DISCRETE AND COMBINATORIAL UTHEMATICSinis.jinr.ru/sl/M_Mathematics/MRef_References/Handbook of...

HANDBOOK OF

DISCRETE AND COMBINATORIAL

UTHEMATICS

KENNETH H. ROSEN AT&T Laboratories

Editor-in-Chief

JOHN G. MICHAELS SUNY Brockport

Project Editor

JONATHAN L. GROSS Columbia University

Associate Editor

JERROLD W. GROSSMAN Oakland University

Associate Editor

DOUGLAS R SHIER Clemson University

Associate Editor

CRC Press

Boca Raton London New York Washington, D.C.

Library of Congress Cataloging-in-Publication Data

Handbook of discrete and combinatorial mathematics / Kenneth H. Rosen, editor in chief,John G. Michaels, project editor...[et al.].

p. c m .Includes bibliographical references and index.ISBN 0-8493-0149-1 (alk. paper)1. Combinatorial analysis-Handbooks, manuals, etc. 2. Computer

science-Mathematics-Handbooks, manuals, etc. I. Rosen, Kenneth H. II. Michaels,John G.

QAl64.H36 19995 I I .6dc21 99-04378

This book contains information obtained from authentic and highIy regarded sources. Reprinted materia1 is quoted withpermission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publishreliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materialsor for the consequences of their use.

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CONTENTS

1. FOUNDATIONS1.1 Propositional and Predicate Logic Jerrold W. Grossman1.2 Set Theory Jerrold W. Grossman1.3 Functions Jerrold W. Grossman1.4 Relations John G. Michaels1.5 Proof Techniques Susanna S. Epp1.6 Axiomatic Program Verification David Riley1.7 Logic-Based Computer Programming Paradigms Mukesh Dalal

2. COUNTING METHODS2.1 Summary of Counting Problems John G. Michaels2.2 Basic Counting Techniques Jay Yellen2.3 Permutations and Combinations Edward W. Packel2.4 Inclusion/Exclusion Robert G. Rieper2.5 Partitions George E. Andrews2.6 Burnside/Polya Counting Formula Alan C. Tucker2.7 Mobius Inversion Counting Edward A. Bender2.8 Young Tableaux Bruce E. Sagan

3. SEQUENCES3.1 Special Sequences Thomas A. Dowling and Douglas R. Shier3.2 Generating Functions Ralph P. Grimaldi3.3 Recurrence Relations Ralph P. Grimaldi3.4 Finite Differences Jay Yellen3.5 Finite Sums and Summation Victor S. Miller3.6 Asymptotics of Sequences Edward A. Bender3.7 Mechanical Summation Procedures Kenneth H. Rosen

4. NUMBER THEORY4.1 Basic Concepts Kenneth H. Rosen4.2 Greatest Common Divisors Kenneth H. Rosen4.3 Congruences Kenneth H. Rosen4.4 Prime Numbers Jon F. Grantham and Carl Pomerance4.5 Factorization Jon F. Grantham and Carl Pomerance4.6 Arithmetic Functions Kenneth H. Rosen4.7 Primitive Roots and Quadratic Residues Kenneth H. Rosen4.8 Diophantine Equations Bart E. Goddard4.9 Diophantine Approximation Jeff Shalit4.10 Quadratic Fields Kenneth H. Rosen

c 2000 by CRC Press LLC

5. ALGEBRAIC STRUCTURES John G. Michaels5.1 Algebraic Models5.2 Groups5.3 Permutation Groups5.4 Rings5.5 Polynomial Rings5.6 Fields5.7 Lattices5.8 Boolean Algebras

6. LINEAR ALGEBRA6.1 Vector Spaces Joel V. Brawley6.2 Linear Transformations Joel V. Brawley6.3 Matrix Algebra Peter R. Turner6.4 Linear Systems Barry Peyton and Esmond Ng6.5 Eigenanalysis R. B. Bapat6.6 Combinatorial Matrix Theory R. B. Bapat

7. DISCRETE PROBABILITY7.1 Fundamental Concepts Joseph R. Barr7.2 Independence and Dependence Joseph R. Barr 4357.3 Random Variables Joseph R. Barr7.4 Discrete Probability Computations Peter R. Turner7.5 Random Walks Patrick Jaillet7.6 System Reliability Douglas R. Shier7.7 Discrete-Time Markov Chains Vidyadhar G. Kulkarni7.8 Queueing Theory Vidyadhar G. Kulkarni7.9 Simulation Lawrence M. Leemis

8. GRAPH THEORY8.1 Introduction to Graphs Lowell W. Beineke8.2 Graph Models Jonathan L. Gross8.3 Directed Graphs Stephen B. Maurer8.4 Distance, Connectivity, Traversability Edward R. Scheinerman8.5 Graph Invariants and Isomorphism Types Bennet Manvel8.6 Graph and Map Coloring Arthur T. White8.7 Planar Drawings Jonathan L. Gross8.8 Topological Graph Theory Jonathan L. Gross8.9 Enumerating Graphs Paul K. Stockmeyer8.10 Algebraic Graph Theory Michael Doob8.11 Analytic Graph Theory Stefan A. Burr8.12 Hypergraphs Andreas Gyarfas

9. TREES9.1 Characterizations and Types of Trees Lisa Carbone9.2 Spanning Trees Uri Peled9.3 Enumerating Trees Paul Stockmeyer


10. NETWORKS AND FLOWS10.1 Minimum Spanning Trees J. B. Orlin and Ravindra K. Ahuja10.2 Matchings Douglas R. Shier10.3 Shortest Paths J. B. Orlin and Ravindra K. Ahuja10.4 Maximum Flows J. B. Orlin and Ravindra K. Ahuja10.5 Minimum Cost Flows J. B. Orlin and Ravindra K. Ahuja10.6 Communication Networks David Simchi-Levi and Sunil Chopra10.7 Difficult Routing and Assignment Problems Bruce L. Golden and Bharat K. Kaku10.8 Network Representations and Data Structures Douglas R. Shier

11. PARTIALLY ORDERED SETS11.1 Basic Poset Concepts Graham Brightwell and Douglas B. West11.2 Poset Properties Graham Brightwell and Douglas B. West

12. COMBINATORIAL DESIGNS12.1 Block Designs Charles J. Colbourn and Jeffrey H. Dinitz12.2 Symmetric Designs & Finite Geometries Charles J. Colbourn and Jeffrey H. Dinitz12.3 Latin Squares and Orthogonal Arrays Charles J. Colbourn and Jeffrey H. Dinitz12.4 Matroids James G. Oxley

13. DISCRETE AND COMPUTATIONAL GEOMETRY13.1 Arrangements of Geometric Objects Ileana Streinu13.2 Space Filling Karoly Bezdek13.3 Combinatorial Geometry Janos Pach13.4 Polyhedra Tamal K. Dey13.5 Algorithms and Complexity in Computational Geometry Jianer Chen13.6 Geometric Data Structures and Searching Dina Kravets 85313.7 Computational Techniques Nancy M. Amato13.8 Applications of Geometry W. Randolph Franklin

14. CODING THEORY AND CRYPTOLOGY Alfred J. Menezes andPaul C. van Oorschot

14.1 Communication Systems and Information Theory14.2 Basics of Coding Theory14.3 Linear Codes14.4 Bounds for Codes14.5 Nonlinear Codes14.6 Convolutional Codes14.7 Basics of Cryptography14.8 Symmetric-Key Systems14.9 Public-Key Systems

15. DISCRETE OPTIMIZATION15.1 Linear Programming Beth Novick15.2 Location Theory S. Louis Hakimi15.3 Packing and Covering Sunil Chopra and David Simchi-Levi15.4 Activity Nets S. E. Elmaghraby15.5 Game Theory Michael Mesterton-Gibbons15.6 Sperners Lemma and Fixed Points Joseph R. Barr


16. THEORETICAL COMPUTER SCIENCE16.1 Computational Models Jonathan L. Gross16.2 Computability William Gasarch16.3 Languages and Grammars Aarto Salomaa16.4 Algorithmic Complexity Thomas Cormen16.5 Complexity Classes Lane Hemaspaandra16.6 Randomized Algorithms Milena Mihail

17. INFORMATION STRUCTURES17.1 Abstract Datatypes Charles H. Goldberg17.2 Concrete Data Structures Jonathan L. Gross17.3 Sorting and Searching Jianer Chen17.4 Hashing Viera Krnanova Proulx17.5 Dynamic Graph Algorithms Joan Feigenbaum and Sampath Kannan

BIOGRAPHIES Victor J. Katz


PREFACEThe importance of discrete and combinatorial mathematics has increased dramaticallywithin the last few years. The purpose of the Handbook of Discrete and CombinatorialMathematics is to provide a comprehensive reference volume for computer scientists,engineers, mathematicians, and others, such as students, physical and social scientists,and reference librarians, who need information about discrete and combinatorial math-ematics.

This book is the first resource that presents such information in a ready-reference formdesigned for use by all those who use aspects of this subject in their work or studies.The scope of this book includes the many areas generally considered to be parts ofdiscrete mathematics, focusing on the information considered essential to its applicationin computer science and engineering. Some of the fundamental topic areas coveredinclude:

logic and set theory graph theoryenumeration treesinteger sequences network sequencesrecurrence relations combinatorial designsgenerating functions computational geometrynumber theory coding theory and cryptographyabstract algebra discrete optimizationlinear algebra automata theorydiscrete probability theory data structures and algorithms.

