TableofContents

TitlePageTableofContentsCopyrightPrefacePartOneNUMBERS1.FromFishtoInfinity2.RockGroups3.TheEnemyofMyEnemy4.Commuting5.DivisionandItsDiscontents6.Location,Location,LocationPartTwoRELATIONSHIPS7.TheJoyofx8.FindingYourRoots9.MyTubRunnethOver10.WorkingYourQuads11.PowerToolsPartThreeSHAPES12.SquareDancing13.SomethingfromNothing14.TheConicConspiracy15.SineQuaNon16.TakeIttotheLimitPartFourCHANGE17.ChangeWeCanBelieveIn18.ItSlices,ItDices19.Allaboute20.LovesMe,LovesMeNot21.StepIntotheLightPartFiveDATA22.TheNewNormal23.ChancesAre24.UntanglingtheWebPartSixFRONTIERS25.TheLoneliestNumbers

26.GroupThink27.TwistandShout28.ThinkGlobally29.AnalyzeThis!30.TheHilbertHotelAcknowledgmentsNotesCreditsIndexAbouttheAuthor

Copyright©2012byStevenStrogatzAllrightsreserved

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Preface

Ihaveafriendwhogetsatremendouskickoutofscience,eventhoughhe’sanartist.Wheneverwegettogetherallhewantstodoischataboutthelatestthinginpsychologyorquantummechanics.Butwhenitcomestomath,hefeelsatsea,anditsaddenshim.Thestrangesymbolskeephimout.Hesayshedoesn’tevenknowhowtopronouncethem.Infact,hisalienationrunsalotdeeper.He’snotsurewhatmathematiciansdo

allday,orwhattheymeanwhentheysayaproofiselegant.SometimeswejokethatIshouldjustsithimdownandteachhimeverything,startingwith1+1=2andgoingasfaraswecan.Crazyasitsounds,that’swhatI’llbetryingtodointhisbook.It’saguided

tourthroughtheelementsofmath,frompreschooltogradschool,foranyoneouttherewho’dliketohaveasecondchanceatthesubject—butthistimefromanadultperspective.It’snotintendedtoberemedial.Thegoalistogiveyouabetterfeelingforwhatmathisallaboutandwhyit’ssoenthrallingtothosewhogetit.We’lldiscoverhowMichaelJordan’sdunkscanhelpexplainthefundamentals

ofcalculus.I’llshowyouasimple—andmind-blowing—waytounderstandthatstapleofgeometry,thePythagoreantheorem.We’lltrytogettothebottomofsomeoflife’smysteries,bigandsmall:DidO.J.doit?Howshouldyouflipyourmattresstogetthemaximumwearoutofit?Howmanypeopleshouldyoudatebeforesettlingdown?Andwe’llseewhysomeinfinitiesarebiggerthanothers.Mathiseverywhere,ifyouknowwheretolook.We’llspotsinewavesin

zebrastripes,hearechoesofEuclidintheDeclarationofIndependence,andrecognizesignsofnegativenumbersintherun-uptoWorldWarI.Andwe’llseehowourlivestodayarebeingtouchedbynewkindsofmath,aswesearchforrestaurantsonlineandtrytounderstand—nottomentionsurvive—thefrighteningswingsinthestockmarket.Byacoincidencethatseemsonlyfittingforabookaboutnumbers,thisone

wasbornonthedayIturnedfifty.DavidShipley,whowasthentheeditoroftheop-edpagefortheNewYorkTimes,hadinvitedmetolunchonthebigday(unawareofitssemicentennialsignificance)andaskedifIwouldeverconsiderwritingaseriesaboutmathforhisreaders.Ilovedthethoughtofsharingthepleasuresofmathwithanaudiencebeyondmyinquisitiveartistfriend.“TheElementsofMath”appearedonlineinlateJanuary2010andranfor

fifteenweeks.Inresponse,lettersandcommentspouredinfromreadersofall

ages.Manywhowrotewerestudentsandteachers.Otherswerecuriouspeoplewho,forwhateverreason,hadfallenoffthetracksomewhereintheirmatheducationbutsensedtheyweremissingsomethingworthwhileandwantedtotryagain.EspeciallygratifyingwerethenotesIreceivedfromparentsthankingmeforhelpingthemexplainmathtotheirkidsand,intheprocess,tothemselves.Evenmycolleaguesandfellowmathaficionadosseemedtoenjoythepieces—whentheyweren’tsuggestingimprovements(orperhapsespeciallythen!).Allinall,theexperienceconvincedmethatthere’saprofoundbutlittle-

recognizedhungerformathamongthegeneralpublic.Despiteeverythingwehearaboutmathphobia,manypeoplewanttounderstandthesubjectalittlebetter.Andoncetheydo,theyfinditaddictive.

TheJoyofxisanintroductiontomath’smostcompellingandfar-reachingideas.Thechapters—somefromtheoriginalTimesseries—arebite-sizeandlargelyindependent,sofeelfreetosnackwhereveryoulike.Ifyouwanttowadedeeperintoanything,thenotesattheendofthebookprovideadditionaldetailsandsuggestionsforfurtherreading.Forthebenefitofreaderswhopreferastep-by-stepapproach,I’vearranged

thematerialintosixmainparts,followingthelinesofthetraditionalcurriculum.Part1,“Numbers,”beginsourjourneywithkindergartenandgrade-school

arithmetic,stressinghowhelpfulnumberscanbeandhowuncannilyeffectivetheyareatdescribingtheworld.Part2,“Relationships,”generalizesfromworkingwithnumberstoworking

withrelationshipsbetweennumbers.Thesearetheideasattheheartofalgebra.Whatmakesthemsocrucialisthattheyprovidethefirsttoolsfordescribinghowonethingaffectsanother,throughcauseandeffect,supplyanddemand,doseandresponse,andsoon—thekindsofrelationshipsthatmaketheworldcomplicatedandrich.Part3,“Shapes,”turnsfromnumbersandsymbolstoshapesandspace—the

provinceofgeometryandtrigonometry.Alongwithcharacterizingallthingsvisual,thesesubjectsraisemathtonewlevelsofrigorthroughlogicandproof.Inpart4,“Change,”wecometocalculus,themostpenetratingandfruitful

branchofmath.Calculusmadeitpossibletopredictthemotionsoftheplanets,therhythmofthetides,andvirtuallyeveryotherformofcontinuouschangeintheuniverseandourselves.Asupportingthemeinthispartistheroleofinfinity.Thedomesticationofinfinitywasthebreakthroughthatmadecalculuswork.Byharnessingtheawesomepoweroftheinfinite,calculuscouldfinallysolvemanylong-standingproblemsthathaddefiedtheancients,andthatultimatelyledtothescientificrevolutionandthemodernworld.

Part5,“Data,”dealswithprobability,statistics,networks,anddatamining,allrelativelyyoungsubjectsinspiredbythemessysideoflife:chanceandluck,uncertainty,risk,volatility,randomness,interconnectivity.Withtherightkindsofmath,andtherightkindsofdata,we’llseehowtopullmeaningfromthemaelstrom.Asweneartheendofourjourneyinpart6,“Frontiers,”weapproachtheedge

ofmathematicalknowledge,theborderlandbetweenwhat’sknownandwhatremainselusive.Thesequenceofchaptersfollowsthefamiliarstructurewe’veusedthroughout—numbers,relationships,shapes,change,andinfinity—buteachofthesetopicsisnowrevisitedmoredeeply,andinitsmodernincarnation.Ihopethatalloftheideasaheadwillprovidejoy—andagoodnumberof

Aha!moments.Butanyjourneyneedstobeginatthebeginning,solet’sstartwiththesimple,magicalactofcounting.

PartOneNUMBERS

1.FromFishtoInfinity

THEBESTINTRODUCTIONtonumbersI’veeverseen—theclearestandfunniestexplanationofwhattheyareandwhyweneedthem—appearsinaSesameStreetvideocalled123CountwithMe.Humphrey,anamiablebutdimwittedfellowwithpinkfurandagreennose,isworkingthelunchshiftattheFurryArmsHotelwhenhetakesacallfromaroomfulofpenguins.Humphreylistenscarefullyandthencallsouttheirordertothekitchen:“Fish,fish,fish,fish,fish,fish.”ThispromptsErnietoenlightenhimaboutthevirtuesofthenumbersix.

Childrenlearnfromthisthatnumbersarewonderfulshortcuts.Insteadof

sayingtheword“fish”exactlyasmanytimesastherearepenguins,Humphreycouldusethemorepowerfulconceptofsix.Asadults,however,wemightnoticeapotentialdownsidetonumbers.Sure,

theyaregreattimesavers,butataseriouscostinabstraction.Sixismoreetherealthansixfish,preciselybecauseit’smoregeneral.Itappliestosixofanything:sixplates,sixpenguins,sixutterancesoftheword“fish.”It’stheineffablethingtheyallhaveincommon.Viewedinthislight,numbersstarttoseemabitmysterious.Theyapparently

existinsomesortofPlatonicrealm,alevelabovereality.Inthatrespecttheyare

morelikeotherloftyconcepts(e.g.,truthandjustice),andlessliketheordinaryobjectsofdailylife.Theirphilosophicalstatusbecomesevenmurkieruponfurtherreflection.Whereexactlydonumberscomefrom?Didhumanityinventthem?Ordiscoverthem?Anadditionalsubtletyisthatnumbers(andallmathematicalideas,forthat

matter)havelivesoftheirown.Wecan’tcontrolthem.Eventhoughtheyexistinourminds,oncewedecidewhatwemeanbythemwehavenosayinhowtheybehave.Theyobeycertainlawsandhavecertainproperties,personalities,andwaysofcombiningwithoneanother,andthere’snothingwecandoaboutitexceptwatchandtrytounderstand.Inthatsensetheyareeerilyreminiscentofatomsandstars,thethingsofthisworld,whicharelikewisesubjecttolawsbeyondourcontrol...exceptthatthosethingsexistoutsideourheads.Thisdualaspectofnumbers—aspartheaven,partearth—isperhapstheir

mostparadoxicalfeature,andthefeaturethatmakesthemsouseful.ItiswhatthephysicistEugeneWignerhadinmindwhenhewroteof“theunreasonableeffectivenessofmathematicsinthenaturalsciences.”Incaseit’snotclearwhatImeanaboutthelivesofnumbersandtheir

uncontrollablebehavior,let’sgobacktotheFurryArms.SupposethatbeforeHumphreyputsinthepenguins’order,hesuddenlygetsacallonanotherlinefromaroomoccupiedbythesamenumberofpenguins,allofthemalsoclamoringforfish.Aftertakingbothcalls,whatshouldHumphreyyellouttothekitchen?Ifhehasn’tlearnedanything,hecouldshout“fish”onceforeachpenguin.Or,usinghisnumbers,hecouldtellthecookheneedssixordersoffishforthefirstroomandsixmoreforthesecondroom.Butwhathereallyneedsisanewconcept:addition.Oncehe’smasteredit,he’llproudlysayheneedssixplussix(or,ifhe’sashowoff,twelve)fish.Thecreativeprocesshereisthesameastheonethatgaveusnumbersinthe

firstplace.Justasnumbersareashortcutforcountingbyones,additionisashortcutforcountingbyanyamount.Thisishowmathematicsgrows.Therightabstractionleadstonewinsight,andnewpower.Beforelong,evenHumphreymightrealizehecankeepcountingforever.Yetdespitethisinfinitevista,therearealwaysconstraintsonourcreativity.

Wecandecidewhatwemeanbythingslike6and+,butoncewedo,theresultsofexpressionslike6+6arebeyondourcontrol.Logicleavesusnochoice.Inthatsense,mathalwaysinvolvesbothinventionanddiscovery:weinventtheconceptsbutdiscovertheirconsequences.Aswe’llseeinthecomingchapters,inmathematicsourfreedomliesinthequestionsweask—andinhowwepursuethem—butnotintheanswersawaitingus.

2.RockGroups

LIKEANYTHINGELSE,arithmetichasitsserioussideanditsplayfulside.Theserioussideiswhatwealllearnedinschool:howtoworkwithcolumns

ofnumbers,addingthem,subtractingthem,grindingthemthroughthespreadsheetcalculationsneededfortaxreturnsandyear-endreports.Thissideofarithmeticisimportant,practical,and—formanypeople—joyless.Theplayfulsideofarithmeticisalotlessfamiliar,unlessyouweretrainedin

thewaysofadvancedmathematics.Yetthere’snothinginherentlyadvancedaboutit.It’sasnaturalasachild’scuriosity.InhisbookAMathematician’sLament,PaulLockhartadvocatesan

educationalapproachinwhichnumbersaretreatedmoreconcretelythanusual:heasksustoimaginethemasgroupsofrocks.Forexample,6correspondstoagroupofrockslikethis:

Youprobablydon’tseeanythingstrikinghere,andthat’sright—unlesswemakefurtherdemandsonnumbers,theyalllookprettymuchthesame.Ourchancetobecreativecomesinwhatweaskofthem.Forinstance,let’sfocusongroupshavingbetween1and10rocksinthem,

andaskwhichofthesegroupscanberearrangedintosquarepatterns.Onlytwoofthemcan:thegroupwith4andthegroupwith9.Andthat’sbecause4=2×2and9=3×3;wegetthesenumbersbysquaringsomeothernumber(actuallymakingasquareshape).

Alessstringentchallengeistoidentifygroupsofrocksthatcanbeneatly

organizedintoarectanglewithexactlytworowsthatcomeouteven.That’spossibleaslongasthereare2,4,6,8,or10rocks;thenumberhastobeeven.Ifwetrytocoerceanyoftheothernumbersfrom1to10—theoddnumbers—intotworows,theyalwaysleaveanoddbitstickingout.

Still,allisnotlostforthesemisfitnumbers.Ifweaddtwoofthemtogether,theirprotuberancesmatchupandtheirsumcomesouteven;Odd+Odd=Even.

Ifweloosentherulesstillfurthertoadmitnumbersgreaterthan10andallow

arectangularpatterntohavemorethantworowsofrocks,someoddnumbersdisplayatalentformakingtheselargerrectangles.Forexample,thenumber15canforma3×5rectangle:

So15,althoughundeniablyodd,atleasthastheconsolationofbeinga

compositenumber—it’scomposedofthreerowsoffiverockseach.Similarly,everyotherentryinthemultiplicationtableyieldsitsownrectangularrockgroup.Yetsomenumbers,like2,3,5,and7,trulyarehopeless.Noneofthemcan

formanysortofrectangleatall,otherthanasimplelineofrocks(asinglerow).Thesestrangelyinflexiblenumbersarethefamousprimenumbers.Soweseethatnumbershavequirksofstructurethatendowthemwith

personalities.Buttoseethefullrangeoftheirbehavior,weneedtogobeyondindividualnumbersandwatchwhathappenswhentheyinteract.Forexample,insteadofaddingjusttwooddnumberstogether,supposewe

addalltheconsecutiveoddnumbers,startingfrom1:

Thesumsabove,remarkably,alwaysturnouttobeperfectsquares.(Wesaw4and9inthesquarepatternsdiscussedearlier,and16=4×4,and25=5×5.)Aquickcheckshowsthatthisrulekeepsworkingforlargerandlargeroddnumbers;itapparentlyholdsallthewayouttoinfinity.Butwhatpossibleconnectioncouldtherebebetweenoddnumbers,withtheirungainlyappendages,andtheclassicallysymmetricalnumbersthatformsquares?Byarrangingourrocksintherightway,wecanmakethissurprisinglinkseemobvious—thehallmarkofanelegantproof.ThekeyistorecognizethatoddnumberscanmakeL-shapes,withtheir

protuberancescastoffintothecorner.AndwhenyoustacksuccessiveL-shapestogether,yougetasquare!

Thisstyleofthinkingappearsinanotherrecentbook,thoughforaltogether

differentliteraryreasons.InYokoOgawa’scharmingnovelTheHousekeeperandtheProfessor,anastutebutuneducatedyoungwomanwithaten-year-oldsonishiredtotakecareofanelderlymathematicianwhohassufferedatraumaticbraininjurythatleaveshimwithonlyeightyminutesofshort-termmemory.Adriftinthepresent,andaloneinhisshabbycottagewithnothingbuthisnumbers,theProfessortriestoconnectwiththeHousekeepertheonlywayheknowshow:byinquiringabouthershoesizeorbirthdayandmakingmathematicalsmalltalkaboutherstatistics.TheProfessoralsotakesaspeciallikingtotheHousekeeper’sson,whomhecallsRoot,becausetheflattopoftheboy’sheadremindshimofthesquarerootsymbol, .OnedaytheProfessorgivesRootalittlepuzzle:Canhefindthesumofallthe

numbersfrom1to10?AfterRootcarefullyaddsthenumbersandreturnswiththeanswer(55),theProfessoraskshimtofindabetterway.Canhefindtheanswerwithoutaddingthenumbers?Rootkicksthechairandshouts,“That’snotfair!”ButlittlebylittletheHousekeepergetsdrawnintotheworldofnumbers,and

shesecretlystartsexploringthepuzzleherself.“I’mnotsurewhyIbecamesoabsorbedinachild’smathproblemwithnopracticalvalue,”shesays.“Atfirst,IwasconsciousofwantingtopleasetheProfessor,butgraduallythatfeelingfadedandIrealizedithadbecomeabattlebetweentheproblemandme.WhenIwokeinthemorning,theequationwaswaiting:

anditfollowedmeallthroughtheday,asthoughithadburneditselfintomyretinaandcouldnotbeignored.”ThereareseveralwaystosolvetheProfessor’sproblem(seehowmanyyou

canfind).TheProfessorhimselfgivesanargumentalongthelineswedevelopedabove.Heinterpretsthesumfrom1to10asatriangleofrocks,with1rockinthefirstrow,2inthesecond,andsoon,upto10rocksinthetenthrow:

Byitsveryappearancethispicturegivesaclearsenseofnegativespace.Itseemsonlyhalfcomplete.Andthatsuggestsacreativeleap.Ifyoucopythetriangle,flipitupsidedown,andadditasthemissinghalftowhat’salreadythere,yougetsomethingmuchsimpler:arectanglewithtenrowsof11rockseach,foratotalof110.

Sincetheoriginaltriangleishalfofthisrectangle,thedesiredsummustbehalfof110,or55.Lookingatnumbersasgroupsofrocksmayseemunusual,butactuallyit’sas

oldasmathitself.Theword“calculate”reflectsthatlegacy—itcomesfromtheLatinwordcalculus,meaningapebbleusedforcounting.Toenjoyworkingwithnumbersyoudon’thavetobeEinstein(Germanfor“onestone”),butitmighthelptohaverocksinyourhead.

3.TheEnemyofMyEnemy

IT’STRADITIONALTOteachkidssubtractionrightafteraddition.Thatmakessense—thesamefactsaboutnumbersgetusedinboth,thoughinreverse.Andtheblackartofborrowing,socrucialtosuccessfulsubtraction,isonlyalittlemorebaroquethanthatofcarrying,itscounterpartforaddition.Ifyoucancopewithcalculating23+9,you’llbereadyfor23–9soonenough.Atadeeperlevel,however,subtractionraisesamuchmoredisturbingissue,

onethatneverariseswithaddition.Subtractioncangeneratenegativenumbers.IfItrytotake6cookiesawayfromyoubutyouhaveonly2,Ican’tdoit—exceptinmymind,whereyounowhavenegative4cookies,whateverthatmeans.Subtractionforcesustoexpandourconceptionofwhatnumbersare.Negative

numbersarealotmoreabstractthanpositivenumbers—youcan’tseenegative4cookiesandyoucertainlycan’teatthem—butyoucanthinkaboutthem,andyouhaveto,inallaspectsofdailylife,fromdebtsandoverdraftstocontendingwithfreezingtemperaturesandparkinggarages.Still,manyofushaven’tquitemadepeacewithnegativenumbers.Asmy

colleagueAndyRuinahaspointedout,peoplehaveconcoctedallsortsoffunnylittlementalstrategiestosidestepthedreadednegativesign.Onmutualfundstatements,losses(negativenumbers)areprintedinredornestledinparentheseswithnaryanegativesigntobefound.ThehistorybookstellusthatJuliusCaesarwasbornin100B.C.,not–100.ThesubterraneanlevelsinaparkinggarageoftenhavedesignationslikeB1andB2.Temperaturesareoneofthefewexceptions:folksdosay,especiallyhereinIthaca,NewYork,thatit’s–5degreesoutside,thougheventhen,manyprefertosay5belowzero.There’ssomethingaboutthatnegativesignthatjustlookssounpleasant,so...negative.Perhapsthemostunsettlingthingisthatanegativetimesanegativeisa

positive.Soletmetrytoexplainthethinkingbehindthat.Howshouldwedefinethevalueofanexpressionlike–1×3,wherewe’re

multiplyinganegativenumberbyapositivenumber?Well,justas1×3means1+1+1,thenaturaldefinitionfor–1×3is(–1)+(–1)+(–1),whichequals–3.Thisshouldbeobviousintermsofmoney:ifyouoweme$1aweek,afterthreeweeksyou’re$3inthehole.Fromthereit’sashorthoptoseewhyanegativetimesanegativeshouldbea

positive.Takealookatthefollowingstringofequations:

Nowlookatthenumbersonthefarrightandnoticetheirorderlyprogression:–3,–2,–1,0,...Ateachstep,we’readding1tothenumberbeforeit.Sowouldn’tyouagreethenextnumbershouldlogicallybe1?That’soneargumentforwhy(–1)×(–1)=1.Theappealofthisdefinitionis

thatitpreservestherulesofordinaryarithmetic;whatworksforpositivenumbersalsoworksfornegativenumbers.Butifyou’reahard-boiledpragmatist,youmaybewonderingifthese

abstractionshaveanyparallelsintherealworld.Admittedly,lifesometimesseemstoplaybydifferentrules.Inconventionalmorality,twowrongsdon’tmakearight.Likewise,doublenegativesdon’talwaysamounttopositives;theycanmakenegativesmoreintense,asin“Ican’tgetnosatisfaction.”(Actually,languagescanbeverytrickyinthisrespect.TheeminentlinguisticphilosopherJ.L.AustinofOxfordoncegavealectureinwhichheassertedthattherearemanylanguagesinwhichadoublenegativemakesapositivebutnoneinwhichadoublepositivemakesanegative—towhichtheColumbiaphilosopherSidneyMorgenbesser,sittingintheaudience,sarcasticallyreplied,“Yeah,yeah.”)Still,thereareplentyofcaseswheretherealworlddoesmirrortherulesof

negativenumbers.Onenervecell’sfiringcanbeinhibitedbythefiringofasecondnervecell.Ifthatsecondnervecellistheninhibitedbyathird,thefirstcellcanfireagain.Theindirectactionofthethirdcellonthefirstistantamounttoexcitation;achainoftwonegativesmakesapositive.Similareffectsoccuringeneregulation:aproteincanturnageneonbyblockinganothermoleculethatwasrepressingthatstretchofDNA.Perhapsthemostfamiliarparalleloccursinthesocialandpoliticalrealmsas

summedupbytheadage“Theenemyofmyenemyismyfriend.”Thistruism,andrelatedonesaboutthefriendofmyenemy,theenemyofmyfriend,andsoon,canbedepictedinrelationshiptriangles.Thecornerssignifypeople,companies,orcountries,andthesidesconnecting

themsignifytheirrelationships,whichcanbepositive(friendly,shownhereassolidlines)ornegative(hostile,shownasdashedlines).

