ENCYCLOPEDIA OF

Mathematicsand Society

ENCYCLOPEDIA OF

Mathematicsand Society

Sarah J. GreenwaldJill E. Thomley

Appalachian State University

VOLUME 1

Salem Press

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Copyright 2012, by Salem Press

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Library of Congress Cataloging-in-Publication Data

Encyclopedia of mathematics and society / Sarah J. Greenwald , Jill E. Thomley, general Editors.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-58765-844-0 (set : alk. paper) -- ISBN 978-1-58765-845-7 (v. 1 : alk. paper) -- ISBN 978-1-58765-846-4

(v. 2 : alk. paper) -- ISBN 978-1-58765-847-1 (v. 3 : alk. paper)

1. Mathematics--Social aspects. I. Greenwald, Sarah J. II. Thomley, Jill E.

QA10.7.E53 2012

303.483--dc23

2011021856

First Printing

Printed in the United States of America

Contents

Volume 1Publishers Note viAbout the Editors viiiIntroduction ixList of Articles xiiiTopic Finder xxiList of Contributors xxviiArticles A to E 1376

Volume 2List of Articles viiArticles F to O 377744

Volume 3List of Articles viiArticles P to Z 7451089Chronology 1091Resource Guide 1109Glossary 1113Index 1127Photo Credits 1191

vi

The Encyclopedia of Mathematics and Society (three volumes) explains how mathematics is at the root of modern civilization, from measuring temperature on a frigid day to driving a car to using a digital camera; enthusiasts might say applied mathematics rules the world. The set includes 478 articles, all of which were written specifically for the work.

Scope of CoverageThe EncyclopediaofMathematicsandSociety is designed to provide students at the high school and under-graduate levels with a convenient source of informa-tion on the fundamental science and the mathematics behind our daily lives, explaining to students how and why mathematics works, and allowing readers to bet-ter understand how disciplines such as algebra, geom-etry, calculus, and others affect what we do every day. This academic, multiauthor reference work serves as a general and nontechnical resource for students and teachers to understand the importance of mathemat-ics; to appreciate the influence of mathematics on soci-eties around the world; to learn the history of applied mathematics; and to initiate educational discussion brought forth by the specific social and topical articles presented in the work.

Publishers Note

The articles in the set fall into one or more of the following broad categories: architecture and engi-neering (35 articles); arts, music, and entertainment (41): business, economics, and marketing (32); com-munication and computers (22); friendship, romance, and religion (18); games, sport, and recreation (42); government, politics, and history (43); history and development of curricular concepts (63); mathemat-ics around the world (21); mathematics culture and identity (27); medicine and health (34); school and society (19); space, time, and distance (25); travel and transportation (18); and weather, nature, and envi-ronment (35).

Rationale for Choice of TopicsMathematics is a fundamental part of society, yet many people may not be aware of the interconnections between what they have learned in school and their everyday lives. In its Curriculum Guide (MAA, 2004), the Mathematical Association of Americas Committee on the Undergraduate Program in Mathematics (CUPM) recommends that mathematics programs lead people to learn mathematics in a way that helps them to better understand its place in society: its meaning, its history, and its uses. In keeping with this

vii

philosophy, the editors chose topics for inclusion based on one or more of the following criteria:

The topic is timely and likely to remain so. The topic can be tied to mathematical

concepts that people likely have been exposed to.

The topic is related to concepts and connections that professional mathematical organizations have suggested are important.

The topic is one that the general public has expressed interest in.

The topic is one we have successfully used or that we know has been successfully used in other contexts.

Article Length and FormatArticles in the encyclopedia range in length from 500 to 3500 words. Each is first presented with the cat-egory to which it belongs (for example, architecture and engineering), an article summary, and fields of study for the article. The fields of study include the following:

Algebra Calculus Communication Connections Data Analysis and Probability Geometry Measurement Number and Operations Problem Solving Reasoning and Proof Representations

Each article is then followed by See Also cross-ref-erences to other relevant articles and Further Read-ing sources that include bibliographic citations. Many articles are richly illustrated with photos and captions, and charts, graphs, and tables. Finally, each article is signed by the contributor to the encyclopedia.

Frontmatter and BackmatterVolume 1 of the EncyclopediaofMathematicsandSoci-etybegins with About the Editors and then presents

their introduction to the encyclopedia. The List of Articles, repeated in all three volumes, features all the articles in alphabetical order with page numbers as they are listed in the encyclopedia. A Topic Finder shows all the articles organized by category to enable readers to find related article by topic. The List of Contribu-tors presents all the writers for the encyclopedia along with their academic or institutional affiliations.

The backmatter of the encyclopedia at the end of Volume 3 has the Chronology of Mathematics, a time-line of major milestones in the discoveries and devel-opment of mathematics. Next is the Resource Guide for further research that includes books that are major works in the history of mathematics as well as current editions of new works, journals in the mathematics field, and Internet sites that pertain to mathematics. A Glossary provides mathematical definitions for terms encountered in the articles. Lastly, a comprehen-sive subject index references all concepts, terms, events, persons, places, and other topics of discussion.

Online AccessSalem provides access to its award-winning content both in traditional printed form and online. Any school or library that purchases this three-volume set is entitled to complimentary access to Salems online ver-sion of the content through our Salem Science Data-base. For more information about our online database, please contact our online customer service representa-tives at (800) 221-1592.

The advantages are clear:

Complimentary with print purchase Fully supported Unlimited users at your library Full access from home or dorm rooms Immediate access via online registration A simple, intuitive interface User profile areas for students and patrons Sophisticated search functionality Complete content, including appendixes Integrated searches with any other Salem

Press product you already have on the Salem Science platform

E-books are also available

viii

About the Editors

Sarah J. Greenwald is a professor of mathematics and a womens studies core faculty member at Appala-chian State University in Boone, North Carolina. She obtained her Ph.D. in mathematics from the University of Pennsylvania in 1998 and since then has published more than 35 articles. Her areas of expertise include Riemannian geometry, popular culture as it pertains to mathematics, and women and minorities in mathemat-ics. Dr. Greenwald has discussed the impacts of scien-tific popular culture representations on NPRs ScienceFriday. She has spoken all over the country, and her interactive mathematics lecture appears on 20th Cen-tury Foxs Futurama movie BendersBigScore.

Dr. Greenwald has won numerous teaching awards including a Mathematical Association of America Alder Award for Distinguished Teaching and an Appalachian State University Wayne D. Duncan Award for Excellence in Teaching in General Education. Dr. Greenwald has also been active in professional service as a member of the Joint Policy Board for Mathematics Advisory Panel for Mathematics Awareness Month and as the associate editor for the Association for Women in Mathematics, just to name a few. Her husband, Joel Landsberg, is the bassist for the Kruger Brothers.

Jill E. Thomley is an associate professor of statistics in the Department of Mathematical Sciences at Appa-lachian State University. Her education and scholarly interests are diverse, generally focusing on mathematics and science applications. She earned a Ph.D. in Deci-sion Sciences from Rensselaer Polytechnic Institute and was awarded the Del and Ruth Karger Dissertation Prize and a Rensselaer Founders Award of Excellence. Additional degrees include an M.S. in industrial/orga-nizational psychology from Rensselaer and an A.B. in psychology from Harvard University.

Along with teaching, Dr. Thomley consults on sta-tistical design and analysis of scientific research and evaluates the results of federal education grants. Areas of focus include computational science, a discipline arising from the intersection of science, mathematics, and computer science, and the adoption and diffu-sion of educational innovations in mathematics and science. She presented at the first Science in Society Conference in 2009 and was published in TheInter-nationalJournalofScienceinSociety. Additional inter-ests include history of statistics and statistics in popu-lar culture.

ixix

Introduction

Mathematics is pervasive in modern society, and on some level we all use mathematics in our daily lives. At the same time, many people are not fully aware of the diverse interactions and connections between mathe-matics and society. Mathematics takes a readily appar-ent starring role in highly technological fields like engi-neering, computer science, and the natural sciences. Outside these fields, however, there are countless ideas, inventions, and advances that cannot be fully realized without the involvement of mathematics.

Organizations like the National Council of Teachers of Mathematics and the Mathematical Association of America recommend that mathematics be explored in the context of contemporary society. To examine these connections, we approach them from different angles. We can look at mathematics through the lens of larger societal structures like nations, cultures, and educa-tional systems, or we can turn this method around to explore the societal structures within mathematics, such as the culture of mathematicians and notions of proof, certainty, and success.

Connections are also found in the countless applica-tions of mathematics to society. Overall, definitions and applications of mathematics are inherently dependent on context: the socio-historical events during which they developed; the people who created or discovered concepts, who built upon the work of others, or who

passed their knowledge on to the next generation; the fundamental connections to daily tasks of living; the ethics, controversies, and philosophies surrounding mathematics; the publics perceptions of mathemat-ics and mathematicians; the way current society uses mathematics to solve problems and educate its citizens; and the way mathematics draws from society in order to grow and evolve.

Mathematics shapes the world in which we live. In the twenty-first century it is almost impossible to find an academic field of study that does not use mathemat-ics, either directly or via tools and technology in which mathematics plays a vital role. The world in turn shapes the discipline of mathematics by inspiring mathemati-cians to formulate new questions, solve new problems, develop new theories, and use new technologies. Each successive generation of mathematicians brings fresh perspectives, expectations, and ways of thinking and working into the culture of mathematics. These math-ematicians are influenced by the home, school, and play environments in which they were raised.

However, despite the mathematics all around us, peoples exposure may be limited. Representations in the media or in popular culture may portray math-ematics and mathematicians in highly stereotypical ways that do not reflect the true depth, breadth, diver-sity, and culture of the mathematics community.

The goal of The Encyclopedia ofMathematics andSociety is to weave multilayered connections between society, history, people, applications, and mathemat-ics. These connections address both mathematical concepts that our readers likely have been exposed to at school, work, or through other sources, as well as advanced topics that are built upon these fundamental ideas. The articles in the Encyclopedia, which were con-tributed by a broad spectrum of authors in many fields, also include connections to multiple disciplines within and outside of mathematics.