Format

The material in the Handbook is presented so that key information can be locatedand used quickly and easily. Each chapter includes a glossary that provides succinctdefinitions of the most important terms from that chapter. Individual topics are cov-ered in sections and subsections within chapters, each of which is organized into clearlyidentifiable parts: definitions, facts, and examples. The definitions included are care-fully crafted to help readers quickly grasp new concepts. Important notation is alsohighlighted in the definitions. Lists of facts include:

information about how material is used and why it is important historical information key theorems the latest results the status of open questions tables of numerical values, generally not easily computed summary tables key algorithms in an easily understood pseudocode information about algorithms, such as their complexity major applications pointers to additional resources, including websites and printed material.


Facts are presented concisely and are listed so that they can be easily found and un-derstood. Extensive crossreferences linking parts of the handbook are also provided.Readers who want to study a topic further can consult the resources listed.

The material in the Handbook has been chosen for inclusion primarily because it isimportant and useful. Additional material has been added to ensure comprehensivenessso that readers encountering new terminology and concepts from discrete mathematicsin their explorations will be able to get help from this book.

Examples are provided to illustrate some of the key definitions, facts, and algorithms.Some curious and entertaining facts and puzzles that some readers may find intriguingare also included.

Each chapter of the book includes a list of references divided into a list of printedresources and a list of relevant websites.

How This Book Was Developed

The organization and structure of the Handbook were developed by a team which in-cluded the chief editor, three associate editors, the project editor, and the editor fromCRC Press. This team put together a proposed table of contents which was then ana-lyzed by members of a group of advisory editors, each an expert in one or more aspectsof discrete mathematics. These advisory editors suggested changes, including the cover-age of additional important topics. Once the table of contents was fully developed, theindividual sections of the book were prepared by a group of more than 70 contributorsfrom industry and academia who understand how this material is used and why it isimportant. Contributors worked under the direction of the associate editors and chiefeditor, with these editors ensuring consistency of style and clarity and comprehensive-ness in the presentation of material. Material was carefully reviewed by authors andour team of editors to ensure accuracy and consistency of style.

The CRC Press Series on Discrete Mathematics and Its Applications

This Handbook is designed to be a ready reference that covers many important distincttopics. People needing information in multiple areas of discrete and combinatorialmathematics need only have this one volume to obtain what they need or for pointersto where they can find out more information. Among the most valuable sources ofadditional information are the volumes in the CRC Press Series on Discrete Mathematicsand Its Applications. This series includes both Handbooks, which are ready references,and advanced Textbooks/Monographs. More detailed and comprehensive coverage inparticular topic areas can be found in these individual volumes:

Handbooks

The CRC Handbook of Combinatorial Designs Handbook of Discrete and Computational Geometry Handbook of Applied Cryptography

Textbooks/Monographs

Graph Theory and its Applications Algebraic Number Theory Quadratics


Design Theory Frames and Resolvable Designs: Uses, Constructions, and Existence Network Reliability: Experiments with a Symbolic Algebra Environment Fundamental Number Theory with Applications Cryptography: Theory and Practice Introduction to Information Theory and Data Compression Combinatorial Algorithms: Generation, Enumeration, and Search

Feedback

To see updates and to provide feedback and errata reports, please consult the Web pagefor this book. This page can be accessed by first going to the CRC website at

http://www.crcpress.com

and then following the links to the Web page for this book.

Acknowledgments

First and foremost, we would like to thank the original CRC editor of this project,Wayne Yuhasz, who commissioned this project. We hope we have done justice to hisoriginal vision of what this book could be. We would also like to thank Bob Stern,who has served as the editor of this project for his continued support and enthusiasmfor this project. We would like to thank Nora Konopka for her assistance with manyaspects in the development of this project. Thanks also go to Susan Fox, for her helpwith production of this book at CRC Press.

We would like to thank the many people who were involved with this project. First,we would like to thank the team of advisory editors who helped make this referencerelevant, useful, unique, and up-to-date. We also wish to thank all the people at thevarious institutions where we work, including the management of AT&T Laboratories fortheir support of this project and for providing a stimulating and interesting atmosphere.

Project Editor John Michaels would like to thank his wife Lois and daughter Margaretfor their support and encouragement in the development of the Handbook. AssociateEditor Jonathan Gross would like to thank his wife Susan for her patient support,Associate Editor Jerrold Grossman would like to thank Suzanne Zeitman for her helpwith computer science materials and contacts, and Associate Editor Douglas Shier wouldlike to thank his wife Joan for her support and understanding throughout the project.


ADVISORY EDITORIAL BOARD

Andrew Odlyzko Chief Advisory EditorAT&T Laboratories

Stephen F. AltschulNational Institutes of Health

George E. AndrewsPennsylvania State University

Francis T. BoeschStevens Institute of Technology

Ernie BrickellCertco

Fan R. K. ChungUniv. of California at San Diego

Charles J. ColbournUniversity of Vermont

Stan DevittWaterloo Maple Software

Zvi GalilColumbia University

Keith GeddesUniversity of Waterloo

Ronald L. GrahamUniv. of California at San Diego

Ralph P. GrimaldiRose-Hulman Inst. of Technology

Frank HararyNew Mexico State University

Alan HoffmanIBM

Bernard KorteRheinische Friedrich-Wilhems-Univ.

Jeffrey C. LagariasAT&T Laboratories

Carl PomeranceUniversity of Georgia

Fred S. RobertsRutgers University

Pierre RosenstiehlCentre dAnalyse et de Math. Soc.

Francis SullivanIDA

J. H. Van LintEindhoven University of Technology

Scott VanstoneUniversity of Waterloo

Peter WinklerBell Laboratories


CONTRIBUTORS

Ravindra K. AhujaUniversity of Florida

Nancy M. AmatoTexas A&M University

George E. AndrewsPennsylvania State University

R. B. BapatIndian Statistical Institute

Joseph R. BarrSPSS

Lowell W. BeinekePurdue University Fort Wayne

Edward A. BenderUniversity of California at San Diego

Karoly BezdekCornell University

Joel V. BrawleyClemson University

Graham BrightwellLondon School of Economics

Stefan A. BurrCity College of New York

Lisa CarboneHarvard University

Jianer ChenTexas A&M University

Sunil ChopraNorthwestern University

Charles J. ColbournUniversity of Vermont

Thomas CormenDartmouth College

Mukesh Dalali2 Technologies

Tamal K. DeyIndian Institute of Technology Kharagpur

Jeffrey H. DinitzUniversity of Vermont

Michael DoobUniversity of Manitoba

Thomas A. DowlingOhio State University

S. E. ElmaghrabyNorth Carolina State University

Susanna S. EppDePaul University

Joan FeigenbaumAT&T Laboratories

W. Randolph FranklinRensselaer Polytechnic Institute

William GasarchUniversity of Maryland

Bart E. GoddardTexas A&M University

Charles H. GoldbergTrenton State College

Bruce L. GoldenUniversity of Maryland

Jon F. GranthamIDA

Ralph P. GrimaldiRose-Hulman Inst. of Technology

Jonathan L. GrossColumbia University


Jerrold W. GrossmanOakland University

Andreas GyarfasHungarian Academy of Sciences

S. Louis HakimiUniversity of California at Davis

Lane HemaspaandraUniversity of Rochester

Patrick JailletUniversity of Texas at Austin

Bharat K. KakuAmerican University

Sampath KannanUniversity of Pennsylvania

Victor J. KatzUniv. of the District of Columbia

Dina KravetsSarnoff Corporation

Vidyadhar G. KulkarniUniversity of North Carolina

Lawrence M. LeemisThe College of William and Mary

Bennet ManvelColorado State University

Stephen B. MaurerSwarthmore College

Alfred J. MenezesUniversity of Waterloo

Michael Mesterton-GibbonsFlorida State University

John G. MichaelsSUNY Brockport

Milena MihailGeorgia Institute of Technology

Victor S. MillerCenter for Communications

Research IDA

Esmond NgLawrence Berkeley National Lab.

Beth NovickClemson University

James B. OrlinMassachusetts Inst. of Technology

James G. OxleyLouisiana State University

Janos PachCity College CUNY, andHungarian Academy of Sciences

Edward W. PackelLake Forest College

Uri PeledUniversity of Illinois at Chicago

Barry PeytonOak Ridge National Laboratory

Carl PomeranceUniversity of Georgia

Viera Krnanova ProulxNortheastern University

Robert G. RieperWilliam Patterson University

David RileyUniversity of Wisconsin

Kenneth H. RosenAT&T Laboratories

Bruce E. SaganMichigan State University

Aarto SalomaaUniversity of Turku, Finland

Edward R. ScheinermanJohns Hopkins University

Jeff ShalitUniversity of Waterloo

Douglas R. ShierClemson University


David Simchi-LeviNorthwestern University

Paul K. StockmeyerThe College of William and Mary

Ileana StreinuSmith College

Alan C. TuckerSUNY Stony Brook

Peter R. TurnerUnited States Naval Academy

Paul C. van OorschotEntrust Technologies

Douglas B. WestUniversity of Illinois at Champaign-

Urbana

Arthur T. WhiteWestern Michigan University

Jay YellenFlorida Institute of Technology


BIOGRAPHIESVictor J. Katz

Niels Henrik Abel (18021829), born in Norway, was self-taught and studied theworks of many mathematicians. When he was nineteen years old, he proved thatthere is no closed formula for solving the general fifth degree equation. He alsoworked in the areas of infinite series and elliptic functions and integrals. The termabelian group was coined in Abels honor in 1870 by Camille Jordan.