Socialscientistsrefertotrianglesliketheoneontheleft,withallsidespositive,asbalanced—there’snoreasonforanyonetochangehowhefeels,sinceit’sreasonabletolikeyourfriends’friends.Similarly,thetriangleontheright,withtwonegativesandapositive,isconsideredbalancedbecauseitcausesnodissonance;eventhoughitallowsforhostility,nothingcementsafriendshiplikehatingthesameperson.Ofcourse,trianglescanalsobeunbalanced.Whenthreemutualenemiessize

upthesituation,twoofthem—oftenthetwowiththeleastanimositytowardeachother—maybetemptedtojoinforcesandganguponthethird.Evenmoreunbalancedisatrianglewithasinglenegativerelationship.For

instance,supposeCarolisfriendlywithbothAliceandBob,butBobandAlicedespiseeachother.Perhapstheywereonceacouplebutsufferedanastybreakup,andeachisnowbadmouthingtheothertoever-loyalCarol.Thiscausespsychologicalstressallaround.Torestorebalance,eitherAliceandBobhavetoreconcileorCarolhastochooseaside.

Inallthesecases,thelogicofbalancematchesthelogicofmultiplication.Inabalancedtriangle,thesignoftheproductofanytwosides,positiveornegative,alwaysagreeswiththesignofthethird.Inunbalancedtriangles,thispatternisbroken.Leavingasidetheverisimilitudeofthemodel,thereareinterestingquestions

hereofapurelymathematicalflavor.Forexample,inaclose-knitnetworkwhereeveryoneknowseveryone,what’sthemoststablestate?Onepossibilityisanirvanaofgoodwill,whereallrelationshipsarepositiveandalltriangleswithinthenetworkarebalanced.Butsurprisingly,thereareotherstatesthatareequallystable.Thesearestatesofintractableconflict,withthenetworksplitintotwohostilefactions(ofarbitrarysizesandcompositions).Allmembersofonefactionarefriendlywithoneanotherbutantagonistictowardeverybodyintheotherfaction.(Soundfamiliar?)Perhapsevenmoresurprisingly,thesepolarizedstatesaretheonlystatesasstableasnirvana.Inparticular,nothree-partysplitcanhaveallitstrianglesbalanced.Scholarshaveusedtheseideastoanalyzetherun-uptoWorldWarI.The

diagramthatfollowsshowstheshiftingalliancesamongGreatBritain,France,Russia,Italy,Germany,andAustria-Hungarybetween1872and1907.

Thefirstfiveconfigurationswereallunbalanced,inthesensethattheyeachcontainedatleastoneunbalancedtriangle.Theresultantdissonancetendedtopushthesenationstorealignthemselves,triggeringreverberationselsewhereinthenetwork.Inthefinalstage,Europehadsplitintotwoimplacablyopposedblocs—technicallybalanced,butonthebrinkofwar.Thepointisnotthatthistheoryispowerfullypredictive.Itisn’t.It’stoo

simpletoaccountforallthesubtletiesofgeopoliticaldynamics.Thepointisthatsomepartofwhatweobserveisduetonothingmorethantheprimitivelogicof“theenemyofmyenemy,”andthispartiscapturedperfectlybythemultiplicationofnegativenumbers.Bysortingthemeaningfulfromthegeneric,thearithmeticofnegativenumberscanhelpusseewheretherealpuzzleslie.

4.Commuting

EVERYDECADEORSOanewapproachtoteachingmathcomesalongandcreatesfreshopportunitiesforparentstofeelinadequate.Backinthe1960s,myparentswereflabbergastedbytheirinabilitytohelpmewithmysecond-gradehomework.They’dneverheardofbase3orVenndiagrams.Nowthetableshaveturned.“Dad,canyoushowmehowtodothese

multiplicationproblems?”Sure,Ithought,untiltheheadshakingbegan.“No,Dad,that’snothowwe’resupposedtodoit.That’stheold-schoolmethod.Don’tyouknowthelatticemethod?No?Well,whataboutpartialproducts?”Thesehumblingsessionshavepromptedmetorevisitmultiplicationfrom

scratch.Andit’sactuallyquitesubtle,onceyoustarttothinkaboutit.Taketheterminology.Does“seventimesthree”mean“sevenaddedtoitself

threetimes”?Or“threeaddedtoitselfseventimes”?Insomeculturesthelanguageislessambiguous.AfriendofminefromBelize

usedtorecitehistimestableslikethis:“Sevenonesareseven,seventwosarefourteen,seventhreesaretwenty-one,”andsoon.Thisphrasingmakesitclearthatthefirstnumberisthemultiplier;thesecondnumberisthethingbeingmultiplied.It’sthesameconventionasinLionelRichie’simmortallyrics“She’sonce,twice,threetimesalady.”(“She’saladytimesthree”wouldneverhavebeenahit.)Maybeallthissemanticfussstrikesyouassilly,sincetheorderinwhich

numbersaremultiplieddoesn’tmatteranyway:7×3=3×7.Fairenough,butthatbegsthequestionI’dliketoexploreinsomedepthhere:Isthiscommutativelawofmultiplication,a×b=b×a,reallysoobvious?Irememberbeingsurprisedbyitasachild;maybeyouweretoo.Torecapturethemagic,imaginenotknowingwhat7×3equals.Soyoutry

countingbysevens:7,14,21.Nowturnitaroundandcountbythreesinstead:3,6,9,...Doyoufeelthesuspensebuilding?Sofar,noneofthenumbersmatchthoseinthesevenslist,butkeepgoing...12,15,18,andthen,bingo,21!Mypointisthatifyouregardmultiplicationasbeingsynonymouswith

repeatedcountingbyacertainnumber(or,inotherwords,withrepeatedaddition),thecommutativelawisn’ttransparent.Butitbecomesmoreintuitiveifyouconceiveofmultiplicationvisually.Think

of7×3asthenumberofdotsinarectangulararraywithsevenrowsandthreecolumns.

Ifyouturnthearrayonitsside,ittransformsintothreerowsandsevencolumns—andsincerotatingthepicturedoesn’tchangethenumberofdots,itmustbetruethat7×3=3×7.

Yetstrangelyenough,inmanyreal-worldsituations,especiallywheremoney

isconcerned,peopleseemtoforgetthecommutativelaw,ordon’trealizeitapplies.Letmegiveyoutwoexamples.Supposeyou’reshoppingforanewpairofjeans.They’reonsalefor20

percentoffthestickerpriceof$50,whichsoundslikeabargain,butkeepinmindthatyoualsohavetopaythe8percentsalestax.Aftertheclerkfinishescomplimentingyouontheflatteringfit,shestartsringingupthepurchasebutthenpausesandwhispers,inaconspiratorialtone,“Hey,letmesaveyousomemoney.I’llapplythetaxfirst,andthentaketwentypercentoffthetotal,soyou’llgetmoremoneyback.Okay?”

Butsomethingaboutthatsoundsfishytoyou.“Nothanks,”yousay.“Couldyoupleasetakethetwentypercentofffirst,thenapplythetaxtothesaleprice?Thatway,I’llpaylesstax.”Whichwayisabetterdealforyou?(Assumebotharelegal.)Whenconfrontedwithaquestionlikethis,manypeopleapproachit

additively.Theyworkoutthetaxandthediscountunderbothscenarios,andthendowhateveradditionsorsubtractionsarenecessarytofindthefinalprice.Doingthingstheclerk’sway,youreason,wouldcostyou$4intax(8percentofthestickerpriceof$50).Thatwouldbringyourtotalto$54.Thenapplyingthe20percentdiscountto$54givesyou$10.80back,soyou’denduppaying$54minus$10.80,whichequals$43.20.Whereasunderyourscenario,the20percentdiscountwouldbeappliedfirst,savingyou$10offthe$50stickerprice.Thenthe8percenttaxonthatreducedpriceof$40wouldbe$3.20,soyou’dstillenduppaying$43.20.Amazing!Butit’smerelythecommutativelawinaction.Toseewhy,think

multiplicatively,notadditively.Applyingan8percenttaxfollowedbya20percentdiscountamountstomultiplyingthestickerpriceby1.08andthenmultiplyingthatresultby0.80.Switchingtheorderoftaxanddiscountreversesthemultiplication,butsince1.08×0.80=0.80×1.08,thefinalpricecomesoutthesame.Considerationslikethesealsoariseinlargerfinancialdecisions.IsaRoth

401(k)betterorworsethanatraditionalretirementplan?Moregenerally,ifyouhaveapileofmoneytoinvestandyouhavetopaytaxesonitatsomepoint,isitbettertotakethetaxbiteatthebeginningoftheinvestmentperiod,orattheend?Onceagain,thecommutativelawshowsit’sawash,allotherthingsbeing

equal(which,sadly,theyoftenaren’t).If,forbothscenarios,yourmoneygrowsbythesamefactorandgetstaxedatthesamerate,itdoesn’tmatterwhetheryoupaythetaxesupfrontorattheend.Pleasedon’tmistakethismathematicalremarkforfinancialadvice.Anyone

facingthesedecisionsinreallifeneedstobeawareofmanycomplicationsthatmuddythewaters:Doyouexpecttobeinahigherorlowertaxbracketwhenyouretire?Willyoumaxoutyourcontributionlimits?Doyouthinkthegovernmentwillchangeitspoliciesaboutthetax-exemptstatusofwithdrawalsbythetimeyou’rereadytotakethemoneyout?Leavingallthisaside(anddon’tgetmewrong,it’sallimportant;I’mjusttryingtofocushereonasimplermathematicalissue),mybasicpointisthatthecommutativelawisrelevanttotheanalysisofsuchdecisions.Youcanfindheateddebatesaboutthisonpersonalfinancesitesonthe

Internet.Evenaftertherelevanceofthecommutativelawhasbeenpointedout,somebloggersdon’tacceptit.It’sthatcounterintuitive.Maybewe’rewiredtodoubtthecommutativelawbecauseindailylife,it

usuallymatterswhatyoudofirst.Youcan’thaveyourcakeandeatittoo.Andwhentakingoffyourshoesandsocks,you’vegottogetthesequencingright.ThephysicistMurrayGell-Manncametoasimilarrealizationonedaywhen

hewasworryingabouthisfuture.AsanundergraduateatYale,hedesperatelywantedtostayintheIvyLeagueforgraduateschool.UnfortunatelyPrincetonrejectedhisapplication.Harvardsaidyesbutseemedtobedraggingitsfeetaboutprovidingthefinancialsupportheneeded.Hisbestoption,thoughhefounditdepressing,wasMIT.InGell-Mann’seyes,MITwasagrubbytechnologicalinstitute,beneathhisrarefiedtaste.Nevertheless,heacceptedtheoffer.YearslaterhewouldexplainthathehadcontemplatedsuicideatthetimebutdecidedagainstitonceherealizedthatattendingMITandkillinghimselfdidn’tcommute.HecouldalwaysgotoMITandcommitsuicidelaterifhehadto,butnottheotherwayaround.Gell-Mannhadprobablybeensensitizedtotheimportanceofnon-

commutativity.Asaquantumphysicisthewouldhavebeenacutelyawarethatatthedeepestlevel,naturedisobeysthecommutativelaw.Andit’sagoodthing,too.Forthefailureofcommutativityiswhatmakestheworldthewayitis.It’swhymatterissolid,andwhyatomsdon’timplode.Specifically,earlyinthedevelopmentofquantummechanics,Werner

HeisenbergandPaulDirachaddiscoveredthatnaturefollowsacuriouskindoflogicinwhichp×q≠q×p,wherepandqrepresentthemomentumandpositionofaquantumparticle.Withoutthatbreakdownofthecommutativelaw,therewouldbenoHeisenberguncertaintyprinciple,atomswouldcollapse,andnothingwouldexist.That’swhyyou’dbettermindyourp’sandq’s.Andtellyourkidstodothe

same.

5.DivisionandItsDiscontents

THERE’SANARRATIVElinethatrunsthrougharithmetic,butmanyofusmisseditinthehazeoflongdivisionandcommondenominators.It’sthestoryofthequestforevermoreversatilenumbers.Thenaturalnumbers1,2,3,andsoonaregoodenoughifallwewanttodois

count,add,andmultiply.Butonceweaskhowmuchremainswheneverythingistakenaway,weareforcedtocreateanewkindofnumber—zero—andsincedebtscanbeowed,weneednegativenumberstoo.Thisenlargeduniverseofnumbers,calledintegers,iseverybitasself-containedasthenaturalnumbersbutmuchmorepowerfulbecauseitembracessubtractionaswell.Anewcrisiscomeswhenwetrytoworkoutthemathematicsofsharing.

Dividingawholenumberevenlyisnotalwayspossible...unlessweexpandtheuniverseoncemore,nowbyinventingfractions.Theseareratiosofintegers—hencetheirtechnicalname,rationalnumbers.Sadly,thisistheplacewheremanystudentshitthemathematicalwall.Therearemanyconfusingthingsaboutdivisionanditsconsequences,but

perhapsthemostmaddeningisthattherearesomanydifferentwaystodescribeapartofawhole.Ifyoucutachocolatelayercakerightdownthemiddleintotwoequalpieces,

youcouldcertainlysaythateachpieceishalfthecake.Oryoumightexpressthesameideawiththefraction1/2,meaning“1of2equalpieces.”(Whenyouwriteitthisway,theslashbetweenthe1andthe2isavisualreminderthatsomethingisbeingsliced.)Athirdwayistosaythateachpieceis50percentofthewhole,meaningliterally“50partsoutof100.”Asifthatweren’tenough,youcouldalsoinvokedecimalnotationanddescribeeachpieceas0.5oftheentirecake.Thisprofusionofchoicesmaybepartlytoblameforthebewildermentmany

ofusfeelwhenconfrontedwithfractions,percentages,anddecimals.AvividexampleappearsinthemovieMyLeftFoot,thetruestoryoftheIrishwriter,painter,andpoetChristyBrown.Bornintoalargeworking-classfamily,hesufferedfromcerebralpalsythatmadeitalmostimpossibleforhimtospeakorcontrolanyofhislimbsexcepthisleftfoot.Asaboyhewasoftendismissedasmentallydisabled,especiallybyhisfather,whoresentedhimandtreatedhimcruelly.Apivotalsceneinthemovietakesplacearoundthekitchentable.Oneof

Christy’soldersistersisquietlydoinghermathhomework,seatednexttoherfather,whileChristy,asusual,isshuntedoffinthecorneroftheroom,twistedin

hischair.Hissisterbreaksthesilence:“What’stwenty-fivepercentofaquarter?”sheasks.Fathermullsitover.“Twenty-fivepercentofaquarter?Uhhh...That’sastupidquestion,eh?Imean,twenty-fivepercentisaquarter.Youcan’thaveaquarterofaquarter.”Sisterresponds,“Youcan.Can’tyou,Christy?”Father:“Ha!Whatwouldheknow?”Writhing,Christystrugglestopickupapieceofchalkwithhisleftfoot.

Positioningitoveraslateonthefloor,hemanagestoscrawla1,thenaslash,thensomethingunrecognizable.It’sthenumber16,butthe6comesoutbackward.Frustrated,heerasesthe6withhisheelandtriesagain,butthistimethechalkmovestoofar,crossingthroughthe6,renderingitindecipherable.“That’sonlyanervoussquiggle,”sneershisfather,turningaway.Christycloseshiseyesandslumpsback,exhausted.Asidefromthedramaticpowerofthescene,what’sstrikingisthefather’s

conceptualrigidity.Whatmakeshiminsistyoucan’thaveaquarterofaquarter?Maybehethinksyoucantakeaquarteronlyoutofawholeorfromsomethingmadeoffourequalparts.Butwhathefailstorealizeisthateverythingismadeoffourequalparts.Inthecaseofanobjectthat’salreadyaquarter,itsfourequalpartslooklikethis:

Since16ofthesethinslicesmaketheoriginalwhole,eachsliceis1/16ofthewhole—theanswerChristywastryingtoscratchout.Aversionofthesamekindofmentalrigidity,updatedforthedigitalage,

madetheroundsontheInternetafewyearsagowhenafrustratedcustomernamedGeorgeVaccarorecordedandpostedhisphoneconversationwithtwoservicerepresentativesatVerizonWireless.Vaccaro’scomplaintwasthathe’dbeenquotedadatausagerateof.002centsperkilobyte,buthisbillshowedhe’dbeencharged.002dollarsperkilobyte,ahundredfoldhigherrate.TheensuingconversationclimbedtothetopfiftyinYouTube’scomedysection.Here’sahighlightthatoccursabouthalfwaythroughtherecording,duringan

exchangebetweenVaccaroandAndrea,theVerizonfloormanager:V:Doyourecognizethatthere’sadifferencebetweenonedollarandonecent?A:Definitely.V:Doyourecognizethere’sadifferencebetweenhalfadollarandhalfacent?A:Definitely.V:Then,doyouthereforerecognizethere’sadifferencebetween.002dollarsand.002cents?A:No.V:No?A:Imeanthere’s...there’sno.002dollars.

AfewmomentslaterAndreasays,“Obviouslyadollaris‘one,decimal,zero,zero,’right?Sowhatwoulda‘pointzerozerotwodollars’looklike?...I’veneverheardof.002dollars...It’sjustnotafullcent.”Thechallengeofconvertingbetweendollarsandcentsisonlypartofthe

problemforAndrea.Therealbarrierisherinabilitytoenvisionaportionofeither.FromfirsthandexperienceIcantellyouwhatit’sliketobemystifiedby

decimals.Ineighthgrade,Ms.Stantonbeganteachingushowtoconvertafractionintoadecimal.Usinglongdivision,wefoundthatsomefractionsgivedecimalsthatterminateinallzeros.Forexample, =.2500...,whichcanberewrittenas.25,sinceallthosezerosamounttonothing.Otherfractionsgivedecimalsthateventuallyrepeat,like

Myfavoritewas ,whosedecimalcounterpartrepeatseverysixdigits:

ThebafflementbeganwhenMs.Stantonpointedoutthatifyoutriplebothsidesofthesimpleequation

you’reforcedtoconcludethat1mustequal.9999...AtthetimeIprotestedthattheycouldn’tbeequal.Nomatterhowmany9s

shewrote,Icouldwritejustasmany0sin1.0000...andthenifwesubtractedhernumberfrommine,therewouldbeateenybitleftover,somethinglike.0000...01.LikeChristy’sfatherandtheVerizonservicereps,Icouldn’tacceptsomething

thathadjustbeenproventome.Isawitbutrefusedtobelieveit.(Thismightremindyouofsomepeopleyouknow.)Butitgetsworse—orbetter,ifyouliketofeelyourneuronssizzle.Backin

Ms.Stanton’sclass,whatstoppedusfromlookingatdecimalsthatneitherterminatenorrepeatperiodically?It’seasytocookupsuchstomach-churners.Here’sanexample:

Bydesign,theblocksof2getprogressivelylongeraswemovetotheright.There’snowaytoexpressthisdecimalasafraction.Fractionsalwaysyielddecimalsthatterminateoreventuallyrepeatperiodically—thatcanbeproven—andsincethisdecimaldoesneither,itcan’tbeequaltotheratioofanywholenumbers.It’sirrational.Givenhowcontrivedthisdecimalis,youmightsupposeirrationalityisrare.

Onthecontrary,itistypical.Inacertainsensethatcanbemadeprecise,almostalldecimalsareirrational.Andtheirdigitslookstatisticallyrandom.Onceyouaccepttheseastonishingfacts,everythingturnstopsy-turvy.Whole

numbersandfractions,sobelovedandfamiliar,nowappearscarceandexotic.Andthatinnocuousnumberlinepinnedtothemoldingofyourgrade-schoolclassroom?Nooneevertoldyou,butit’schaosupthere.

6.Location,Location,Location

I’DWALKEDPASTEzraCornell’sstatuehundredsoftimeswithoutevenglancingathisgreenishlikeness.ButthenonedayIstoppedforacloserlook.

Ezraappearsoutdoorsyandruggedlydignifiedinhislongcoat,vest,andboots,hisrighthandrestingonawalkingstickandholdingarumpled,wide-brimmedhat.Themonumentcomesacrossasunpretentiousanddisarminglydirect—muchlikethemanhimself,byallaccounts.