In general, the articles do not teach or present detailed mathematical theory, derivations, and equa-tions. There is already a vast array of textbooks and other works better able to accomplish that important task. Instead, we intended them to serve as a founda-tion and jumping-off point for additional explorations. As mathematics professor and educator Art Johnson has noted in other settings, we hope that this type of contextualization helps people to see mathematics as a discipline that transcends culture, time, and gender, and as a discipline for everyone, everywhere.

In keeping with this focus on linkages and inter-disciplinarity, we have organized the articles not by mathematics topic but according to various connect-ing themes. For example, there are few stand-alone articles about individual people within the encyclope-dia. Instead, we encouraged our authors to include sig-nificant mathematical contributors within the associ-ated context of one or more topics or applications. The people we did choose to include as stand-alone articles serve to highlight the diversity of individuals who have produced great achievements with mathematics.

Further, our intent was to discuss, via these indi-viduals and other articles in the Mathematics Culture and Identity theme, the community of mathematicians today: who mathematicians are, as professionals and people; the type of work mathematicians do; the dif-ferent ways in which mathematicians describe math-ematics and where their ideas come from; and math-ematicians personal processes when working with mathematics. We also wanted to address in these arti-cles how the mathematics community perceives itself and how it is in turn perceived by society.

Articles within the History and Development of Curricular Topics theme highlight many of the earliest known uses, both ancient and modern advances, and people who have contributed to the development and

spread of the concept or field. In contrast, the articles within the School and Society theme examine the importance of broad fields inside and outside of school, primarily in the United States. These articles showcase, for example, what jobs use particular skills and why the field is a fundamental part of current school curricula and society. The Mathematics Around the World theme extends the discussion of cross-cultural attitudes and perspectives on mathematics, with geographic regions grouped according to current United Nations stan-dards. Other themes that center on mathematics appli-cation are Games, Sport, and Recreation; Government, Politics, and History; and Space, Time, and Distance.

Why did we choose to focus on connections? In modern society, widespread Internet access has placed data about a broad spectrum of people, objects, and events essentially at our fingertips, yet mathematics content may be buried among other discussions rather than brought to the forefront.

Both Internet and other types of library searches can result in a potentially overwhelming number of results, many of which contain almost nothing of mathematical relevance, though important connec-tions exist. Too often, regardless of the amount of data or sources returned, connections between mathemat-ics, people, objects, and events are missing, or they are presented in isolation from their broader historical context. Such connections are critical components of knowledge acquisition, creation, and dissemination. They are what allow people to extrapolate from what they already know to new situations, to create new knowledge or new applications, to overcome existing negative stereotypes about mathematics, and to fully understand the timeline of human events from mul-tiple perspectives.

Even several hundred articles cannot provide an exhaustive examination of mathematics and society. At best, we can perhaps provide a snapshot of the history, people, applications, and mathematical connections as they exist at the time of publication, with some discus-sion of the rich history and speculations about future directions. Hopefully, this encyclopedia is a representa-tive sampling of articles that, with the accompanying further readings, will allow a reader to follow the path to related topics of interest.

In making the very difficult decision regarding what topics to include, given that time and space were not unlimited, we used an array of selection criteria, such

x Introduction

as: the topic was timely and likely to remain so for a reasonable period of time; the topic was tied to math-ematical concepts to which people likely have been exposed; the topic was related to concepts and con-nections that professional mathematical organizations suggested are important; the topic is one that the gen-eral public has expressed interest in; or the topic was one that we ourselves have successfully used or that we know has been successfully used in other educational or professional contexts.

When embarking on this work, we already knew in a general sense how pervasive mathematics is in soci-ety, and we were eager to share these ideas with oth-ers. However, even though we are mathematicians with diverse interdisciplinary backgrounds, research and

Introduction xi

teaching interests, we were surprised to discover so many interesting and amazing connections. We learned more than we ever imagined we would. It was regret-tably impossible to include everything we thought was interesting or important, and we have accumulated a long list of items that we want to explore in the future, on our own or with our students and colleagues.

The creation of this encyclopedia has been an intel-lectual pleasure and a profound learning experience, and we hope that our readers find the same kind of enjoyment and wonder that we experienced.

Sarah J. GreenwaldJill E. ThomleyGeneralEditors

xiii

List of Articles

AAccident Reconstruction 1Accounting 2Acrostics, Word Squares, and Crosswords 5Actors, seeWriters,Producers,andActors 1081Addition and Subtraction 7Advertising 10Africa, Central 13Africa, Eastern 15Africa, North 17Africa, Southern 19Africa, West 20African Mathematics 23AIDS, seeHIV/AIDS 478Aircraft Design 25Airplanes/Flight 28Algebra and Algebra Education 31Algebra in Society 36Analytic Geometry, seeCoordinateGeometry 247Anesthesia 42Animals 43Animation and CGI 49Apgar Scores 51Arabic/Islamic Mathematics 53Archery 55Archimedes 57Arenas, Sports 61Artillery 63

Asia, Central and Northern 66Asia, Eastern 68Asia, Southeastern 70Asia, Southern 72Asia, Western 74Astronomy 76Atomic Bomb (Manhattan Project) 79Auto Racing 81Axiomatic Systems 84

BBabylonian Mathematics 87Ballet 90Ballroom Dancing 91Bankruptcy, Business 92Bankruptcy, Personal 94Bar Codes 96Baseball 97Basketball 99Basketry 102Bees 103Betting and Fairness 105Bicycles 107Billiards 110Binomial Theorem 111Birthday Problem 113Black Holes 115

xiv List of Articles

Blackmun, Harry A. 117Blackwell, David 118Board Games 119Body Mass Index 122Brain 124Bridges 129Budgeting 130Burns, Ursula 132Bus Scheduling 133

CCalculators in Classrooms 137Calculators in Society 139Calculus and Calculus Education 142Calculus in Society 148Cameras, seeDigitalCameras 304Calendars 153Canals 155Carbon Dating 157Carbon Footprint 159Careers 162Caribbean America 166Carpentry 167Castillo-Chvez, Carlos 169Castles 171Caves and Caverns 172Cell Phone Networks 174Census 176Central America 178Cerf, Vinton 180Cheerleading 181Chemotherapy 183Chinese Mathematics 184Circumference, seePerimeter

andCircumference 761City Planning 188Civil War, U.S. 191Climate Change 194Climbing 200Clocks 201Closed-Box Collecting 204Clouds 206Clubs and Honor

Societies 207Cochlear Implants 209Cocktail Party Problem 210Coding and Encryption 212Cold War 214

Combinations, seePermutationsandCombinations 763

Comic Strips 218Communication in Society 219Comparison Shopping 225Competitions and Contests 227Composing 229Computer-Generated Imagery (CGI),

seeAnimationandCGI 49Congressional Representation 231Conic Sections 235Connections in Society 238Continuity, seeLimitsandContinuity 552Contra and Square Dancing 243Cooking 244Coordinate Geometry 247Coral Reefs 250Counterintelligence,

seeIntelligenceandCounterintelligence 508Coupons and Rebates 252Credit Cards 253Crime Scene Investigation 255Crochet and Knitting 257Crosswords,

seeAcrostics,WordSquares,andCrosswords 5Crystallography 259Cubes and Cube Roots 260Currency Exchange 262Curricula, International 264Curriculum, College 267Curriculum, K12 274Curves 280

DDams 283Data Analysis and Probability in Society 284Data Mining 290Daubechies, Ingrid 292Deep Submergence Vehicles 294Deforestation 295Deming, W. Edwards 298Diagnostic Testing 299Dice Games 301Digital Book Readers 303Digital Cameras 304Digital Images 306Digital Storage 308Disease Survival Rates 311

List of Articles xv

Diseases, Tracking Infectious 312Division, seeMultiplicationandDivision 685Domes 315Doppler Radar 316Drug Dosing 317DVR Devices 319

EEarthquakes 323Educational Manipulatives 324Educational Testing 326EEG/EKG 329Egyptian Mathematics 330Einstein, Albert 333Elections 335Electricity 340Elementary Particles 342Elevation 344Elevators 346Encryption, seeCodingandEncryption 212Energy 348Energy, Geothermal, seeGeothermalEnergy 441Engineering Design 351Equations, Polar 353Escher, M.C. 354Ethics 356Europe, Eastern 358Europe, Northern 361Europe, Southern 363Europe, Western 365Expected Values 368Exponentials and Logarithms 370Extinction 372Extreme Sports 373

FFantasy Sports Leagues 377Farming 379Fax Machines 382Fertility 384Fibonacci Tuning, seePythagorean

andFibonacciTuning 823FICO Score 386File Downloading and Sharing 387Fingerprints 388Firearms 390Fireworks 392Fishing 394

Floods 395Football 398Forecasting 400Forecasting, Weather, seeWeatherForecasting 1052Forest Fires 402Fuel Consumption 404Function Rate of Change 405Functions 408Functions, Recursive 410

GGame Theory 413Games, seeBoardGames;VideoGames 119, 1032Garfield, Richard 415Genealogy 417Genetics 419Geometry and Geometry Education 422Geometry in Society 427Geometry of Music 433Geometry of the Universe 436Geothermal Energy 441Gerrymandering 443Global Warming, seeClimateChange 194Golden Ratio 445Government and State Legislation 448GPS 450Graham, Fan Chung 453Graphs 454Gravity 456Greek Mathematics 458Green Design 461Green Mathematics 463Gross Domestic Product (GDP) 466Growth Charts 468Guns,seeFirearms 390Gymnastics 469