Abraham ibn Ezra (10891164) was a Spanish-Jewish poet, philosopher, astrologer,and biblical commentator who was born in Tudela, but spent the latter part ofhis life as a wandering scholar in Italy, France, England, and Palestine. It was inan astrological text that ibn Ezra developed a method for calculating numbers ofcombinations, in connection with determining the number of possible conjunctions ofthe seven planets (including the sun and the moon). He gave a detailed argumentfor the cases n = 7, k = 2 to 7, of a rule which can easily be generalize to the modernformula C(n, k) =

n1i=k1 C(i, k 1). Ibn Ezra also wrote a work on arithmetic in

which he introduced the Hebrew-speaking community to the decimal place-valuesystem. He used the first nine letters of the Hebrew alphabet to represent the firstnine numbers, used a circle to represent zero, and demonstrated various algorithmsfor calculation in this system.

Aristotle (384322 B.C.E.) was the most famous student at Platos academy in Athens.After Platos death in 347 B.C.E., he was invited to the court of Philip II of Mace-don to educate Philips son Alexander, who soon thereafter began his successfulconquest of the Mediterranean world. Aristotle himself returned to Athens, wherehe founded his own school, the Lyceum, and spent the remainder of his life writingand lecturing. He wrote on numerous subjects, but is perhaps best known for hisworks on logic, including the Prior Analytics and the Posterior Analytics. In theseworks, Aristotle developed the notion of logical argument, based on several explicitprinciples. In particular, he built his arguments out of syllogisms and concluded thatdemonstrations using his procedures were the only certain way of attaining scientificknowledge.

Emil Artin (18981962) was born in Vienna and in 1921 received a Ph.D. from the Uni-versity of Leipzig. He held a professorship at the University of Hamburg until 1937,when he came to the United States. In the U.S. he taught at the University of NotreDame, Indiana University, and Princeton. In 1958 he returned to the Universityof Hamburg. Artins mathematical contributions were in number theory, algebraictopology, linear algebra, and especially in many areas of abstract algebra.

Charles Babbage (17921871) was an English mathematician best known for his in-vention of two of the earliest computing machines, the Difference Engine, designedto calculate polynomial functions, and the Analytical Engine, a general purpose cal-culating machine. The Difference Engine was designed to use the idea that the nthorder differences in nth degree polynomials were always constant and then to workbackwards from those differences to the original polynomial values. Although Bab-


bage received a grant from the British government to help in building the Engine, henever was able to complete one because of various difficulties in developing machineparts of sufficient accuracy. In addition, Babbage became interested in his moreadvanced Analytical Engine. This latter device was to consist of a store, in whichthe numerical variables were kept, and a mill, in which the operations were per-formed. The entire machine was to be controlled by instructions on punched cards.Unfortunately, although Babbage made numerous engineering drawings of sectionsof the Analytical Engine and gave a series of seminars in 1840 on its workings, hewas never able to build a working model.

Paul Gustav Heinrich Bachmann (18371920) studied mathematics at the Univer-sity of Berlin and at Gottingen. In 1862 he received a doctorate in group theory andheld positions at the universities at Breslau and Munster. He wrote several volumeson number theory, introducing the big-O notation in his 1892 book.

John Backus (born 1924) received bachelors and masters degrees in mathematicsfrom Columbia University. He led the group at IBM that developed FORTRAN.He was a developer of ALGOL, using the Backus-Naur form for the syntax of thelanguage. He received the National Medal of Science in 1974 and the Turing Awardin 1977.

Abu-l-Abbas Ahmad ibn Muhammad ibn al-Banna al-Marrakushi (12561321) was an Islamic mathematician who lived in Marrakech in what is now Morocco.Ibn al-Banna developed the first known proof of the basic combinatorial formulas,beginning by showing that the number of permutations of a set of n elements was n!and then developing in a careful manner the multiplicative formula to compute thevalues for the number of combinations of k objects in a set of n. Using these tworesults, he also showed how to calculate the number of permutations of k objects froma set of n. The formulas themselves had been known in the Islamic world for manyyears, in connection with specific problems like calculating the number of words ofa given length which could be formed from the letters of the Arabic alphabet. Ibnal-Bannas main contribution, then, was to abstract the general idea of permutationsand combinations out of the various specific problem situations considered earlier.

Thomas Bayes (17021761) an English Nonconformist, wrote an Introduction to theDoctrine of Fluxions in 1736 as a response to Berkeleys Analyst with its severe crit-icism of the foundations of the calculus. He is best known, however, for attemptingto answer the basic question of statistical inference in his An Essay Towards Solvinga Problem in the Doctrine of Chances, published three years after his death. Thatbasic question is to determine the probability of an event, given empirical evidencethat it has occurred a certain number of times in a certain number of trials. To dothis, Bayes gave a straightforward definition of probability and then proved that fortwo events E and F , the probability of E given that F has happened is the quo-tient of the probability of both E and F happening divided by the probability of Falone. By using areas to model probability, he was then able to show that, if x is theprobability of an event happening in a single trial, if the event has happened p timesin n trials, and if 0 < r < s < 1, then the probability that x is between r and s isgiven by the quotient of two integrals. Although in principle these integrals can becalculated, there has been a great debate since Bayes time about the circumstancesunder which his formula gives an appropriate answer.

James Bernoulli (Jakob I) (16541705) was one of eight mathematicians in threegenerations of his family. He was born in Basel, Switzerland, studied theology inaddition to mathematics and astronomy, and entered the ministry. In 1682 be began


to lecture at the University of Basil in natural philosophy and mechanics. He becameprofessor at the University of Basel in 1687, and remained there until his death. Hisresearch included the areas of the calculus of variations, probability, and analyticgeometry. His most well-known work is Ars Conjectandi, in which he describedresults in combinatorics and probability, including applications to gambling and thelaw of large numbers; this work also contained a reprint of the first formal treatisein probability, written in 1657 by Christiaan Huygens.

Bhaskara (11141185), the most famous of medieval Indian mathematicians, gave acomplete algorithmic solution to the Pell equation Dx21 = y2. That equation hadbeen studied by several earlier Indian mathematicians as well. Bhaskara served muchof his adult life as the head of the astronomical observatory at Ujjain, some 300 milesnortheast of Bombay, and became widely respected for his skills in astronomy and themechanical arts, as well as mathematics. Bhaskaras mathematical contributions arechiefly found in two chapters, the Lilavati and the Bijaganita, of a major astronomicalwork, the Siddhantasiromani. These include techniques of solving systems of linearequations with more unknowns than equations as well as the basic combinatorialformulas, although without any proofs.

George Boole (18151864) was an English mathematician most famous for his workin logic. Born the son of a cobbler, he had to struggle to educate himself whilesupporting his family. But he was so successful in his self-education that he was ableto set up his own school before he was 20 and was asked to give lectures on the workof Isaac Newton. In 1849 he applied for and was appointed to the professorship inmathematics at Queens College, Cork, despite having no university degree. In 1847,Boole published a small book, The Mathematical Analysis of Logic, and seven yearslater expanded it into An Investigation of the Laws of Thought. In these books, Booleintroduced what is now called Boolean algebra as part of his aim to investigate thefundamental laws of those operations of the mind by which reasoning is performed;to give expression to them in the symbolical language of a Calculus, and upon thisfoundation to establish the science of Logic and construct its method. In additionto his work on logic, Boole wrote texts on differential equations and on differenceequations that were used in Great Britain until the end of the nineteenth century.

William Burnside (18521927), born in London, graduated from Cambridge in 1875,and remained there as lecturer until 1885. He then went to the Royal Naval Collegeat Greenwich, where he stayed until he retired. Although he published much inapplied mathematics, probability, and elliptic functions, he is best known for hisextensive work in group theory (including the classic book Theory of Groups). Hisconjecture that groups of odd order are solvable was proved by Walter Feit and JohnThompson and published in 1963.

Georg Ferdinand Ludwig Philip Cantor (18451918) was born in Russia to Danishparents, received a Ph.D. in number theory in 1867 at the University of Berlin, andin 1869 took a position at Halle University, where he remained until his retirement.He is regarded as a founder of set theory. He was interested in theology and thenature of the infinite. His work on the convergence of Fourier series led to his studyof certain types of infinite sets of real numbers, and ultimately to an investigationof transfinite numbers.

Augustin-Louis Cauchy (17891857) the most prolific mathematician of the nine-teenth century, is most famous for his textbooks in analysis written in the 1820s foruse at the Ecole Polytechnique, textbooks which became the model for calculus textsfor the next hundred years. Although born in the year the French Revolution began,


Cauchy was a staunch conservative. When the July Revolution of 1830 led to theoverthrow of the last Bourbon king, Cauchy refused to take the oath of allegiance tothe new king and went into a self-imposed exile in Italy and then in Prague. He didnot return to his teaching posts until the Revolution of 1848 led to the removal ofthe requirement of an oath of allegiance. Among the many mathematical subjectsto which he contributed besides calculus were the theory of matrices, in which hedemonstrated that every symmetric matrix can be diagonalized by use of an orthog-onal substitution, and the theory of permutations, in which he was the earliest toconsider these from a functional point of view. In fact, he used a single letter, say S,to denote a permutation and S1 to denote its inverse and then noted that thepowers S, S2, S3, . . . of a given permutation on a finite set must ultimately resultin the identity. He also introduced the current notation (a1a2 . . . an) to denote thecyclic permutation on the letters a1, a2, . . . , an.