WhichiswhyitseemssodiscordantthatEzra’sdatesareinscribedonthepedestalinpompousRomannumerals:

EZRACORNELLMDCCCVII–MDCCCLXXIV

Whynotwritesimply1807–1874?Romannumeralsmaylookimpressive,butthey’rehardtoreadandcumbersometouse.Ezrawouldhavehadlittlepatienceforthat.Findingagoodwaytorepresentnumbershasalwaysbeenachallenge.Since

thedawnofcivilization,peoplehavetriedvarioussystemsforwritingnumbersandreckoningwiththem,whetherfortrading,measuringland,orkeepingtrackoftheherd.Whatnearlyallthesesystemshaveincommonisthatourbiologyisdeeply

embeddedinthem.Throughthevagariesofevolution,wehappentohavefivefingersoneachoftwohands.Thatpeculiaranatomicalfactisreflectedintheprimitivesystemoftallying;forexample,thenumber17iswrittenas

Here,eachoftheverticalstrokesineachgroupmusthaveoriginallymeantafinger.Maybethediagonalslashwasathumb,foldedacrosstheotherfourfingerstomakeafist?Romannumeralsareonlyslightlymoresophisticatedthantallies.Youcan

spotthevestigeoftalliesinthewayRomanswrote2and3,asIIandIII.Likewise,thediagonalslashisechoedintheshapeoftheRomansymbolfor5,V.But4isanambiguouscase.Sometimesit’swrittenasIIII,tallystyle(you’lloftenseethisonfancyclocks),thoughmorecommonlyit’swrittenasIV.Thepositioningofasmallernumber(I)totheleftofalargernumber(V)indicatesthatyou’resupposedtosubtractI,ratherthanaddit,asyouwouldifitwerestationedontheright.ThusIVmeans4,whereasVImeans6.

TheBabylonianswerenotnearlyasattachedtotheirfingers.Theirnumeralsystemwasbasedon60—aclearsignoftheirimpeccabletaste,for60isanexceptionallypleasantnumber.Itsbeautyisintrinsicandhasnothingtodowithhumanappendages.Sixtyisthesmallestnumberthatcanbedividedevenlyby1,2,3,4,5,and6.Andthat’sjustforstarters(there’salso10,12,15,20,and30).Becauseofitspromiscuousdivisibility,60ismuchmorecongenialthan10foranysortofcalculationormeasurementthatinvolvescuttingthingsintoequalparts.Whenwedivideanhourinto60minutes,oraminuteinto60seconds,orafullcircleinto360degrees,we’rechannelingthesagesofancientBabylon.ButthegreatestlegacyoftheBabyloniansisanideathat’ssocommonplace

todaythatfewofusappreciatehowsubtleandingeniousitis.Toillustrateit,let’sconsiderourownHindu-Arabicsystem,which

incorporatesthesameideainitsmodernform.Insteadof60,thissystemisbasedontensymbols:1,2,3,4,5,6,7,8,9,and,mostbrilliant,0.Thesearecalleddigits,naturally,fromtheLatinwordforafingeroratoe.Thegreatinnovationhereisthateventhoughthissystemisbasedonthe

number10,thereisnosinglesymbolreservedfor10.Tenismarkedbyaposition—thetensplace—insteadofasymbol.Thesameistruefor100,or1,000,oranyotherpowerof10.Theirdistinguishedstatusissignifiednotbyasymbolbutbyaparkingspot,areservedpieceofrealestate.Location,location,location.Contrasttheeleganceofthisplace-valuesystemwiththemuchcruder

approachusedinRomannumerals.Youwant10?We’vegot10.It’sX.We’vealsogot100(C)and1,000(M),andwe’lleventhrowinspecialsymbolsforthe5family:V,L,andD,for5,50,and500.TheRomanapproachwastoelevateafewfavorednumbers,givethemtheir

ownsymbols,andexpressalltheother,second-classnumbersascombinationsofthose.Unfortunately,Romannumeralscreakedandgroanedwhenfacedwith

anythinglargerthanafewthousand.Inaworkaroundsolutionthatwouldnowadaysbecalledakludge,thescholarswhowerestillusingRomannumeralsintheMiddleAgesresortedtopilingbarsontopoftheexistingsymbolstoindicatemultiplicationbyathousand.Forinstance, meanttenthousand,and

meantathousandthousandsor,inotherwords,amillion.Multiplyingbyabillion(athousandmillion)wasrarelynecessary,butifyoueverhadto,youcouldalwaysputasecondbarontopofthe .Asyoucansee,thefunneverstopped.ButintheHindu-Arabicsystem,it’sasnaptowriteanynumber,nomatter

howbig.Allnumberscanbeexpressedwiththesametendigits,merelybyslottingthemintotherightplaces.Furthermore,thenotationisinherentlyconcise.Everynumberlessthanamillion,forexample,canbeexpressedinsixsymbolsorfewer.Trydoingthatwithwords,tallies,orRomannumerals.Bestofall,withaplace-valuesystem,ordinarypeoplecanlearntodo

arithmetic.Youjusthavetomasterafewfacts—themultiplicationtableanditscounterpartforaddition.Onceyougetthosedown,that’sallyou’lleverneed.Anycalculationinvolvinganypairofnumbers,nomatterhowbig,canbeperformedbyapplyingthesamesetsoffacts,overandoveragain,recursively.Ifitallsoundsprettymechanical,that’spreciselythepoint.Withplace-value

systems,youcanprogramamachinetodoarithmetic.Fromtheearlydaysofmechanicalcalculatorstothesupercomputersoftoday,theautomationofarithmeticwasmadepossiblebythebeautifulideaofplacevalue.Buttheunsungherointhisstoryis0.Without0,thewholeapproachwould

collapse.It’stheplaceholderthatallowsustotell1,10,and100apart.Allplace-valuesystemsarebasedonsomenumbercalled,appropriately

enough,thebase.Oursystemisbase10,ordecimal(fromtheLatinrootdecem,meaning“ten”).Aftertheonesplace,thesubsequentconsecutiveplacesrepresenttens,hundreds,thousands,andsoon,eachofwhichisapowerof10:

GivenwhatIsaidearlieraboutthebiological,asopposedtothelogical,originofourpreferenceforbase10,it’snaturaltoask:Wouldsomeotherbasebemoreefficient,oreasiertomanipulate?Astrongcasecanbemadeforbase2,thefamousandnowubiquitousbinary

systemusedincomputersandallthingsdigital,fromcellphonestocameras.Ofallthepossiblebases,itrequiresthefewestsymbols—justtwoofthem,0and1.Assuch,itmeshesperfectlywiththelogicofelectronicswitchesoranythingelsethatcantogglebetweentwostates—onoroff,openorclosed.Binarytakessomegettingusedto.Insteadofpowersof10,itusespowersof

2.Itstillhasaonesplacelikethedecimalsystem,butthesubsequentplacesnowstandfortwos,fours,andeights,because

Ofcourse,wewouldn’twritethesymbol2,becauseitdoesn’texistinbinary,justasthere’snosinglenumeralfor10indecimal.Inbinary,2iswrittenas10,meaningone2andzero1s.Similarly,4wouldbewrittenas100(one4,zero2s,andzero1s),and8wouldbe1000.Theimplicationsreachfarbeyondmath.Ourworldhasbeenchangedbythe

powerof2.Inthepastfewdecadeswe’vecometorealizethatallinformation—notjustnumbers,butalsolanguage,images,andsound—canbeencodedinstreamsofzerosandones.WhichbringsusbacktoEzraCornell.Tuckedattherearofhismonument,andalmostcompletelyobscured,isa

telegraphmachine—amodestreminderofhisroleinthecreationofWesternUnionandthetyingtogetheroftheNorthAmericancontinent.

Asacarpenterturnedentrepreneur,CornellworkedforSamuelMorse,whose

namelivesoninthecodeofdotsanddashesthroughwhichtheEnglishlanguagewasreducedtotheclicksofatelegraphkey.Thosetwolittlesymbolsweretechnologicalforerunnersoftoday’szerosandones.MorseentrustedCornelltobuildthenation’sfirsttelegraphline,alinkfrom

BaltimoretotheU.S.Capitol,inWashington,D.C.Fromtheverystartitseemsthathehadaninklingofwhathisdotsanddasheswouldbring.Whenthelinewasofficiallyopened,onMay24,1844,Morsesentthefirstmessagedownthewire:“WhathathGodwrought.”

PartTwoRELATIONSHIPS

7.TheJoyofx

NOWIT’STIMEtomoveonfromgrade-schoolarithmetictohigh-schoolmath.Overthenexttenchapterswe’llberevisitingalgebra,geometry,andtrig.Don’tworryifyou’veforgottenthemall—therewon’tbeanyteststhistimearound.Insteadofworryingaboutthedetailsofthesesubjects,wehavetheluxuryofconcentratingontheirmostbeautiful,important,andfar-reachingideas.Algebra,forexample,mayhavestruckyouasadizzyingmixofsymbols,

definitions,andprocedures,butintheendtheyallboildowntojusttwoactivities—solvingforxandworkingwithformulas.Solvingforxisdetectivework.You’researchingforanunknownnumber,x.

You’vebeenhandedafewcluesaboutit,eitherintheformofanequationlike2x+3=7or,lessconveniently,inaconvolutedverbaldescriptionofit(asinthosescarywordproblems).Ineithercase,thegoalistoidentifyxfromtheinformationgiven.Workingwithformulas,bycontrast,isablendofartandscience.Insteadof

dwellingonaparticularx,you’remanipulatingandmassagingrelationshipsthatcontinuetoholdevenasthenumbersinthemchange.Thesechangingnumbersarecalledvariables,andtheyarewhattrulydistinguishesalgebrafromarithmetic.Theformulasinquestionmightexpresselegantpatternsaboutnumbersfortheirownsake.Thisiswherealgebrameetsart.Ortheymightexpressrelationshipsbetweennumbersintherealworld,astheydointhelawsofnatureforfallingobjectsorplanetaryorbitsorgeneticfrequenciesinapopulation.Thisiswherealgebrameetsscience.Thisdivisionofalgebraintotwograndactivitiesisnotstandard(infact,Ijust

madeitup),butitseemstoworkprettywell.InthenextchapterI’llhavemoretosayaboutsolvingforx,sofornowlet’sfocusonformulas,startingwithsomeeasyexamplestoclarifytheideas.Afewyearsago,mydaughterJorealizedsomethingaboutherbigsister,

Leah.“Dad,there’salwaysanumberbetweenmyageandLeah’s.RightnowI’msixandLeah’seight,andsevenisinthemiddle.Andevenwhenwe’reold,likewhenI’mtwentyandshe’stwenty-two,therewillstillbeanumberinthemiddle!”Jo’sobservationqualifiesasalgebra(thoughnoonebutaproudfatherwould

seeitthatway)becauseshewasnoticingarelationshipbetweentwoever-changingvariables:herage,x,andLeah’sage,y.Nomatterhowoldbothofthemare,Leahwillalwaysbetwoyearsolder:y=x+2.

Algebraisthelanguageinwhichsuchpatternsaremostnaturallyphrased.Ittakessomepracticetobecomefluentinalgebra,becauseit’sloadedwithwhattheFrenchcallfauxamis,“falsefriends”:apairofwords,eachfromadifferentlanguage(inthiscase,Englishandalgebra),thatsoundrelatedandseemtomeanthesamethingbutthatactuallymeansomethinghorriblydifferentfromeachotherwhentranslated.Forexample,supposethelengthofahallwayisywhenmeasuredinyards,

andfwhenmeasuredinfeet.Writeanequationthatrelatesytof.MyfriendGrantWiggins,aneducationconsultant,hasbeenposingthis

problemtostudentsandfacultyforyears.Hesaysthatinhisexperience,studentsgetitwrongmorethanhalfthetime,eveniftheyhaverecentlytakenandpassedanalgebracourse.Ifyouthinktheanswerisy=3f,welcometotheclub.Itseemslikesuchastraightforwardtranslationofthesentence“Oneyard

equalsthreefeet.”Butassoonasyoutryafewnumbers,you’llseethatthisformulagetseverythingbackward.Saythehallwayis10yardslong;everyoneknowsthat’s30feet.Yetwhenyoupluginy=10andf=30,theformuladoesn’twork!Thecorrectformulaisf=3y.Here3reallymeans“3feetperyard.”When

youmultiplyitbyyinyards,theunitsofyardscanceloutandyou’releftwithunitsoffeet,asyoushouldbe.Checkingthattheunitscancelproperlyhelpsavoidthiskindofblunder.For

example,itcouldhavesavedtheVerizoncustomerservicereps(discussedinchapter5)fromconfusingdollarsandcents.Anotherkindofformulaisknownasanidentity.Whenyoufactoredor

multipliedpolynomialsinalgebraclass,youwereworkingwithidentities.Youcanusethemnowtoimpressyourfriendswithnumericalparlortricks.Here’sonethatimpressedthephysicistRichardFeynman,noslouchhimselfatmentalmath:WhenIwasatLosAlamosIfoundoutthatHansBethewasabsolutelytopnotchatcalculating.Forexample,onetimewewereputtingsomenumbersintoaformula,andgotto48squared.IreachfortheMarchantcalculator,andhesays,“That’s2,300.”Ibegintopushthebuttons,andhesays,“Ifyouwantitexactly,it’s2,304.”Themachinesays2,304.“Gee!That’sprettyremarkable!”Isay.“Don’tyouknowhowtosquarenumbersnear50?”hesays.“Yousquare

50—that’s2,500—andsubtract100timesthedifferenceofyournumberfrom50(inthiscaseit’s2),soyouhave2,300.Ifyouwantthecorrection,

squarethedifferenceandadditon.Thatmakes2,304.”

Bethe’strickisbasedontheidentity

Hehadmemorizedthatequationandwasapplyingitforthecasewherexis–2,correspondingtothenumber48=50–2.Foranintuitiveproofofthisformula,imagineasquarepatchofcarpetthat

measures50+xoneachside.

Thenitsareais(50+x)squared,whichiswhatwe’relookingfor.Butthediagramaboveshowsthatthisareaismadeofa50-by-50square(thiscontributesthe2,500totheformula),tworectanglesofdimensions50byx(eachcontributesanareaof50x,foracombinedtotalof100x),andfinallythelittlex-by-xsquaregivesanareaofxsquared,thefinalterminBethe’sformula.Relationshipslikethesearenotjustfortheoreticalphysicists.Anidentity

similartoBethe’sisrelevanttoanyonewhohasmoneyinvestedinthestockmarket.Supposeyourportfoliodropscatastrophicallyby50percentoneyearandthengains50percentthenext.Evenafterthatdramaticrecovery,you’dstillbedown25percent.Toseewhy,observethata50percentlossmultipliesyourmoneyby0.50,anda50percentgainmultipliesitby1.50.Whenthosehappenbacktoback,yourmoneymultipliesby0.50times1.50,whichequals0.75—inotherwords,a25percentloss.Infact,younevergetbacktoevenwhenyouloseandgainbythesame

percentageinconsecutiveyears.Withalgebrawecanunderstandwhy.Itfollows

fromtheidentity

Inthedownyeartheportfolioshrinksbyafactor1–x(wherex=0.50intheexampleabove),andthengrowsbyafactor1+xthefollowingyear.Sothenetchangeisafactorof

andaccordingtotheformulaabove,thisequals

Thepointisthatthisexpressionisalwayslessthan1foranyxotherthan0.Soyounevercompletelyrecoupyourlosses.Needlesstosay,noteveryrelationshipbetweenvariablesisasstraightforward

asthoseabove.Yettheallureofalgebraisseductive,andingulliblehandsitspawnssuchsillinessasaformulaforthesociallyacceptableagedifferenceinaromance.AccordingtosomesitesontheInternet,ifyourageisx,politesocietywilldisapproveifyoudatesomeoneyoungerthanx/2+7.Inotherwords,itwouldbecreepyforanyoneovereighty-twotoeyemy

forty-eight-year-oldwife,evenifshewereavailable.Buteighty-one?Noproblem.Ick.Ick.Ick...

8.FindingYourRoots

FORMORETHAN2,500years,mathematicianshavebeenobsessedwithsolvingforx.Thestoryoftheirstruggletofindtheroots—thesolutions—ofincreasinglycomplicatedequationsisoneofthegreatepicsinthehistoryofhumanthought.OneoftheearliestsuchproblemsperplexedthecitizensofDelosaround430

B.C.Desperatetostaveoffaplague,theyconsultedtheoracleofDelphi,whoadvisedthemtodoublethevolumeoftheircube-shapedaltartoApollo.Unfortunately,itturnsoutthatdoublingacube’svolumerequiredthemtoconstructthecuberootof2,ataskthatisnowknowntobeimpossible,giventheirrestrictiontousenothingbutastraightedgeandcompass,theonlytoolsallowedinGreekgeometry.Laterstudiesofsimilarproblemsrevealedanotherirritant,anagginglittle

thingthatwouldn’tgoaway:evenwhensolutionswerepossible,theyofteninvolvedsquarerootsofnegativenumbers.Suchsolutionswerelongderidedassophisticorfictitiousbecausetheyseemednonsensicalontheirface.Untilthe1700sorso,mathematiciansbelievedthatsquarerootsofnegative

numberssimplycouldn’texist.Theycouldn’tbepositivenumbers,afterall,sinceapositivetimesapositive

isalwayspositive,andwe’relookingfornumberswhosesquareisnegative.Norcouldnegativenumberswork,sinceanegativetimesanegativeis,again,positive.Thereseemedtobenohopeoffindingnumbersthatwhenmultipliedbythemselveswouldgivenegativeanswers.We’veseencriseslikethisbefore.Theyoccurwheneveranexistingoperation

ispushedtoofar,intoadomainwhereitnolongerseemssensible.Justassubtractingbiggernumbersfromsmalleronesgaverisetonegativenumbers(chapter3)anddivisionspawnedfractionsanddecimals(chapter5),thefreewheelinguseofsquarerootseventuallyforcedtheuniverseofnumberstoexpand...again.Historically,thisstepwasthemostpainfulofall.Thesquarerootof–1still

goesbythedemeaningnameofi,for“imaginary.”Thisnewkindofnumber(orifyou’dratherbeagnostic,callitasymbol,nota

number)isdefinedbythepropertythat

It’struethatican’tbefoundanywhereonthenumberline.Inthatrespectit’smuchstrangerthanzero,negativenumbers,fractions,orevenirrationalnumbers,allofwhich—weirdastheyare—stillhavetheirplacesinline.Butwithenoughimagination,ourmindscanmakeroomforiaswell.Itlives

offthenumberline,atrightanglestoit,onitsownimaginaryaxis.Andwhenyoufusethatimaginaryaxistotheordinary“real”numberline,youcreatea2-Dspace—aplane—whereanewspeciesofnumberslives.

Thesearethecomplexnumbers.Here“complex”doesn’tmean“complicated”;itmeansthattwotypesofnumbers,realandimaginary,havebondedtogethertoformacomplex,ahybridnumberlike2+3i.Complexnumbersaremagnificent,thepinnacleofnumbersystems.They

enjoyallthesamepropertiesasrealnumbers—youcanaddandsubtractthem,multiplyanddividethem—buttheyarebetterthanrealnumbersbecausetheyalwayshaveroots.Youcantakethesquarerootorcuberootoranyrootofacomplexnumber,andtheresultwillstillbeacomplexnumber.Betteryet,agrandstatementcalledthefundamentaltheoremofalgebrasays

thattherootsofanypolynomialarealwayscomplexnumbers.Inthatsensethey’retheendofthequest,theholygrail.Theuniverseofnumbersneedneverexpandagain.Complexnumbersaretheculminationofthejourneythatbeganwith1.

Youcanappreciatetheutilityofcomplexnumbers(orfinditmoreplausible)ifyouknowhowtovisualizethem.Thekeyistounderstandwhatmultiplyingbyilookslike.Supposewemultiplyanarbitrarypositivenumber,say3,byi.Theresultistheimaginarynumber3i.

Somultiplyingbyiproducesarotationcounterclockwisebyaquarterturn.Ittakesanarrowoflength3pointingeastandchangesitintoanewarrowofthesamelengthbutnowpointingnorth.Electricalengineerslovecomplexnumbersforexactlythisreason.Having

suchacompactwaytorepresent90-degreerotationsisveryusefulwhenworkingwithalternatingcurrentsandvoltages,orwithelectricandmagneticfields,becausetheseofteninvolveoscillationsorwavesthatareaquartercycle(i.e.,90degrees)outofphase.Infact,complexnumbersareindispensabletoallengineers.Inaerospace

engineeringtheyeasedthefirstcalculationsoftheliftonanairplanewing.Civilandmechanicalengineersusethemroutinelytoanalyzethevibrationsoffootbridges,skyscrapers,andcarsdrivingonbumpyroads.The90-degreerotationpropertyalsoshedslightonwhati²=–1reallymeans.

Ifwemultiplyapositivenumberbyi²,thecorrespondingarrowrotates180degrees,flippingfromeasttowest,becausethetwo90-degreerotations(oneforeachfactorofi)combinetomakea180-degreerotation.

Butmultiplyingby–1producestheverysame180-degreeflip.That’sthesenseinwhichi²=–1.Computershavebreathednewlifeintocomplexnumbersandtheage-old

problemofrootfinding.Whenthey’renotbeingusedforWebsurfingore-mail,themachinesonourdeskscanrevealthingstheancientsneverdreamedof.In1976,myCornellcolleagueJohnHubbardbeganlookingatthedynamics

ofNewton’smethod,apowerfulalgorithmforfindingrootsofequationsinthecomplexplane.Themethodtakesastartingpoint(anapproximationtotheroot)anddoesacertaincomputationthatimprovesit.Bydoingthisrepeatedly,alwaysusingthepreviouspointtogenerateabetterone,themethodbootstrapsitswayforwardandrapidlyhomesinonaroot.Hubbardwasinterestedinproblemswithmultipleroots.Inthatcase,which

rootwouldthemethodfind?Heprovedthatiftherewerejusttworoots,thecloseronewouldalwayswin.Butiftherewerethreeormoreroots,hewasbaffled.Hisearlierproofnolongerapplied.SoHubbarddidanexperiment.Anumericalexperiment.HeprogrammedacomputertorunNewton’smethod.Thenhetoldittocolor-

codemillionsofdifferentstartingpointsaccordingtowhichroottheyapproachedandtoshadethemaccordingtohowfasttheygotthere.Beforehepeekedattheresults,heanticipatedthattherootswouldmost

quicklyattractthepointsnearbyandthusshouldappearasbrightspotsinasolidpatchofcolor.Butwhatabouttheboundariesbetweenthepatches?Thosehecouldn’tpicture,atleastnotinhismind’seye.Thecomputer’sanswerwasastonishing.

