Table Of Contenti ©2003TheTeachingCompanyLimitedPartnership ©2003TheTeachingCompanyLimitedPartnership ii
Edward B. Burger,PH.D. MichaelStarbird, Ph.D.
ProfessorofMathematicsandChair,
UniversityDistinguished TeachingProfessor in Mathematics,
DepartmentofMathematicsandStatistics,WilliamsCollege
TheUniversityofTexasatAustin
EdwardBurgerisProfessorofMathematicsandChairoftheDepartmentof
Mathematics and Statistics at Williams College. He graduated Summa Michael Starbird is a Professor of Mathematics and is a University
CumLaude from Connecticut College in 1985 where he earned a DistinguishedTeachingProfessoratTheUniversityofTexasatAustin.He
B.A. with distinctioninmathematics.HereceivedhisPh.D.inmathematics receivedhisB.A.degreefromPomonaCollegein1970andhisPh.D.degreein
fromTheUniversityofTexasatAustinin1990.In1990,hejoinedthefacultyat mathematicsfromtheUniversityofWisconsin,Madisonin1974.In1974,he
theMathematics Departmentat Williams College. For theacademic year joinedthefacultyoftheDepartmentofMathematicsofTheUniversityofTexas
19901991,hewasaPostdoctoralFellowattheUniversityofWaterlooin atAustinwherehehasstayedexceptforleavesasaVisitingMemberofthe
Canada.Duringhissabbaticalyears1994-1995;1998-1999;and2002-2003he InstituteforAdvancedStudyinPrinceton,NewJersey;aVisitingAssociate
wastheStanislawM.UlamVisitingProfessorofMathematicsattheUniversity ProfessorattheUniversityofCalifornia,SanDiego;andaMemberofthe
ofColoradoatBoulder. TechnicalStaffattheJetPropulsionLaboratoryinPasadena,California.
In1987,ProfessorBurgerreceivedtheLeFevere.TeachingAwardatThe HeservedasAssociateDeanintheCollegeofNaturalSciencesatThe
UniversityofTexasatAustin.Hereceived:the2000NortheasternSectionofthe UniversityofTexasatAustinfrom1989to1997.Heisamemberofthe
MathematicalAssociationofAmericaAwardforDistinguishedCollegeor AcademyofDistinguishedTeachersatUT.Hehaswonmanyteachingawards
UniversityTeachingofMathematics,andin2001,hereceivedtheMathematical includingaMinnieStevensPiperProfessorship,whichisawardedeachyearto
Association ofAmerica Deborah and FranklinTepper HaimoNational tenprofessorsfromanysubjectatanycollegeoruniversityinthestateofTexas;
AwardforDistinguishedCollegeorUniversityTeachingofMathematics.In the inaugural awarding of the Dad's Association Centennial Teaching
2003,hereceivedtheResidenceLifeAcademicTeachingAwardatthe Fellowship;theExcellenceAwardfromtheEyesofTexas;thePresident's
UniversityofColoradoat Boulder. Burger was named the2001-2003 AssociatesTeachingExcellenceAward;theJeanHollowayAwardforTeaching
GeorgePolya Lecturer bytheMathematicalAssociation of America. He Excellence,whichistheoldestteachingawardatUTandisawardedtoone
was also the2001 Genevieve W. GoreDistinguished Residentat professoreachyear;theChadOliverPlanIITeachingAward,whichisstudent-
Westminster Collegeandthe 2001 Ceciland IdaGreenHonorsProfessorat selectedandawardedeachyeartooneprofessorinthePlan11liberalartshonors
TexasChristianUniversity. program;andtheFriarSocietyCentennialTeachingFellowship,whichis
awardedtooneprofessoratUTannuallyandincludesthelargestmonetary
Burger'sresearchinterests areinnumbertheory,andheis theauthor ofover
teachingprizegivenatUT.Also,in1989,hewastheRecreationalSportsSuper
25papersappearinginscholarlyjournals.Heistheauthorofseveralbooks.
RacquetsChampion.
TogetherwithMichaelStarbird,heco-authoredTheHeartofMathematics:An
invitationtoeffectivethinkingwhichwona2001RobertW.HamiltonBook Hismathematicalresearchisinthefieldoftopology. Herecentlyservedasa
Award.HealsopublishedExploringtheNumberJungle:Ajourneyinto Member-at-LargeoftheCounciloftheAmericanMathematicalSociety.