HHarmonics 471Hawking, Stephen 473Helicopters 475Highways 476Hitting a Home Run 477HIV/AIDS 478Hockey 480Home Buying 482Houses of Worship 485HOV Lane Management 488

xvi List of Articles

Hunt, Fern 489Hurricanes and Tornadoes 490

IIncan and Mayan Mathematics 493Income Tax 496Individual Retirement Accounts (IRAs),

seePensions,IRAs,andSocialSecurity 757Industrial Revolution 499Infantry (Aerial and Ground Movements) 501Infinity 504Insurance 506Intelligence and Counterintelligence 508Intelligence Quotients 511Interdisciplinary Mathematics Research,

seeMathematicsResearch,Interdisciplinary 632Interior Design 514Internet 515Interplanetary Travel 520Inventory Models 523Investments, seeMutualFunds 691Irrational Numbers, seeNumbers,Rational

andIrrational 724Islamic Mathematics, seeArabic/Islamic

Mathematics 53

JJackson, Shirley Ann 525Joints 526

KKicking a Field Goal 529King, Ada (Countess of Lovelace),

seeLovelace,Ada 565Knitting, seeCrochetandKnitting 257Knots 531

LLandscape Design 533LD50/Median Lethal Dose 535Learning Exceptionalities 536Learning Models and Trajectories 540Legislation, seeGovernment

andStateLegislation 448Levers 544Life Expectancy 545Light 547Light Bulbs 549

Lightning 550Limits and Continuity 552Linear Concepts 553Literature 556Loans 561Logarithms, seeExponentialsandLogarithms 370Lotteries 563Lovelace, Ada 565

MMagic 567Mapping Coastlines 570Maps 571Marine Navigation 574Market Research 578Marriage 580Martial Arts 582Math Gene 584Mathematical Certainty 586Mathematical Friendships and Romances 588Mathematical Modeling 589Mathematical Puzzles 593Mathematician Defined 595Mathematicians, Amateur 597Mathematicians, Religious 600Mathematics, African, seeAfricanMathematics 23Mathematics, Applied 603Mathematics, Arabic/Islamic,

seeArabic/IslamicMathematics 53Mathematics, Babylonian,

seeBabylonianMathematics 87Mathematics, Defined 608Mathematics, Chinese, seeChineseMathematics 184Mathematics, Egyptian,

seeEgyptianMathematics 330Mathematics, Elegant 610Mathematics, Greek, seeGreekMathematics 458Mathematics, Green, seeGreenMathematics 463Mathematics, Incan and Mayan,

seeIncanandMayanMathematics 493Mathematics, Native American,

seeNativeAmericanMathematics 697Mathematics, Roman, seeRomanMathematics 878Mathematics, Theoretical 613Mathematics, Utility of 618Mathematics, Vedic, seeVedicMathematics 1029Mathematics: Discovery or Invention 620Mathematics and Religion 622

List of Articles xvii

Mathematics Genealogy Project 628Mathematics Literacy and Civil Rights 630Mathematics Research, Interdisciplinary 632Mathematics Software,

seeSoftware,Mathematics 926Matrices 634Mattresses 636Mayan Mathematics,

seeIncanandMayanMathematics 493Measurement, Systems of 637Measurement in Society 640Measurements, Area 645Measurements, Length 647Measurements, Volume 651Measures of Center 653Measuring Time 655Measuring Tools 657Medical Imaging 659Medical Simulations 660Microwave Ovens 661Middle Ages 663Military Draft 665Minorities 667Missiles 671Molecular Structure 672Money 674Moon 677Movies, Making of 679Movies, Mathematics in 681MP3 Players 684Multiplication and Division 685Music, Geometry of,

seeGeometryofMusic 433Music, Popular, seePopularMusic 786Musical Theater 689Mutual Funds 691

NNanotechnology 693National Debt 695Native American Mathematics 697Nervous System 700Neural Networks 701Newman, Ryan 703Nielsen Ratings 704Normal Distribution 706North America 708Number and Operations 710

Number and Operations in Society 714Number Theory 719Numbers, Complex 721Numbers, Rational and Irrational 724Numbers, Real 727Numbers and God 729Nutrition 731

OOcean Tides and Waves, seeTidesandWaves 993Oceania, Australia and New Zealand 735Oceania, Pacific Islands 737Operations, seeNumberandOperations;

NumberandOperationsinSociety 714Optical Illusions 739Orbits, Planetary, seePlanetaryOrbits 771Origami 741

PPacemakers 745Packing Problems 746Painting 748Parallel Postulate 750Parallel Processing 752Payroll 754Pearl Harbor, Attack on 755Pensions, IRAs, and Social Security 757Percussion Instruments 760Perimeter and Circumference 761Permutations and Combinations 763Perry, William J. 766Personal Computers 767Pi 770Planetary Orbits 771Plate Tectonics 773Plays 774Poetry 777Polygons 779Polyhedra 782Polynomials 784Popular Music 786PredatorPrey Models 788Predicting Attacks 790Predicting Divorce 792Predicting Preferences 793Pregnancy 796Prehistory 798Probability 800

xviii List of Articles

Probability in Society, seeDataAnalysisandProbabilityinSociety 284

Problem Solving in Society 804Producers, seeWriters,Producers,andActors 1081Professional Associations 809Proof 812Proof in Society,

seeReasoningandProofinSociety 845Psychological Testing 815Pulleys 818Puzzles 819Puzzles, Mathematical, seeMathematicalPuzzles 593Pythagorean and Fibonacci Tuning 823Pythagorean School 825Pythagorean Theorem 827

QQuality Control 831Quilting 833

RRacquet Games 835Radar, seeDopplerRadar 316Radiation 836Radio 838Raghavan, Prabhakar 840Randomness 841Rankings 843Rational Numbers, seeNumbers,

RationalandIrrational 724Reasoning and Proof in Society 845Recycling 850Relativity 853Religion, Mathematics and,

seeMathematicsandReligion 622Religious Mathematicians

seeMathematicians,Religious 600Religious Symbolism 855Religious Writings 857Renaissance 860Representations in Society 863Revolutionary War, U.S. 868Ride, Sally 870Risk Management 872Robots 874Roller Coasters 877Roman Mathematics 878Ross, Mary G. 880

Ruler and Compass Constructions 881

SSacred Geometry 885Sales Tax and Shipping Fees 886Sample Surveys 888Satellites 890Scales 892Scatterplots 894Scheduling 896Schools 897Science Fiction 899Sculpture 903Search Engines 905Segway 906Sequences and Series 908Servers 910Shipping 912Similarity 914Six Degrees of Kevin Bacon 915Skating, Figure 916Skydiving 918Skyscrapers 919SMART Board 920Smart Cars 922Soccer 923Social Networks 924Social Security, seePensions,IRAs,

andSocialSecurity 757Software, Mathematics 926Solar Panels 930South America 931Space Travel, seeInterplanetaryTravel 520Spaceships 933Spam Filters 935Sport Handicapping 936Sports Arenas, seeArenas,Sports 61Squares and Square Roots 938Stalactites and Stalagmites 940State Legislation, seeGovernment

andStateLegislation 448Statistics Education 941Step and Tap Dancing 946Stethoscopes 947Stock Market Indices 949Strategy and Tactics 951Street Maintenance 954String Instruments 956

Stylometry 957Submarines, seeDeepSubmergenceVehicles 294Succeeding in Mathematics 958Sudoku 962Sunspots 963Subtraction, seeAdditionandSubtraction 7Surfaces 964Surgery 966Swimming 969Symmetry 970Synchrony and Spontaneous Order 972

TTao, Terence 975Tax,seeIncomeTax;SalesTax

andShippingFees 496, 886Telephones 976Telescopes 978Television, Mathematics in 981Televisions 985Temperature 987Textiles 989Thermostat 990Tic-Tac-Toe 992Tides and Waves 993Time, Measuring, seeMeasuringTime 655Time Signatures 995Toilets 997Tools, Measuring, seeMeasuringTools 657Tornadoes, seeHurricanesandTornadoes 490Tournaments 998Traffic 1000Trains 1001Trajectories, seeLearningModelsandTrajectories 540Transformations 1004Transplantation 1006Travel Planning 1007Traveling Salesman Problem 1009Trigonometry 1010Tunnels 1014

UUltrasound 1017Unemployment, Estimating 1018

Units of Area 1020Units of Length 1021Units of Mass 1023Units of Volume 1024Universal Constants 1026Universal Language 1027

VVectors 1029Vedic Mathematics 1031Vending Machines 1033Video Games 1034Vietnam War 1037Viruses 1038Vision Correction 1039Visualization 1041Volcanoes 1044Volleyball 1046Voting,seeElections 335Voting Methods 1047

WWater Distribution 1051Water Quality 1053Waves, seeTidesandWaves 993Weather Forecasting 1054Weather Scales 1057Weightless Flight 1059Wheel 1060Wiles, Andrew 1061Wind and Wind Power 1063Wind Instruments 1065Windmills 1066Wireless Communication 1068Women 1069World War I 1073World War II 1075Wright, Frank Lloyd 1080Writers, Producers, and

Actors 1081

ZZero 1087

List of Articles xix

xxi

Topic Finder

The following list is provided for readers to find articles related by topic.