Theborderlandslookedlikepsychedelichallucinations.Thecolorsintermingledthereinanalmostimpossiblypromiscuousmanner,touchingeachotheratinfinitelymanypointsandalwaysinathree-way.Inotherwords,wherevertwocolorsmet,thethirdwouldalwaysinsertitselfandjointhem.Magnifyingtheboundariesrevealedpatternswithinpatterns.

Thestructurewasafractal—anintricateshapewhoseinnerstructurerepeatedatfinerandfinerscales.Furthermore,chaosreignedneartheboundary.Twopointsmightstartvery

closetogether,bouncesidebysideforawhile,andthenveerofftodifferentroots.Thewinningrootwasasunpredictableasthewinningnumberinagameofroulette.Littlethings—tiny,imperceptiblechangesintheinitialconditions—couldmakeallthedifference.Hubbard’sworkwasanearlyforayintowhat’snowcalledcomplexdynamics,

avibrantblendofchaostheory,complexanalysis,andfractalgeometry.Inawayitbroughtgeometrybacktoitsroots.In600B.C.amanualwritteninSanskritfortemplebuildersinIndiagavedetailedgeometricinstructionsforcomputingsquareroots,neededinthedesignofritualaltars.Morethan2,500yearslater,in1976,mathematicianswerestillsearchingforroots,butnowtheinstructions

werewritteninbinarycode.Someimaginaryfriendsyouneveroutgrow.

9.MyTubRunnethOver

UNCLEIRVWASmydad’sbrotheraswellashispartnerinashoestoretheyownedinourtown.Hehandledthebusinessendofthingsandmostlystayedintheofficeupstairs,becausehewasgoodwithnumbersandnotsogoodwiththecustomers.WhenIwasabouttenoreleven,UncleIrvgavememyfirstwordproblem.It

stickswithmetothisday,probablybecauseIgotitwrongandfeltembarrassed.Ithadtodowithfillingabathtub.Ifthecold-waterfaucetcanfillthetubina

half-hour,andthehot-waterfaucetcanfillitinanhour,howlongwillittaketofillthetubwhenthey’rerunningtogether?

I’mprettysureIguessedforty-fiveminutes,asmanypeoplewould.UncleIrvshookhisheadandgrinned.Then,inhishigh-pitchednasalvoice,heproceededtoschoolme.“Steven,”hesaid,“figureouthowmuchwaterpoursintothetubinaminute.”

Thecoldwaterfillsthetubinthirtyminutes,soinoneminuteitfills ofthetub.Butthehotwaterrunsslower—ittakessixtyminutes,whichmeansitfillsonly ofthetubperminute.Sowhenthey’rebothrunning,theyfill

ofthetubinaminute.Toaddthosefractions,observethat60istheirlowestcommondenominator.

Then,rewriting as ,weget

whichmeansthatthetwofaucetsworkingtogetherfill ofthetubperminute.Sotheyfillthewholetubintwentyminutes.Overtheyearssincethen,I’vethoughtaboutthisbathtubproblemmany

times,alwayswithaffectionforbothUncleIrvandthequestionitself.Therearebroaderlessonstobelearnedhere—lessonsabouthowtosolveproblemsapproximatelywhenyoucan’tsolvethemexactly,andhowtosolvethemintuitively,forthepleasureoftheAha!moment.Considermyinitialguessofforty-fiveminutes.Bylookingatanextreme,or

limiting,case,wecanseethatthatanswercan’tpossiblyberight.Infact,it’sabsurd.Tounderstandwhy,supposethehotwaterwasn’tturnedon.Thenthecoldwater—onitsown—wouldfillthetubinthirtyminutes.SowhatevertheanswertoUncleIrv’squestionis,ithastobelessthanthis.Afterall,runningthehotwateralongwiththecoldcanonlyhelp.Admittedly,thisconclusionisnotasinformativeastheexactansweroftwenty

minuteswefoundbyUncleIrv’smethod,butithastheadvantageofnotrequiringanycalculation.Adifferentwaytosimplifytheproblemistopretendthetwofaucetsrunatthe

samerate.Sayeachcanfillthetubinthirtyminutes(meaningthatthehotwaterrunsjustasfastasthecold).Thentheanswerwouldbeobvious.Becauseofthesymmetryofthenewsituation,thetwoperfectlymatchedfaucetswouldtogetherfillthetubinfifteenminutes,sinceeachdoeshalfthework.ThisinstantlytellsusthatUncleIrv’sscenariomusttakelongerthanfifteen

minutes.Why?Becausefastplusfastbeatsslowplusfast.Ourmake-believesymmetricalproblemhastwofastfaucets,whereasUncleIrv’shasoneslow,onefast.Andsincefifteenminutesistheanswerwhenthey’rebothfast,UncleIrv’stubcanonlytakelonger.Theupshotisthatbyconsideringtwohypotheticalcases—onewiththehot

wateroff,andanotherwithitmatchedtothecoldwater—welearnedthattheanswerliessomewherebetweenfifteenandthirtyminutes.Inmuchharderproblemswhereitmaybeimpossibletofindanexactanswer—notjustinmathbutinotherdomainsaswell—thissortofpartialinformationcanbeveryvaluable.Evenifwe’reluckyenoughtocomeupwithanexactanswer,that’sstillno

causeforcomplacency.Theremaybeeasierorclearerwaystofindthesolution.Thisisoneplacewheremathallowsforcreativity.Forexample,insteadofUncleIrv’stextbookmethod,withitsfractionsand

commondenominators,here’samoreplayfulroutetothesameresult.Itdawnedonmesomeyearslater,whenItriedtopinpointwhytheproblemissoconfusinginthefirstplaceandrealizedit’sbecauseofthefaucets’differentspeeds.Thatmakesitaheadachetokeeptrackofwhateachfaucetcontributes,especiallyifyoupicturethehotandcoldwatersloshingtogetherandmixinginthetub.Solet’skeepthetwotypesofwaterapart,atleastinourminds.Insteadofa

singlebathtub,imaginetwoassemblylinesofthem,twoseparateconveyorbeltsshuttlingbathtubspastahot-waterfaucetononesideandacold-waterfaucetontheother.

Eachfaucetstandsinplaceandfillsitsowntubs—nomixingallowed.Andassoonasatubfillsup,itmovesondowntheline,makingwayforthenextone.Noweverythingbecomeseasy.Inonehour,thehot-waterfaucetfillsonetub,

whilethecold-waterfaucetfillstwo(sinceeachtakesahalf-hour).Thatamountstothreetubsperhour,oronetubeverytwentyminutes.Eureka!Sowhydosomanypeople,includingmychildhoodself,blunderinto

guessingforty-fiveminutes?Whyisitsotemptingtosplitthedifferenceofthirtyandsixtyminutes?I’mnotsure,butitseemstobeacaseoffaultypatternrecognition.Maybethebathtubproblemisbeingconflatedwithotherswheresplittingthedifferencewouldmakesense.Mywifeexplainedittomebyanalogy.Imagineyou’rehelpingalittleoldladycrossthestreet.Withoutyourhelp,itwouldtakehersixtyseconds,whileyou’dzipacrossinthirtyseconds.Howlong,then,wouldittakethetwoofyou,walkingarminarm?Acompromisearoundforty-fivesecondsseemsplausiblebecausewhengrannyisclingingtoyourelbow,sheslowsyoudownandyouspeedherup.

Thedifferencehereisthatyouandgrannyaffecteachother’sspeeds,butthefaucetsdon’t.They’reindependent.Apparentlyoursubconsciousmindsdon’tspotthisdistinction,atleastnotwhenthey’releapingtothewrongconclusion.Thesilverliningisthatevenwronganswerscanbeeducational...aslongas

yourealizethey’rewrong.Theyexposemisguidedanalogiesandotherwoollythinking,andbringthecruxoftheproblemintosharperrelief.Otherclassicwordproblemsareexpresslydesignedtotricktheirvictimsby

misdirection,likeamagician’ssleightofhand.Thephrasingofthequestionsetsatrap.Ifyouanswerbyinstinct,you’llprobablyfallforit.Trythisone.Supposethreemencanpaintthreefencesinthreehours.How

longwouldittakeonemantopaintonefence?It’stemptingtoblurtout“onehour.”Thewordsthemselvesnudgeyouthat

way.Thedrumbeatinthefirstsentence—threemen,threefences,threehours—catchesyourattentionbyestablishingarhythm,sowhenthenextsentencerepeatsthepatternwithoneman,onefence,____hours,it’shardtoresistfillingintheblankwith“one.”Theparallelconstructionsuggestsananswerthat’slinguisticallyrightbutmathematicallywrong.Thecorrectansweristhreehours.Ifyouvisualizetheproblem—mentallypicturethreemenpaintingthree

fencesandallfinishingafterthreehours,justastheproblemstates—therightanswerbecomesclear.Forallthreefencestobedoneafterthreehours,eachmanmusthavespentthreehoursonhis.

Theundistractedreasoningthatthisproblemrequiresisoneofthemost

valuablethingsaboutwordproblems.Theyforceustopauseandthink,ofteninunfamiliarways.Theygiveuspracticeinbeingmindful.Perhapsevenmoreimportant,wordproblemsgiveuspracticeinthinkingnot

justaboutnumbers,butaboutrelationshipsbetweennumbers—howtheflowratesofthefaucetsaffectthetimerequiredtofillthetub,forexample.Andthatistheessentialnextstepinanyone’smatheducation.Understandably,alotofushavetroublewithit;relationshipsaremuchmoreabstractthannumbers.Butthey’realsomuchmorepowerful.Theyexpresstheinnerlogicoftheworldaroundus.Causeandeffect,supplyanddemand,inputandoutput,doseandresponse—allinvolvepairsofnumbersandtherelationshipsbetweenthem.Wordproblemsinitiateusintothiswayofthinking.However,KeithDevlinraisesaninterestingcriticisminhisessay“The

problemwithwordproblems.”Hispointisthattheseproblemstypicallyassumeyouunderstandtherulesofthegameandagreetoplaybythem,eventhoughthey’reoftenartificial,sometimesabsurdlyso.Forexample,inourproblemaboutthreemenpaintingthreefencesinthreehours,itwasimplicitthat(1)allthreemenpaintatthesamerateand(2)theyallpaintsteadily,neverslowingdownorspeedingup.Bothassumptionsareunrealistic.You’resupposedtoknownottoworryaboutthatandgoalongwiththegag,becauseotherwisetheproblemwouldbetoocomplicatedandyouwouldn’thaveenoughinformationtosolveit.You’dneedtoknowexactlyhowmucheachpainterslowsdownashegetstiredinthethirdhour,howofteneachonestopsforasnack,andsoon.Thoseofuswhoteachmathshouldtrytoturnthisbugintoafeature.We

shouldbeupfrontaboutthefactthatwordproblemsforceustomakesimplifyingassumptions.That’savaluableskill—it’scalledmathematicalmodeling.Scientistsdoitallthetimewhentheyapplymathtotherealworld.Butthey,unliketheauthorsofmostwordproblems,areusuallycarefultostatetheirassumptionsexplicitly.Sothanks,UncleIrv,forthatfirstlesson.Humiliating?Yes.Unforgettable?

Yes,thattoo...butinagoodway.

10.WorkingYourQuads

THEQUADRATICFORMULAistheRodneyDangerfieldofalgebra.Eventhoughit’soneoftheall-timegreats,itdon’tgetnorespect.Professionalscertainlyaren’tenamoredofit.Whenmathematiciansand

physicistsareaskedtolistthetoptenmostbeautifulorimportantequationsofalltime,thequadraticformulanevermakesthecut.Ohsure,everybodyswoonsover1+1=2,andE=mc²,andthepertlittlePythagoreantheorem,struttinglikeit’sallthatjustbecausea²+b²=c².Butthequadraticformula?Notachance.Admittedly,it’sunsightly.Somestudentsprefertosounditout,treatingitasa

ritualincantation:“xequalsnegativeb,plusorminusthesquarerootofbsquaredminusfourac,allovertwoa.”Othersmadeofsternerstufflooktheformulastraightintheface,confrontingahodgepodgeoflettersandsymbolsmoreformidablethananythingthey’veencountereduptothatpoint:

It’sonlywhenyouunderstandwhatthequadraticformulaistryingtodothat

youcanbegintoappreciateitsinnerbeauty.InthischapterIhopetogiveyouafeelingfortheclevernesspackedintothatporcupineofsymbols,alongwithabettersenseofwhattheformulameansandwhereitarises.Therearemanysituationsinwhichwe’dliketofigureoutthevalueofsome

unknownnumber.Whatdoseofradiationtherapyshouldbegiventoshrinkathyroidtumor?Howmuchmoneywouldyouhavetopayeachmonthtocoverathirty-yearmortgageof$200,000atafixedannualinterestrateof5percent?HowfastdoesarockethavetogotoescapetheEarth’sgravity?Algebraistheplacewherewecutourteethonthesimplestproblemsofthis

type.ThesubjectwasdevelopedbyIslamicmathematiciansaroundA.D.800,buildingonearlierworkbyEgyptian,Babylonian,Greek,andIndianscholars.OnepracticalimpetusatthattimewasthechallengeofcalculatinginheritancesaccordingtoIslamiclaw.Forexample,supposeawidowerdiesandleaveshisentireestateof10

dirhamstohisdaughterandtwosons.Islamiclawrequiresthatboththesonsmustreceiveequalshares.Moreover,eachsonmustreceivetwiceasmuchasthe

daughter.Howmanydirhamswilleachheirreceive?Let’susetheletterxtodenotethedaughter’sinheritance.Eventhoughwe

don’tknowwhatxisyet,wecanreasonaboutitasifitwereanordinarynumber.Specifically,weknowthateachsongetstwiceasmuchasthedaughterdoes,sotheyeachreceive2x.Thus,takentogether,theamountthatthethreeheirsinheritisx+2x+2x,foratotalof5x,andthismustequalthetotalvalueoftheestate,10dirhams.Hence5x=10dirhams.Finally,bydividingbothsidesoftheequationby5,weseethatx=2dirhamsisthedaughter’sshare.Andsinceeachofthesonsinherits2x,theybothget4dirhams.Noticethattwotypesofnumbersappearedinthisproblem:knownnumbers,

like2,5,and10,andunknownnumbers,likex.Oncewemanagedtoderivearelationshipbetweentheunknownandtheknown(asencapsulatedintheequation5x=10),wewereabletochipawayattheequation,dividingbothsidesby5toisolatetheunknownx.Itwasabitlikeasculptorworkingthemarble,tryingtoreleasethestatuefromthestone.Aslightlydifferenttacticwouldhavebeenneededifwehadencountereda

knownnumberbeingsubtractedfromanunknown,asinanequationlikex–2=5.Tofreexinthiscase,wewouldpareawaythe2byaddingittobothsidesoftheequation.Thisyieldsanunencumberedxontheleftand5+2=7ontheright.Thusx=7,whichyoumayhavealreadyrealizedbycommonsense.Althoughthistacticisnowfamiliartoallstudentsofalgebra,theymaynot

realizetheentiresubjectisnamedafterit.Intheearlypartoftheninthcentury,MuhammadibnMusaal-Khwarizmi,amathematicianworkinginBaghdad,wroteaseminaltextbookinwhichhehighlightedtheusefulnessofrestoringaquantitybeingsubtracted(like2,above)byaddingittotheothersideofanequation.Hecalledthisprocessal-jabr(Arabicfor“restoring”),whichlatermorphedinto“algebra.”Then,longafterhisdeath,hehittheetymologicaljackpotagain.Hisownname,al-Khwarizmi,livesontodayintheword“algorithm.”Inhistextbook,beforewadingintotheintricaciesofcalculatinginheritances,

al-Khwarizmiconsideredamorecomplicatedclassofequationsthatembodyrelationshipsamongthreekindsofnumbers,notthemeretwoconsideredabove.Alongwithknownnumbersandanunknown(x),theseequationsalsoincludedthesquareoftheunknown(x²).Theyarenowcalledquadraticequations,fromtheLatinquadratus,for“square.”AncientscholarsinBabylonia,Egypt,Greece,China,andIndiahadalreadytackledsuchbrainteasers,whichoftenaroseinarchitecturalorgeometricalproblemsinvolvingareasorproportions,andhadshownhowtosolvesomeofthem.Anexamplediscussedbyal-Khwarizmiis

Inhisday,however,suchproblemswereposedinwords,notsymbols.Heasked:“Whatmustbethesquarewhich,whenincreasedbytenofitsownroots,amountstothirty-nine?”(Here,theterm“root”referstotheunknownx.)Thisproblemismuchtougherthanthetwoweconsideredabove.Howcanwe

isolatexnow?Thetricksusedearlierareinsufficient,becausethex²and10xtermstendtosteponeachother’stoes.Evenifyoumanagetoisolatexinoneofthem,theotherremainstroublesome.Forinstance,ifwedividebothsidesoftheequationby10,the10xsimplifiestox,whichiswhatwewant,butthenthex²becomesx²/10,whichbringsusnoclosertofindingxitself.Thebasicobstacle,inanutshell,isthatwehavetodotwothingsatonce,andtheyseemalmostincompatible.Thesolutionthatal-Khwarizmipresentsisworthdelvingintoinsomedetail,

firstbecauseit’ssoslick,andsecondbecauseit’ssopowerful—itallowsustosolveallquadraticequationsinasinglestroke.BythatImeanthatiftheknownnumbers10and39abovewerechangedtoanyothernumbers,themethodwouldstillwork.Theideaistointerpreteachofthetermsintheequationgeometrically.Think

ofthefirstterm,x²,astheareaofasquarewithdimensionsxbyx.

Similarly,regardthesecondterm,10x,astheareaofarectangleofdimensions10byxor,moreingenious,astheareaoftwoequalrectangles,eachmeasuring5byx.(Splittingtherectangleintotwopiecessetsthestageforthekeymaneuverthatfollows,knownascompletingthesquare.)

Attachthetwonewrectanglesontothesquaretoproduceanotchedshapeofareax²+10x:

Viewedinthislight,al-Khwarizmi’spuzzleamountstoasking:Ifthenotched

shapeoccupies39squareunitsofarea,howlargewouldxhavetobe?

Thepictureitselfsuggestsanalmostirresistiblenextstep.Lookatthatmissingcorner.Ifonlyitwerefilledin,thenotchedshapewouldturnintoaperfectsquare.Solet’stakethehintandcompletethesquare.

Supplyingthemissing5×5squareadds25squareunitstotheexistingareaofx²+10x,foratotalofx²+10x+25.Equivalently,thatcombinedareacanbeexpressedmoreneatlyas(x+5)²,sincethecompletedsquareisx+5unitslongoneachside.Theupshotisthatx²and10xarenowmovinggracefullyasacouple,rather

thansteppingoneachother’stoes,bybeingpairedwithinthesingleexpression(x+5)².That’swhatwillsoonenableustosolveforx.Meanwhile,becauseweadded25unitsofareatotheleftsideoftheequation

x²+10x=39,wemustalsoadd25totherightside,tokeeptheequationbalanced.Since39+25=64,ourequationthenbecomes

Butthat’sacinchtosolve.Takingsquarerootsofbothsidesgivesx+5=8,sox=3.Loandbehold,3reallydoessolvetheequationx²+10x=39.Ifwesquare3

(giving9)andthenadd10times3(giving30),thesumis39,asdesired.There’sonlyonesnag.Ifal-Khwarizmiweretakingalgebratoday,he

wouldn’treceivefullcreditforthisanswer.Hefailstomentionthatanegativenumber,x=–13,alsoworks.Squaringitgives169;addingittentimesgives–130;andtheytooaddupto39.Butthisnegativesolutionwasignoredinancienttimes,sinceasquarewithasideofnegativelengthisgeometricallymeaningless.

Today,algebraislessbeholdentogeometryandweregardthepositiveandnegativesolutionsasequallyvalid.Inthecenturiesafteral-Khwarizmi,scholarscametorealizethatallquadratic

equationscouldbesolvedinthesameway,bycompletingthesquare—aslongasonewaswillingtoallowthenegativenumbers(andtheirbewilderingsquareroots)thatoftencameupintheanswers.Thislineofargumentrevealedthatthesolutionstoanyquadraticequation

(wherea,b,andcareknownbutarbitrarynumbers,andxisunknown)couldbeexpressedbythequadraticformula,

What’ssoremarkableaboutthisformulaishowbrutallyexplicitandcomprehensiveitis.There’stheanswer,rightthere,nomatterwhata,b,andchappentobe.Consideringthatthereareinfinitelymanypossiblechoicesforeachofthem,that’salotforasingleformulatomanage.Inourowntime,thequadraticformulahasbecomeanirreplaceabletoolfor

practicalapplications.Engineersandscientistsuseittoanalyzethetuningofaradio,theswayingofafootbridgeoraskyscraper,thearcofabaseballoracannonball,theupsanddownsofanimalpopulations,andcountlessotherreal-worldphenomena.Foraformulabornofthemathematicsofinheritance,that’squitealegacy.

11.PowerTools

IFYOUWEREanavidtelevisionwatcherinthe1980s,youmayrememberaclevershowcalledMoonlighting.Knownforitssnappydialogueandtheromanticchemistrybetweenitscostars,itfeaturedCybillShepherdandBruceWillisasMaddieHayesandDavidAddison,acoupleofwisecrackingprivatedetectives.