Arthur Cayley (18211895), although graduating from Trinity College, Cambridgeas Senior Wrangler, became a lawyer because there were no suitable mathematicspositions available at that time in England. He produced nearly 300 mathematicalpapers during his fourteen years as a lawyer, and in 1863 was named Sadlerian profes-sor of mathematics at Cambridge. Among his numerous mathematical achievementsare the earliest abstract definition of a group in 1854, out of which he was able tocalculate all possible groups of order up to eight, and the basic rules for operatingwith matrices, including a statement (without proof) of the Cayley-Hamilton theo-rem that every matrix satisfies its characteristic equation. Cayley also developed themathematical theory of trees in an article in 1857. In particular, he dealt with thenotion of a rooted tree, a tree with a designated vertex called a root, and developeda recursive formula for determining the number of different rooted trees in terms ofits branches (edges). In 1874, Cayley applied his results on trees to the study ofchemical isomers.

Pafnuty Lvovich Chebyshev (18211894) was a Russian who received his mastersdegree in 1846 from Moscow University. From 1860 until 1882 he was a professor atthe University of St. Petersburg. His mathematical research in number theory dealtwith congruences and the distribution of primes; he also studied the approximationof functions by polynomials.

Avram Noam Chomsky (born 1928) received a Ph.D. in linguistics at the Universityof Pennsylvania. For many years he has been a professor of foreign languages andlinguistics at M.I.T. He has made many contributions to the study of linguisticsand the study of grammars.

Chrysippus (280206 B.C.E.) was a Stoic philosopher who developed some of the ba-sic principles of the propositional logic, which ultimately replaced Aristotles logic ofsyllogisms. He was born in Cilicia, in what is now Turkey, but spent most of his lifein Athens, and is said to have authored more than 700 treatises. Among his otherachievements, Chrysippus analyzed the rules of inference in the propositional calcu-lus, including the rules of modus ponens, modus tollens, the hypothetical syllogism,and the alternative syllogism.

Alonzo Church (19031995) studied under Hilbert at Gottingen, was on the facultyat Princeton from 1927 until 1967, and then held a faculty position at UCLA. Heis a founding member of the Association for Symbolic Logic. He made many con-tributions in various areas of logic and the theory of algorithms, and stated theChurch-Turing thesis (if a problem can be solved with an effective algorithm, thenthe problem can be solved by a Turing machine).


George Dantzig (born 1914) is an American mathematician who formulated the gen-eral linear programming problem of maximizing a linear objective function subjectto several linear constraints and developed the simplex method of solution in 1947.His study of linear programming grew out of his World War II service as a mem-ber of Air Force Project SCOOP (Scientific Computation of Optimum Programs),a project chiefly concerned with resource allocation problems. After the war, linearprogramming was applied to numerous problems, especially military and economicones, but it was not until such problems could be solved on a computer that the realimpact of their solution could be felt. The first successful solution of a major linearprogramming problem on a computer took place in 1952 at the National Bureau ofStandards. After he left the Air Force, Dantzig worked for the Rand Corporationand then served as a professor of operations research at Stanford University.

Richard Dedekind (18311916) was born in Brunswick, in northern Germany, andreceived a doctorate in mathematics at Gottingen under Gauss. He held positionsat Gottingen and in Zurich before returning to the Polytechnikum in Brunswick.Although at various times he could have received an appointment to a major Ger-man university, he chose to remain in his home town where he felt he had sufficientfreedom to pursue his mathematical research. Among his many contributions washis invention of the concept of ideals to resolve the problem of the lack of uniquefactorization in rings of algebraic integers. Even though the rings of integers them-selves did not possess unique factorization, Dedekind showed that every ideal is eitherprime or uniquely expressible as the product of prime ideals. Dedekind publishedthis theory as a supplement to the second edition (1871) of Dirichlets Vorlesungenuber Zahlentheorie, of which he was the editor. In the supplement, he also gave oneof the first definitions of a field, confining this concept to subsets of the complexnumbers.

Abraham deMoivre (16671754) was born into a Protestant family in Vitry, France,a town about 100 miles east of Paris, and studied in Protestant schools up to the ageof 14. Soon after the revocation of the Edict of Nantes in 1685 made life very difficultfor Protestants in France, however, he was imprisoned for two years. He then leftFrance for England, never to return. Although he was elected to the Royal Societyin 1697, in recognition of a paper on A method of raising an infinite Multinomialto any given Power or extracting any given Root of the same, he never achieved auniversity position. He made his living by tutoring and by solving problems arisingfrom games of chance and annuities for gamblers and speculators. DeMoivres majormathematical work was The Doctrine of Chances (1718, 1736, 1756), in which hedevised methods for calculating probabilities by use of binomial coefficients. Inparticular, he derived the normal approximation to the binomial distribution and,in essence, invented the notion of the standard deviation.

Augustus DeMorgan (18061871) graduated from Trinity College, Cambridge in1827. He was the first mathematics professor at University College in London, wherehe remained on the faculty for 30 years. He founded the London Mathematical Soci-ety. He wrote over 1000 articles and textbooks in probability, calculus, algebra, settheory, and logic (including DeMorgans laws, an abstraction of the duality principlefor sets). He gave a precise definition of limit, developed tests for convergence ofinfinite series, and gave a clear explanation of the Principle of Mathematical Induc-tion.

Rene Descartes (15961650) left school at 16 and went to Paris, where he studiedmathematics for two years. In 1616 he earned a law degree at the University ofPoitiers. In 1617 he enlisted in the army and traveled through Europe until 1629,


when he settled in Holland for the next 20 years. During this productive period ofhis life he wrote on mathematics and philosophy, attempting to reduce the sciencesto mathematics. In 1637 his Discours was published; this book contained the devel-opment of analytic geometry. In 1649 he has invited to tutor the Queen Christinaof Sweden in philosophy. There he soon died of pneumonia.

Leonard Eugene Dickson (18741954) was born in Iowa and in 1896 received thefirst Ph.D. in mathematics given by the University of Chicago, where he spent muchof his faculty career. His research interests included abstract algebra (including thestudy of matrix groups and finite fields) and number theory.

Diophantus (c. 250) was an Alexandrian mathematician about whose life little isknown except what is reported in an epigram of the Greek Anthology (c. 500), fromwhich it can calculated that he lived to the age of 84. His major work, however,the Arithmetica, has been extremely influential. Despite its title, this is a book onalgebra, consisting mostly of an organized collection of problems translatable intowhat are today called indeterminate equations, all to be solved in rational numbers.Diophantus introduced the use of symbolism into algebra and outlined the basic rulesfor operating with algebraic expressions, including those involving subtraction. Itwas in a note appended to Problem II-8 of the 1621 Latin edition of the Arithmetica to divide a given square number into two squares that Pierre de Fermat firstasserted the impossibility of dividing an nth power (n > 2) into the sum of two nthpowers. This result, now known as Fermats Last Theorem, was finally proved in1994 by Andrew Wiles.

Charles Lutwidge Dodgson (18321898) is more familiarly known as Lewis Carroll,the pseudonym he used in writing his famous childrens works Alice in Wonderlandand Through the Looking Glass. Dodgson graduated from Oxford University in 1854and the next year was appointed a lecturer in mathematics at Christ Church College,Oxford. Although he was not successful as a lecturer, he did contribute to fourareas of mathematics: determinants, geometry, the mathematics of tournaments andelections, and recreational logic. In geometry, he wrote a five-act comedy, Euclidand His Modern Rivals, about a mathematics lecturer Minos in whose dreams Eucliddebates his Elements with various modernizers but always manages to demolish theopposition. He is better known, however, for his two books on logic, SymbolicLogic and The Game of Logic. In the first, he developed a symbolical calculus foranalyzing logical arguments and wrote many humorous exercises designed to teachhis methods, while in the second, he demonstrated a game which featured variousforms of the syllogism.

Eratosthenes (276194 B.C.E) was born in Cyrene (North Africa) and studied atPlatos Academy in Athens. He was tutor of the son of King Ptolemy III Euergetesin Alexandria and became chief librarian at Alexandria. He is recognized as theforemost scholar of his time and wrote in many areas, including number theory (hissieve for obtaining primes) and geometry. He introduced the concepts of meridiansof longitude and parallels of latitude and used these to measure distances, includingan estimation of the circumference of the earth.

Paul Erdos (19131996) was born in Budapest. At 21 he received a Ph.D. in math-ematics from Eotvos University. After leaving Hungary in 1934, he traveled exten-sively throughout the world, with very few possessions and no permanent home,working with other mathematicians in combinatorics, graph theory, number theory,and many other areas. He was author or coauthor of approximately 1500 paperswith 500 coauthors.


Euclid (c. 300 B.C.E.) is responsible for the most famous mathematics text of all time,the Elements. Not only does this work deal with the standard results of planegeometry, but it also contains three chapters on number theory, one long chapteron irrational quantities, and three chapters on solid geometry, culminating with theconstruction of the five regular solids. The axiom-definition-theorem-proof style ofEuclids work has become the standard for formal mathematical writing up to thepresent day. But about Euclids life virtually nothing is known. It is, however,generally assumed that he was among the first mathematicians at the Museum andLibrary of Alexandria, which was founded around 300 B.C.E by Ptolemy I Soter,the Macedonian general of Alexander the Great who became ruler of Egypt afterAlexanders death in 323 B.C.E.