Architecture and EngineeringAircraft DesignBridges CanalsCarpentry CastlesCity Planning DamsDomes Electricity Elevators Engineering Design Fireworks Green Design Highways Jackson, Shirley AnnLandscape DesignLeversLight Bulbs Mattresses Microwave OvensNanotechnology Packing Problems PulleysRobots

Schools SkyscrapersSolar Panels Symmetry Televisions ThermostatToiletsTunnels Vending MachinesWater DistributionWindmills Wright, Frank Lloyd

Arts, Music, and EntertainmentAnimation and CGIBalletBallroom DancingClosed-Box Collecting Comic StripsComposing Contra and Square DancingCooking Crochet and Knitting Digital Cameras Digital Images Escher, M.C. Geometry of Music Golden Ratio

xxii Topic Finder

Harmonics Literature Magic Movies, Making of Movies, Mathematics in Musical TheaterNielsen Ratings Optical Illusions Origami Painting Percussion InstrumentsPoetry PlaysPopular MusicPythagorean and Fibonacci Tuning Quilting Scales Science Fiction Sculpture Step and Tap DancingString InstrumentsTelevisions Textiles Time SignaturesWind InstrumentsWriters, Producers, and Actors

Business, Economics, and MarketingAccounting Advertising Bankruptcy, Business Bankruptcy, PersonalBar Codes Budgeting Burns, UrsulaComparison Shopping Coupons and RebatesCredit Cards Currency ExchangeData Mining Deming, W. EdwardsFICO ScoreForecasting Gross Domestic Product (GDP)Home BuyingIndustrial RevolutionInsurance Inventory Models

Loans Market Research Money Mutual Funds Payroll Pensions, IRAs, and Social SecurityPredicting Preferences Quality Control Sales Tax and Shipping Fees Scheduling ShippingStock Market Indices

Communication and ComputersCalculators in Society Cell Phone Networks Cerf, VintonCoding and Encryption Digital Book Readers Digital Storage DVR Devices Fax Machines File Downloading and Sharing Internet Lovelace, AdaMP3 Players Neural Networks Parallel Processing Personal Computers Radio Satellites Search Engines Servers SMART Board Spam FiltersWireless Communication

Friendship, Romance, and ReligionBirthday Problem Cocktail Party Problem Genealogy Houses of Worship Marriage Mathematical Friendships and RomancesMathematicians, Religious Mathematics and Religion Numbers and God Predicting Divorce

Topic Finder xxiii

Pythagorean School Raghavan, PrabhakarReligious Symbolism Religious WritingsSacred Geometry Six Degrees of Kevin Bacon Social Networks

Games, Sport, and RecreationAcrostics, Word Squares, and CrosswordsArchery Arenas, SportsAuto RacingBaseballBasketballBasketryBetting and Fairness BilliardsBoard Games Cheerleading Climbing Dice Games Extreme Sports Fantasy Sports Leagues Fishing FootballGame Theory Garfield, RichardGymnastics Hitting a Home Run HockeyKicking a Field Goal KnotsLotteries Martial Arts Mathematical PuzzlesNewman, RyanPuzzles Racquet Games Rankings Roller CoastersSkating, Figure Skydiving SoccerSport Handicapping Sudoku SwimmingTic-Tac-Toe

Tournaments Video Games Volleyball

Government, Politics, and HistoryAfrican MathematicsArabic/Islamic Mathematics ArchimedesArtilleryAtomic Bomb (Manhattan Project)Babylonian Mathematics Blackmun, Harry A.Census Chinese Mathematics Civil War, U.S.Cold WarCongressional RepresentationCrime Scene InvestigationDeep Submergence Vehicles Egyptian Mathematics Elections FirearmsGerrymandering Greek Mathematics Incan and Mayan Mathematics Income Tax Infantry (Aerial and Ground Movements)Intelligence and CounterintelligenceMiddle AgesMilitary Draft MissilesNational DebtNative American Mathematics Pearl Harbor, Attack onPensions, IRAs, and Social Security Perry, William J.Predicting AttacksPrehistoryRenaissanceRevolutionary War, U.S.Roman MathematicsStrategy and TacticsUnemployment, EstimatingVedic Mathematics Vietnam WarVoting Methods World War IWorld War II

xxiv Topic Finder

History and Development ofCurricular ConceptsAddition and SubtractionAlgebra and Algebra EducationAxiomatic SystemsBinomial TheoremCalculators in ClassroomsCalculus and Calculus EducationConic SectionsCoordinate GeometryCubes and Cube RootsCurvesEquations, PolarExpected ValuesExponentials and LogarithmsFunction Rate of ChangeFunctionsFunctions, Recursive Geometry and Geometry EducationGraphsInfinityLimits and ContinuityLinear ConceptsMathematical ModelingMatricesMeasurement, Systems ofMeasurement in SocietyMeasurements, AreaMeasurements, LengthMeasurements, VolumeMeasures of CenterMeasuring TimeMeasuring ToolsMultiplication and DivisionNormal DistributionNumber and OperationsNumber TheoryNumbers, ComplexNumbers, RealParallel PostulatePerimeter and CircumferencePermutations and CombinationsPiPolygonsPolyhedraPolynomialsProbabilityProof

Pythagorean TheoremRandomnessRational NumbersRuler and Compass ConstructionsSample SurveysScatterplotsSequences and SeriesSimilaritySquares and Square RootsStatistics EducationSurfacesTransformationsTrigonometryVectorsVisualizationZero

Mathematics Around the WorldAfrica, CentralAfrica, EasternAfrica, NorthAfrica, SouthernAfrica, WestAsia, Central and NorthernAsia, EasternAsia, SoutheasternAsia, SouthernAsia, WesternCaribbean AmericaCentral AmericaCurricula, International Europe, EasternEurope, Northern Europe, SouthernEurope, WesternNorth AmericaOceania, Australia and New ZealandOceania, Pacific IslandsSouth America

Mathematics Culture and IdentityBlackwell, David Careers Castillo-Chvez, CarlosClubs and Honor SocietiesCompetitions and ContestsDaubechies, IngridEthics

Topic Finder xxv

Graham, Fan ChungHunt, FernMath Gene Mathematical Certainty Mathematician DefinedMathematicians, Amateur Mathematics, AppliedMathematics, Defined Mathematics, ElegantMathematics, TheoreticalMathematics, Utility ofMathematics: Discovery or InventionMathematics Genealogy ProjectMathematics Research, InterdisciplinaryMinorities Professional Associations Ross, Mary G.Tao, TerenceWiles, Andrew Women

Medicine and HealthAnesthesiaApgar ScoresBody Mass Index Brain ChemotherapyCochlear ImplantsDiagnostic Testing Disease Survival Rates Diseases, Tracking Infectious Drug Dosing EEG/EKG FertilityFingerprintsGenetics Growth Charts HIV/AIDSIntelligence Quotients JointsLD50/Median Lethal DoseLife ExpectancyMedical Imaging Medical Simulations Molecular StructureNervous System Nutrition Pacemakers

PregnancyPsychological TestingStethoscopesSurgeryTransplantation UltrasoundViruses Vision Correction

School and SocietyAlgebra in SocietyCalculus in SocietyCommunication in SocietyConnections in SocietyCurriculum, CollegeCurriculum, K12Data Analysis and Probability in SocietyEducational TestingGeometry in SocietyGovernment and State LegislationLearning ExceptionalitiesLearning Models and TrajectoriesMathematics Literacy and Civil RightsMeasurement in SocietyNumber and Operations in SocietyProblem Solving in SocietyReasoning and Proof in SocietyRepresentations in SocietySucceeding in Mathematics

Space, Time, and DistanceAstronomyBlack HolesCalendars ClocksEinstein, AlbertElementary Particles Elevation Geometry of the Universe Gravity Hawking, StephenInterplanetary Travel Mapping CoastlinesMeasuring Time MoonPlanetary Orbits RelativityRide, Sally

TelescopesTemperature Units of Area Units of Length Units of Mass Units of Volume Universal Constants Universal Language

Travel and Transportation Accident ReconstructionAirplanes/Flight Bicycles Bus Scheduling Fuel Consumption GPS Helicopters HOV Lane Management Maps SegwaySmart Cars Spaceships Traffic TrainsTravel Planning Traveling Salesman Problem Weightless Flight Wheel

Weather, Nature, and EnvironmentAnimalsBeesCarbon Dating

Carbon FootprintCaves and CavernsClimate Change CloudsCoral Reefs CrystallographyDeforestation Doppler Radar Earthquakes Energy Extinction FarmingFloodsForest FiresGeothermal EnergyGreen Mathematics Hurricanes and TornadoesLight Lightning Plate TectonicsPredatorPrey Models Radiation Recycling Stalactites and StalagmitesSunspots Synchrony and Spontaneous OrderTides and Waves Volcanoes Water QualityWeather Forecasting Weather Scales Wind and Wind Power

xxvi Topic Finder

xxvii

List of Contributors

Stephen Abbott MiddleburyCollegeJohn G. Alford SamHoustonStateUniversityMicah Altman HarvardUniversityMohamed Amezziane DePaulUniversityOr Syd Amit BostonCollegeJim Austin IndependentScholarSukantadev Bag UniversityCollegeCorkZenia C. Bahorski EasternMichiganUniversityHyungryul Baik CornellUniversityThomas E. Baker UniversityofScrantonRon Barnes UniversityofHouston,DowntownEric Barth KalamazooCollegeJohn Beam UniversityofWisconsin,Oshkosh

Judith E. Beauford UniversityoftheIncarnateWordLinda Becerra UniversityofHouston,DowntownRobert A. Beeler EasternTennesseeStateUniversityKimberly Edginton Bigelow UniversityofDaytonBonnie Ellen Blustein WestLosAngelesCollegeNorma Boakes RichardStocktonCollegeofNewJerseyMatt Boelkins GrandValleyStateUniversityMark Bollman AlbionCollegeVladimir E. Bondarenko GeorgiaStateUniversityCasey Borch UniversityofAlabamaatBirminghamSarah Boslaugh WashingtonUniversitySchoolofMedicineMarek Brabec AcademyofSciencesoftheCzechRepublicMurray R. Bremner UniversityofSaskatchewan

David Brink UniversityCollegeDublin,IrelandPatrick L. Brockett UniversityofTexasatAustinJohn N. A. Brown IndependentScholarChris D. Cantwell ImperialCollegeLondonPeter J. Carrington UniversityofWaterlooJen-Mei Chang CaliforniaStateUniversity,LongBeachDarrah Chavey BeloitCollegeJohn T. Chen BowlingGreenStateUniversityDiana Cheng MiddleTennesseeStateUniversityKa-Luen Cheung TheHongKongInstituteofEducationJason L. Churchill CleoResearchAssociatesLoren Cobb UniversityofColorado,DenverShirley Coleman NewcastleUniversityDogan Comez NorthDakotaStateUniversityJustin Corfield GeelongGrammarSchoolBeth Cory SamHoustonStateUniversityVesta Coufal GonzagaUniversityKumer Pial Das LamarUniversityRichard De Veaux WilliamsCollegeMarilena Di Bucchianico RutgersUniversityDaniel Disegni ColumbiaUniversityMaria Droujkova NaturalMathLeigh H. Edwards FloridaStateUniversitySteven R. Edwards SouthernPolytechnicStateUniversity