diophantineanalysis,and,withRobertTubbs,co-authoredMaking
Heisinterestedinbringingauthenticunderstandingofsignificantideasin
TranscendenceTransparent:Anintuitiveapproachtoclassicaltranscendental
mathematicstopeoplewhoarenotnecessarilymathematicallyoriented.Hehas
number theory. He has also authored five virtual video CD-ROM textbooks
developedandtaughtanacclaimedclassthatpresentshigher-levelmathematics
withThinkwell.BurgerhasservedasChaironvariousnationalprogram
toliberalartsstudents.Recently,co-authorEdwardB.BurgerandhewroteThe
committees for theMathematical Association ofAmerica; serves as
HeartofMathematics:Aninvitationtoeffectivethinkingwhichwona2001
AssociateEditoroftheAmericanMathematicalMonthly andarefereeformany
RobertW.HamiltonBookAward.AreviewintheAmericanMathematical
mathematical journals;and was amember oftheCommittee on Committees
Monthly(June-July,2001)ofthisbooksaidamongmuchmore,"Thisisvery
for theAmericanMathematicalSociety.
possiblythebest'mathematicsforthenon-mathematician'bookthatIhave
ProfessorBurgerisanotedspeakerandhasgivenover300lecturesaroundthe seen—andthatincludespopular(non-textbook)booksthatonewouldfindina
world.Hislecturesrangefromkeynoteaddressesatinternationalmathematical generalbookstore."Helovestoseerealpeoplefindtheintrigueandfascination
conferencesinCanada,France,Hungary,Japan,andtheUnitedStates;to thatmathematicscanbring.
mathematicalcolloquiaandseminarsatcollegesanduniversities;topresentations
HispreviousTeachingCompanycourseisentitledChangeandMotion:
atprimaryandsecondaryschools;toentertainingperformancesforgeneral
CalculusMadeClear.
audiences;totelevisionandradioappearancesincludingNationalPublicRadio.
iii ©2003TheTeachingCompanyLimitedPartnership ©2003TheTeachingCompanyLimitedPartnership iv
Thiscoursewasmadepossibleincooperationwith:
KeyCollegePublishing,Emeryville,California Table of Contents
www.keycollege.com
The Joy ofThinking:
Publishersof
TheHeartofMathematics:Aninvitationtoeffectivethinking The Beautyand Powerof Classical Mathematical Ideas
byEdwardB.BurgerandMichaelStarbird Part I
http://www.heartofmath.com
ProfessorBiography.......................................................................................ii
CourseScope..................................................................................................1
Introduction:
LectureOne GreatIdeasThatBringOurWorldintoFocus......... 4
NumberContemplation:
LectureTwo HowMany?CountingSurprises..............................9
LectureThree Fermat'sLastTheoremandtheAllureof
Number.................................................................15
LectureFour PiningforNature'sNumbers................................. 20
LectureFive SizinguptheFibonacciNumbers..........................24
The VisualWorldof Geometry:
LectureSix TheSexiestRectangle...........................................29
LectureSeven TheHiddenBeautyoftheGoldenRectangle........ 33
LectureEight ThePythagoreanTheoremand
GeometryofEllipses............................................36
LectureNine Not-so-PlatonicRelationshipsinPlatonicSolids... 40
LectureTen HuntingforaSixthPlatonicSolid.........................45
LectureEleven IsThereaFourthDimension?CanWeSeeIt?.......50
LectureTwelve TheInvisibleArtoftheFourthDimension.............54
Timeline........................................................................................................58
Glossary........................................................................................................62
BiographicalNotes.......................................................................................66
Bibliography.................................................................................................72
v ©2003TheTeachingCompanyLimitedPartnership ©2003TheTeachingCompanyLimitedPartnership vi
The Joy ofThinking: Ourexplorationofmathematicalmagicandmajestywillbeginmost
appropriatelywiththecontemplationofthenotionofnumber.Simplecounting
The Beautyand Powerof Classical MathematicalIdeas
willleadustosurprisingoutcomesandthediscoveryofanincrediblepatternof
ScopeFun,joy,pleasure,aesthetics,intrigue,beauty,richness,texture,power, numbersinnature.Unexpectedly,exploringthisnaturalnumberpatternwilltake
andlifearewordsnotnormallyassociatedwithmathematics.Inthiscourse, usdirectlytoageometricalrealmthatinfluencestheartsandournotionof
however,wewilldiscovermathematicsasanartisticandcreativerealmthat beauty.
containssomeofthegreatestideasofhumanhistory—ideasthathaveshaped Withouraestheticsasourguide,wewilldiscovernewmathematicalinsights
cultures.Wewillexplorethefourthdimension,coincidences,fractals,theallure througharchitecture,art,andeveninsnails.Issuesofgraceandelegance
ofnumberandgeometry,andbringtheseweightynotionsbackdowntoearth. naturallygiverisetoanexplorationoftheideaofsymmetry.Lookingforthe
Everylecturewilldevelopincredibleideasbystartingfromcommonplace mostsymmetricalobjectsisanage-oldquestwitharichreward.Wewill
observationsandbuildingfromthere.Bycountingthespiralsontheprickly appreciatethesewondrousobjectsnotonlyfortheirindividualcharacter,butwe
facadesofpineapplesandpinecones,wewilldiscoveranumberpatternthathas willalsoseetheirintimate,dualrelationshipswithoneanother.Eventhe
alifeofitsownandexpressesitselfwithinpaintings,architecture,andmusic. mundanesoccerballwillbecomearicherobjectwhenweseeitina
Bytracingtheedgeofatwistedstripofpaper,wewilldevelopinsightsintothe mathematicallight.