Leonhard Euler (17071783) was born in Basel, Switzerland and became one of theearliest members of the St. Petersburg Academy of Sciences. He was the most pro-lific mathematician of all time, making contributions to virtually every area of thesubject. His series of analysis texts established many of the notations and methodsstill in use today. He created the calculus of variations and established the theory ofsurfaces in differential geometry. His study of the Konigsberg bridge problem led tothe formulation and solution of one of the first problems in graph theory. He madenumerous discoveries in number theory, including a detailed study of the propertiesof residues of powers and the first statement of the quadratic reciprocity theorem.He developed an algebraic formula for determining the number of partitions of aninteger n into m distinct parts, each of which is in a given set A of distinct positiveintegers. And in a paper of 1782, he even posed the problem of the existence of apair of orthogonal latin squares: If there are 36 officers, one of each of six ranks fromeach of six different regiments, can they be arranged in a square in such a way thateach row and column contains exactly one officer of each rank and one from eachregiment?

Kamal al-Din al-Farisi (died 1320) was a Persian mathematician most famous for hiswork in optics. In fact, he wrote a detailed commentary on the great optical work ofIbn al-Haytham. But al-Farisi also made major contributions to number theory. Heproduced a detailed study of the properties of amicable numbers (pairs of numbersin which the sum of the proper divisors of each is equal to the other). As part of thisstudy, al-Farisi developed and applied various combinatorial principles. He showedthat the classical figurate numbers (triangular, pyramidal, etc.) could be interpretedas numbers of combinations and thus helped to found the theory of combinatoricson a more abstract basis.

Pierre de Fermat (16011665) was a lawyer and magistrate for whom mathematicswas a pastime that led to contributions in many areas: calculus, number theory,analytic geometry, and probability theory. He received a bachelors degree in civillaw in 1631, and from 1648 until 1665 was Kings Counsellor. He suffered an attackof the plague in 1652, and from then on he began to devote time to the studyof mathematics. He helped give a mathematical basis to probability theory when,together with Blaise Pascal, he solved Meres paradox: why is it less likely to roll a 6at least once in four tosses of one die than to roll a double 6 in 24 tosses of two dice.He was a discoverer of analytic geometry and used infinitesimals to find tangentlines and determine maximum and minimum values of curves. In 1657 he publisheda series of mathematical challenges, including the conjecture that xn + yn = zn hasno solution in positive integers if n is an integer greater than 2. He wrote in themargin of a book that he had a proof, but the proof would not fit in the margin. Hisconjecture was finally proved by Andrew Wiles in 1994.


Fibonacci (Leonardo of Pisa) (c. 1175c. 1250) was the son of a Mediterranean mer-chant and government worker named Bonaccio (hence his name filius Bonaccio, sonof Bonaccio). Fibonacci, born in Pisa and educated in Bougie (on the north coastof Africa where his father was administrator of Pisas trading post), traveled exten-sively around the Mediterranean. He is regarded as the greatest mathematician ofthe Middle Ages. In 1202 he wrote the book Liber Abaci, an extensive treatmentof topics in arithmetic and algebra, and emphasized the benefits of Arabic numerals(which he knew about as a result of his travels around the Mediterranean). In thisbook he also discussed the rabbit problem that led to the sequence that bears hisname: 1, 1, 2, 3, 5, 8, 13, . . . . In 1225 he wrote the book Liber Quadratorum, studyingsecond degree diophantine equations.

Joseph Fourier (17681830), orphaned at the age of 9, was educated in the militaryschool of his home town of Auxerre, 90 miles southeast of Paris. Although he hopedto become an army engineer, such a career was not available to him at the timebecause he was not of noble birth. He therefore took up a teaching position. Dur-ing the Revolution, he was outspoken in defense of victims of the Terror of 1794.Although he was arrested, he was released after the death of Robespierre and wasappointed in 1795 to a position at the Ecole Polytechnique. After serving in variousadministrative posts under Napoleon, he was elected to the Academie des Sciencesand from 1822 until his death served as its perpetual secretary. It was in connectionwith his work on heat diffusion, detailed in his Analytic Theory of Heat of 1822,and, in particular, with his solution of the heat equation vt =

2vx2 +

2vy2 , that he

developed the concept of a Fourier series. Fourier also analyzed the relationshipbetween the series solution of a partial differential equation and an appropriate inte-gral representation and thereby initiated the study of Fourier integrals and Fouriertransforms.

Georg Frobenius (18491917) organized and analyzed the central ideas of the theory ofmatrices in his 1878 memoir On linear substitutions and bilinear forms. Frobeniusthere defined the general notion of equivalent matrices. He also dealt with thespecial cases of congruent and similar matrices. Frobenius showed that when twosymmetric matrices were similar, the transforming matrix could be taken to beorthogonal, one whose inverse equaled its transpose. He then made a detailed studyof orthogonal matrices and showed that their eigenvalues were complex numbersof absolute value 1. He also gave the first complete proof of the Cayley-Hamiltontheorem that a matrix satisfies its characteristic equation. Frobenius, a full professorin Zurich and later in Berlin, made his major mathematical contribution in the areaof group theory. He was instrumental in developing the concept of an abstract group,as well as in investigating the theory of finite matrix groups and group characters.

Evariste Galois (18111832) led a brief, tragic life which ended in a duel fought undermysterious circumstances. He was born in Bourg-la-Reine, a town near Paris. Hedeveloped his mathematical talents early and submitted a memoir on the solvabil-ity of equations of prime degree to the French Academy in 1829. Unfortunately,the referees were never able to understand this memoir nor his revised version sub-mitted in 1831. Meanwhile, Galois became involved in the revolutionary activitiessurrounding the July revolution of 1830 and was arrested for threatening the lifeof King Louis-Phillipe and then for wearing the uniform of a National Guard divi-sion which had been dissolved because of its perceived threat to the throne. Hismathematics was not fully understood until fifteen years after his death when hismanuscripts were finally published by Liouville in the Journal des mathematique.But Galois had in fact shown the relationship between subgroups of the group of


permutations of the roots of a polynomial equation and the various extension fieldsgenerated by these roots, the relationship at the basis of what is now known as Galoistheory . Galois also developed the notion of a finite field in connection with solvingthe problem of finding solutions to congruences F (x) 0 (mod p), where F (x) is apolynomial of degree n and no residue modulo the prime p is itself a solution.

Carl Friedrich Gauss (17771855), often referred to as the greatest mathematicianwho ever lived, was born in Brunswick, Germany. He received a Ph.D. from theUniversity of Helmstedt in 1799, proving the Fundamental Theorem of Algebra aspart of his dissertation. At age 24 Gauss published his important work on numbertheory, the Disquisitiones Arithmeticae, a work containing not only an extensivediscussion of the theory of congruences, culminating in the quadratic reciprocitytheorem, but also a detailed treatment of cyclotomic equations in which he showedhow to construct regular n-gons by Euclidean techniques whenever n is prime andn1 is a power of 2. Gauss also made fundamental contributions to the differentialgeometry of surfaces as well as to complex analysis, astronomy, geodesy, and statisticsduring his long tenure as a professor at the University of Gottingen. It was inconnection with using the method of least squares to solve an astronomical problemthat Gauss devised the systematic procedure for solving a system of linear equationstoday known as Gaussian elimination. (Unknown to Gauss, the method appeared inChinese mathematics texts 1800 years earlier.) Gauss notebooks, discovered afterhis death, contained investigations in numerous areas of mathematics in which hedid not publish, including the basics of non-Euclidean geometry.

Sophie Germain (17761831) was forced to study in private due to the turmoil ofthe French Revolution and the opposition of her parents. She nevertheless mas-tered mathematics through calculus and wanted to continue her study in the EcolePolytechnique when it opened in 1794. But because women were not admitted asstudents, she diligently collected and studied the lecture notes from various mathe-matics classes and, a few years later, began a correspondence with Gauss (under thepseudonym Monsieur LeBlanc, fearing that Gauss would not be willing to recognizethe work of a woman) on ideas in number theory. She was, in fact, responsible forsuggesting to the French general leading the army occupying Brunswick in 1807 thathe insure Gauss safety. Germains chief mathematical contribution was in connec-tion with Fermats Last Theorem. She showed that xn + yn = zn has no positiveinteger solution where xyz is not divisible by n for any odd prime n less than 100.She also made contributions in the theory of elasticity and won a prize from theFrench Academy in 1815 for an essay in this field.

Kurt Godel (19061978) was an Austrian mathematician who spent most of his life atthe Institute for Advanced Study in Princeton. He made several surprising contribu-tions to set theory, demonstrating that Hilberts goal of showing that a reasonableaxiomatic system for set theory could be proven to be complete and consistent was infact impossible. In several seminal papers published in the 1930s, Godel proved thatit was impossible to prove internally the consistency of the axioms of any reasonablesystem of set theory containing the axioms for the natural numbers. Furthermore,he showed that any such system was inherently incomplete, that is, that there arepropositions expressible in the system for which neither they nor their negations areprovable. Godels investigations were stimulated by the problems surrounding theaxiom of choice, the axiom that for any set S of nonempty disjoint sets, there isa subset T of the union of S that has exactly one element in common with eachmember of S. Since that axiom led to many counterintuitive results, it was impor-tant to show that the axiom could not lead to contradictions. But given his initial


results, the best Godel could do was to show that the axiom of choice was relativelyconsistent, that its addition to the Zermelo-Fraenkel axiom set did not lead to anycontradictions that would not already have been implied without it.