Whileinvestigatingoneparticularlytoughcase,Davidasksacoroner’s

assistantforhisbestguessaboutpossiblesuspects.“Beatsme,”saystheassistant.“ButyouknowwhatIdon’tunderstand?”Davidreplies,“Logarithms?”Then,reactingtoMaddie’slook:“What?Youunderstoodthose?”Thatprettywellsumsuphowmanypeoplefeelaboutlogarithms.Their

peculiarnameisjustpartoftheirimageproblem.Mostfolksneverusethem

againafterhighschool,atleastnotconsciously,andareoblivioustothelogarithmshidingbehindthescenesoftheirdailylives.ThesameistrueofmanyoftheotherfunctionsdiscussedinalgebraIIand

precalculus.Powerfunctions,exponentialfunctions—whatwasthepointofallthat?Mygoalinthischapteristohelpyouappreciatethefunctionofallthosefunctions,evenifyouneverhaveoccasiontopresstheirbuttonsonyourcalculator.Amathematicianneedsfunctionsforthesamereasonthatabuilderneeds

hammersanddrills.Toolstransformthings.Sodofunctions.Infact,mathematiciansoftenrefertothemastransformationsbecauseofthis.Butinsteadofwoodandsteel,thematerialsthatfunctionspoundawayonarenumbersandshapesand,sometimes,evenotherfunctions.ToshowyouwhatImean,let’splotthegraphoftheequationy=4–x².You

mayrememberhowthissortofactivitygoes:Youdrawapictureofthexyplanewiththex-axisrunninghorizontallyandthey-axisvertically.Thenforeachxyoucomputethecorrespondingyandplotthemtogetherasasinglepointinthexyplane.Forexample,whenxis1,theequationsaysy=4–1²,whichis4–1,or3.So(x,y)=(1,3)isapointonthegraph.Afteryoucalculateandplotafewmorepoints,thefollowingpictureemerges.

Thebowedshapeofthecurveisduetotheactionofmathematicalpliers.In

theequationfory,thefunctionthattransformsxintox²behavesalotlikethecommontoolforbendingandpullingthings.Whenit’sappliedtoeverypointonapieceofthex-axis(whichyoucouldvisualizeasastraightpieceofwire),the

pliersbendandelongatethatpieceintothedownward-curvingarchshownabove.Andwhatroledoesthe4playintheequationy=4–x²?Itactslikeanailfor

hangingapictureonawall.Itliftsupthebentwirearchby4units.Sinceitraisesallpointsbythesameamount,it’sknownasaconstantfunction.Thisexampleillustratesthedualnatureoffunctions.Ontheonehand,they’re

tools:thex²bendsthepieceofthex-axis,andthe4liftsit.Ontheotherhand,they’rebuildingblocks:the4andthe–x²canberegardedascomponentpartsofamorecomplicatedfunction,4–x²,justaswires,batteries,andtransistorsarecomponentpartsofaradio.Onceyoustarttolookatthingsthisway,you’llnoticefunctionseverywhere.

Thearchingcurveabove—technicallyknownasaparabola—isthesignatureofthesquaringfunctionx²operatingbehindthescenes.Lookforitwhenyou’retakingasipfromawaterfountainorwatchingabasketballarctowardthehoop.AndifyoueverhaveafewminutestospareonalayoverinDetroit’sinternationalairport,besuretostopbythewaterfeatureintheDeltaterminaltoenjoytheworld’smostbreathtakingparabolasatplay.

Parabolasandconstantsareassociatedwithawiderclassoffunctions—power

functionsoftheformxn,inwhichavariablexisraisedtoafixedpowern.Foraparabola,n=2;foraconstant,n=0.Changingthevalueofnyieldsotherhandytools.Forexample,raisingxtothe

firstpower(n=1)givesafunctionthatworkslikearamp,asteadyinclineofgrowthordecay.It’scalledalinearfunctionbecauseitsxygraphisaline.Ifyou

leaveabucketoutinasteadyrain,thewatercollectingatthebottomriseslinearlyintime.Anotherusefultoolistheinversesquarefunction,1/x²,correspondingtothe

casen=–2.(Thepowerbecomes–2becausethefunctionisaninversesquare;thex²appearsinthedenominator.)Thisfunctionisgoodfordescribinghowwavesandforcesattenuateastheyspreadoutinthreedimensions—forinstance,howasoundsoftensasitmovesawayfromitssource.Powerfunctionslikethesearethebuildingblocksthatscientistsandengineers

usetodescribegrowthanddecayintheirmildestforms.Butwhenyouneedmathematicaldynamite,it’stimetounpackthe

exponentialfunctions.Theydescribeallsortsofexplosivegrowth,fromnuclearchainreactionstotheproliferationofbacteriainapetridish.Themostfamiliarexampleisthefunction10ˣ,inwhich10israisedtothepowerx.Makesurenottoconfusethiswiththeearlierpowerfunctions.Heretheexponent(thepowerx)isavariable,andthebase(thenumber10)isaconstant—whereasinapowerfunctionlikex²,it’stheotherwayaround.Thisswitchmakesahugedifference:asxgetslargerandlarger,anexponentialfunctionofxeventuallygrowsfasterthananypowerfunction,nomatterhowlargethepower.Exponentialgrowthisalmostunimaginablyrapid.That’swhyit’ssohardtofoldapieceofpaperinhalfmorethansevenor

eighttimes.Eachfoldingapproximatelydoublesthethicknessofthewad,causingittogrowexponentially.Meanwhile,thewad’slengthshrinksinhalfeverytime,andthusdecreasesexponentiallyfast.Forastandardsheetofnotebookpaper,aftersevenfoldsthewadbecomesthickerthanitislong,soitcan’tbefoldedagain.Itdoesn’tmatterhowstrongthepersondoingthefoldingis.Forasheettobeconsideredlegitimatelyfoldedntimes,theresultingwadisrequiredtohave2ⁿlayersinastraightline,andthiscan’thappenifthewadisthickerthanitislong.ThechallengewasthoughttobeimpossibleuntilBritneyGallivan,thena

juniorinhighschool,solveditin2002.Shebeganbyderivingaformula

thatpredictedthemaximumnumberoftimes,n,thatpaperofagiventhicknessTandlengthLcouldbefoldedinonedirection.Noticetheforbiddingpresenceoftheexponentialfunction2ⁿintwoplaces—oncetoaccountforthedoublingofthewad’sthicknessateachfold,andanothertimetoaccountforthehalvingof

itslength.Usingherformula,Britneyconcludedthatshewouldneedtouseaspecialroll

oftoiletpapernearlythree-quartersofamilelong.Sheboughtthepaper,andinJanuary2002,shewenttoashoppingmallinherhometownofPomona,California,andunrolledthepaper.Sevenhourslater,andwiththehelpofherparents,shesmashedtheworldrecordbyfoldingthepaperinhalftwelvetimes!Intheory,exponentialgrowthisalsosupposedtograceyourbankaccount.If

yourmoneygrowsatanannualinterestrateofr,afteroneyearitwillbeworth(1+r)timesyouroriginaldeposit;aftertwoyears,(1+r)²;andafterxyears,(1+r)ˣtimesyourinitialdeposit.Thusthemiracleofcompoundingthatwesooftenhearaboutiscausedbyexponentialgrowthinaction.Whichbringsusbacktologarithms.Weneedthembecauseit’susefultohave

toolsthatcanundotheactionsofothertools.Justaseveryofficeworkerneedsbothastaplerandastapleremover,everymathematicianneedsexponentialfunctionsandlogarithms.They’reinverses.Thismeansthatifyoutypeanumberxintoyourcalculatorandthenpunchthe10ˣbuttonfollowedbythelogxbutton,you’llgetbacktothenumberyoustartedwith.Forexample,ifx=2,then10ˣwouldbe10²,whichequals100.Takingthelogofthatthenbringstheresultbackto2;thelogbuttonundoestheactionofthe10ˣbutton.Hencelog(100)equals2.Likewise,log(1,000)=3andlog(10,000)=4,because1,000=103and10,000=104.Noticesomethingmagicalhere:asthenumbersinsidethelogarithmsgrew

multiplicatively,increasingtenfoldeachtimefrom100to1,000to10,000,theirlogarithmsgrewadditively,increasingfrom2to3to4.Ourbrainsperformasimilartrickwhenwelistentomusic.Thefrequenciesofthenotesinascale—do,re,mi,fa,sol,la,ti,do—soundtouslikethey’rerisinginequalsteps.Butobjectivelytheirvibrationalfrequenciesarerisingbyequalmultiples.Weperceivepitchlogarithmically.Ineveryplacewheretheyarise,fromtheRichterscaleforearthquake

magnitudestopHmeasuresofacidity,logarithmsmakewonderfulcompressors.They’reidealfortakingquantitiesthatvaryoverawiderangeandsqueezingthemtogethersotheybecomemoremanageable.Forinstance,100and100milliondifferamillionfold,agulfthatmostofusfindincomprehensible.Buttheirlogarithmsdifferonlyfourfold(theyare2and8,because100=10²and100million=108).Inconversation,wealluseacrudeversionoflogarithmicshorthandwhenwerefertoanysalarybetween$100,000and$999,999asbeingsixfigures.That“six”isroughlythelogarithmofthesesalaries,whichinfactspantherangefromfivetosix.

Asimpressiveasallthesefunctionsmaybe,amathematician’stoolboxcanonlydosomuch—whichiswhyIstillhaven’tassembledmyIkeabookcases.

PartThreeSHAPES

12.SquareDancing

IBETICANguessyourfavoritemathsubjectinhighschool.Itwasgeometry.SomanypeopleI’vemetovertheyearshaveexpressedaffectionforthat

subject.Isitbecausegeometrydrawsontherightsideofthebrain,andthatappealstovisualthinkerswhomightotherwisecringeatitscoldlogic?Maybe.Butsomepeopletellmetheylovedgeometrypreciselybecauseitwassological.Thestep-by-stepreasoning,witheachnewtheoremrestingfirmlyonthosealreadyestablished—that’sthesourceofsatisfactionformany.Butmybesthunch(and,fulldisclosure,Ipersonallylovegeometry)isthat

peopleenjoyitbecauseitmarrieslogicandintuition.Itfeelsgoodtousebothhalvesofthebrain.Toillustratethepleasuresofgeometry,let’srevisitthePythagoreantheorem,

whichyouprobablyrememberasa²+b²=c².Partofthegoalhereistoseewhyit’strueandappreciatewhyitmatters.Beyondthat,byprovingthetheoremintwodifferentways,we’llcometoseehowoneproofcanbemoreelegantthananother,eventhoughbotharecorrect.ThePythagoreantheoremisconcernedwithrighttriangles—meaningthose

witharight(90-degree)angleatoneofthecorners.Righttrianglesareimportantbecausethey’rewhatyougetifyoucutarectangleinhalfalongitsdiagonal:

Andsincerectanglescomeupofteninallsortsofsettings,sodorighttriangles.Theyarise,forinstance,insurveying.Ifyou’remeasuringarectangularfield,

youmightwanttoknowhowfaritisfromonecornertothediagonallyoppositecorner.(Bytheway,thisiswheregeometrystarted,historically—inproblemsoflandmeasurement,ormeasuringtheearth:geo=“earth,”andmetry=“measurement.”)ThePythagoreantheoremtellsyouhowlongthediagonaliscomparedtothe

sidesoftherectangle.Ifonesidehaslengthaandtheotherhaslengthb,thetheoremsaysthediagonalhaslengthc,where

Forsomereason,thediagonalistraditionallycalledthehypotenuse,though

I’venevermetanyonewhoknowswhy.(AnyLatinorGreekscholars?)Itmusthavesomethingtodowiththediagonalsubtendingarightangle,butasjargongoes,“subtending”isaboutasopaqueas“hypotenuse.”Anyway,here’showthetheoremworks.Tokeepthenumberssimple,let’ssay

a=3yardsandb=4yards.Thentofigureouttheunknownlengthc,wedonourblackhoodsandintonethatc²isthesumof3²plus4²,whichis9plus16.(Keepinmindthatallofthesequantitiesarenowmeasuredinsquareyards,sincewesquaredtheyardsaswellasthenumbersthemselves.)Finally,since9+16=25,wegetc²=25squareyards,andthentakingsquarerootsofbothsidesyieldsc=5yardsasthelengthofthehypotenuse.ThiswayoflookingatthePythagoreantheoremmakesitseemlikea

statementaboutlengths.Buttraditionallyitwasviewedasastatementaboutareas.Thatbecomesclearerwhenyouhearhowtheyusedtosayit:Thesquareonthehypotenuseisthesumofthesquaresontheothertwosides.

Noticetheword“on.”We’renotspeakingofthesquareofthehypotenuse—that’sanewfangledalgebraicconceptaboutmultiplyinganumber(thelengthofthehypotenuse)byitself.No,we’rereferringheretoasquareliterallysittingonthehypotenuse,likethis:

Let’scallthisthelargesquare,todistinguishitfromthesmallandmediumsquareswecanbuildontheothertwosides:

Thenthetheoremsaysthatthelargesquarehasthesameareaasthesmallandmediumsquarescombined.Forthousandsofyears,thismarvelousfacthasbeenexpressedinadiagram,

aniconicmnemonicofdancingsquares:

Viewingthetheoremintermsofareasmakesitalotmorefuntothinkabout.

Forexample,youcantestit—andtheneatit—bybuildingthesquaresoutofmanylittlecrackers.Oryoucantreatthetheoremlikeachild’spuzzle,withpiecesofdifferentshapesandsizes.Byrearrangingthesepuzzlepieces,wecanprovethetheoremverysimply,asfollows.Let’sgobacktothetiltedsquaresittingonthehypotenuse.

Ataninstinctivelevel,youshouldfeelabitdisquietedbythisimage.Thesquarelookspotentiallyunstable,likeitmighttoppleorslidedowntheramp.Andthere’salsoanunpleasantarbitrarinessaboutwhichofthefoursidesofthesquaregetstotouchthetriangle.Guidedbytheseintuitivefeelings,let’saddthreemorecopiesofthetriangle

aroundthesquaretomakeamoresolidandsymmetricalpicture:

Nowrecallwhatwe’retryingtoprove:thatthetiltedwhitesquareinthepictureabove(whichisjustourearlierlargesquare—it’sstillsittingrightthereonthehypotenuse)hasthesameareaasthesmallandmediumsquaresputtogether.Butwherearethoseothersquares?Well,wehavetoshiftsometrianglesaroundtofindthem.Thinkofthepictureaboveasdepictingapuzzle,withfourtriangularpieces

wedgedintothecornersofarigidpuzzleframe.

Inthisinterpretation,thetiltedsquareistheemptyspaceinthemiddleofthepuzzle.Therestoftheareainsidetheframeisoccupiedbythepuzzlepieces.Nowlet’strymovingthepiecesaroundinvariousways.Ofcourse,nothing

wedocaneverchangethetotalamountofemptyspaceinsidetheframe—it’salwayswhateverarealiesoutsidethepieces.Thebrainstorm,then,istorearrangethepieceslikethis:

Allofasuddentheemptyspacehaschangedintothetwoshapeswe’relookingfor—thesmallsquareandthemediumsquare.Andsincethetotalareaofemptyspacealwaysstaysthesame,we’vejustproventhePythagoreantheorem!Thisproofdoesfarmorethanconvince;itilluminates.That’swhatmakesit

elegant.Forcomparison,here’sanotherproof.It’sequallyfamous,andit’sperhapsthe

simplestproofthatavoidsusingareas.Asbefore,considerarighttrianglewithsidesoflengthaandband

hypotenuseoflengthc,asshownbelowontheleft.

Now,bydivineinspirationorastrokeofgenius,somethingtellsustodrawalinesegmentperpendiculartothehypotenuseanddowntotheoppositecorner,asshowninthetriangleontheright.Thiscleverlittleconstructioncreatestwosmallertrianglesinsidetheoriginal

one.It’seasytoprovethatallthesetrianglesaresimilar—whichmeanstheyhaveidenticalshapesbutdifferentsizes.Thatinturnimpliesthatthelengthsoftheircorrespondingpartshavethesameproportions,whichtranslatesintothefollowingsetofequations:

Wealsoknowthat

becauseourconstructionmerelysplittheoriginalhypotenuseoflengthcintotwosmallersidesoflengthsdande.Atthispointyoumightbefeelingabitlost,oratleastunsureofwhattodo

next.There’samorassoffiveequationsabove,andwe’retryingtowhittlethemdowntodeducethat

Tryitforafewminutes.You’lldiscoverthattwooftheequationsareirrelevant.That’sugly;anelegantproofshouldinvolvenothingsuperfluous.Withhindsight,ofcourse,youwouldn’thavelistedthoseequationstobeginwith.But

thatwouldjustbeputtinglipstickonap...(themissingwordhereis“proof”).Nevertheless,bymanipulatingtherightthreeequations,youcangetthe

theoremtopopout.Seethenoteson[>]forthemissingsteps.Wouldyouagreewithmethat,onaestheticgrounds,thisproofisinferiorto

thefirstone?Foronething,itdragsneartheend.Andwhoinvitedallthatalgebratotheparty?Thisissupposedtobeageometryevent.Butamoreseriousdefectistheproof’smurkiness.Bythetimeyou’redone

sloggingthroughit,youmightbelievethetheorem(grudgingly),butyoustillmightnotseewhyit’strue.Leavingproofsaside,whydoesthePythagoreantheoremevenmatter?

Becauseitrevealsafundamentaltruthaboutthenatureofspace.Itimpliesthatspaceisflat,asopposedtocurved.Forthesurfaceofaglobeorabagel,forexample,thetheoremneedstobemodified.Einsteinconfrontedthischallengeinhisgeneraltheoryofrelativity(wheregravityisnolongerviewedasaforce,butratherasamanifestationofthecurvatureofspace),andsodidBernhardRiemannandothersbeforehimwhenlayingthefoundationsofnon-Euclideangeometry.It’salongroadfromPythagorastoEinstein.Butatleastit’sastraightline...

formostoftheway.

13.SomethingfromNothing

EVERYMATHCOURSEcontainsatleastonenotoriouslydifficulttopic.Inarithmetic,it’slongdivision.Inalgebra,it’swordproblems.Andingeometry,it’sproofs.Moststudentswhotakegeometryhaveneverseenaproofbefore.The

experiencecancomeasashock,soperhapsawarninglabelwouldbeinorder,somethinglikethis:Proofscancausedizzinessorexcessivedrowsiness.Sideeffectsofprolongedexposuremayincludenightsweats,panicattacks,and,inrarecases,euphoria.Askyourdoctorifproofsarerightforyou.Disorientingasproofscanbe,learningtodothemhaslongbeenthought

essentialtoaliberaleducation—moreessentialthanthesubjectmatteritself,somewouldsay.Accordingtothisview,geometryisgoodforthemind;ittrainsyoutothinkclearlyandlogically.It’snotthestudyoftriangles,circles,andparallellinespersethatmatters.What’simportantistheaxiomaticmethod,theprocessofbuildingarigorousargument,stepbystep,untiladesiredconclusionhasbeenestablished.EuclidlaiddownthisdeductiveapproachintheElements(nowthemost

reprintedtextbookofalltime)about2,300yearsago.Eversince,Euclideangeometryhasbeenamodelforlogicalreasoninginallwalksoflife,fromscienceandlawtophilosophyandpolitics.Forexample,IsaacNewtonchanneledEuclidinthestructureofhismasterworkTheMathematicalPrinciplesofNaturalPhilosophy.Usinggeometricalproofs,hededucedGalileo’sandKepler’slawsaboutprojectilesandplanetsfromhisowndeeperlawsofmotionandgravity.Spinoza’sEthicsfollowsthesamepattern.ItsfulltitleisEthicaOrdineGeometricoDemonstrata(EthicsDemonstratedinGeometricalOrder).YoucanevenhearechoesofEuclidintheDeclarationofIndependence.WhenThomasJeffersonwrote,“Weholdthesetruthstobeself-evident,”hewasmimickingthestyleoftheElements.Euclidhadbegunwiththedefinitions,postulates,andself-evidenttruthsofgeometry—theaxioms—andfromthemerectedanedificeofpropositionsanddemonstrations,onetruthlinkedtothenextbyunassailablelogic.JeffersonorganizedtheDeclarationinthesamewaysothatitsradicalconclusion,thatthecolonieshadtherighttogovernthemselves,wouldseemasinevitableasafactofgeometry.Ifthatintellectuallegacyseemsfar-fetched,keepinmindthatJefferson

reveredEuclid.Afewyearsafterhefinishedhissecondtermaspresidentandsteppedoutofpubliclife,hewrotetohisoldfriendJohnAdamsonJanuary12,

1812,aboutthepleasuresofleavingpoliticsbehind:“IhavegivenupnewspapersinexchangeforTacitusandThucydides,forNewtonandEuclid;andIfindmyselfmuchthehappier.”Still,what’smissinginallthisworshipofEuclideanrationalityisan

appreciationofgeometry’smoreintuitiveaspects.Withoutinspiration,there’dbenoproofs—ortheoremstoproveinthefirstplace.Likecomposingmusicorwritingpoetry,geometryrequiresmakingsomethingfromnothing.Howdoesapoetfindtherightwords,oracomposerahauntingmelody?Thisisthemysteryofthemuse,andit’snolessmysteriousinmaththanintheothercreativearts.Asanillustration,considertheproblemofconstructinganequilateraltriangle

—atrianglewithallthreesidesthesamelength.Therulesofthegamearethatyou’regivenonesideofthetriangle,thelinesegmentshownhere:

Yourtaskistousethatsegment,somehow,toconstructtheothertwosides,andtoprovethattheyeachhavethesamelengthastheoriginal.Theonlytoolsatyourdisposalareastraightedgeandacompass.Astraightedgeallowsyoutodrawastraightlineofanylength,ortodrawastraightlinebetweenanytwopoints.Acompassallowsyoutodrawacircleofanyradius,centeredatanypoint.Keepinmind,however,thatastraightedgeisnotaruler.Ithasnomarkings

onitandcan’tbeusedtomeasurelengths.(Specifically,youcan’tuseittocopyormeasuretheoriginalsegment.)Norcanacompassserveasaprotractor;allitcandoisdrawcircles,notmeasureangles.Ready?Begin.Thisisthemomentofparalysis.Wheretostart?Logicwon’thelpyouhere.Skilledproblemsolversknowthatabetter

approachistorelaxandplaywiththepuzzle,hopingtogetafeelforit.Forinstance,maybeitwouldhelptousethestraightedgetodrawtiltedlinesthroughtheendsofthesegment,likethis:

Noluck.Althoughthelinesformatriangle,there’snoguaranteeit’sanequilateraltriangle.Anotherstabinthedarkwouldbetodrawsomecircleswiththecompass.But

where?Aroundoneoftheendpoints?