Caleb Emmons PacificUniversityJonathan L. Entin CaseWesternReserveUniversityGisela Ernst-Slavit WashingtonStateUniversity,VancouverAmy Everton IndependentScholarJonathan David Farley UniversityofOxfordLee Anne Flagg UniversityofAlabamaatBirminghamFrancesco Flammini UniversityofNaplesFredericoIIRobert D. Foreman UniversityofOklahomaDaniel J. Galiffa PennStateErie,TheBehrendCollegeAngela Gallegos OccidentalCollegeCatherine C. Galley IndependentScholarJoseph A. Gallian UniversityofMinnesota,DuluthJoaquim Alves Gaspar UniversidadedeLisboaSommer Gentry UnitedStatesNavalAcademyMark Ginn AppalachianStateUniversityDarren Glass GettysburgCollegeDeborah L. Gochenaur ShippensburgUniversityChristopher Goff UniversityofthePacificLidia Gonzalez CityUniversityofNewYorkJeff Goodman AppalachianStateUniversityRick Gorvett UniversityofIllinoisatUrbana-ChampaignJudith V. Grabiner PitzerCollegeMichael K. Green StateUniversityofNewYork,OneontaSarah J. Greenwald AppalachianStateUniversity

xxviii List of Contributors

William Griffiths SouthernPolytechnicStateUniversityAlexander A. Gurshtein MesaStateCollegeJuan B. Gutierrez UniversityofMiamiSimone Gyorfi O.GogaHighSchool,Jibou,RomaniaGareth Hagger-Johnson TheUniversityofLeedsThomas W. Hair FloridaGulfCoastUniversityAnthony Harkin RochesterInstituteofTechnologyZiaul Hasan UniversityofIllinois,ChicagoDeborah J. Hilton IndependentScholarHolly Hirst AppalachianStateUniversityCalli A. Holaway UniversityofAlabamaLiang Hong BradleyUniversityBrian Hopkins SaintPetersCollegeLinda Hutchison UniversityofWyomingYih-Kuen Jan UniversityofOklahomaJerry Johnson WesternWashingtonUniversityPete Johnson EasternConnecticutStateUniversityPhillip Johnson AppalachianStateUniversityD. Keith Jones UniversityofSouthampton Ugur Kaplan KadirHasUniversity,IstanbulDavid I. Kennedy ShippensburgUniversityofPennsylvaniaCathy Kessel IndependentScholarMichael Cap Khoury UniversityofMichigan,AnnArborChristine Klein IndependentScholar

Michael Klucznik St.BonaventureUniversityRick Kreminski ColoradoStateUniversity,PuebloMatt Kretchmar DenisonUniversityBill Ktepi IndependentScholarMaria Elizete Kunkel UniversityofUlm,GermanyKonnie G. Kustron EasternMichiganUniversityAlistair Kwan YaleUniversityJames Landau IndependentScholarCarmen M. Latterell UniversityofMinnesota,DuluthMichele LeBlanc CaliforniaLutheranUniversityStephen Lee MathematicsinEducationandIndustryEddie Leung HongKongInstituteofEducationFuyuan Liao UniversityofOklahomaSilvia Liverani UniversityofBristolMichael G. Lovorn UniversityofAlabamaChad T. Lower PennsylvaniaCollegeofTechnologyMargaret MacDougall UniversityofEdinburghMedicalSchoolYiu-Kwong Man TheHongKongInstituteofEducationPhilip McCartney NorthernKentuckyUniversityElizabeth A. McMillan-McCartney NorthernKentuckyUniversityLiliana Monteiro AffiliationTKMariana Montiel GeorgiaStateUniversityDeborah Moore-Russo StateUniversityofNewYork,UniversityatBuffaloAshwin Mudigonda UniversalRoboticsInc.

List of Contributors xxix

Andrew Nevai UniversityofCentralFloridaSamuel Obara TexasStateUniversityEoin OConnell DeakinUniversitySerkan Ozel BogaziciUniversityZeynep Ebrar Yetkiner Ozel FatihUniversityJulian Palmore UniversityofIllinoisatUrbana-ChampaignRobert W. Peck LouisianaStateUniversitySchoolofMusicJosipa G. Petrunic UniversityCollegeLondonThomas J. Pfaff IthacaCollegeBiljana Popovic UniversityofNisZoran Petrovic UniversityofBelgradeMichael Qaissaunee BrookdaleCommunityCollegeMatina J. Rassias UniversityCollegeofLondonGregory Rhoads AppalachianStateUniversityMark Roddy SeattleUniversityMaria Elizabeth S. Rodrigues UniversityofUlm,GermanyDavid C. Royster UniversityofKentuckyDouglas Rugh IndependentScholarKarim Salim IndependentScholarAlun Salt UniversityofLeicesterKady Schneiter UtahStateUniversityRichard Schugart WesternKentuckyUniversityCarl R. Seaquist TexasTechUniversityDorry Segev JohnsHopkinsUniversitySchoolofMedicine

Abhijit Sen SuriVidyasagarCollegePadmanabhan Seshaiyer GeorgeMasonUniversityShahriar Shahriari PomonaCollegeBarbara A. Shipman UniversityofTexasatArlingtonKevin L. Shirley AppalachianStateUniversityLawrence H. Shirley TowsonUniversityDaniel Showalter OhioUniversityJorge Nuno Silva UniversityofCalifornia,BerkeleyFlorence Mihaela Singer UniversityofPloiesti,RomaniaKelli M. Slaten UniversityofNorthCarolina,WilmingtonDavid Slavit WashingtonStateUniversityMark R. Snavely CarthageCollegeHenrik Sorensen AarhusUniversityRavi Sreenivasan UniversityofMysoreChristopher J. Stapel UniversityofKentuckyCatherine Stenson JuniataCollegeKristi L. Stringer UniversityofAlabamaatBirminghamStephen Szydlik UniversityofWisconsin,OshkoshTristan Tager IndianaUniversityCourtney K. Taylor NorthGreenvilleUniversityJill E. Thomley AppalachianStateUniversityTodd Timmons UniversityofArkansas,FortSmithElena Toneva EasternWashingtonUniversityMarcella Bush Trevino IndependentScholar

xxx List of Contributors

Juliana Utley OklahomaStateUniversityK. G. Valente ColgateUniversityDaniela Velichova SlovakTechnicalUniversityinBratislavaCarlos J. Vilalta CenterforEconomicResearchandTeachingEliseo Vilalta-Perdomo TecnologicodeMonterreyJiri Wackerman InstituteforFrontierAreasof PsychologyandMentalHealthKaren Doyle Walton DeSalesUniversityChristopher J. Weinmann IndependentScholarMatthew West UniversityofAlabamaatBirmingham

Bethany White UniversityofWesternOntarioSharon Whitton HofstraUniversityConnie Wilmarth NorthwestChristianUniversityElizabeth L. Wilmer OberlinCollegeDaniel P. Wisniewski DeSalesUniversityTodd Wittman UniversityofCalifornia, LosAngelesQiang Zhao TexasStateUniversityLinda Reichwein Zientek SamHoustonStateUniversity

List of Contributors xxxi

Accident Reconstruction Category: Travel and Transportation.Fields of Study: Algebra; Data Analysis and Probability; Measurement.Summary: Accidents can be mathematically reconstructed to model accident risk and to improve safety equipment designs.

Accident reconstruction is important for understand-ing how accidents happen and for preventing accidents in the future. Principles and techniques from physics, mathematics, engineering, and other sciences are used to quantify critical variables and calculate others. For example, the initial speed of a suddenly braking vehi-cle can be determined by mathematically analyzing tire skid and yaw marks. The length of skid marks is a function of vehicle velocity and the amount of friction between the wheels and the road surface. In the case of yaw or circular motion, the radius of the yaw mark is also a factor in the calculation, as well as the eleva-tion of the road. Speed can also be calculated from the trajectories, angles, and other characteristics of objects struck by a speeding vehicle, or between two or more colliding vehicles. Investigators may use distances and angles to determine the original positions of passengers

who have been ejected from a vehicle. For more com-plex modeling, mathematicians, engineers, and other accident reconstructors rely on principles and equa-tions from physics, such as those governing energy and momentum, as well as vehicle specifications, mechani-cal failure analyses, geometric characteristics of high-ways, and quantification of visibility, perception, and reaction. Data from both real accidents and staged col-lisions, along with statistically designed safety analyses and other methods such as stochastic modeling, are often used to construct accident simulations and visu-alizations for use in a wide variety of contexts, including legal proceedings. Actuaries use accident data to model accident risk, which in turn influences insurance rates and public policy, such as seat belt and helmet laws.

Modeling Accident ReconstructionsAccidents related to travel and transportation can have a variety of negative consequences including personal injury and death. The analysis of accidents can lead to improved designs of vehicles and reduced fatali-ties as well as warning travelers about potential risks of travel. In reconstructing accidents, evidence from photographs, videos, eyewitnesses, or police reports is collected. Decision trees are used to ask questions at each stage of reconstruction and help decide the closest accident scenario dictated by the available evi-dence. In such reconstructions, probability must be

A

assigned for the likely cause of the accident and for the particular accident type among the possible accident scenarios based on the available evidence. Stochastic modeling is used to help solve such problems in acci-dent reconstructions.