William Rowan Hamilton (18051865), born in Dublin, was a child prodigy whobecame the Astronomer Royal of Ireland in 1827 in recognition of original workin optics accomplished during his undergraduate years at Trinity College, Dublin.In 1837, he showed how to introduce complex numbers into algebra axiomaticallyby considering a+ ib as a pair (a, b) of real numbers with appropriate computationalrules. After many years of seeking an appropriate definition for multiplication rulesfor triples of numbers which could be applied to vector analysis in 3-dimensionalspace, he discovered that it was in fact necessary to consider quadruplets of numbers,which Hamilton named quaternions. Although quaternions never had the influenceHamilton forecast for them in physics, their noncommutative multiplication providedthe first significant example of a mathematical system which did not obey one of thestandard arithmetical laws of operation and thus opened the way for more freedomin the creation of mathematical systems. Among Hamiltons other contributions wasthe development of the Icosian game, a graph with 20 vertices on which pieces wereto be placed in accordance with various conditions, the overriding one being that apiece was always placed at the second vertex of an edge on which the previous piecehad been placed. One of the problems Hamilton set for the game was, in essence, todiscover a cyclic path on his game board which passed through each vertex exactlyonce. Such a path in a more general setting is today called a Hamilton circuit.

Richard W. Hamming (19151998) was born in Chicago and received a Ph.D. inmathematics from the University of Illinois in 1942. He was the author of the firstmajor paper on error correcting and detecting codes (1950). His work on this problemhad been stimulated in 1947 when he was using an early Bell System relay computeron weekends only. During the weekends the machine was unattended and woulddump any work in which it discovered an error and proceed to the next problem.Hamming realized that it would be worthwhile for the machine to be able not onlyto detect an error but also to correct it, so that his jobs would in fact be completed.In his paper, Hamming used a geometric model by considering an n-digit code wordto be a vertex in the unit cube in the n-dimensional vector space over the field oftwo elements. He was then able to show that the relationship between the wordlength n and the number m of digits which carry the information was 2m 2nn+1 .(The remaining k = nm digits are check digits which enable errors to be detectedand corrected.) In particular, Hamming presented a particular type of code, todayknown as a Hamming code, with n = 7 and m = 4. In this code, the set of actualcode words of 4 digits was a 4-dimensional vector subspace of the 7-dimensionalspace of all 7-digit binary strings.

Godfrey Harold Hardy (18771947) graduated from Trinity College, Cambridge in1899. From 1906 until 1919 he was lecturer at Trinity College, and, recognizing thegenius of Ramanujan, invited Ramanujan to Cambridge in 1914. Hardy held theSullivan chair of geometry at Oxford from 1919 until 1931, when he returned toCambridge, where he was Sadlerian professor of pure mathematics until 1942. Hedeveloped the Hardy-Weinberg law which predicts patterns of inheritance. His mainareas of mathematical research were analysis and number theory, and he publishedover 100 joint papers with Cambridge colleague John Littlewood. Hardys book ACourse in Pure Mathematics revolutionized mathematics teaching, and his book AMathematicians Apology gives his view of what mathematics is and the value of itsstudy.


Abu Ali al-Hasan ibn al-Haytham (Alhazen) (9651039) was one of the mostinfluential of Islamic scientists. He was born in Basra (now in Iraq) but spent mostof his life in Egypt, after he was invited to work on a Nile control project. Althoughthe project, an early version of the Aswan dam project, never came to fruition, ibnal-Haytham did produce in Egypt his most important scientific work, the Optics.This work was translated into Latin in the early thirteenth century and was studiedand commented on in Europe for several centuries thereafter. Although there wasmuch mathematics in the Optics, ibn al-Haythams most interesting mathematicalwork was the development of a recursive procedure for producing formulas for thesum of any integral powers of the integers. Formulas for the sums of the integers,squares, and cubes had long been known, but ibn al-Haytham gave a consistentmethod for deriving these and used this to develop the formula for the sum of fourthpowers. Although his method was easily generalizable to the discovery of formulasfor fifth and higher powers, he gave none, probably because he only needed the fourthpower rule in his computation of the volume of a paraboloid of revolution.

Hypatia (c. 370415), the first woman mathematician on record, lived in Alexandria.She was given a very thorough education in mathematics and philosophy by herfather Theon and became a popular and respected teacher. She was responsible fordetailed commentaries on several important Greek works, including Ptolemys Al-magest, Apollonius Conics, and Diophantus Arithmetica. Unfortunately, Hypatiawas caught up in the pagan-Christian turmoil of her times and was murdered by anenraged mob.

Leonid Kantorovich (19121986) was a Soviet economist responsible for the develop-ment of linear optimization techniques in relation to planning in the Soviet economy.The starting point of this development was a set of problems posed by the Leningradtimber trust at the beginning of 1938 to the Mathematics Faculty at the Universityof Leningrad. Kantorovich explored these problems in his 1939 book MathematicalMethods in the Organization and Planning of Production. He believed that oneway to increase productivity in a factory or an entire industrial organization wasto improve the distribution of the work among individual machines, the orders tovarious suppliers, the different kinds of raw materials, the different types of fuels,and so on. He was the first to recognize that these problems could all be put into thesame mathematical language and that the resulting mathematical problems couldbe solved numerically, but for various reasons his work was not pursued by Sovieteconomists or mathematicians.

Abu Bakr al-Karaji (died 1019) was an Islamic mathematician who worked in Bagh-dad. In the first decade of the eleventh century he composed a major work onalgebra entitled al-Fakhri (The Marvelous), in which he developed many algebraictechniques, including the laws of exponents and the algebra of polynomials, with theaim of systematizing methods for solving equations. He was also one of the earlyoriginators of a form of mathematical induction, which was best expressed in hisproof of the formula for the sum of integral cubes.

Stephen Cole Kleene (19091994) studied under Alonzo Church and received hisPh.D. from Princeton in 1934. His research has included the study of recursive func-tions, computability, decidability, and automata theory. In 1956 he proved KleenesTheorem, in which he characterized the sets that can be recognized by finite-stateautomata.

Felix Klein (18491925) received his doctorate at the University of Bonn in 1868.In 1872 he was appointed to a position at the University of Erlanger, and in his


opening address laid out the Erlanger Programm for the study of geometry based onthe structure of groups. He described different geometries in terms of the propertiesof a set that are invariant under a group of transformations on the set and gavea program of study using this definition. From 1875 until 1880 he taught at theTechnische Hochschule in Munich, and from 1880 until 1886 in Leipzig. In 1886Klein became head of the mathematics department at Gottingen and during histenure raised the prestige of the institution greatly.

Donald E.Knuth (born 1938) received a Ph.D. in 1963 from the California Instituteof Technology and held faculty positions at the California Institute of Technology(19631968) and Stanford (19681992). He has made contributions in many areas,including the study of compilers and computational complexity. He is the designerof the mathematical typesetting system TEX. He received the Turing Award in 1974and the National Medal of Technology in 1979.

Kazimierz Kuratowski (18961980) was the son of a famous Warsaw lawyer who be-came an active member of the Warsaw School of Mathematics after World War I. Hetaught both at Lwow Polytechnical University and at Warsaw University until theoutbreak of World War II. During that war, because of the persecution of educatedPoles, he went into hiding under an assumed name and taught at the clandestineWarsaw University. After the war, he helped to revive Polish mathematics, servingas director of the Polish National Mathematics Institute. His major mathemati-cal contributions were in topology; he formulated a version of a maximal principleequivalent to the axiom of choice. This principle is today known as Zorns lemma.Kuratowski also contributed to the theory of graphs by proving in 1930 that anynon-planar graph must contain a copy of one of two particularly simple non-planargraphs.

Joseph Louis Lagrange (17361813) was born in Turin into a family of French de-scent. He was attracted to mathematics in school and at the age of 19 became amathematics professor at the Royal Artillery School in Turin. At about the sametime, having read a paper of Eulers on the calculus of variations, he wrote to Eu-ler explaining a better method he had recently discovered. Euler praised Lagrangeand arranged to present his paper to the Berlin Academy, to which he was laterappointed when Euler returned to Russia. Although most famous for his AnalyticalMechanics, a work which demonstrated how problems in mechanics can generally bereduced to solutions of ordinary or partial differential equations, and for his Theoryof Analytic Functions, which attempted to reduce the ideas of calculus to those ofalgebraic analysis, he also made contributions in other areas. For example, he un-dertook a detailed review of solutions to quadratic, cubic, and quartic polynomialsto see how these methods might generalize to higher degree polynomials. He was ledto consider permutations on the roots of the equations and functions on the rootsleft unchanged by such permutations. As part of this work, he discovered a versionof Lagranges theorem to the effect that the order of any subgroup of a group dividesthe order of the group. Although he did not complete his program and produce amethod of solving higher degree polynomial equations, his methods were applied byothers early in the nineteenth century to show that such solutions were impossible.