Oraroundapointinsidethesegment?

Thatsecondchoicelooksunpromising,sincethere’snoreasontopickoneinteriorpointoveranother.Solet’sgobacktodrawingcirclesaroundendpoints.

Unfortunatelythere’sstillalotofarbitrarinesshere.Howbigshouldthecirclesbe?Nothing’spoppingoutatusyet.Afterafewmoreminutesofflailingaroundlikethis,frustration(andan

impendingheadache)maytemptustogiveup.Butifwedon’t,wemightgetluckyandrealizethere’sonlyonenaturalcircletodraw.Let’sseewhathappensifweputthesharppointofthecompassatoneendofthesegmentandthepencilattheother,andthentwirlthecompassthroughafullcircle.We’dgetthis:

Ofcourse,ifwe’dusedtheotherendpointasthecenter,we’dgetthis:

Howaboutdrawingbothcirclesatthesametime—fornogoodreason,justnoodling?

Diditjusthityou?Ashiverofinsight?Lookatthediagramagain.There’sa

curvyversionofanequilateraltrianglestaringatus.Itstopcorneristhepointwherethecirclesintersect.

Sonowlet’sturnthatintoarealtriangle,withstraightsides,bydrawinglinesfromtheintersectionpointtotheendpointsoftheoriginalsegment.Theresultingtrianglesurelooksequilateral.

Havingallowedintuitiontoguideusthisfar,nowandonlynowisittimefor

logictotakeoverandfinishtheproof.Forclarity,let’spanbacktothefulldiagramandlabelthepointsofinterestA,B,andC.

Theproofalmostprovesitself.ThesidesACandBChavethesamelengthastheoriginalsegmentAB,sincethat’showweconstructedthecircles;weusedABastheradiusforboth.SinceACandBCarealsoradii,theytoohavethislength,soallthreelengthsareequal,andthetriangleisequilateral.QED.Thisargumenthasbeenaroundforcenturies.Infact,it’sEuclid’sopening

shot—thefirstpropositioninBookIoftheElements.Butthetendencyhasalwaysbeentopresentthefinaldiagramwiththeartfulcirclesalreadyinplace,whichrobsthestudentofthejoyofdiscoveringthem.That’sapedagogicalmistake.Thisisaproofthatanyonecanfind.Itcanbenewwitheachgeneration,ifweteachitright.Thekeytothisproof,clearly,wastheinspiredconstructionofthetwocircles.

Amorefamousresultingeometrycanbeprovenbyasimilarlydeftconstruction.It’sthetheoremthattheanglesofatriangleaddupto180degrees.Inthiscase,thebestproofisnotEuclid’sbutanearlieroneattributedtothe

Pythagoreans.Itgoeslikethis.Consideranytriangle,andcallitsanglesa,b,c.

Drawalineparalleltothebase,goingthroughthetopcorner.

Nowweneedtodigressforasecondtorecallapropertyofparallellines:if

anotherlinecutsacrosstwoparallellineslikeso,

theangleslabeledhereasa(knowninthetradeasalternateinteriorangles)areequal.Let’sapplythisfacttotheconstructionaboveinwhichwedrewalinethrough

thetopcornerofatriangleparalleltoitsbase.

Byinvokingtheequalityofalternateinteriorangles,weseethattheanglejusttotheleftofthetopcornermustbeequaltoa.Likewisetheangleatthetoprightisequaltob.Sotheanglesa,b,andctogetherformastraightangle—anangleof180degrees—whichiswhatwesoughttoprove.Thisisoneofthemostbracingargumentsinallofmathematics.Itopenswith

aboltoflightning,theconstructionoftheparallelline.Oncethatlinehasbeendrawn,theproofpracticallyrisesoffthetableandwalksbyitself,likeDr.Frankenstein’screation.Andwhoknows?Ifwehighlightthisothersideofgeometry—itsplayful,

intuitiveside,whereasparkofimaginationcanbequicklyfannedintoaproof—maybesomedayallstudentswillremembergeometryastheclasswheretheylearnedtobelogicalandcreative.

14.TheConicConspiracy

WHISPERINGGALLERIESAREremarkableacousticspacesfoundbeneathcertaindomes,vaults,orcurvedceilings.AfamousoneislocatedoutsidetheOysterBarrestaurantinNewYorkCity’sGrandCentralStation.It’safunplacetotakeadate:thetwoofyoucanexchangesweetnothingswhileyou’refortyfeetapartandseparatedbyabustlingpassageway.You’llheareachotherclearly,butthepassersbywon’thearawordyou’resaying.

Toproducethiseffect,thetwoofyoushouldstandatdiagonallyopposite

cornersofthespace,facingthewall.Thatputsyoueachnearafocus,aspecialpointatwhichthesoundofyourvoicegetsfocusedasitreflectsoffthepassageway’scurvedwallsandceiling.Ordinarily,thesoundwavesyouproducetravelinalldirectionsandbounceoffthewallsatdisparatetimesandplaces,scramblingthemsomuchthattheyareinaudiblewhentheyarriveattheearofalistenerfortyfeetaway(whichiswhythepassersbycan’thearwhatyou’resaying).Butwhenyouwhisperatafocus,thereflectedwavesallarriveatthesametimeattheotherfocus,thusreinforcingoneanotherandallowingyour

wordstobeheard.Ellipsesdisplayasimilarflairforfocusing,thoughinamuchsimplerform.If

wefashionareflectorintheshapeofanellipse,twoparticularpointsinsideit(markedasF1andF2inthefigurebelow)willactasfociinthefollowingsense:alltheraysemanatingfromalightsourceatoneofthosepointswillbereflectedtotheother.

Letmetrytoconveyhowamazingthatisbyrestatingitinafewways.SupposethatDarthandLukeenjoyplayinglasertaginanellipticalarenawith

mirroredwalls.Bothhaveagreednottoaimdirectlyattheother—theyhavetozapeachotherwithbankshots.Darth,notbeingparticularlyastuteaboutgeometryoroptics,suggeststheyeachstandatafocalpoint.“Fine,”saysLuke,“aslongasIgettotakethefirstshot.”Well,itwouldn’tbemuchofaduel,becauseLukecan’tmiss!Nomatterhowfoolishlyheaimshislaser,he’llalwaystagDarth.Everyshot’sawinner.Orifpoolisyourgame,imagineplayingbilliardsonanellipticaltablewitha

pocketatonefocus.Tosetupatrickshotthatisguaranteedtoscratcheverytime,placethecueballattheotherfocus.Nomatterhowyoustriketheballandnomatterwhereitcaromsoffthecushion,it’llalwaysgointothepocket.

Paraboliccurvesandsurfaceshaveanimpressivefocusingpoweroftheir

own:eachcantakeparallelincomingwavesandfocusthematasinglepoint.Thisfeatureoftheirgeometryhasbeenveryusefulinsettingswherelightwaves,soundwaves,orothersignalsneedtobeamplified.Forinstance,parabolicmicrophonescanbeusedtopickuphushedconversationsandarethereforeofinterestinsurveillance,espionage,andlawenforcement.Theyalsocomeinhandyinnaturerecording,tocapturebirdsongsoranimalgrunts,andintelevisedsports,tolisteninonacoachcursingatanofficial.Parabolicantennascanamplifyradiowavesinthesameway,whichiswhysatellitedishesforTVreceptionandgiantradiotelescopesforastronomyhavetheircharacteristicallycurvedshapes.Thisfocusingpropertyofparabolasisjustasusefulwhendeployedinreverse.

Supposeyouwanttomakeastronglydirectionalbeamoflight,likethatneededforspotlightsorcarheadlights.Onitsown,abulb—evenapowerfulone—typicallywouldn’tbegoodenough;itwastestoomuchlightbyshiningitinalldirections.Butnowplacethatbulbatthefocusofaparabolicreflector,andvoilà!Theparabolacreatesadirectionalbeamautomatically.Ittakesthebulb’sraysand,byreflectingthemofftheparabola’ssilveredinnersurface,makesthemallparallel.

Onceyoubegintoappreciatethefocusingabilitiesofparabolasandellipses,

youcan’thelpbutwonderifsomethingdeeperisatworkhere.Arethesecurvesrelatedinsomemorefundamentalway?Mathematiciansandconspiracytheoristshavethismuchincommon:we’re

suspiciousofcoincidences—especiallyconvenientones.Therearenoaccidents.Thingshappenforareason.Whilethismindsetmaybejustatouchparanoidwhenappliedtoreallife,it’saperfectlysanewaytothinkaboutmath.Intheidealworldofnumbersandshapes,strangecoincidencesusuallyarecluesthatwe’remissingsomething.Theysuggestthepresenceofhiddenforcesatwork.Solet’slookmoredeeplyintothepossiblelinksbetweenparabolasand

ellipses.Atfirstglancetheyseemanunlikelycouple.Parabolasarearch-shapedandexpansive,stretchingoutatbothends.Ellipsesareoval-shaped,likesquashedcircles,closedandcompact.

Butassoonasyoumovebeyondtheirappearancesandprobetheirinneranatomy,youstarttonoticehowsimilartheyare.Theybothbelongtoaroyalfamilyofcurves,agenetictiethatbecomesobvious—onceyouknowwhattolookfor.Toexplainhowthey’rerelated,wehavetorecallwhat,exactly,thesecurves

are.Aparabolaiscommonlydefinedasthesetofallpointsequidistantfroma

givenpointandagivenlinenotcontainingthatpoint.That’samouthfulofadefinition,butit’sactuallyprettyeasytounderstandonceyoutranslateitintopictures.CallthegivenpointF,for“focus,”andcallthelineL.

Now,accordingtothedefinition,aparabolaconsistsofallthepointsthatliejustasfarfromFastheydofromL.Forexample,thepointPlyingstraightdownfromF,halfwaytoL,wouldcertainlyqualify:

InfinitelymanyotherpointsP1,P2,...worktoo.Theylieofftoeithersidelikethis:

Here,thepointP1liesatthesamedistance,d1,fromthelineasitdoesfromthefocus.ThesameistrueforthepointP2,exceptnowtheshareddistanceissomeothernumber,d2.Takentogether,allthepointsPwiththispropertyformthe

parabola.ThereasonforcallingFthefocusbecomesclearifwethinkoftheparabolaas

acurvedmirror.Itturnsout(thoughIwon’tproveit)thatifyoushineabeamoflightstraightataparabolicmirror,allthereflectedrayswillintersectsimultaneouslyatthepointF,producinganintenselyfocusedspotoflight.

Itworkssomethinglikethoseoldsuntanningreflectorsthatfriedagenerationoffaces,backinthedaysbeforeanybodyworriedaboutskincancer.Nowlet’sturntothecorrespondingstoryforellipses.Anellipseisdefinedas

thesetofpointsthesumofwhosedistancesfromtwogivenpointsisaconstant.Whenrephrasedinmoredown-to-earthlanguage,thisdescriptionprovidesarecipefordrawinganellipse.Getapencil,asheetofpaper,acorkboard,twopushpins,andapieceofstring.Laythepaperontheboard.Pintheendsofthestringdownthroughthepaper,beingcarefultoleavesomeslack.Thenpullthestringtautwiththepenciltoformanangle,asshownbelow.Begindrawing,keepingthestringtaut.Afterthepencilhasgoneallthewayaroundbothpinsandreturnedtoitsstartingpoint,theresultingcurveisanellipse.

Noticehowthisrecipeimplementsthedefinitionabove,wordforword.The

pinsplaytheroleofthetwogivenpoints.Andthesumofthedistancesfromthemtoapointonthecurvealwaysremainsconstant,nomatterwherethepencilis—becausethosedistancesalwaysadduptothelengthofthestring.

Sowherearetheellipse’sfociinthisconstruction?Atthepins.Again,Iwon’tproveit,butthosearethepointsthatallowLukeandDarthtoplaytheirgameofcan’t-misslasertagandthatgiverisetoascratch-every-timepooltable.Parabolasandellipses:Whyisitthatthey,andonlythey,havesuchfantastic

powersoffocusing?What’sthesecrettheyshare?They’rebothcross-sectionsofthesurfaceofacone.Acone?Ifyoufeellikethatjustcameoutofnowhere,that’spreciselythe

point.Thecone’sroleinallthishasbeenhiddensofar.Toseehowit’simplicated,imagineslicingthroughaconewithameat

cleaver,somewhatlikecuttingthroughasalami,atprogressivelysteeperangles.Ifthecutislevel,thecurveofintersectionisacircle.

Ifinsteadtheconeisslicedonagentlebias,theresultingcurvebecomesanellipse.

Asthecutsbecomesteeper,theellipsegetslongerandslimmerinitsproportions.Atacriticalangle,whenthebiasgetssosteepthatitmatchestheslopeoftheconeitself,theellipseturnsintoaparabola.

Sothat’sthesecret:aparabolaisanellipseindisguise,inacertainlimitingsense.Nowonderitsharestheellipse’smarvelousfocusingability.It’sbeenpasseddownthroughthebloodline.Infact,circles,ellipses,andparabolasareallmembersofalarger,tight-knit

family.They’recollectivelyknownasconicsections—curvesobtainedbycuttingthesurfaceofaconewithaplane.Plusthere’soneadditionalsibling:iftheconeisslicedverysteeply,onabiasgreaterthanitsownslope,theresultingincisionformsacurvecalledahyperbola.Unlikealltheothers,itcomesintwopieces,notone.

Thesefourtypesofcurvesappearevenmoreintimatelyrelatedwhenviewed

fromothermathematicalperspectives.Inalgebra,theyariseasthegraphsofsecond-degreeequations

wheretheconstantsA,B,C,...determinewhetherthegraphisacircle,ellipse,parabola,orhyperbola.Incalculus,theyariseastrajectoriesofobjectstuggedbytheforceofgravity.Soit’snoaccidentthatplanetsmoveinellipticalorbitswiththesunatone

focus;orthatcometssailthroughthesolarsystemonelliptic,parabolic,orhyperbolictrajectories;orthatachild’sballtossedtoaparentfollowsaparabolicarc.They’reallmanifestationsoftheconicconspiracy.Focusonthatthenexttimeyouplaycatch.

15.SineQuaNon

MYDADHADafriendnamedDavewhoretiredtoJupiter,Florida.WevisitedhimforafamilyvacationwhenIwasabouttwelveorso,andsomethingheshowedusmadeanindelibleimpressiononme.Davelikedtochartthetimesoftheglorioussunrisesandsunsetshecould

watchfromhisdeckallyearlong.Everydayhemarkedtwodotsonhischart,andaftermanymonthshenoticedsomethingcurious.Thetwocurveslookedlikeopposingwaves.Oneusuallyrosewhiletheotherfell;whensunrisewasearlier,sunsetwaslater.

Buttherewereexceptions.ForthelastthreeweeksofJuneandformostofDecemberandearlyJanuary,sunriseandsunsetbothcamelatereachday,givingthewavesaslightlylopsidedappearance.Still,themessageofthecurveswasunmistakable:theoscillatinggapbetween

themshowedthedaysgrowinglongerandshorterwiththechangingoftheseasons.Bysubtractingthelowercurvefromtheupperone,Davealsofiguredouthowthenumberofhoursofdaylightvariedthroughouttheyear.Tohisamazement,thiscurvewasn’tlopsidedatall.Itwasbeautifullysymmetrical.

Whathewaslookingatwasanearlyperfectsinewave.Youmayremember

havingheardaboutsinewavesifyoutooktrigonometryinhighschool,althoughyourteacherprobablytalkedmoreaboutthesinefunction,afundamentaltoolforquantifyinghowthesidesandanglesofatrianglerelatetooneanother.Thatwastrigonometry’soriginalkillerapp,ofgreatutilitytoancientastronomersandsurveyors.Yettrigonometry,belyingitsmuchtoomodestname,nowgoesfarbeyondthe

measurementoftriangles.Byquantifyingcirclesaswell,ithaspavedthewayfortheanalysisofanythingthatrepeats,fromoceanwavestobrainwaves.It’sthekeytothemathematicsofcycles.Toseehowtrigconnectscircles,triangles,andwaves,imaginealittlegirl

goingroundandroundonaFerriswheel.

Sheandhermom,whobothhappentobemathematicallyinclined,havedecidedthisisaperfectopportunityforanexperiment.ThegirltakesaGPSgadgetwithherontheridetorecordheraltitude,momentbymoment,asthewheelcarriesherup,thenthrillinglyoverthetopandbacktowardtheground,thenupandaroundagain,andsoon.Theresultslooklikethis:

Thisshapeisasinewave.Itariseswheneveronetracksthehorizontalorverticalexcursionsofsomething—orsomeone—movinginacircle.Howisthissinewaverelatedtothesinefunctiondiscussedintrigclass?Well,

supposeweexamineasnapshotofthegirl.Atthemomentshown,she’satsomeangle,callita,relativetothedashedlineinthediagram.

Forconvenience,let’ssupposethehypotenuseoftherighttriangleshown—whichisalsothewheel’sradius—is1unitlong.Thensina(pronounced“sineofa”)tellsushowhighthegirlis.Moreprecisely,sinaisdefinedasthegirl’saltitudemeasuredfromthecenterofthewheelwhenshe’slocatedattheanglea.Asshegoesroundandround,herangleawillprogressivelyincrease.

Eventuallyitexceeds90degrees,atwhichpointwecannolongerregardaasanangleinarighttriangle.Doesthatmeantrignolongerapplies?No.Undeterredasusual,mathematicianssimplyenlargethedefinitionofthe

sinefunctiontoallowforanyangle,notjustthoselessthan90degrees,andthendefine“sina”asthegirl’sheightaboveorbelowthecenterofthecircle.Thecorrespondinggraphofsina,asakeepsincreasing(orevengoesnegative,ifthewheelreversesdirection),iswhatwemeanbyasinewave.Itrepeatsitselfeverytimeachangesby360degrees,correspondingtoafullrevolution.Thissamesortofconversionofcircularmotionintosinewavesisapervasive,

thoughoftenunnoticed,partofourdailyexperience.Itcreatesthehumofthefluorescentlightsoverheadinouroffices,areminderthatsomewhereinthepowergrid,generatorsarespinningatsixtycyclespersecond,convertingtheirrotarymotionintoalternatingcurrent,theelectricalsinewavesonwhichmodernlifedepends.WhenyouspeakandIhear,bothourbodiesareusingsinewaves—yoursinthevibrationsofyourvocalcordstoproducethesounds,andmineintheswayingofthehaircellsinmyearstoreceivethem.Ifweopenourheartstothesesinewavesandtuneintotheirsilentthrumming,theyhavethepowertomoveus.There’ssomethingalmostspiritualaboutthem.Whenaguitarstringispluckedorwhenchildrenjiggleajumprope,theshape

thatappearsisasinewave.Theripplesonapond,theridgesofsanddunes,the

stripesofazebra—allaremanifestationsofnature’smostbasicmechanismofpatternformation:theemergenceofsinusoidalstructurefromabackgroundofblanduniformity.

Therearedeepmathematicalreasonsforthis.Wheneverastateoffeaturelessequilibriumlosesstability—forwhateverreason,andbywhateverphysical,biological,orchemicalprocess—thepatternthatappearsfirstisasinewave,oracombinationofthem.Sinewavesaretheatomsofstructure.They’renature’sbuildingblocks.

Withoutthemthere’dbenothing,givingnewmeaningtothephrase“sinequanon.”Infact,thewordsareliterallytrue.Quantummechanicsdescribesrealatoms,

andhenceallofmatter,aspacketsofsinewaves.Evenatthecosmologicalscale,sinewavesformtheseedsofallthatexists.Astronomershaveprobedthespectrum(thepatternofsinewaves)ofthecosmicmicrowavebackgroundandfoundthattheirmeasurementsmatchthepredictionsofinflationarycosmology,theleadingtheoryforthebirthandgrowthoftheuniverse.SoitseemsthatoutofafeaturelessBigBang,primordialsinewaves—ripplesinthedensityofmatterandenergy—emergedspontaneouslyandspawnedthestuffofthecosmos.Stars.Galaxies.And,ultimately,littlekidsridingFerriswheels.

16.TakeIttotheLimit

INMIDDLESCHOOLmyfriendsandIenjoyedchewingontheclassicconundrums.Whathappenswhenanirresistibleforcemeetsanimmovableobject?Easy—theybothexplode.Philosophy’strivialwhenyou’rethirteen.Butonepuzzlebotheredus:Ifyoukeepmovinghalfwaytothewall,willyou

evergetthere?Somethingaboutthisonewasdeeplyfrustrating,thethoughtofgettingcloserandcloserandyetneverquitemakingit.(There’sprobablyametaphorforteenageangstintheresomewhere.)Anotherconcernwasthethinlyveiledpresenceofinfinity.Toreachthewallyou’dneedtotakeaninfinitenumberofsteps,andbytheendthey’dbecomeinfinitesimallysmall.Whoa.Questionslikethishavealwayscausedheadaches.Around500B.C.,Zenoof

Eleaposedfourparadoxesaboutinfinitythatpuzzledhiscontemporariesandthatmayhavebeenpartlytoblameforinfinity’sbanishmentfrommathematicsforcenturiesthereafter.InEuclideangeometry,forexample,theonlyconstructionsallowedwerethosethatinvolvedafinitenumberofsteps.Theinfinitewasconsideredtooineffable,toounfathomable,andtoohardtomakelogicallyrigorous.ButArchimedes,thegreatestmathematicianofantiquity,realizedthepower

oftheinfinite.Heharnessedittosolveproblemsthatwereotherwiseintractableandintheprocesscameclosetoinventingcalculus—nearly2,000yearsbeforeNewtonandLeibniz.Inthecomingchapterswe’lldelveintothegreatideasattheheartofcalculus.

ButfornowI’dliketobeginwiththefirstbeautifulhintsofthem,visibleinancientcalculationsaboutcirclesandpi.Let’srecallwhatwemeanby“pi.”It’saratiooftwodistances.Oneofthemis

thediameter,thedistanceacrossthecirclethroughitscenter.Theotheristhecircumference,thedistancearoundthecircle.“Pi”isdefinedastheirratio,thecircumferencedividedbythediameter.