shapeofouruniverse.Byhavingmonkeysrandomlytypingonakeyboardto Thefourthdimension,atfirst,appearsinexplicableandinaccessible.However
createHamlet,wewilldiscovertheunderpinningsofmolecularmotion.Thusin herewewillapplythelessonsofmathematicalthinkingtopenetratethehaze
thiscourse,sometimesfrivolous,inauspiciousbeginningsquicklyleadusto andlookintoauniversewiththeeyesofthemind.Indeed,developing
fundamentalinsightsintoourlivesandourworld. intuitionaboutthisinvisiblerealmisarichexampleofthepowerofanalogy
Mathematicalthinkingleadsnotonlytoinsightsaboutoureverydaylivesand andabstraction.Theabstractnotionsofthefourthdimensioncaneventouch
everydayworld,butalsopointsustoworldsfarbeyondourown.Oneofthe ourheartsandmindsthroughthebeautifulworksofarttheyinspired.
joysoflifecomesfromplayingwithideasthatarenotboundedbymerereality. Aswewillsee,seekingbeautyandsurpriseneednottakeusoutsideour
Thereisnoreasonwhyourmindscannotgowherenomindhasgonebeforeand world.Asimplecreationfromastripofpaperwillofferusaglimpseintoa
livetotelltheexcitingtale.Amathematicalpointofviewcantakeusfarand worldofbendingandtwistingspace.Ourexplorationswilltakeusfarbeyond
bringusbackhomewithanewperspectiveoneverything. ourownworldandintuitionaswerethinkthebasicnotionsofinsideand
Whetherviewersalreadyenjoymathematicsorfranklyhatemathematics,these outside.
excursionswillenableeveryonetobringgreatmathematicalideastolife.The Otherobjectsofbeautycanexistcompletelyinourminds,butcanonlypartially
lectureswillcreatealivelyandentertainingintellectualtour-de-forcethatopens berenderedowingtothelimitationsofreality.Wewilldiscoverthatfractal
eyesandopensminds. picturesareliterallyinfinitelyintricate.Theyarisefromrepeatingasimple
Oneoftheobstaclestraditionallyassociatedwithmathematicsisitshierarchical processinfinitelyoften,andtheyseemtocapturethecomplexityofnature.
structure.Onegoaloftheselecturesistodemonstratethatmathematicalideas Repetitionisattheheartoffractalsandperhapsattheheartofournaturalworld.
arewithinthereachofpeoplewhomaynothaveanextensivebackgroundin Surprisingly,theseincrediblevistaswillthencarryusfarafieldtothebirthof
mathematics.Theuniverseofmathematicscontainssomeofthegreatestideasof computingmachines.Infact,thesimpleactoffoldingasheetofpaperwilloffer
humankind—ideascomparabletotheworksofShakespeare,Plato,and usabridgefromthewondrouspatternsoffractalstotheveryfoundationsofall
Michelangelo.Thesemathematicalideashelpedshapehistory,andtheycanadd programming.
texture,beauty,andwondertoourlives.
1 ©2003TheTeachingCompanyLimitedPartnership ©2003TheTeachingCompanyLimitedPartnership 2
Afinalthemeofthiscourseisthatmuchofourworldexperienceinvolves Lecture One
chanceandrandomness.Althoughcounterintuitiveatfirst,wewillseethat
Great IdeasThat BringOurWorldintoFocus
amazingcoincidencesarenearlycertaintohappentoallofus.Afterthinking
aboutchanceandrandomness,surprisesappearinawholenewlight.
EdwardB.Burger, Ph.D.andMichaelStarbird,Ph.D.
