy on Limits

#### Fourier's Series

#### The Father of Modern Analysis, Weierstrass

#### Sonya Kovalevsky

#### The Axiomatic Movement: Pasch and Hilbert

#### 11.4

#### Arithmetic Generalized

#### Babbage and the Analytical Engine

#### Peacock's Treatise on Algebra

#### The Representations of Complex Numbers

#### Hamilton's Discovery of Quaternions

#### Matrix Algebra: Cayley and Sylvester

#### Boole's Algebra of Logic

## 12 Transition to the Twenthieth Century: Cantor and Kronecker

#### 12.1

#### The Emergence of American Mathematics

#### Ascendency of the German Universities

#### American Mathematics Takes Root: 1800-1900

#### The Twentieth Century Consolidation

#### 12.2

#### Counting the Infinite

#### The Last Universalist: Poincaré

#### Cantor's Theory of Infinite Sets

#### Kronecker's View of Set Theory

#### Countable and Uncountable Sets

#### Transcendental Numbers

#### The Continuum Hypothesis

#### 12.3

#### The Paradoxes of Set Theory

#### The Early Paradoxes

#### Zermelo and the Axiom of Choice

#### The Logistic School: Frege, Peano and Russell

#### Hilbert's Formalistic Approach

#### Brouwer's Intuitionism

## 13 Extensions and Generalizations: Hardy, Hausdorff, and Noether

#### 13.1

#### Hardy and Ramanujan

#### The Tripos Examination

#### The Rejuvenation of English Mathematics

#### A Unique Collaboration: Hardy and Littlewood

#### India's Prodigy, Ramanujan

#### 13.2

#### The Beginnings of Point-Set Topology

#### Frechet's Metric Spaces

#### The Neighborhood Spaces of Hausdorff

#### Banach and Normed Linear Spaces

#### 13.3

#### Some Twentieth-Century Developments

#### Emmy Noether's Theory of Rings

#### Von Neumann and the Computer

#### Women in Modern Mathematics

#### A Few Recent Advances

## General Bibliography

## Additional Reading

## The Greek Alphabet

## Solutions to Selected Problems

## Index

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