Ifyou’reacarefulthinker,youmightbeworriedaboutsomethingalready.

Howdoweknowthatpiisthesamenumberforallcircles?Coulditbedifferentforbigcirclesandlittlecircles?Theanswerisno,buttheproofisn’ttrivial.Here’sanintuitiveargument.Imagineusingaphotocopiertoreduceanimageofacircleby,say,50percent.

Thenalldistancesinthepicture—includingthecircumferenceandthediameter—wouldshrinkinproportionby50percent.Sowhenyoudividethenewcircumferencebythenewdiameter,that50percentchangewouldcancelout,leavingtheratiobetweenthemunaltered.Thatratioispi.Ofcourse,thisdoesn’ttellushowbigpiis.Simpleexperimentswithstrings

anddishesaregoodenoughtoyieldavaluenear3,orifyou’remoremeticulous,.Butsupposewewanttofindpiexactlyoratleastapproximateittoany

desiredaccuracy.Whatthen?Thiswastheproblemthatconfoundedtheancients.BeforeturningtoArchimedes’sbrilliantsolution,weshouldmentionone

otherplacewherepiappearsinconnectionwithcircles.Theareaofacircle(theamountofspaceinsideit)isgivenbytheformula

HereAisthearea,πistheGreekletterpi,andristheradiusofthecircle,definedashalfthediameter.

Allofusmemorizedthisformulainhighschool,butwheredoesitcomefrom?It’snotusuallyproveningeometryclass.Ifyouwentontotakecalculus,youprobablysawaproofofitthere,butisitreallynecessarytousecalculustoobtainsomethingsobasic?Yes,itis.Whatmakestheproblemdifficultisthatcirclesareround.Iftheyweremade

ofstraightlines,there’dbenoissue.Findingtheareasoftrianglesandsquaresiseasy.Butworkingwithcurvedshapeslikecirclesishard.Thekeytothinkingmathematicallyaboutcurvedshapesistopretendthey’re

madeupoflotsoflittlestraightpieces.That’snotreallytrue,butitworks...aslongasyoutakeittothelimitandimagineinfinitelymanypieces,eachinfinitesimallysmall.That’sthecrucialideabehindallofcalculus.Here’sonewaytouseittofindtheareaofacircle.Beginbychoppingthe

areaintofourequalquarters,andrearrangethemlikeso.

Thestrangescallopedshapeonthebottomhasthesameareaasthecircle,thoughthatmightseemprettyuninformativesincewedon’tknowitsareaeither.

Butatleastweknowtwoimportantfactsaboutit.First,thetwoarcsalongitsbottomhaveacombinedlengthequaltohalfthecircumferenceoftheoriginalcircle(becausetheotherhalfofthecircumferenceisaccountedforbythetwoarcsontop).Sincethewholecircumferenceispitimesthediameter,halfofitispitimeshalfthediameteror,equivalently,pitimestheradiusr.That’swhythediagramaboveshowsπrasthecombinedlengthofthearcsatthebottomofthescallopedshape.Second,thestraightsidesofthesliceshavealengthofr,sinceeachofthemwasoriginallyaradiusofthecircle.Next,repeattheprocess,butthistimewitheightslices,stackedalternatelyas

before.

Thescallopedshapelooksabitlessbizarrenow.Thearcsonthetopandthebottomarestillthere,butthey’renotaspronounced.Anotherimprovementistheleftandrightsidesofthescallopedshapedon’ttiltasmuchastheyusedto.Despitethesechanges,thetwofactsabovecontinuetohold:thearcsonthebottomstillhaveanetlengthofπrandeachsidestillhasalengthofr.Andofcoursethescallopedshapestillhasthesameareaasbefore—theareaofthecirclewe’reseeking—sinceit’sjustarearrangementofthecircle’seightslices.Aswetakemoreandmoreslices,somethingmarveloushappens:the

scallopedshapeapproachesarectangle.Thearcsbecomeflatterandthesidesbecomealmostvertical.

Inthelimitofinfinitelymanyslices,theshapeisarectangle.Justasbefore,thetwofactsstillhold,whichmeansthisrectanglehasabottomofwidthπrandasideofheightr.

Butnowtheproblemiseasy.Theareaofarectangleequalsitswidthtimesitsheight,somultiplyingπrtimesryieldsanareaofπr²fortherectangle.Andsincetherearrangedshapealwayshasthesameareaasthecircle,that’stheanswerforthecircletoo!What’ssocharmingaboutthiscalculationisthewayinfinitycomestothe

rescue.Ateveryfinitestage,thescallopedshapelooksweirdandunpromising.Butwhenyoutakeittothelimit—whenyoufinallygettothewall—itbecomessimpleandbeautiful,andeverythingbecomesclear.That’showcalculusworksatitsbest.Archimedesusedasimilarstrategytoapproximatepi.Hereplacedacircle

withapolygonwithmanystraightsidesandthenkeptdoublingthenumberofsidestogetclosertoperfectroundness.Butratherthansettlingforanapproximationofuncertainaccuracy,hemethodicallyboundedpibysandwichingthecirclebetweeninscribedandcircumscribedpolygons,asshown

belowfor6-,12-,and24-sidedfigures.

ThenheusedthePythagoreantheoremtoworkouttheperimetersoftheseinnerandouterpolygons,startingwiththehexagonandbootstrappinghiswayupto12,24,48,and,ultimately,96sides.Theresultsforthe96-gonsenabledhimtoprovethat

Indecimalnotation(whichArchimedesdidn’thave),thismeanspiisbetween3.1408and3.1429.Thisapproachisknownasthemethodofexhaustionbecauseofthewayit

trapstheunknownnumberpibetweentwoknownnumbersthatsqueezeitfromeitherside.Theboundstightenwitheachdoubling,thusexhaustingthewiggleroomforpi.Inthelimitofinfinitelymanysides,boththeupperandlowerboundswould

convergetopi.Unfortunately,thislimitisn’tassimpleastheearlierone,wherethescallopedshapemorphedintoarectangle.Sopiremainsaselusiveasever.Wecandiscovermoreandmoreofitsdigits—thecurrentrecordisover2.7trilliondecimalplaces—butwe’llneverknowitcompletely.Asidefromlayingthegroundworkforcalculus,Archimedestaughtusthe

powerofapproximationanditeration.Hebootstrappedagoodestimateintoabetterone,usingmoreandmorestraightpiecestoapproximateacurvedobjectwithincreasingaccuracy.Morethantwomillennialater,thisstrategymaturedintothemodernfieldof

numericalanalysis.Whenengineersusecomputerstodesigncarsthatareoptimallystreamlined,orwhenbiophysicistssimulatehowanewchemotherapy

druglatchesontoacancercell,theyareusingnumericalanalysis.Themathematiciansandcomputerscientistswhopioneeredthisfieldhavecreatedhighlyefficient,repetitivealgorithms,runningbillionsoftimespersecond,thatenablecomputerstosolveproblemsineveryaspectofmodernlife,frombiotechtoWallStreettotheInternet.Ineachcase,thestrategyistofindaseriesofapproximationsthatconvergetothecorrectanswerasalimit.Andthere’snolimittowherethat’lltakeus.

PartFourCHANGE

17.ChangeWeCanBelieveIn

LONGBEFOREIknewwhatcalculuswas,Isensedtherewassomethingspecialaboutit.Mydadhadspokenaboutitinreverentialtones.Hehadn’tbeenabletogotocollege,beingachildoftheDepression,butsomewherealongtheline,maybeduringhistimeintheSouthPacificrepairingB-24bomberengines,he’dgottenafeelforwhatcalculuscoulddo.Imagineamechanicallycontrolledbankofantiaircraftgunsautomaticallyfiringatanincomingfighterplane.Calculus,hesupposed,couldbeusedtotellthegunswheretoaim.EveryyearaboutamillionAmericanstudentstakecalculus.Butfarfewer

reallyunderstandwhatthesubjectisaboutorcouldtellyouwhytheywerelearningit.It’snottheirfault.Therearesomanytechniquestomasterandsomanynewideastoabsorbthattheoverallframeworkiseasytomiss.Calculusisthemathematicsofchange.Itdescribeseverythingfromthespread

ofepidemicstothezigsandzagsofawell-throwncurveball.Thesubjectisgargantuan—andsoareitstextbooks.Manyexceedathousandpagesandworknicelyasdoorstops.Butwithinthatbulkyou’llfindtwoideasshiningthrough.Alltherest,as

RabbiHillelsaidofthegoldenrule,isjustcommentary.Thosetwoideasarethederivativeandtheintegral.Eachdominatesitsownhalfofthesubject,named,respectively,differentialandintegralcalculus.Roughlyspeaking,thederivativetellsyouhowfastsomethingischanging;

theintegraltellsyouhowmuchit’saccumulating.Theywereborninseparatetimesandplaces:integralsinGreecearound250B.C.;derivativesinEnglandandGermanyinthemid-1600s.YetinatwiststraightoutofaDickensnovel,they’veturnedouttobebloodrelatives—thoughittookalmosttwomillenniaforanyonetoseethefamilyresemblance.Thenextchapterwillexplorethatastonishingconnection,aswellasthe

meaningofintegrals.Butfirst,tolaythegroundwork,let’slookatderivatives.Derivativesareallaroundus,evenifwedon’trecognizethemassuch.For

example,theslopeofarampisaderivative.Likeallderivatives,itmeasuresarateofchange—inthiscase,howfaryou’regoingupordownforeverystepyoutake.Asteepramphasalargederivative.Awheelchair-accessibleramp,withitsgentlegradient,hasasmallderivative.Everyfieldhasitsownversionofaderivative.Whetheritgoesbymarginal

returnorgrowthrateorvelocityorslope,aderivativebyanyothernamestillsmellsassweet.Unfortunately,manystudentsseemtocomeawayfromcalculus

withamuchnarrowerinterpretation,regardingthederivativeassynonymouswiththeslopeofacurve.Theirconfusionisunderstandable.It’scausedbyourrelianceongraphsto

expressquantitativerelationships.Byplottingyversusxtovisualizehowonevariableaffectsanother,allscientiststranslatetheirproblemsintothecommonlanguageofmathematics.Therateofchangethatreallyconcernsthem—aviralgrowthrate,ajet’svelocity,orwhatever—thengetsconvertedintosomethingmuchmoreabstractbuteasiertopicture:aslopeonagraph.Likeslopes,derivativescanbepositive,negative,orzero,indicatingwhether

somethingisrising,falling,orlevelingoff.ConsiderMichaelJordanflyingthroughtheairbeforemakingoneofhisthunderousdunks.

Justafterliftoff,hisverticalvelocity(therateatwhichhiselevationchangesin

timeand,thus,anotherderivative)ispositive,becausehe’sgoingup.Hiselevationisincreasing.Onthewaydown,thisderivativeisnegative.Andatthehighestpointofhisjump,whereheseemstohangintheair,hiselevationismomentarilyunchangingandhisderivativeiszero.Inthatsensehetrulyishanging.There’samoregeneralprincipleatworkhere—thingsalwayschangeslowest

atthetoporthebottom.It’sespeciallynoticeablehereinIthaca.Duringthedarkestdepthsofwinter,thedaysarenotjustunmercifullyshort;theybarelyimprovefromonetothenext.Whereasoncespringstartspopping,thedayslengthenrapidly.Allofthismakessense.Changeismostsluggishattheextremespreciselybecausethederivativeiszerothere.Thingsstandstill,momentarily.Thiszero-derivativepropertyofpeaksandtroughsunderliessomeofthemost

practicalapplicationsofcalculus.Itallowsustousederivativestofigureoutwhereafunctionreachesitsmaximumorminimum,anissuethatariseswheneverwe’relookingforthebestorcheapestorfastestwaytodosomething.Myhigh-schoolcalculusteacher,Mr.Joffray,hadaknackformakingsuch

“max-min”questionscomealive.Onedayhecameboundingintoclassandbegantellingusabouthishikethroughasnow-coveredfield.Thewindhadapparentlyblownalotofsnowacrosspartofthefield,blanketingitheavilyandforcinghimtowalkmuchmoreslowlythere,whiletherestofthefieldwasclear,allowinghimtostridethroughiteasily.Inasituationlikethat,hewonderedwhatpathahikershouldtaketogetfrompointAtopointBasquicklyaspossible.

Onethoughtwouldbetotrudgestraightacrossthedeepsnow,tocutdownon

theslowestpartofthehike.Thedownside,though,istherestofthetripwilltakelongerthanitwouldotherwise.

AnotherstrategyistoheadstraightfromAtoB.That’scertainlytheshortestdistance,butitdoescostextratimeinthemostarduouspartofthetrip.

Withdifferentialcalculusyoucanfindthebestpath.It’sacertainspecificcompromisebetweenthetwopathsconsideredabove.

Theanalysisinvolvesfourmainsteps.(Forthosewho’dliketoseethedetails,

referencesaregiveninthenoteson[>].)First,noticethatthetotaltimeoftravel—whichiswhatwe’retryingto

minimize—dependsonwherethehikeremergesfromthesnow.Hecouldchoosetoemergeanywhere,solet’sconsiderallhispossibleexitpointsasavariable.Eachoftheselocationscanbecharacterizedsuccinctlybyspecifyingasinglenumber:thedistancexwherethehikeremergesfromthesnow.

(Implicitly,thetraveltimealsodependsonthelocationsofAandBandonthehiker’sspeedsinbothpartsofthefield,butallthoseparametersaregiven.Theonlythingunderthehiker’scontrolisx.)Second,givenachoiceofxandtheknownlocationsofthestartingpointA

andthedestinationB,wecancalculatehowmuchtimethehikerspendswalkingthroughthefastandslowpartsofthefield.Foreachlegofthetrip,thiscalculationrequiresthePythagoreantheoremandtheoldalgebramantra“distanceequalsratetimestime.”Addingthetimesforbothlegstogetherthenyieldsaformulaforthetotaltraveltime,T,asafunctionofx.Third,wegraphTversusx.Thebottomofthecurveisthepointwe’reseeking

—itcorrespondstotheleasttimeoftravelandhencethefastesttrip.

Fourth,tofindthislowestpoint,weinvokethezero-derivativeprinciple

mentionedabove.WecalculatethederivativeofT,setitequaltozero,andsolveforx.Thesefourstepsrequireacommandofgeometry,algebra,andvarious

derivativeformulasfromcalculus—skillsequivalenttofluencyinaforeignlanguageand,therefore,stumblingblocksformanystudents.Butthefinalanswerisworththestruggle.Itrevealsthatthefastestpathobeys

arelationshipknownasSnell’slaw.What’sspookyisthatnatureobeysittoo.Snell’slawdescribeshowlightraysbendwhentheypassfromairintowater,

astheydowhenthesunshinesintoaswimmingpool.Lightmovesmoreslowlyinwater,muchlikethehikerinthesnow,anditbendsaccordinglytominimizeitstraveltime.Similarly,lightbendswhenittravelsfromairintoglassorplastic,aswhenitrefractsthroughyoureyeglasslenses.Theeeriepointisthatlightbehavesasifitwereconsideringallpossiblepaths

andthentakingthebestone.Nature—cuethethemefromTheTwilightZone—somehowknowscalculus.

18.ItSlices,ItDices

MATHEMATICALSIGNSANDsymbolsareoftencryptic,butthebestofthemoffervisualcluestotheirownmeaning.Thesymbolsforzero,one,andinfinityaptlyresembleanemptyhole,asinglemark,andanendlessloop:0,1,∞.Andtheequalsign,=,isformedbytwoparallellinesbecause,asitsoriginator,WelshmathematicianRobertRecorde,wrotein1557,“notwothingscanbemoreequal.”Incalculusthemostrecognizableiconistheintegralsign:

Itsgracefullinesareevocativeofamusicalcleforaviolin’sf-hole—afittingcoincidence,giventhatsomeofthemostenchantingharmoniesinmathematicsareexpressedbyintegrals.ButtherealreasonthatthemathematicianGottfriedLeibnizchosethissymbolismuchlesspoetic.It’ssimplyalong-neckedS,for“summation.”Asforwhat’sbeingsummed,thatdependsoncontext.Inastronomy,the

gravitationalpullofthesunontheEarthisdescribedbyanintegral.ItrepresentsthecollectiveeffectofalltheminusculeforcesgeneratedbyeachsolaratomattheirvaryingdistancesfromtheEarth.Inoncology,thegrowingmassofasolidtumorcanbemodeledbyanintegral.Socanthecumulativeamountofdrugadministeredduringthecourseofachemotherapyregimen.Tounderstandwhysumsliketheserequireintegralcalculusandnotthe

ordinarykindofadditionwelearnedingradeschool,let’sconsiderwhatchallengeswe’dfaceifweactuallytriedtocalculatethesun’sgravitationalpullontheEarth.Thefirstdifficultyisthatthesunisnotapoint...andneitheristheEarth.Bothofthemaregiganticballsmadeupofstupendousnumbersofatoms.EveryatominthesunexertsagravitationaltugoneveryatomintheEarth.Ofcourse,sinceatomsaretiny,theirmutualattractionsarealmostinfinitesimallysmall,yetbecausetherearealmostinfinitelymanyofthem,inaggregatetheycanstillamounttosomething.Somehowwehavetoaddthemallup.Butthere’sasecondandmoreseriousdifficulty:Thosepullsaredifferentfor

differentpairsofatoms.Somearestrongerthanothers.Why?Becausethe

strengthofgravitychangeswithdistance—theclosertwothingsare,themorestronglytheyattract.TheatomsonthefarsidesofthesunandtheEarthfeeltheleastattraction;thoseonthenearsidesfeelthestrongest;andthoseinbetweenfeelforcesofmiddlingstrength.Integralcalculusisneededtosumallthosechangingforces.Amazingly,itcanbedone—atleastintheidealizedlimitwherewetreattheEarthandthesunassolidspherescomposedofinfinitelymanypointsofcontinuousmatter,eachexertinganinfinitesimalattractionontheothers.Asinallofcalculus:infinityandlimitstotherescue!Historically,integralsarosefirstingeometry,inconnectionwiththeproblem

offindingtheareasofcurvedshapes.Aswesawinchapter16,theareaofacirclecanbeviewedasthesumofmanythinpieslices.Inthelimitofinfinitelymanyslices,eachofwhichisinfinitesimallythin,thoseslicescouldthenbecunninglyrearrangedintoarectanglewhoseareawasmucheasiertofind.Thatwasatypicaluseofintegrals.They’reallabouttakingsomethingcomplicatedandslicinganddicingittomakeiteasiertoaddup.Ina3-Dgeneralizationofthismethod,Archimedes(andbeforehim,Eudoxus,

around400B.C.)calculatedthevolumesofvarioussolidshapesbyreimaginingthemasstacksofmanywafersordisks,likeasalamislicedthin.Bycomputingthechangingvolumesofthevaryingslicesandtheningeniouslyintegratingthem—addingthembacktogether—theywereabletodeducethevolumeoftheoriginalwhole.Todaywestillaskbuddingmathematiciansandscientiststosharpentheir

skillsatintegrationbyapplyingthemtotheseclassicgeometryproblems.They’resomeofthehardestexercisesweassign,andalotofstudentshatethem,butthere’snosurerwaytohonethefacilitywithintegralsneededforadvancedworkineveryquantitativedisciplinefromphysicstofinance.Onesuchmind-benderconcernsthevolumeofthesolidcommontotwo

identicalcylinderscrossingatrightangles,likestovepipesinakitchen.

Ittakesanunusualgiftofimaginationtovisualizethisthree-dimensionalshape.Sothere’snoshameinadmittingdefeatandlookingforawaytomakeitmorepalpable.Todoso,youcanresorttoatrickmyhigh-schoolcalculusteacherused—takeatincanandcutthetopoffwithmetalshearstoformacylindricalcoringtool.ThencorealargeIdahopotatoorapieceofStyrofoamfromtwomutuallyperpendiculardirections.Inspecttheresultingshapeatyourleisure.Computergraphicsnowmakeitpossibletovisualizethisshapemoreeasily.

Remarkably,ithassquarecross-sections,eventhoughitwascreatedfromroundcylinders.