Randomnessisapowerfulforce—itpermitsustodevelopafeelformeasuring
ourexpectationsaboutfutureeventswhoseoutcomesarebasedonchance.We
willbeabletoseelotteries,insurance,andothergamesofchancewithamore
Scope:Whenwesetouttounderstandourworldmoredeeply,westartwith
criticalandinformedeye
ourexperienceandtrytoseeitwithgreaterdetailandnuance.Abasic
Aswewillseethroughthecourse,themathematicsofferedherewillnot waytorefineourworldviewistobecomemorepreciseindescribing
remindusofschool—therewillhenoformulas;noproblems;noequations;no whatwesee.Wecanmovefromaqualitativeestimateofamountto
techniques;nodrills;andno,therewon'tbeanytests.Somepeoplemightnot measuringourworldbycounting.Wecandescribeandexplorethe
evenwanttocallitmath,butwewillexperienceawayofthinkingthatopens shapeswesee.Wecansensiblyreasonabouttheuncertainandthe
doors,opensminds,andleavesussmilingwhileponderingsomeofthegreatest unknown. Andwecangobeyondconcreteexperiencebyexpanding
conceptseverconceived. ourinnerworldtoincludeabstractideas.Mathematicscontainsideas
andapproachesthatcrystallizeandextendourreach.
Oneofthegreatfeaturesaboutmathematicsisthatithasanendlessfrontier.
Thefartherwetravel,themoreweseeovertheemerginghorizon.Themorewe Theclassicaltheoriesofnumbers,geometry,topology,fractals,and
discover, themoreweunderstandwhatwe'vealreadyseen,andthemorewesee probabilityoffertoolstohelpusbetterappreciateandunderstandour
ahead.Deepideastrulyarewithinthereachofusall.Howmanymoreideasare world.Eachtopichassurprisesofitsownandeachilluminatesthe
thereforyoutoexploreandenjoy?Howlongisyourlife? worldthroughitsinsightsandthroughpotentstrategiesofthoughtthat
transcendmathematicalissues.Mathematicsiseverywhere andcan
improveourlivesifweembraceit.
Outline
I. Oneofthebasicchallengesofthinking,bothforindividualsandfor
society,istounderstandourworldmoredeeply.Lessonsfrom
mathematicscanhelpusrefineandexpandourappreciationoftheworld.
Todeepenourexperienceoftheworld,weanalyzeitinseveralways.
A. Wemeasureitandcountthingsinit.Wequantifytheworld.
B. Weidentifyanddescribeshapesandformsthatmakeupourvisualand
physicalimpressionsoftheworld.
C. Wedealwithuncertaintyandtheunknown.
II. Ineachofthesebroadviewsoftheworld,mathematicsbringsusthe
lightofrefinement.
A. Formeasuringandcounting,wehavedevelopedtheconcept
ofnumber.
B. Tounderstandshapes,wehavedevelopedthemathematicalworld
ofgeometry.
C. Todealwiththeindeterminate,mathematicsofferstheconceptof
probabilitytoprovideuswithaquantitativesenseofour
uncertainty.
3 ©2003TheTeachingCompanyLimitedPartnership ©2003TheTeachingCompanyLimitedPartnership 4
D. Thiscoursewillfollowthesethemesinthreegroupsoflectures 3. Naturalnumbersprovidethetoolsforthemostbasicmeansof
treating,respectively,number,geometry,andprobability.Eachof quantifyingtheworld,namely,counting.Wewillfindsome
thelectureswillpresentintriguingissuesfromeachtopic. startlingpossibilitiesinthismundanepractice.
III.Oneofthesourcesofthepowerofmathematicalthinkingisthatitallows 4. Naturalnumbershavecapturedtheimaginationsofpeople
ustogobeyondourconcreteexperienceoftheworldintoabstractideas. fromaroundtheworldandthroughouthistory.Thestudyof
naturalnumbersbeganthousandsofyearsagoandcontinues
A. Oneofthestandardprocessesofmathematicaldiscoveryistoidentify
withincreasingvigortothisday.
essentialfeaturesofthephysicalworldandreasonabouttheminthe
abstract.Thisprocessoftenleadstonewideasintheabstractthatare, 5. Totheeyeofthemathematician,individualnumbershavetheir
inturn,reflectedinthephysicalworld.Ourworldseemstoconformto ownpersonalitiesanduniquecharacteristicsthatdifferentiateeach
thesamelogicalcoherencethatunderliesabstractmathematical numberfromtheothers.
reasoning.
B. LecturesFourandFivewillspanthespectrumfromnumbertonature.
B. Ontheotherhand,ifweviewmathematicssolelyasatoolfor Wewilllearnthatnumbershelpusuncoverhiddenpatternsandbring
understandingourworld,thenwemissoneofitsmostfundamental oureverydayworldintofocus.
features—itsaestheticappeal.
1. Often, we see beauty in nature because we subconsciously sense
1. Thebeautyofmathematicsarisesfromitsconceptualelegance. hidden structure and order. That order itself has an independent
richnessandbeauty.
2. Atfirst,thisartisticsideofmathematicsmayseemforeign,butwe
hopethatyouwillcometoviewmathematicalideasas 2. Hiddenwithinthespiralpatternofapinecone,forexample,we
aestheticallypleasing. willfindtheFibonaccinumbers.Thissimplesequenceleadsto
bothmathematicalandaestheticinsights.