Uses of Accident ReconstructionsAnother important aspect of accident reconstructions is to estimate the probability of occurrence of various types of injuries one may suffer in accidents. Such prob-ability estimates are used to help calculate travel insur-ance. By nature, accidents happen randomly andsince the types of injuries suffered in accidents also vary randomlyit is important to model accident types and predict the kinds of injuries one may suffer in different accident types. Such models can help prepare commu-nities with the optimal number of emergency services and also help doctors prepare for any unique types of injuries they are likely to deal with.

A typical problem is determining the types of spe-cial medical facilities that should be established to deal with travel-related accidents in a city. Such problems require stochastic modeling based on past data, which will help in simulating different types of accidents. Simulations help in planning emergency services to deal with accidents. Accident reconstructions may also help in forecasting the number of accidents of different types likely to happen in the near future, which may lead to better planning of the health, emergency, and disaster management facilities in the city.

Safety and Design Using Accident ReconstructionsAccident reconstructions also may help in improving vehicle design. Incorporating safety devices in vehicles is also a very important aspect of design. Safety devices, which help in avoiding severe injuries to passengers because of accidents, are designed with the help of accident reconstruction and are always a matter of high priority. Simulations can be used to develop sen-sors that can give an early warning about impending accidents or reduce the speeds of vehiclesthereby reducing the severity of an accident. In creating such designs, mathematical optimization methods are used to determine the optimal cost and space to be allotted. Another crucial application of accident reconstruction and accident modeling is driver training. Sophisticated simulators can be used to simulate different accident

scenarios and train drivers to react appropriately to each situation in real time. These simulators are based on algorithms and use random number generators to simulate accident situations. Well-developed algo-rithms that closely simulate real accidents are needed to reduceor even eliminatemajor accidents.

Further ReadingBrach, Raymond, and R. Matthew Branch. Vehicle

Accident Analysis and Reconstruction Methods. Warrendale, PA: SAE International, 2005.

Franck, Harold, and Darren Franck. Mathematical Methods for Accident Reconstruction: A Forensic Engineering Perspective. Boca Raton, FL: CRC Press, 2009.

Ravi Sreenivasan

See Also: Animation and CGI; Crime Scene Investigation; Data Mining; Insurance; Probability.

AccountingCategory: Business, Economics, and Marketing.Fields of Study: Algebra; Number and Operations.Summary: Accounting applies mathematics to the recording and analysis of a businesss financial status.

Accounting is the recording, interpretation, and presen-tation of financial information about a business entity, typically with the goal of producing financial state-ments that describe the businesss economic resources in standardized terms. Formal accounting began with the work of Franciscan friar Luca Pacioli, who intro-duced accounting techniques in his 1494 mathematical work Summa de Arithmetica, Geometria, Proportioni et Proportionalita. During the Industrial Revolution, Josiah Wedgwood introduced cost accounting, a tech-nique to ensure a profit margin by calculating the costs of materials and labor at every stage of production and setting the price accordingly. The needs of stockholders and other interested parties within the business, and an increasingly complex business environment, have increased the need for financial record-keeping tech-niques that are thorough and produce useful financial

Accounting

and to decrease assets, the transaction is recorded in the right column (crediting the account). Similarly, since liabilities and owners equity are listed on the right-hand side of the equation, to increase liabilities and owners equity, the transaction is recorded in the right column (crediting the account) and to decrease liabilities and owners equity, the transaction is recorded in the left column (debiting the account). For example, suppose a company needed to purchase $100 worth of office supplies. Furthermore, suppose the company pays $40 with cash and puts the remaining $60 on account (store credit). The general ledger may look like the following:

Figure 1. Purchased office supplies.

04-31-2017 Office supplies $100

Cash $40

Accounts payable $60

statements. Modern accounting is assisted by a variety of software packages, but the accountant must still be well-versed in mathematics in order to interpret the information. The fundamental accounting equation can be stated as the following:

Assets = Liabilities + Owners Equity.

For any given company, assets can be thought of as what the company owns. This includes cash (actual cash and bank accounts), money that is owed to the business (called accounts receivables), inventory, build-ings, land, equipment, and intangibles like patents and goodwill. Liabilities are what the company owes. This includes money owed to a bank (notes payable), suppli-ers (accounts payable), or the government (taxes pay-able). Owners equity can take several forms depending on who the owners are: a single person (sole propri-etor), a few people (partnership), or shareholders (cor-poration). Each method of ownership has advantages and disadvantages, but regardless of the method, the owners equity can be thought of as a net asset since it can be found by subtracting liabilities from assets.

Accounting as Record KeepingWhenever a financial transaction takes place, it must be recorded in at least three locations. First, it will be recorded in the general ledger (a book of entry sum-marizing a companys financial transactions). When recorded, the entry should contain the date of the transaction, a brief description of the transaction, and the monetary changes to all accounts affected (which will be at least two).

From there, the transaction gets recorded a second time in a secondary (or subsidiary) ledger for each of the accounts affected. When the amounts are recorded, they are put into the left (debit) column or the right (credit) column of the ledger. (In bookkeeping, debit and credit mean left and right, respectively; they are not related to debit or credit cards in this situation.) The total of each column of the general ledger record must add to the same sum. In that manner, all money can be accounted for as going into or out of an account.

In order to determine whether to credit or debit an account, a general rule that works for most accounts is to first look at the fundamental accounting equation. Since assets are listed on the left, to increase assets, the transac-tion is recorded in the left column (debiting the account)

Accounting

Accounting is the process of keeping track of the operations and financial status of a business.

In Figure 1, notice that both the right and left columns add up to $100; this shows that no money was lost in the process. Office supplies are considered an asset, so since the company increased the amount of office sup-plies, that account was recorded on the leftin other words, it debited office supplies for $100. Cash is also an asset, but the company decreased the amount of cash it had. As a result, cash was credited (the transaction was recorded on the right for that account). Accounts payable is a liability the company owes to the retailer it purchased the products from. Since the company increased the amount it owed the retailer, that account was recorded on the right as an increase to the compa-nys liabilitiesaccounts payable was credited.

Once this transaction was recorded in the general ledger, the company would also need to record this transaction in the Office Supplies ledger, the Cash ledger, and the Accounts Payable ledger. Accounts are debited or credited in their specific ledgers in the exact same manner that they are debited or credited in the general ledger. In a similar manner, the retailer who sold the office supplies would need to record this same transaction into his or her general and secondary led-gers. However, the retailers transaction would use the opposite side to denote the sale as follows:

Figure 2. Sold office supplies.

04-31-2017 Cash $40Account Receivable $60Inventory $100

Again, the right and left columns add up to the same amount. Contrary to the purchasing company, the receiving company lists three assets to record the trans-action. Cash and accounts receivable are both being increased, so debited. The asset inventory is being decreased and results in a credit to inventory. If this were a large company, rather than record each individ-ual transaction, the retailer would most likely record an entire days transactions as a single entry at the end of each business day. Once the general ledger has been recorded, the secondary ledgers need adjusting entries as well to denote the transaction(s).

Accounting as Record SharingIn addition to keeping records of transactions for a busi-ness, accounting is responsible for creating reports that

summarize the journals to share with others. To learn about the reports and how to create reports intended for people outside the business (such as shareholders, creditors, or government agencies), a person can take a class in financial accounting. To learn about the reports and how to create reports intended for people inside the business (such as managers), a person can take a class in managerial accounting.

The most common reports created for people outside the business are balance sheets, income statements, cash flow statements, and retained earnings statements. Of the four statement types, the balance sheet is written as a snapshot of the company at a point in time. In contrast, the other three statements are created to show what hap-pened over a period of time such as a month, quarter, or year. When creating these reports, the income state-ment is usually completed first. As its name implies, the income statement is created to determine the companys income during a specific time period. The income state-ment is also known as a profit and loss statement (P&L) or earnings statement. Information from the income statement is then used to create the retained earnings statement. Finally, the information from the retained earnings statement is used on the balance sheet.

The balance sheet first lists all of the companys assets in order of liquidity (the ability to turn the asset into

Accounting

Benfords Law

B enfords law,namedafterphysicist FrankBenford, gives the probability with whichthenumbers1through9willoccurasthefirstdigit inmanytypesofreal-lifedata.Forexam-ple, ina listofactualbankaccountdepositsinagivenday,about30%ofthetimethefirstdigitofthedepositamountwillbea1.Fraudu-lentdatathathasbeencreatedbypeopleoftendoesnotmatchtheexpectedprobabilities.

In very large modern data sets, highlyfocusedtestsuse thisprinciple tofinddevia-tions in selected subsets; for example, theoccurrenceofasuspiciouslylargefrequencyof$24receiptssubmittedinacompanythathasa$25maximummealallowance.

cash easily) from the most liquid to the least liquid. The assets are then added together to find the total assets of the company. The balance sheet next lists all of the com-panys liabilities in order of due date from the soonest due to the latest due. Below the liabilities is listed the owners equity (which includes retained earnings from the retained earnings statement). The liabilities and owners equity are added together. Referring back to the fundamental accounting equation, both of these amounts (the total assets and the sum of the liabilities and owners equity) should equal one another.

Reports created for internal users vary widely depending on the reasoning and the need for the report. Internal reports are usually created and specifically designed for making decisions within the company. For example, manufacturers could use internal reports to determine the optimal price of their product.

Manufacturers may also use internal reports to determine if it is more cost effective to create a needed part or to purchase the part from another company. They may need to consider continuing or eliminating a division of their company. Managerial accounting is also responsible for budgeting and forecasting.

Mathematical ModelsMany areas in financial accounting rely on mathemati-cal models for explanation and prediction. For exam-ple, models have played important roles in applications such as understanding the consequences of public dis-closure, formalizing market efficiency or competition, measuring income, and evaluating equilibrium pricing for goods and services. Some important mathematical techniques used in accounting models include linear regression, systems of simultaneous equations, equi-librium notions, and stochastic analysis. In the lat-ter, random rather than constant inputs are used to model scenarios where decisions must be made under realistic conditions of uncertainty. The data used in these models may be cross-sectional (representing a single snapshot in time) or longitudinal (one or more variables are measured repeatedly to detect trends and patterns). Probability theory is also used to detect instances of accounting fraud.