Gabriel Lame (17951870) was educated at the Ecole Polytechnique and the Ecoledes Mines before going to Russia to direct the School of Highways and Transporta-tion in St. Petersburg. After his return to France in 1832, he taught at the EcolePolytechnique while also working as an engineering consultant. Lame contributedoriginal work to number theory, applied mathematics, and thermodynamics. Hisbest-known work is his proof of the case n = 5 of Fermats Last Theorem in 1839.


Eight years later, he announced that he had found a general proof of the theorem,which began with the factorization of the expression xn + yn over the complex num-bers as (x + y)(x + y)(x + 2y) . . . (x + n1y), where is a primitive root ofxn 1 = 0. He planned to show that the factors in this expression are all relativelyprime and therefore that if xn + yn = zn, then each of the factors would itself be annth power. He would then use the technique of infinite descent to find a solution insmaller numbers. Unfortunately Lames idea required that the ring of integers in thecyclotomic field of the nth roots of unity be a unique factorization domain. And, asKummer had already proved three years earlier, unique factorization in fact fails inmany such domains.

Edmund Landau (18771938) received a doctorate under Frobenius and taught atthe University of Berlin and at Gottingen. His research areas were analysis andanalytic number theory, including the distribution of primes. He used the big-Onotation (also called a Landau symbol) in his work to estimate the growth of variousfunctions.

Pierre-Simon de Laplace (17491827) entered the University of Caen in 1766 tobegin preparation for a career in the church. He soon discovered his mathematicaltalents, however, and in 1768 left for Paris to continue his studies. He later taughtmathematics at the Ecole Militaire to aspiring cadets. Legend has it that he exam-ined, and passed, Napoleon there in 1785. He was later honored by both Napoleonand King Louis XVIII. Laplace is best known for his contributions to celestial me-chanics, but he was also one of the founders of probability theory and made manycontributions to mathematical statistics. In fact, he was one of the first to apply histheoretical results in statistics to a genuine problem in statistical inference, whenhe showed from the surplus of male to female births in Paris over a 25-year periodthat it was morally certain that the probability of a male birth was in fact greaterthan 12 .

Gottfried Wilhelm Leibniz (16461716), born in Leipzig, developed his version ofthe calculus some ten years after Isaac Newton, but published it much earlier. Hebased his calculus on the inverse relationship of sums and differences, generalizedto infinitesimal quantities called differentials. Leibniz hoped that his most origi-nal contribution to philosophy would be the development of an alphabet of humanthought, a way of representing all fundamental concepts symbolically and a methodof combining these symbols to represent more complex thoughts. Although he nevercompleted this project, his interest in finding appropriate symbols ultimately ledhim to the d and

symbols for the calculus that are used today. Leibniz spent much

of his life in the diplomatic service of the Elector of Mainz and later was a Counsel-lor to the Duke of Hanover. But he always found time to pursue his mathematicalideas and to carry on a lively correspondence on the subject with colleagues all overEurope.

Levi ben Gerson (12881344) was a rabbi as well as an astronomer, philosopher,biblical commentator, and mathematician. He lived in Orange, in southern France,but little is known of his life. His most famous mathematical work is the MaaseiHoshev (The Art of the Calculator) (1321), which contains detailed proofs of thestandard combinatorial formulas, some of which use the principle of mathematicalinduction. About a dozen copies of this medieval manuscript are extant, but it isnot known whether the work had any direct influence elsewhere in Europe.

Augusta Ada Byron King Lovelace (18151852) was the child of the famous poetGeorge Gordon, the sixth Lord Byron, who left England five weeks after his daugh-


ters birth and never saw her again. She was raised by her mother, Anna IsabellaMillbanke, a student of mathematics herself, so she received considerably more math-ematics education than was usual for girls of her time. She was tutored privately bywell-known mathematicians, including William Frend and Augustus DeMorgan. Herhusband, the Earl of Lovelace, was made a Fellow of the Royal Society in 1840, andthrough this connection, Ada was able to gain access to the books and papers sheneeded to continue her mathematical studies and, in particular, to understand theworkings of Babbages Analytical Engine. Her major mathematical work is a heav-ily annotated translation of a paper by the Italian mathematician L. F. Menabreadealing with the Engine, in which she gave explicit descriptions of how it wouldsolve specific problems and described, for the first time in print, what would todaybe called a computer program, in this case a program for computing the Bernoullinumbers. Interestingly, only her initials, A.A.L., were used in the published ver-sion of the paper. It was evidently not considered proper in mid-nineteenth centuryEngland for a woman of her class to publish a mathematical work.

Jan 5Lukasiewicz (18781956) studied at the University of Lwow and taught at theUniversity of Lwow, the University of Warsaw, and the Royal Irish Academy. Alogician, he worked in the area of many-valued logic, writing papers on three-valuedand m-valued logics, He is best known for the parenthesis-free notation he developedfor propositions, called Polish notation.

Percy Alexander MacMahon (18541929) was born into a British army family andjoined the army himself in 1871, reaching the rank of major in 1889. Much ofhis army service was spent as an instructor at the Royal Military Academy. Hisearly mathematical work dealt with invariants, following on the work of Cayleyand Sylvester, but a study of symmetric functions eventually led to his interestin partitions and to his extension of the idea of a partition to higher dimensions.MacMahons two volume treatise Combinatorial Analysis (191516) is a classic inthe field. It identified and clarified the basic results of combinatorics and showedthe way toward numerous applications.

Mahavira (ninth century) was an Indian mathematician of the medieval period whosemajor work, the Ganitasarasangraha, was a compilation of problems solvable by var-ious algebraic techniques. For example, the work included a version of the hundredfowls problem: Doves are sold at the rate of 5 for 3 coins, cranes at the rate of 7for 5, swans at the rate of 9 for 7, and peacocks at the rate of 3 for 9. A certain manwas told to bring at these rates 100 birds for 100 coins for the amusement of thekings son and was sent to do so. What amount does he give for each? Mahaviraalso presented, without proof and in words, the rule for calculating the number ofcombinations of r objects out of a set of n. His algorithm can be easily translated intothe standard formula. Mahavira then applied the rule to two problems, one aboutcombinations of tastes and another about combinations of jewels on a necklace.

Andrei Markov (18561922) was a Russian mathematician who first defined whatare now called Markov chains in a paper of 1906 dealing with the Law of LargeNumbers and subsequently proved many of the standard results about them. Hisinterest in these chains stemmed from the needs of probability theory. Markov neverdealt with their application to the sciences, only considering examples from literarytexts, where the two possible states in the chain were vowels and consonants. Markovtaught at St. Petersburg University from 1880 to 1905 and contributed to such fieldsas number theory, continued fractions, and approximation theory. He was an activeparticipant in the liberal movement in pre-World War I Russia and often criticizedpublicly the actions of state authorities. In 1913, when as a member of the Academy


of Sciences he was asked to participate in the pompous ceremonies celebrating the300th anniversary of the Romanov dynasty, he instead organized a celebration of the200th anniversary of Jacob Bernoullis publication of the Law of Large Numbers.

Marin Mersenne (15881648) was educated in Jesuit schools and in 1611 joined theOrder of Minims. From 1619 he lived in the Minim Convent de lAnnonciade near thePlace Royale in Paris and there held regular meetings of a group of mathematiciansand scientists to discuss the latest ideas. Mersenne also served as the unofficialsecretary of the republic of scientific letters in Europe. As such, he receivedmaterial from various sources, copied it, and distributed it widely, thus serving asa walking scientific journal. His own contributions were primarily in the areaof music theory as detailed in his two great works on the subject, the Harmonieuniverselle and the Harmonicorum libri, both of which appeared in 1636. As part ofhis study of music, he developed the basic combinatorial formulas by considering thepossible tunes one could create out of a given number of notes. Mersenne was alsogreatly interested in the relationship of theology to science. He was quite concernedwhen he learned that Galileo could not publish one of his works because of theInquisition and, in fact, offered his assistance in this matter.

Hermann Minkowski (18641909) was a German Jewish mathematician who receivedhis doctorate at the University of Konigsberg. He became a lifelong friend of DavidHilbert and, on Hilberts suggestion, was called to Gottingen in 1902. In 1883, heshared the prize of the Paris Academy of Sciences for his essay on the topic of therepresentations of an integer as a sum of squares. In his essay, he reconstructedthe entire theory of quadratic forms in n variables with integral coefficients. Infurther work on number theory, he brought to bear geometric ideas beginning withthe realization that a symmetric convex body in n-space defines a notion of distanceand hence a geometry in that space. The connection with number theory dependson the representation of forms by lattice points in space.

Muhammad ibn Muhammad al-Fullani al-Kishnawi (died 1741) was a nativeof northern Nigeria and one of the few African black scholars known to have madecontributions to pure mathematics before the modern era. Muhammads mostimportant work, available in an incomplete manuscript in the library of the Schoolof Oriental and African Studies in London, deals with the theory of magic squares.He gave a clear treatment of the standard construction of magic squares and alsostudied several other constructions using knights moves, borders added to a magicsquare of lower order, and the formation of a square from a square number of smallermagic squares.

Peter Naur (born 1928) was originally an astronomer, using computers to calculateplanetary motion. In 1959 he became a full-time computer scientist; he was a de-veloper of the programming language ALGOL and worked on compilers for ALGOLand COBOL. In 1969 he took a computer science faculty position at the Universityof Copenhagen.