3. Thefirstplacethatwethinkaboutfindingaestheticsin
mathematicsisingeometry.Wewillalsodiscoverrelationships 3. Wewillseethatapowerfulmethodofdiscoveringanewideais
amongnumbersandamazinginsightsintocoincidenceand toisolatethepattern,thenexplorethepatterninitsownright.
probabilitythatgiveusthesamesenseofbeautythatwegetfroma
4. Thatjourneyintotheabstractcaninformourunderstandingof
geometricconcept.
theworld,whichissometimesfuzzy,eventhoughwedon't
4. Weseeaneleganceofproportionandconnectionamongideas realizethisfact.
themselves,evenwhentheyhavenogeometricappeal.
C. Thenextsetoflectureswillmovefromthenumberworldintothe
C. Anotherattractivefeatureofmathematicsisitstimelessness. geometricworld.
Mathematicalideasconnectushistoricallywithpeoplefromancient
1. TheaestheticappealofboththeParthenonandamusical
timestothepresent.
compositionbyDebussymightbeexplainedbyasingle
IV. Towhetyourappetitesfortheideastocome,wewillbrieflydescribe underlyingmathematicalrelationship.
theoutlineofthiscourse.
2. LecturesSixandSevenopenourexplorationofthegeometric
A. LecturesTwoandThreewillintroducethestudyofnumbers. universebyconsideringissuesofbeauty.
1. Lifeisfullofnumbers.Fromthemomentababyisborn,its 3. Wewillbeginwithabasicquestionofgraceandproportions:
parentsrecordthedate,thetime,itslength,itsweight,andso Whatarethedimensionsofthemostpleasingrectangle?The
on. answerleadsustothe GoldenRectangle,famedforitsrecurring
roleinart,architecture,andmusic.
2. Inthesamewaythatnumbersplaysuchafundamentalroleinour
dailylives,theyarealsofoundationalintherealmof 4. Wemightalsoaskourselveswhetheraestheticsinforms
mathematics. mathematicalthinkingormathematicalthinkinginfluences
aesthetics.
5 ©2003TheTeachingCompanyLimitedPartnership ©2003TheTeachingCompanyLimitedPartnership 6
D. Wethenturntootherexamplesofclassicalbeautyin LecturesEight, I. Thethirdsegmentofthecoursetakesonthechallengeofprobability.
Nine,andTen,asweexplorethegeometrythatwelearnedinour
1. InLecturesNineteenandTwenty,wewillseehowtodescribe
youth.
quantitativelytheleveloflikelihoodthatanuncertainfutureevent
1. Ifwehadtoselectjustonetheoremtobestrepresentallof mayhave.
mathematics,wemightchoosethePythagoreantheorem.In
2. InLecturesTwenty-OnethroughTwenty-Three,we'lllearnhowto
LectureEight,we'llexaminethreeproofsofthistheoremthatspan
measurethevalueofanuncertainfuture,discoverthecertaintyin
aperiodfromancienttimestothelate19thcentury.
randomness,andexplainthereasonfortheomnipresentlife
2. Wealsoexamineconicsectionstoillustratecertainbasic experienceofcoincidences.
definitionsingeometry.
3. Probabilityandstatisticsenableustobetterunderstandourworld.
3. LecturesNineandTencontinuethisclassicalthemebyexploring
Theyarethemathematicalfoundationsforcommonsense,
theregularorPlatonicsolidswiththeirmanifoldsymmetry.
wisdom,andgoodjudgment.Wewillexplainhowtoplaceavalue
E. LecturesElevenandTwelvetakeustoaworldthatisvisibleonly ondecisionsconcerninginvestments,gamesofchance,andlife
throughtheeyeofthemind,thefourthdimension. insurance.
1. Theveryphraseconjuresupimagesofsciencefictionoreventhe 4. Finally,we'llseethatrandombehaviorandcoincidencesdooccur
supernatural.Itsounds,allatonce,eerie,romantic,andmystical. withpredictablefrequency.
2. Becausethefourthdimensionliesbeyondourdailyexperience, J. Theworldisfullofquestions,interest,andsurprises.Mathematical
explainingandunderstandingitrequiresustodevelopanintuition ideasandthemathematicalwayoflookingatthingscanopenoureyes
aboutaworldthatweareunabletosee.Nevertheless,that topreviouslyunobservedconnections,insights,andwonder.Wehope
understandinglieswithinourreach. toshowthejoy,thecreativity,theaestheticappeal,andthewonderof
someofthemostintriguingideaseverconceived.