Further Reading Davis, Morton D. The Math of Money: Making

Mathematical Sense of Your Personal Finances. New York: Copernicus, 2001.

Hoyle, Joe Ben, Thomas F. Schaefer, and Timothy S. Doupnik. Fundamentals of Advanced Accounting. New York: McGraw-Hill, 2010.

Kimmel, Paul D., Jerry J. Weygandt, and Donald E. Keiso. Financial Accounting: Tools for Business Decision Making. 5th ed. Hoboken, NJ: Wiley, 2009.

Mullis, Darrell, and Judith Handler Orloff. The Accounting Game: Basic Accounting Fresh From the Lemonade Stand. Naperville, IL: Sourcebooks, 2008.

Verrecchia, Robert. The Use of Mathematical Models in Financial Accounting. Journal of Accounting Research 20 (1982).

Weygandt, Jerry J., Paul D. Kimmel, and Donald E. Keiso. Managerial Accounting: Tools for Business Decision Making. 4th ed. Hoboken, NJ: Wiley, 2008.

Chad T. Lower

See Also: Budgeting; Payroll; Shipping.

Acrostics, Word Squares, and CrosswordsCategory: Games, Sport, and Recreation.Fields of Study: Geometry; Number and Operations; Problem Solving.Summary: Mathematics and symmetry come into play in creating and solving word puzzles.

Acrostics, word squares, and crossword puzzles are the most common forms of word puzzles in English. Acros-tics and word squares are over 2000 years old and call for the solver to discover words hidden either covertly (acrostics) or overtly (word squares). The crossword puzzle premiered in 1913 and is similar to a word square expanded onto a larger grid, with gaps. Word puzzles have been used as mnemonics, ciphers, literary devices, educational exercises, and as simple games. Their con-struction, especially in the case of crossword puzzles, is informed by geometry; their solution can be pursued through probability theory. In a sense, the construc-tion and solving of word puzzles provide pleasures very similar to those of doing mathematics.

Acrostics, Word Squares, and Crosswords

Historic ExamplesThe earliest examples of acrostics are in the Old Testa-ment of the Bible. The Lamentations of Jeremiah and 12 Psalms are arranged so that the first letters of each verse spell out the Hebrew alphabet.

In Greece in 400 b.c.e., Dionysius forged a Sopho-clean text titled Parthenopaeus with the intention of mocking his rival, Heraclides. Having declared the author to be Sophocles, Heraclides was referred to in one of the several acrostics that Dionysius had included, which read, Heraclides is ignorant of letters.

In more contemporary times, novelist Vladimir Nabokov enjoyed chess problems, and one can find acrostics, number puzzles, cryptic references, and puns in several of his novels and stories. The last paragraph of his 1951 short story The Vane Sisters, for example, can be read both as the narrators confusion and acros-tically (taking the first letter of each word) as a message from the dead sisters.

Acrostics are often found in poetry because of its greater flexibility in syntax and phrasing. Former U.S. President George Washington is known to have con-structed at least one acrostic when he was 15a love poem for a girl about whom nothing is known other than her name, Frances.

Another good example of an acrostic poem is to be found at the end of Lewis Carrolls 1871 book Alice Through the Looking Glass; each letter of the name Alice Pleasance Liddell begins a new line in the poem about childhood innocence.

Word SquaresIf the first acrostics appeared in the Old Testament, word squares were not far behind. One of the most well known is a Latin word square from about 2000 years ago:

S A T O R A R E P O T E N E T O P E R A R O T A S

This word square is called a 5-by-5 symmetric word square because there are five words that can be read either down or across. The words TENET, OPERA, and ROTAS will be familiar to speakers of lan-guages descended from Latin. SATOR is a Latin word

for planter or creator. AREPO is a contentious word; it can be assumed that it was at some time used in Latin. This particular word square is unique in another waySATOR reversed is ROTAS, AREPO is OPERA reversed, and TENET is palindromic (reads the same forward and backward).

Below is an example of an ordinary symmetrical 4-by-4 word square using English words

B A S E A W A Y S A L E E Y E S

Many 5-by-5 and 6-by-6 squares exist in English. There are even a few 9-by-9 word squares, though many of the constituent words are extremely unfamiliar.

Those with an interest in algebra will notice that symmetry in word squares is equivalent to symmetry in matrices. If one transposesswaps the rows and columnsa symmetrical word square, the resulting word square is the same as the original. A non-symmetrical word square does not have this property. A 4-by-4 double word square, like the one below, is not symmetrical. It is a double word square because it contains twice the number of words of a 4-by-4 symmetrical square, that is, eight:

D A R T O B O E C L A M K E M P

CrosswordsWord squares can be entertaining in themselves. How-ever, simply by expanding a word square onto a larger grid and using gaps to section long words into shorter ones, one can create a puzzle of an altogether different kind. By doing so, puzzle creator Arthur Wynne turned the largely esoteric practice of crafting word squares into a puzzle for the massesthe crossword.

The first published crossword appeared in December 1913 in the newspaper New York World. Wynne wrote definitions for each of the words he had used to complete a diamond-shaped grid, and it was up to the solvers of the newspapers puzzle page to fill in the blanks.

Wynnes grid was almost fully checked, which means that most letters were part of two wordsa white square

Acrostics, Word Squares, and Crosswords

is unchecked when it is part of only one word. In U.S. crosswords, it remains the norm to have very heavilyif not fullychecked grids. For other crossword types, particularly cryptic crosswords, grids may be only 50% to 60% checked. Having a fully checked grid means that it is possible to complete the crossword by entering only the across (or down) words. As the number of unchecked squares increases, however, the ability to build on ones correct answers decreases. Most crosswords have a 15-by-15 grid and twofold rotational symmetry (they look the same after 180 degrees of rotation), but differences in the number of checked squares can produce as many as 80 words or as few as 30.

PROVERB, a computer program designed to solve crosswords, relies on the heavily checked nature of American-style grids. Computer scientist Michael Littman and others report that PROVERB averaged more than 95% correct answers in less than 15 minutes per puzzle on a sample of 370 puzzles. This result is better than average human solvers but not better than the best. If nothing else, the complexity of the PROVERB program serves to highlight the vast computing power humans naturally possess.

Instinctively, many people may not be aware that the five most frequently used letters in the English language are E, T, A, O, and I. Crosswords setters (and PROV-ERB), on the other hand, are acutely aware of this and aim to use letters in their longer words that will be easy to intersect with the shorter ones. It is therefore worth bearing in mind that, for example, Erie and Taoist will appear in crosswords much more often than jazz and Quixote. Incidentally, the five least frequently used letters are K, J, X, Z, and Q.

Estimates suggest that fewer than 100 people con-struct crossword puzzles for a living in the United States. Mathematician Byron Walden has been called one of the best by a New York Times crossword editor. For some, he may be most well known for writing the puzzle that was used in the championship round of the Ameri-can Crossword Puzzle Tournament, later featured in the film Wordplay. He has also analyzed and given talks on symmetry and patterns associated with conventional crossword construction, with the aim of helping people become more skilled puzzle solvers.

Mathematician Kiran Kedlaya is also a well-known puzzle solver and creator. He believes that the brain processes required for computer science, mathemat-ics, music, and crossword puzzles are similar, and he

pursues all of these activities professionally and recre-ationally. One puzzle he created was published on the well-known New York Times crossword page, and he regularly contributes mathematics puzzles to compe-titions like the USA Mathematical Olympiad. He has been quoted as saying, Its important to tell kids who are interested in math as a career that there are many venues to do it, not just in the academic area within math departments.

Further ReadingBalfour, Sandy. Pretty Girl in Crimson Rose (8).

Sirlingshire, UK: Palimpset Book Production, 2003.Littman, M., et al. A Probabilistic Approach to Solving

Crossword Puzzles. Artificial Intelligence 134 (2002).MacNutt, Derrick Somerset. Ximenes on the Art of the

Crossword. London: Methuen & Co., 1966.

Eoin OConnell

See Also: Literature; Poetry; Puzzles; Religious Writings; Sudoku.

ActorsSee Writers, Producers, and Actors

Addition and Subtraction Category: History and Development of Curricular Concepts. Fields of Study: Communication; Connections; Number and Operations; Representations.Summary: Addition and subtraction are binary mathematical operations, each the inverse of the other, and are among the oldest mathematical concepts.

Addition can be thought of as a process of accumulation. For example, if a flock of 3 sheep is joined with a flock

Addition and Subtraction

of 4 sheep, the combined flock will have 7 sheep. Thus, 7 is the sum that results from the addition of the numbers 3 and 4. This can be written as 3 + 4 = 7 where the sign + is read plus and the sign = is read equals. Both 3 and 4 are called addends. Addition is commutative; that is, the order of the addends is irrelevant to how the sum is formed. Subtraction finds the remainder after a quantity is diminished by a certain amount. If from a flock containing 5 sheep, 3 sheep are removed, then 2 sheep remain. In this example, 5 is the minu-end, 3 is the subtrahend, and 2 is the remainder or difference. This can be written as 5 3 = 2 where is read minus. Subtraction is not commutative and therefore the ordering of the minuend and subtrahend affects the result: 5 3 = 2 , but 3 5 = 2.

The concept of addition can be extended to have meaning for fractions, negative numbers, real num-bers, measurements, and other mathematical entities. The algorithms used for computing the sum or differ-ence, some of which have been taught for millennia, ultimately depend on the representation used for the numbers. For example, the approach used for adding Roman numerals is different from that used to add Hindu-Arabic numbers. Computers perform subtrac-tion using the same circuits they use for addition.

History and Development of Addition and SubtractionHuman beings ability to add and subtract small whole numbers is probably innate. Some of the ear-liest descriptions of techniques for handling large numbers come from ancient China during the War-ring States period (475221 b.c.e.), when arithmetic operations were performed by manipulating rods on a flat surface that was partitioned by vertical and horizontal lines. The numbers were represented by a positional base-10 system. Some scholars believe that this systemafter moving westward through India and the Islamic Empirebecame the modern system of representing numbers.