Amalie Emmy Noether (18821935) received her doctorate from the University ofErlangen in 1908 and a few years later moved to Gottingen to assist Hilbert inthe study of general relativity. During her eighteen years there, she was extremelyinfluential in stimulating a new style of thinking in algebra by always emphasizingits structural rather than computational aspects. In 1934 she became a professorat Bryn Mawr College and a member for the Institute for Advanced Study. She ismost famous for her work on Noetherian rings, and her influence is still evident intodays textbooks in abstract algebra.


Blaise Pascal (16231662) showed his mathematical precocity with his Essay on Con-ics of 1640, in which he stated his theorem that the opposite sides of a hexagoninscribed in a conic section always intersect in three collinear points. Pascal is bet-ter known, however, for his detailed study of what is now called Pascals triangleof binomial coefficients. In that study Pascal gave an explicit description of math-ematical induction and used that method, although not quite in the modern sense,to prove various properties of the numbers in the triangle, including a method ofdetermining the appropriate division of stakes in a game interrupted before its con-clusion. Pascal had earlier discussed this matter, along with various other ideas inthe theory of probability, in correspondence with Fermat in the 1650s. These letters,in fact, can be considered the beginning of the mathematization of probability.

Giuseppe Peano (18581932) studied at the University of Turin and then spent theremainder of his life there as a professor of mathematics. He was originally known asan inspiring teacher, but as his studies turned to symbolic logic and the foundationsof mathematics and he attempted to introduce some of these notions in his elemen-tary classes, his teaching reputation changed for the worse. Peano is best knownfor his axioms for the natural numbers, first proposed in the Arithmetices prin-cipia, nova methodo exposita of 1889. One of these axioms describes the principleof mathematical induction. Peano was also among the first to present an axiomaticdescription of a (finite-dimensional) vector space. In his Calcolo geometrico of 1888,Peano described what he called a linear system, a set of quantities provided withthe operations of addition and scalar multiplication which satisfy the standard prop-erties. He was then able to give a coherent definition of the dimension of a linearsystem as the maximum number of linearly independent quantities in the system.

Charles Sanders Peirce (18391914) was born in Massachusetts, the son of a Harvardmathematics professor. He received a masters degree from Harvard in 1862 and anadvanced degree in chemistry from the Lawrence Scientific School in 1863. He madecontributions to many areas of the foundations and philosophy of mathematics. Hewas a prolific writer, leaving over 100,000 pages of unpublished manuscript at hisdeath.

George Polya (18871985) was a Hungarian mathematician who received his doctor-ate at Budapest in 1912. From 1914 to 1940 he taught in Zurich, then emigrated tothe United States where he spent most of the rest of his professional life at StanfordUniversity. Polya developed some influential enumeration ideas in several papers inthe 1930s, in particular dealing with the counting of certain configurations that arenot equivalent under the action of a particular permutation group. For example,there are 16 ways in which one can color the vertices of a square using two colors,but only six are non-equivalent under the various symmetries of the square. In 1937,Polya published a major article in the field, Combinatorial Enumeration of Groups,Graphs and Chemical Compounds, in which he discussed many mathematical as-pects of the theory of enumeration and applied it to various problems. Polyas workon problem solving and heuristics, summarized in his two volume work Mathematicsand Plausible Reasoning , insured his fame as a mathematics educator; his ideas areat the forefront of recent reforms in mathematics education at all levels.

Qin Jiushao (12021261), born in Sichuan, published a general procedure for solvingsystems of linear congruences the Chinese remainder theorem in his Shushujiuzhang (Mathematical Treatise in Nine Sections) in 1247, a procedure which makesessential use of the Euclidean algorithm. He also gave a complete description of amethod for numerically solving polynomial equations of any degree. Qins methodhad been developed in China over a period of more than a thousand years; it is


similar to a method used in the Islamic world and is closely related to what is nowcalled the Horner method of solution, published by William Horner in 1819. Qinstudied mathematics at the Board of Astronomy, the Chinese agency responsiblefor calendrical computations. He later served the government in several offices, butbecause he was extravagant and boastful, he was several times relieved of his dutiesbecause of corruption. These firings notwithstanding, Qin became a wealthy manand developed an impressive reputation in love affairs.

Srinivasa Ramanujan (18871920) was born near Madras into the family of a book-keeper. He studied mathematics on his own and soon began producing results incombinatorial analysis, some already known and others previously unknown. At theurging of friends, he sent some of his results to G. H. Hardy in England, who quicklyrecognized Ramanujans genius and invited him to England to develop his untrainedmathematical talent. During the war years from 1914 to 1917, Hardy and Ramanu-jan collaborated on a number of papers, including several dealing with the theoryof partitions. Unfortunately, Ramanujan fell ill during his years in the unfamiliarclimate of England and died at age 32 soon after returning to India. Ramanujanleft behind several notebooks containing statements of thousands of results, enoughwork to keep many mathematicians occupied for years in understanding and provingthem.

Frank Ramsey (19031930), son of the president of Magdalene College, Cambridge,was educated at Winchester and Trinity Colleges. He was then elected a fellow ofKings College, where he spent the remainder of his life. Ramsey made importantcontributions to mathematical logic. What is now called Ramsey theory began withhis clever combinatorial arguments to prove a generalization of the pigeonhole prin-ciple, published in the paper On a Problem of Formal Logic. The problem of thatpaper was the Entscheidungsproblem (the decision problem), the problem of search-ing for a general method of determining the consistency of a logical formula. Ramseyalso made contributions to the mathematical theory of economics and introduced thesubjective interpretation to probability. In that interpretation, Ramsey argues thatdifferent people when presented with the same evidence, will have different degreesof belief. And the way to measure a persons belief is to propose a bet and see whatare the lowest odds the person will accept. Ramseys death at the age of 26 deprivedthe mathematical community of a brilliant young scholar.

Bertrand Arthur William Russell (18721970) was born in Wales and studied atTrinity College, Cambridge. A philosopher/mathematician, he is one of the foundersof modern logic and wrote over 40 books in different areas. In his most famouswork, Principia Mathematica, published in 191013 with Alfred North Whitehead,he attempted to deduce the entire body of mathematics from a single set of primitiveaxioms. A pacifist, he fought for progressive causes, including womens suffrage inGreat Britain and nuclear disarmament. In 1950 he won a Nobel Prize for literature.

al-Samawal ibn Yahya ibn Yahuda al-Maghribi (11251180) was born in Bagh-dad to well-educated Jewish parents. Besides giving him a religious education, theyencouraged him to study medicine and mathematics. He wrote his major mathemat-ical work, Al-Bahir (The Shining), an algebra text that dealt extensively with thealgebra of polynomials. In it, al-Samawal worked out the laws of exponents, bothpositive and negative, and showed how to divide polynomials even when the divisionwas not exact. He also used a form of mathematical induction to prove the binomialtheorem, that (a + b)n =

nk=0 C(n, k)a

nkbk, where the C(n, k) are the entries inthe Pascal triangle, for n 12. In fact, he showed why each entry in the trianglecan be formed by adding two numbers in the previous row. When al-Samawal was


about 40, he decided to convert to Islam. To justify his conversion to the world,he wrote an autobiography in 1167 stating his arguments against Judaism, a workwhich became famous as a source of Islamic polemics against the Jews.

Claude Elwood Shannon (born 1916) applied Boolean algebra to switching circuitsin his masters thesis at M.I.T in 1938. Shannon realized that a circuit can berepresented by a set of equations and that the calculus necessary for manipulatingthese equations is precisely the Boolean algebra of logic. Simplifying these equationsfor a circuit would yield a simpler, equivalent circuit. Switches in Shannons calculuswere either open (represented by 1) or closed (represented by 0); placing switchesin parallel was represented by the Boolean operation +, while placing them inparallel was represented by . Using the basic rules of Boolean algebra, Shannonwas, for example, able to construct a circuit which would add two numbers given inbinary representation. He received his Ph.D. in mathematics from M.I.T. in 1940and spent much of his professional life at Bell Laboratories, where he worked onmethods of transmitting data efficiently and made many fundamental contributionsto information theory.

James Stirling (16921770) studied at Glasgow University and at Balliol College,Oxford and spent much of his life as a successful administrator of a mining companyin Scotland. His mathematical work included an exposition of Newtons theory ofcubic curves and a 1730 book entitled Methodus Differentialis which dealt withsummation and interpolation formulas. In dealing with the convergence of series,Stirling found it useful to convert factorials into powers. By considering tables offactorials, he was able to derive the formula for log n!, which leads to what is nowknown as Stirlings approximation: n! (ne )n

2n. Stirling also developed the

Stirling numbers of the first and second kinds, sequences of numbers important inenumeration.

Sun Zi (4th century) is the author of Sunzi suanjing (Master Suns MathematicalManual), a manual on arithmetical operations which eventually became part of therequired course of study for Chinese civil servants. The most famous problem inthe work is one of the first examples of what is today called the Chinese remainderproblem: We have things of which we do not know the number; if we count them bythrees, the remainder is 2; if we count them by fives, the remainder is 3; if we countthem by sevens, the remainder is 2. How many things are there? Sun Zi gives theanswer, 23, along with some explanation of how the problem should be solved. Butsince this is the only problem of its type in the book, it is not known whether SunZi had developed a general method of solving simultaneous linear congruences.

James Joseph Sylvester (18141897), who was born into a

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