3. We'lllearnthatexplorationsofunfamiliarrealmsoftenbeginby
delvingintothedepthsofthefamiliar. SuggestedReading:
F. In Lectures Thirteen and Fourteen, we change perspectives from the EdwardB.BurgerandMichaelStarbird,TheHeartofMathematics:
fourth dimension. Here, werelax the rigid structuresof both classical Aninvitationtoeffectivethinking,KeyCollegePublishing.
and abstractgeometry and construct twisted surfacesthat, amazingly,
QuestionstoConsider:
haveonlyoneside.
G. InLecturesFifteenandSixteen,we'llmovefromthesimplicityof 1. Do you think that mathematics contains ideas comparable in philosophical
foldingpapertothesubtleandsurprisingpatternsofinfinitefractalart and cultural significance to the best ideas and products in art, philosophy,
andtheconceptofautomatatheory.We'llclosetheseexplorations music,literature,andhistory?
withtherealizationthatthesimplepaper-foldingprocessisthekeyto 2. Doyouthinkthatthegreatestmathematicalideasaregenuinelyaccessible
unlockingthesecretsofthebeautifulfractalimageknownasthe andintelligibletopeoplewhodonothaveextensivemathematical
DragonCurve. education?
H. Havingexaminedsimplegeometricshapesandregularsolidstothis
pointinthecourse,weexplore,inLecturesSeventeenandEighteen,
irregularobjects,suchasclouds,rocks,andmountains.We'lllearn
thatnotallgeometricshapesareelegantbecauseoftheirsimplicity.
Fractalshaveinfinitecomplexityandtantalizeuswiththeirproperties
ofrepeatedself-similarity.
7 ©2003TheTeachingCompanyLimitedPartnership ©2003TheTeachingCompanyLimitedPartnership 8
Lecture Two A. Weunderstandtheideaofone,forexample,aswellastwo'scompany,
three'sacrowd,andsoon.Wecantakethisideaupto24hoursina
How Many?CountingSurprises
day,28daysinamonth,and365daysinayear.Thesenumbersare
MichaelStarbird,Ph.D. familiarbecausewedealwiththemallthetime.
Scope:Lifeisfullofnumbers.Themomentwewereborn,ourparentsnoted B. Butmostnumbersaremuchlessfamiliarbecauseoftheproperty
thateverynaturalnumberissucceededbyanothernaturalnumber—
thetime,ourweight,andourlengthand,mostimportant,countedour
infinitely.
toes.Numbersaccompanyusthroughoutourlives.Justasnumbers
playafundamentalroleinourdailylives,theyalsoplayafoundational 1. Todemonstratethefactthatmostnaturalnumbersarenotfamiliar,
roleintherealmofmathematics. wecanshowa500-digitnaturalnumberthatnohumanbeinghas
everseenbeforewatchingthislecture.Infact,noonehasseena
Webeginwiththenaturalnumbers(1,2,3,4,...)andwiththemost
numberwithinabillionofitorevenatrillionofit.
basicuseofnumbers—tocountthings.Soon,wefindstartling
possibilitiesinthismundanepracticewhenweestablishirrefutablythat 2. Howdoweknowthatnoonehaseverseenthisnumber?The
somewhereonEarth,therearetwopeoplewithpreciselythesame answeristhatthenumberofnumbersthatcanbemadewith500
numberofhairsontheirbodies.Thisinsightisnotbiological;itisa digitsissovastthatevenifeveryhumanbeingwerecounting
mathematicalcertaintybasedoncountingandlogic.Thecompelling numbersforeverysecondoftheageoftheuniverse,theywould
argumentforthisfactdisplaysitselfrepeatedlyinourdailylives,as stillnotevenbeclosetothisnumber.
wellasindeepmathematicaltreatises;hence,itisreferredtobyits
C. Ifweweretryingtounderstandnumbersbynamingthem,we
formalname—thePigeonholeprinciple.
wouldfindthatwewouldsoonrunoutofnames.
Outline
1. Thinkofthenumbersthatweknowthathavenames,suchas
I. Ourlectureserieshastwoparallelgoals:topresentsometrulyintriguing billion,trillion,orevenquadrillion.Thesenumbershavejustafew
ideasofmathematicsandtopointoutandcelebrateeffectivestrategiesof digits;theydon'tevencomeclosetohaving500digits.
thinkingthatarebeautifullyillustratedbythemathematicalwayof
2. Thenumbersfrom1-9haveonedigit.Wecannamenumbers
discoveringanddevelopingconcepts.
thatgoupto12or15digits,butthenwerunoutofnames.