The Greeks in the fifth century b.c.e., in addition to using a complex ciphered system for representing numbers, used a system that is very similar to Roman numerals. It is possible that the Greeks performed arithmetic operations by manipulating small stones on a large, flat surface partitioned by lines. A similar stone tablet was found on the island of Salamis in the 1800s and is believed to date from the fourth century b.c.e.

The word calculate was derived from the Latin word for little stone.

The Romans had arithmetic devices similar in appearance to the typical Chinese abacus. It is difficult to use modern paper-and-pencil techniques for adding and subtracting Roman numerals (with I as one, II as two, V as five, X as ten, L as fifty, C as one hundred, D as five hundred, M as one thousand)but it worked well in its time, since it was devised for use with an abacus.

During the Middle Ages, counting boards were used to perform arithmetic. A counting board con-sisted of a series of actual or virtual horizontal lines that were labeled from the bottom by I, X, C, M, and so on. The system borrowed the symbols used for core numbers from the Roman system. The spaces between the lines were labeled starting from the bottom by V, L, and D. A number like MMDCCXXXVIIII (2739) would be represented by placing the appropriate number of counters on each line. The line labeled M would have 2 counters (for 2000, or two thousands). The space just below, labeled D, would have 1 coun-ter (500, or one five-hundreds); the line labeled C, 2 counters (200, or two hundreds); the space labeled L, 0 counters; the line labeled X, 3 counters (for 30, or three tens); the line labeled V, 1 counter (5); and the line labeled I, 4 counters (4, or four ones). The total of all these numbers is 2739. Note that accountants used VIIII (denoting five plus four) to represent 9, whereas stonemasons used IX(denoting 10 less 1). To compute the sum MMDCCXXXVIIII + MCLXI, a person would simply transcribe the numbers to the counting board and then combine the counters fol-lowing rules of carrying to ensure that no more than 4 counters were on any line and 1 counter on any space. This representation was then easily transcribed back into Roman numerals.

Many early books on arithmetic claim that this method of performing arithmetic was especially pre-ferred by women, who at times had the responsibility for keeping the books for small family businesses. Hindu-Arabic numerals and paper-and-pencil methods for performing arithmetic began to appear in Europe in the twelfth century and replaced Roman numerals and the counting board by the nineteenth century.

Two Methods for Subtracting by HandTwo popular methods for handling borrowing that are taught today are shown below. The method shown

Addition and Subtraction

in the figure below on the left is popular in Italy, Eng-land, and the United States, while the one on the right is popular in Spain, France, and parts of Latin America. The example is to compute 30471964. Starting with the method on the left, first begin with the rightmost column and subtract 4 from 7. Write the result, 3, below the 4. Moving one column to the left, try to subtract 6 from 4, which cannot be done without using negative numbers. The method is thus to attempt to borrow 1 from 0, which is the digit to the left of the 4. Again, this cannot be done without using negative numbers. Therefore, the method is to borrow 1 from 3, which is the digit to the left of the 0 resulting in crossing out the 3 and replacing it with a 2. Then the zero becomes a 10, and it in turn can be replaced by a 9 so the borrowed 1 can be placed in front of the 4 to make it 14. Now, one can subtract 6 from 14 to get 8, which is written below the 6. Moving left to the next column, one can subtract 9 from 9 to get a 0, which is written below the 9. Finally, 1 is subtracted from 2 to get a 1, which is written below.

3 0 4 71 9 6 4

1 0 8 3

3 0 4 71 9 6 4

1 0 8 3

2 9 1

1 1

To solve the problem using subtraction with carry, use the example on the right. The carrying numbers (the small 1s) affect the numbers on a diagonal, as shown in the example. The number 1 adds 10 to the integer in the top row and adds 1 to the integer in the bottom row. Starting from the rightmost column, 4 is subtracted from 7, resulting in 3, which is written below. Then, try to subtract 6 from 4, which cannot be done, so insert a small 1 to the left of the space between the 4 and the 6. This is interpreted to mean that the 4 has become 14. Subtract 6 from 14 and record the answer, 8, below. Move left to the next column con-taining 0 and 9. The small 1, written above and to the right of the 9, is added to the 9 to get 10. Attempt to subtract the 10 from the 0 above, which cannot be done. Instead, write a small 1 just to the left of the space between the 0 and 9, and interpret this to mean that the 0 has become a 10. Now, 10 minus 10 is 0, which is written below. Move left to the next column. The small 1, written above and to the right of the 1, is added to the 1 giving 2, which is subtracted from 3 resulting in 1, which is written below.

Adding and Subtracting on a ComputerAt the most basic level, whole numbers are represented in a computer in base-two by a sequence of the binary states Hi and Lo interpreted as 1 and 0. The circuits that perform addition are implemented by sequences of logi-cal gates. Typically a 1 in the leftmost bit indicates that the number is negative, with the remaining bits indicat-ing the magnitude of the number. Subtraction can be performed by the same circuits that perform addition. Two popular approaches are designated as ones com-plement and twos complement. Ones complement can best be explained by performing subtraction in base-10 using nines complement. Assume a computation of 30471964. To find the nines complement of 1964, subtract each digit from 9 to obtain 8035. This is added to 3047 resulting in 11,082. The leftmost 1 is viewed as a carry and brought around and added to the right-most digit in an operation called end-around carry to obtain the final result: 1083.

Generalizing Addition and SubtractionThe sum of two fractions a/b and c/d is defined to be

ad + bc

bd.

The sum of irrational numbers (numbers that can-not be represented as fractions of whole numbers) can be approximated only by adding their approximating rationals. The exact sum of two irrational lengths, a and b, can be found exactly using geometry by first extending the segment representing a sufficiently on one end so that the length b can be marked off from that end with a compass.

Addition can be generalized to other mathematical objects, such as complex numbers and matrices. One of these objects, typically called the additive identity and denoted by 0, has the property such that if a is any object then the sum of 0 and a is a. The addi-tive reciprocal of an object a is denoted by a and is defined to the object so that the sum a + a( ) is 0. The difference a b is defined to be a + b( ).

Further ReadingFlegg, G. Numbers: Their History and Meaning. New

York: Schocken Books, 1983.Karpinski, L. C. The History of Arithmetic. New York:

Russell & Russell, 1965.

Addition and Subtraction

Pullan, J. M. The History of the Abacus. New York: F. A. Praeger, 1969.

Rafiquzzaman, M. Fundamentals of Digital Logic and Microcomputer Design. Hoboken, NJ: Wiley, 2005.

Yong, L. L., and A. T. Se. Fleeting Footsteps. Singapore: Word Scientific Publishers, 2004.

Carl R. SeaquistCatherine C. Galley

See Also: Multiplication and Division; Number and Operations; Number and Operations in Society.

AdvertisingCategory: Business, Economics, and Marketing.Fields of Study: Algebra; Data Analysis and Probability; Number and Operations.Summary: Mathematics is used to weigh the costs and gains of advertising and to profile and target consumers.

Advertising delivers product information from suppli-ers to consumerssuppliers may be manufacturers, hospitals, software developers, educatorsand is criti-cal to the success of a business in marketing develop-ment. Advertising media may be traditional (such as television, newspapers, and posters) or technological (via Internet and e-mail), as well as commercial (to sell products for profit) or noncommercial (in politi-cal campaigns or for religious purposes). The annual advertising cost in the United States amounts to more than $100 billion.

Advertising includes two stages: the planning stage for marketing strategies, whose goal is business devel-opment, and the analysis stage of cost analysis involved with the forms and the contents of communication between suppliers and potential customers. Math-ematics and statistics play critical roles in both stages of advertising.

Market SharesIn the planning stage, the analysis of market shares for advertising necessitates matrix operations and multivariate probability inequalities to portray the

dynamics of market shares over time. The follow-ing is an example of matrix operations, which bridge advertising with market shares. Consider the market shares of General Motors (GM) and Ford in the U.S. automobile industry. Assume that the current market shares distribute as follows:

General Motors: 21% Ford: 17% Other Manufacturers: 62%

If GM starts an advertising campaign with the goal of increasing the market share to 29% in three years, GM may count on customers to switch from Ford or other manufacturers to GM. However, in reality, some of the GM customers may switch to Ford or to other manufacturers.

Let a1, a

2, a

3 be the percentages of original GM users

who, at the end of the advertising campaign, remain with GM, who switch to Ford, and who switch to other manufacturers, respectively. Let b

1, b

2, b

3 be the

percentages of original Ford users who switch to GM, who remain with Ford, and who switch to other man-ufacturers, respectively. Let c

1, c

2, c

3 be the percentages

of the other customers who switch to GM, who switch to Ford, and who remain with their manufacturers, respectively. Then, the market shares x

GM, x

Ford, and

xOthers

at the end of the three years are determined by the following simple matrix equation:

a b c

a b c

a b c

1 1 1

2 2 2

3 3 3

21

17

62

%%%

x

x

x

GM

Ford

Others

=

.

If GM intends to increase xGM

to 29%, GM should advertise specifically to different groups of customers. This is mathematically equivalent to manipulating the elements in the 3 3 matrix above within plausible ranges of the elements.

The foregoing scenario is a simplified example to illustrate the role of matrix operations in advertising. In reality, the story is more complex. For example, the 3 3 matrix above will become an n n matrix, where n is the number of competing suppliers in the market. Also, the stochastic feature of the supply-demand mar-ket, the market shares, and the corresponding elements

0 Advertising

for the n n matrix change constantly under the influ-ence of the advertising campaign.

Thus, it is more appropriate to treat the market shares as a vector consisting of random variables. In this case, one of the convenient approaches to evaluating the market shares is the method of multivariate probabil-ity inequalities in conjunction with the construction of Hamilton-type circ