A. Webeginwiththesimpleprocessofcounting,oneofthemost
3. These12-or15-digitnumbersaresosmallcomparedtothe
powerfulandfundamentalideaseverconceivedbyhumanbeings,and
500-digitnumberthattheyarealmostnothing.
theconceptofnaturalnumbers,whicharebuildingblocksbothfor
understandingmathematicsandforunderstandingourworld. D. Togetsomesenseofthevariousordersofmagnitudeandgive
meaningtolargenumbers,wecanassociatethesenumberswith
B. Theconceptofthecountingnumbers;theideathatforeachnatural
familiarexamplesfromoureverydayexperience.Forexample:
number,thereisanextnumber;andthenotionthatnumberscanbe
thoughtofindependentlyofobjectsareintellectualadvancesfor 1. Rightnow,thepopulationoftheEarthis6.3billion,onlya7-
understandingtheworld. digitnumber.
C. Onewaytobecomeabetterthinkeristoaskthequestion"How 2. Howmanystarsarethereinthenightsky?Intheyear150A.D.,
many?"inanydiscussion.Thatquestionforcesthediscussiontofocus Ptolemy counted 1,022 stars in the sky. In contrast, I was
onquantitativemeasuresofanyissue. brought up in the bright lights of the Los Angeles area, and I
neversawastarinthesky.
D. Thenumbersthatweusetomakesuchmeasuresarecalledthenatural
numbers,thatis,thenumbersthatstartwith1,2,3,4andgoon 3. Thenumberofmealsweeatinayearisabout1,000.The
forever. Themostbasicpropertyofnaturalnumbersisthattheydogo numberofhoursofclassroominstructioninastandardcollege
onforever;foreachnaturalnumber,thereisanextnumber. educationrangesfrom1,000to2,000.
II. First,let'stalkaboutnumberswithwhichwearefamiliarbyvirtueof 4. Thenumberofhoursofworkinayearisabout2,000.
understandingcollectionsthatcontainthem.
9 ©2003TheTeachingCompanyLimitedPartnership ©2003TheTeachingCompanyLimitedPartnership 10
III. Gettingintothehabitofestimatingquantitiesorfindingtheorderof E. Thenationaldebtisnow$6.6trillion.
magnitude(ageneral size)for collectionscanputvariousideasinto
1. Supposealegislatorpresentedabilltopaydownthenational
perspective.
debt,claimingthatthiscost-savingmeasurewouldreducethe
A. For example, the number of people in large cities is measured in debtby$1millionperhour.Howlongwouldittaketopayoffthe
smallnumbersofmillions.Thenumberofpixelsonahighresolution nationaldebt?
digital
2. Thereare24hoursineachdayand365daysintheyearso,ata
pictureisreferredtoasafewmegapixels(withmegameaning milliondollarsperhour,itwouldtakeabout660yearstopaydown
roughlyonemillion). thedebt.
B. Supposesomeoneofferedyou$1millionin$1billsifyoucould IV. Wecanfindseveralcommonplaceexamplestoillustratehowwecan
carryitaway.Willyoubeabletodosoandgetrich? thinkquantitatively.
1. Estimatingcanhelpusanswerthisquestion. A. Oneofthewayswecanunderstandhownumbersgrowistoconduct
asimpleexperimentinfoldingpaper.
2. Howbigwouldastackofonemillion$1billsbe?Wecancover
onepieceofpaperwithaboutfive$1bills.Tworeamsofpaper 1.Wecanaskwhatwouldhappenifwefoldedapieceofpaper50
stackedup,then,wouldaccountfor$5,000andwouldweigh times.Ifwefoldthepaperonce,itistwolayersthick;ifwemake
about9pounds. threefolds,thepaperisnoweightlayersthick. Continuingto
foldthepaper,weseethateachfolddoublesthenumberof
3. Onemilliondollarswouldbeequalto200timesourstackof
layersofthickness(fourfolds-16layers;fivefolds-32layers;6
tworeamsofpaperandwouldweighabout1,800pounds.
folds-64layers,andsoon).
C. Let'sthinkaboutbillionsbylookingatBillGates,president
2. Ultimately,afterwehavefoldedthepaper42times,its
ofMicrosoft.Oneyear,hispersonalwealthgrewby$20
thicknesswouldreachalmosttothemoonand,after51folds,
billion.
pastthesun.Thisstoryservesasanexampleofexponential
1. Ifheworked40hoursaweek,50weeksayear,thenhewould growth.
workapproximately2,000hoursayear.Ifheearned$20billion
B. Anotherexampleofexponentialgrowthrelatestopopulationgrowth.
ayear,hewasmakingabout$10millionanhour.
1. Duringthe20thcentury,theworldpopulationgrewataratethat
2. IfGatessawa$100billontheground,woulditbeworthhis
washigherthan1%ayear.Couldthatrateofgrowthbe
whiletopickitup,orshouldhejustkeepworking?
sustainedforlongperiodsoftime?
3. DividingtheS10millionGatesearnsinanhourbythe
2. Supposethatatyear0,theworldpopulationwas225million.
3,600secondsinanhour,wefindthatheearns5100every
0.036seconds.Heisprobablybetterofftojustkeep 3. Ifthepopulationgrewattherateof1.34%annually,after
working. 2,000years,thepopulationwouldbeabout100quintillion
people(1followedbytwenty0's).
D. Wehearagreatdealaboutoverpopulation,butifairandelbowroom
werenotanissue,wecouldputtheentireworldpopulationof 4. The earth has about two quadrillion square feet of land, so
humans,6.3billionpeople,intoacubicmile. dividingthenumberofpeoplebythenumberofsquarefeetshows
that, on average, every single square foot of land would have
1. Alinearmileis5,280feet;acubicmileis5,2803,whichis
about50,000peopleonit.
150billioncubicfeet.
V. Finally,let'suseamethodoflogicalreasoningtoanswerthefollowing
2. ThatareaissufficienttoalloweverypersononEarth2feetx
question:Aretheretwohumanbeingsalivetodaywhohaveprecisely
2feetx6feetofspace.
thesamenumberofhairsontheirbodies?
11 ©2003TheTeachingCompanyLimitedPartnership ©2003TheTeachingCompanyLimitedPartnership 12
A. Wecananswerthisquestionbyreasoningthroughaprooftoshowthat SuggestedReading:
theremustbepairsofpeoplewhohaveexactlythesamenumberof
EdwardB.BurgerandMichaelStarbird,TheHeartofMathematics:An
hairs.Thisprocess,inturn,willshowusaprincipleofreasoningthat
invitationtoeffectivethinking,KeyCollegePublishing,Section2.1,"Counting:
canbecommonlyappliedintheworld.
HowthePigeonholeprincipleleadstoprecisionthroughestimation."
B. Wecouldfirstestimatethenumberofhairsononeperson'sbodyby
QuestionstoConsider:
closelyexaminingonesquareinchofhairandcomingupwithan
upperlimitforthenumberofhairsinonesquareinch. 1. Twothousandyearsago,anobleArabiankingwishedtorewardhis
ministerofscience.Althoughthemodestministerresistedanyrewardfrom
C. Wecouldthenestimatethenumberofsquareinchesonthe
theking,thekingfinallyforcedhimtostatearewardthattheminister
humanbodybyenvisioningthebodyasacylinder.
desired.Impishlytheministersaidthathewouldbecontentwiththe
D. Ifwemultiplythesetwonumbers,wewouldgetanestimateof followingtoken."Letustakeacheckerboard.OnthefirstsquareIwouldbe
thenumberofhairsonahumanbody. mostgratefulifyouwouldplaceonepieceofgold.Thenonthenextsquare
twiceasmuchasbefore,thusplacingtwopieces,andoneachsubsequent
E. We might then multiplythat number by10 to get an upper limit. In
square,placingtwiceasmanypiecesofgoldasintheprevioussquare.1
this way, we could reason that no one human being has more than,
wouldbemostcontentwithallthegoldthatisontheboardonceyour
say,100millionhairs.(infact,wehavefarfewerhairsthanthat.)
majestyhasfinished."Thissoundedextremelyreasonableandtheking
F. Howdoesthatfactguaranteeusthattheremustbetwopeopleon agreed.Giventhatthereare64squaresonacheckerboard,roughlyhow
Earthwhohavethesamenumberofhairs?Accordingtothe manypiecesofgolddidthekinghavetogivetoour"modest"ministerof
Pigeonholeprinciple,with6.3billionpeopleonEarth,allhaving science?Whydidthekinghavehimexecuted?
fewerthan100millionhairs,wemusthavepairsofpeoplewhohave
2. Howdoyouknowthatinsomeyearsoon,morethan50millionpeoplewill
theexactsamenumbersofhairs.
die?
G. OtherexamplesinwhichweseethePigeonholeprinciplein
actionincludethefollowing:
1. Therearetwotreesintheworldthathavethesamenumberof
leaves.
2. Ifweworkinanofficewithmorethan30employeeswhoarrive
duringthesamehalf-hourperiod,twoofthemwillarriveatwork
attheexactsameminute.
3. Ifweareinagroupof370people,twoarecertaintosharethe
samebirthday.
VI. Thestepofquantificationhelpsusunderstandeveryaspectofmathematics
andtheworld.
A. Whenwelookatgeometricalstructures,wewillfindthatcountingand
quantifyingourobservationswillleadustoappreciatethemmore
clearly.
B. Whenweexploreprobability,wewillseethatputtinganumerical
valueonuncertaintyallowsustoputsomeorderevenonrandomness.
13 ©2003TheTeachingCompanyLimitedPartnership ©2003TheTeachingCompanyLimitedPartnership 14