It’sastackofinfinitelymanylayers,eachawafer-thinsquare,taperingfromalargesquareinthemiddletoprogressivelysmalleronesandfinallytosinglepointsatthetopandbottom.Still,picturingtheshapeismerelythefirststep.Whatremainsistodetermine

itsvolume,bytallyingthevolumesofalltheseparateslices.Archimedesmanagedtodothis,butonlybyvirtueofhisastoundingingenuity.Heusedamechanicalmethodbasedonleversandcentersofgravity,ineffectweighingtheshapeinhismindbybalancingitagainstothershealreadyunderstood.Thedownsideofhisapproach,besidestheprohibitivebrillianceitrequired,wasthatitappliedtoonlyahandfulofshapes.Conceptualroadblockslikethisstumpedtheworld’sfinestmathematiciansfor

thenextnineteencenturies...untilthemid-1600s,whenJamesGregory,IsaacBarrow,IsaacNewton,andGottfriedLeibnizestablishedwhat’snowknownasthefundamentaltheoremofcalculus.Itforgedapowerfullinkbetweenthetwotypesofchangebeingstudiedincalculus:thecumulativechangerepresentedbyintegrals,andthelocalrateofchangerepresentedbyderivatives(thesubjectofchapter17).Byexposingthisconnection,thefundamentaltheoremgreatlyexpandedtheuniverseofintegralsthatcouldbesolved,anditreducedtheircalculationtogruntwork.Nowadayscomputerscanbeprogrammedtouseit—andsocanstudents.Withitshelp,eventhestovepipeproblemthatwasonceaworld-classchallengebecomesanexercisewithincommonreach.(ForthedetailsofArchimedes’sapproachaswellasthemodernone,consultthereferencesinthenoteson[>].)Beforecalculusandthefundamentaltheoremcamealong,onlythesimplest

kindsofnetchangecouldbeanalyzed.Whensomethingchangessteadily,ataconstantrate,algebraworksbeautifully.Thisisthedomainof“distanceequalsratetimestime.”Forexample,acarmovingatanunchangingspeedof60milesperhourwillsurelytravel60milesinthefirsthour,and120milesbytheendofthesecondhour.Butwhataboutchangethatproceedsatachangingrate?Suchchanging

changeisallaroundus—intheacceleratingdescentofapennydroppedfromatallbuilding,intheebbandflowofthetides,intheellipticalorbitsoftheplanets,inthecircadianrhythmswithinus.Onlycalculuscancopewiththecumulativeeffectsofchangesasnon-uniformasthese.FornearlytwomillenniaafterArchimedes,justonemethodexistedfor

predictingtheneteffectofchangingchange:addupthevaryingslices,bitbybit.Youweresupposedtotreattherateofchangeasconstantwithineachslice,theninvoketheanalogof“distanceequalsratetimestime”toinchforwardtotheend

ofthatslice,andrepeatuntilalltheslicesweredealtwith.Mostofthetimeitcouldn’tbedone.Theinfinitesumsweretoohard.Thefundamentaltheoremenabledalotoftheseproblemstobesolved—not

allofthem,butmanymorethanbefore.Itoftengaveashortcutforsolvingintegrals,atleastfortheelementaryfunctions(sumsandproductsofpowers,exponentials,logarithms,andtrigfunctions)thatdescribesomanyofthephenomenainthenaturalworld.Here’sananalogythatIhopewillshedsomelightonwhatthefundamental

theoremsaysandwhyit’ssohelpful.(MycolleagueCharliePeskinatNewYorkUniversitysuggestedit.)Imagineastaircase.Thetotalchangeinheightfromthetoptothebottomisthesumoftherisesofallthestepsinbetween.That’strueevenifsomeofthestepsrisemorethanothersandnomatterhowmanystepsthereare.Thefundamentaltheoremofcalculussayssomethingsimilarforfunctions—if

youintegratethederivativeofafunctionfromonepointtoanother,yougetthenetchangeinthefunctionbetweenthetwopoints.Inthisanalogy,thefunctionisliketheelevationofeachstepcomparedtogroundlevel.Therisesofindividualstepsarelikethederivative.Integratingthederivativeislikesummingtherises.Andthetwopointsarethetopandthebottom.Whyisthissohelpful?Supposeyou’regivenahugelistofnumberstosum,

asoccurswheneveryou’recalculatinganintegralbyslices.Ifyoucansomehowmanagetofindthecorrespondingstaircase—inotherwords,ifyoucanfindanelevationfunctionforwhichthosenumbersaretherises—thencomputingtheintegralisasnap.It’sjustthetopminusthebottom.That’sthegreatspeedupmadepossiblebythefundamentaltheorem.Andit’s

whywetortureallbeginningcalculusstudentswithmonthsoflearninghowtofindelevationfunctions,technicallycalledantiderivativesorindefiniteintegrals.Thisadvanceallowedmathematicianstoforecasteventsinachangingworldwithmuchgreaterprecisionthanhadeverbeenpossible.Fromthisperspective,thelastinglegacyofintegralcalculusisaVeg-O-Matic

viewoftheuniverse.Newtonandhissuccessorsdiscoveredthatnatureitselfunfoldsinslices.Virtuallyallthelawsofphysicsfoundinthepast300yearsturnedouttohavethischaracter,whethertheydescribethemotionsofparticlesortheflowofheat,electricity,air,orwater.Togetherwiththegoverninglaws,theconditionsineachsliceoftimeorspacedeterminewhatwillhappeninadjacentslices.Theimplicationswereprofound.Forthefirsttimeinhistory,rational

predictionbecamepossible...notjustonesliceatatimebut,withthehelpofthefundamentaltheorem,byleapsandbounds.

Sowe’relongoverduetoupdateoursloganforintegrals—from“Itslices,itdices”to“Recalculating.Abetterrouteisavailable.”

19.Allaboute

AFEWNUMBERSARESUCHCELEBRITIESthattheygobysingle-letterstagenames,somethingnotevenMadonnaorPrincecanmatch.Themostfamousisπ,thenumberformerlyknownas3.14159...Closebehindisi,theit-numberofalgebra,theimaginarynumbersoradicalit

changedwhatitmeanttobeanumber.NextontheAlist?Sayhellotoe.Nicknamedforitsbreakoutroleinexponentialgrowth,eis

nowtheZeligofadvancedmathematics.Itpopsupeverywhere,peekingoutfromthecornersofthestage,teasingusbyitspresenceinincongruousplaces.Forexample,alongwiththeinsightsitoffersaboutchainreactionsandpopulationbooms,ehasathingortwotosayabouthowmanypeopleyoushoulddatebeforesettlingdown.Butbeforewegettothat,whatise,exactly?Itsnumericalvalueis2.71828

...butthat’snotterriblyenlightening.Icouldtellyouthateequalsthelimitingnumberapproachedbythesum

aswetakemoreandmoreterms.Butthat’snotparticularlyhelpfuleither.Instead,let’slookateinaction.Imaginethatyou’vedeposited$1,000inasavingsaccountatabankthatpays

anincrediblygenerousinterestrateof100percent,compoundedannually.Ayearlater,youraccountwouldbeworth$2,000—theinitialdepositof$1,000plusthe100percentinterestonit,equaltoanother$1,000.Knowingasuckerwhenyouseeone,youaskthebankforevenmore

favorableterms:Howwouldtheyfeelaboutcompoundingtheinterestsemiannually,meaningthatthey’dbepayingonly50percentinterestforthefirstsixmonths,followedbyanother50percentforthesecondsixmonths?You’dclearlydobetterthanbefore—sinceyou’dgaininterestontheinterest—buthowmuchbetter?Theansweristhatyourinitial$1,000wouldgrowbyafactorof1.50overthe

firsthalfoftheyear,andthenbyanotherfactorof1.50overthesecondhalf.Andsince1.50times1.50is2.25,yourmoneywouldamounttoacool$2,250afteroneyear,substantiallymorethanthe$2,000yougotfromtheoriginaldeal.

Whatifyoupushedevenharderandpersuadedthebanktodividetheyearintomoreandmoreperiods—daily,bythesecond,orevenbythenanosecond?Wouldyoumakeasmallfortune?Tomakethenumberscomeoutnicely,here’stheresultforayeardividedinto

100equalperiods,aftereachofwhichyou’dbepaid1percentinterest(the100percentannualratedividedevenlyinto100installments):yourmoneywouldgrowbyafactorof1.01raisedtothe100thpower,andthatcomesouttobeabout2.70481.Inotherwords,insteadof$2,000or$2,250,you’dhave$2,704.81.Finally,theultimate:iftheinterestwascompoundedinfinitelyoften—thisis

calledcontinuouscompounding—yourtotalafteroneyearwouldbebiggerstill,butnotbymuch:$2,718.28.Theexactansweris$1,000timese,whereeisdefinedasthelimitingnumberarisingfromthisprocess:

Thisisaquintessentialcalculusargument.Aswediscussedinthelastfew

chapterswhenwecalculatedtheareaofacircleorponderedthesun’sgravitationalpullontheEarth,whatdistinguishescalculusfromtheearlierpartsofmathisitswillingnesstoconfront—andharness—theawesomepowerofinfinity.Whetherwe’relookingatlimits,derivatives,orintegrals,wealwayshavetosidleuptoinfinityinonewayoranother.Inthelimitingprocessthatledtoeabove,weimaginedslicingayearinto

moreandmorecompoundingperiods,windowsoftimethatbecamethinnerandthinner,ultimatelyapproachingwhatcanonlybedescribedasinfinitelymany,infinitesimallythinwindows.(Thismightsoundparadoxical,butit’sreallynoworsethantreatingacircleasthelimitofaregularpolygonwithmoreandmoresides,eachofwhichgetsshorterandshorter.)Thefascinatingthingisthatthemoreoftentheinterestiscompounded,thelessyourmoneygrowsduringeachperiod.Yetitstillamountstosomethingsubstantialafterayear,becauseit’sbeenmultipliedoversomanyperiods!Thisisacluetotheubiquityofe.Itoftenariseswhensomethingchanges

throughthecumulativeeffectofmanytinyevents.Consideralumpofuraniumundergoingradioactivedecay.Momentby

moment,everyatomhasacertainsmallchanceofdisintegrating.Whetherandwheneachonedoesiscompletelyunpredictable,andeachsucheventhasaninfinitesimaleffectonthewhole.Nevertheless,inensemblethesetrillionsof

eventsproduceasmooth,predictable,exponentiallydecayinglevelofradioactivity.Orthinkabouttheworld’spopulation,whichgrowsapproximately

exponentially.Allaroundtheworld,childrenarebeingbornatrandomtimesandplaces,whileotherpeoplearedying,alsoatrandomtimesandplaces.Eacheventhasaminusculeimpact,percentagewise,ontheworld’soverallpopulation—yetinaggregatethatpopulationgrowsexponentiallyataverypredictablerate.Anotherrecipeforecombinesrandomnesswithenormousnumbersof

choices.Letmegiveyoutwoexamplesinspiredbyeverydaylife,thoughinhighlystylizedform.Imaginethere’saverypopularnewmovieshowingatthelocaltheater.It’sa

romanticcomedy,andhundredsofcouples(manymorethanthetheatercanaccommodate)arelinedupattheboxoffice,desperatetogetin.Oncealuckycouplegettheirtickets,theyscrambleinsideandchoosetwoseatsrightnexttoeachother.Tokeepthingssimple,let’ssupposetheychoosetheseseatsatrandom,whereverthere’sroom.Inotherwords,theydon’tcarewhethertheysitclosetothescreenorfaraway,ontheaisleorinthemiddleofarow.Aslongasthey’retogether,sidebyside,they’rehappy.Also,let’sassumenocouplewilleverslideovertomakeroomforanother.

Onceacouplesitsdown,that’sit.Nocourtesywhatsoever.Knowingthis,theboxofficestopssellingticketsassoonasthereareonlysingleseatsleft.Otherwisebrawlswouldensue.Atfirst,whenthetheaterisprettyempty,there’snoproblem.Everycouple

canfindtwoadjacentseats.Butafterawhile,theonlyseatsleftaresingles—solitary,uninhabitabledeadspacesthatacouplecan’tuse.Inreallife,peopleoftencreatethesebuffersdeliberately,eitherfortheircoatsortoavoidsharinganarmrestwitharepulsivestranger.Inthismodel,however,thesedeadspacesjusthappenbychance.Thequestionis:Whenthere’snoroomleftforanymorecouples,what

fractionofthetheater’sseatsareunoccupied?Theanswer,inthecaseofatheaterwithmanyseatsperrow,turnsoutto

approach

soabout13.5percentoftheseatsgotowaste.Althoughthedetailsofthecalculationaretoointricatetopresenthere,it’s

easytoseethat13.5percentisintherightballparkbycomparingitwithtwoextremecases.Ifallcouplessatnexttoeachother,packedinwithperfectefficiencylikesardines,there’dbenowastedseats.

However,ifthey’dpositionedthemselvesasinefficientlyaspossible,alwayswithanemptyseatbetweenthem(andleavinganemptyaisleseatononeendortheotherofeachrow,asinthediagrambelow),one-thirdoftheseatswouldbewasted,becauseeverycoupleusesthreeseats:twoforthemselves,andoneforthedeadspace.

Guessingthattherandomcaseshouldliesomewherebetweenperfectefficiencyandperfectinefficiency,andtakingtheaverageof0and ,we’dexpectthatabout ,or16.7percent,oftheseatswouldbewasted,notfarfromtheexactanswerof13.5percent.Herethelargenumberofchoicescameaboutbecauseofallthewaysthat

couplescouldbearrangedinahugetheater.Ourfinalexampleisalsoaboutarrangingcouples,exceptnowintime,notspace.WhatI’mreferringtoisthevexingproblemofhowmanypeopleyoushoulddatebeforechoosingamate.Thereal-lifeversionofthisproblemistoohardformath,solet’sconsiderasimplifiedmodel.Despiteitsunrealisticassumptions,itstillcapturessomeoftheheartbreakinguncertaintiesofromance.Let’ssupposeyouknowhowmanypotentialmatesyou’regoingtomeet

duringyourlifetime.(Theactualnumberisnotimportantaslongasit’sknownaheadoftimeandit’snottoosmall.)Alsoassumeyoucouldrankthesepeopleunambiguouslyifyoucouldsee

themallatonce.Thetragedy,ofcourse,isthatyoucan’t.Youmeetthemoneatatime,inrandomorder.SoyoucanneverbesureifDreamboat—who’dranknumber1onyourlist—isjustaroundthecorner,orwhetheryou’vealreadymetandparted.Andthewaythisgameworksis,onceyouletsomeonego,heorsheisgone.

Nosecondchances.Finally,assumeyoudon’twanttosettle.IfyouendupwithSecondBest,or

anyoneelsewho,inretrospect,wouldn’thavemadethetopofyourlist,you’llconsideryourlovelifeafailure.

Isthereanyhopeofchoosingyouronetruelove?Ifso,whatcanyoudotogiveyourselfthebestodds?Agoodstrategy,thoughnotthebestone,istodivideyourdatinglifeintotwo

equalhalves.Inthefirsthalf,you’rejustplayingthefield;inthesecond,you’rereadytogetserious,andyou’regoingtograbthefirstpersonyoumeetwho’sbetterthaneveryoneelseyou’vedatedsofar.Withthisstrategy,there’satleasta25percentchanceofsnaggingDreamboat.

Here’swhy:Youhavea50-50chanceofmeetingDreamboatinthesecondhalfofyourdatinglife,your“getserious”phase,anda50-50chanceofmeetingSecondBestinthefirsthalf,whileyou’replayingthefield.Ifbothofthosethingshappen—andthereisa25percentchancethattheywill—thenyou’llendupwithyouronetruelove.That’sbecauseSecondBestraisedthebarsohigh.Nooneyoumeetafter

you’rereadytogetseriouswilltemptyouexceptDreamboat.Soeventhoughyoucan’tbesureatthetimethatDreamboatis,infact,TheOne,that’swhoheorshewillturnouttobe,sincenooneelsecanclearthebarsetbySecondBest.Theoptimalstrategy,however,istostopplayingthefieldalittlesooner,after

only1/e,orabout37percent,ofyourpotentialdatinglifetime.Thatgivesyoua1/echanceofendingupwithDreamboat.AslongasDreamboatisn’tplayingtheegametoo.

20.LovesMe,LovesMeNot

“INTHESPRING,”wroteTennyson,“ayoungman’sfancylightlyturnstothoughtsoflove.”Alas,hiswould-bepartnerhasthoughtsofherown—andtheinterplaybetweenthemcanleadtothetumultuousupsanddownsthatmakenewlovesothrilling,andsopainful.Toexplaintheseswings,manylovelornsoulshavesoughtanswersindrink;othershaveturnedtopoetry.We’llconsultcalculus.Theanalysisbelowisofferedtongue-in-cheek,butittouchesonaserious

point:Whilethelawsoftheheartmayeludeusforever,thelawsofinanimatethingsarenowwellunderstood.Theytaketheformofdifferentialequations,whichdescribehowinterlinkedvariableschangefrommomenttomoment,dependingontheircurrentvalues.Asforwhatsuchequationshavetodowithromance—well,attheveryleasttheymightshedalittlelightonwhy,inthewordsofanotherpoet,“thecourseoftrueloveneverdidrunsmooth.”Toillustratetheapproach,supposeRomeoisinlovewithJulietbutthat,in

ourversionofthestory,Julietisaficklelover.ThemoreRomeolovesher,themoreshewantstorunawayandhide.Butwhenhetakesthehintandbacksoff,shebeginstofindhimstrangelyattractive.He,however,tendstomirrorher:hewarmsupwhensheloveshimandcoolsdownwhenshehateshim.Whathappenstoourstar-crossedlovers?Howdoestheirloveebbandflow

overtime?That’swherecalculuscomesin.BywritingequationsthatsummarizehowRomeoandJulietrespondtoeachother’saffectionsandthensolvingthoseequationswithcalculus,wecanpredictthecourseoftheiraffair.Theresultingforecastforthiscoupleis,tragically,anever-endingcycleofloveandhate.Atleasttheymanagetoachievesimultaneousloveaquarterofthetime.

Toreachthisconclusion,I’veassumedthatRomeo’sbehaviorcanbemodeled

bythedifferentialequation

whichdescribeshowhislove(representedbyR)changesinthenextinstant(representedbydt).Accordingtothisequation,theamountofchange(dR)isjustamultiple(a)ofJuliet’scurrentlove(J)forhim.Thisreflectswhatwealreadyknow—thatRomeo’slovegoesupwhenJulietloveshim—butitassumessomethingmuchstronger.ItsaysthatRomeo’sloveincreasesindirectlinearproportiontohowmuchJulietloveshim.Thisassumptionoflinearityisnotemotionallyrealistic,butitmakestheequationsmucheasiertosolve.Juliet’sbehavior,bycontrast,canbemodeledbytheequation

ThenegativesigninfrontoftheconstantbreflectshertendencytocooloffwhenRomeoishotforher.Theonlyremainingthingweneedtoknowishowtheloversfeltabouteach

otherinitially(RandJattimet=0).Theneverythingabouttheiraffairispredetermined.WecanuseacomputertoinchRandJforward,instantbyinstant,changingtheirvaluesasprescribedbythedifferentialequations.Actually,withthehelpofthefundamentaltheoremofcalculus,wecandomuchbetterthanthat.Becausethemodelissosimple,wedon’thavetotrudgeforwardonemomentatatime.CalculusyieldsapairofcomprehensiveformulasthattellushowmuchRomeoandJulietwilllove(orhate)eachotheratanyfuturetime.Thedifferentialequationsaboveshouldberecognizabletostudentsof

physics:RomeoandJulietbehavelikesimpleharmonicoscillators.SothemodelpredictsthatR(t)andJ(t)—thefunctionsthatdescribethetimecourseoftheirrelationship—willbesinewaves,eachwaxingandwaningbutpeakingatdifferenttimes.Themodelcanbemademorerealisticinvariousways.Forinstance,Romeo

mightreacttohisownfeelingsaswellastoJuliet’s.Hemightbethetypeofguywhoissoworriedaboutthrowinghimselfatherthatheslowshimselfdownashisloveforhergrows.Orhemightbetheothertype,onewholovesfeelinginlovesomuchthathelovesherallthemoreforit.AddtothosepossibilitiesthetwowaysRomeocouldreacttoJuliet’s

affections—eitherincreasingordecreasinghisown—andyouseethattherearefourpersonalitytypes,eachcorrespondingtoadifferentromanticstyle.MystudentsandthoseinPeterChristopher’sclassatWorcesterPolytechnicInstitutehavesuggestedsuchdescriptivenamesasHermitandMalevolentMisanthropefortheparticularkindofRomeowhodampsdownhisownloveandalsorecoilsfromJuliet’s.WhereasthesortofRomeowhogetspumpedbyhisownardorbutturnedoffbyJuliet’shasbeencalledNarcissisticNerd,BetterLatentThanNever,andaFlirtingFink.(Feelfreetocomeupwithyourownnamesforthesetwotypesandtheothertwopossibilities.)Althoughtheseexamplesarewhimsical,thekindsofequationsthatarisein

themareprofound.Theyrepresentthemostpowerfultoolhumanityhasevercreatedformakingsenseofthematerialworld.SirIsaacNewtonuseddifferentialequationstosolvetheancientmysteryofplanetarymotion.Insodoing,heunifiedtheearthlyandcelestialspheres,showingthatthesamelawsofmotionappliedtoboth.Inthenearly350yearssinceNewton,mankindhascometorealizethatthe

lawsofphysicsarealwaysexpressedinthelanguageofdifferentialequations.Thisistruefortheequationsgoverningtheflowofheat,air,andwater;forthe

lawsofelectricityandmagnetism;evenfortheunfamiliarandoftencounterintuitiveatomicrealm,wherequantummechanicsreigns.Inallcases,thebusinessoftheoreticalphysicsboilsdowntofindingtheright

differentialequationsandsolvingthem.WhenNewtondiscoveredthiskeytothesecretsoftheuniverse,hefeltitwassopreciousthathepublisheditonlyasananagraminLatin.Looselytranslated,itreads:“Itisusefultosolvedifferentialequations.”Thesillyideathatloveaffairsmightlikewisebedescribedbydifferential

equationsoccurredtomewhenIwasinloveforthefirsttime,tryingtounderstandmygirlfriend’sbafflingbehavior.Itwasasummerromanceattheendofmysophomoreyearincollege.IwasalotlikethefirstRomeoabove,andshewasevenmorelikethefirstJuliet.ThecyclingofourrelationshipdrovemecrazyuntilIrealizedthatwewerebothactingmechanically,followingsimplerulesofpushandpull.Butbytheendofthesummermyequationsstartedtobreakdown,andIwasmoremystifiedthanever.Asitturnedout,therewasanimportantvariablethatI’dleftoutoftheequations—heroldboyfriendwantedherback.Inmathematicswecallthisathree-bodyproblem.It’snotoriouslyintractable,

especiallyintheastronomicalcontextwhereitfirstarose.AfterNewtonsolvedthedifferentialequationsforthetwo-bodyproblem(thusexplainingwhytheplanetsmoveinellipticalorbitsaroundthesun),heturnedhisattentiontothethree-bodyproblemforthesun,Earth,andmoon.Hecouldn’tsolveit,andneithercouldanyoneelse.Itlaterturnedoutthatthethree-bodyproblemcontainedtheseedsofchaos,renderingitsbehaviorunpredictableinthelongrun.Newtonknewnothingaboutchaoticdynamics,butaccordingtohisfriend

EdmundHalley,hecomplainedthatthethree-bodyproblemhad“madehisheadache,andkepthimawakesooften,thathewouldthinkofitnomore.”I’mwithyouthere,SirIsaac.

21.StepIntotheLight

MR.DICURCIOWASmymentorinhighschool.Hewasdisagreeableanddemanding,withnerdyblack-rimmedglassesandapenchantforsarcasm,sohischarmswereeasytomiss.ButIfoundhispassionforphysicsirresistible.OnedayImentionedtohimthatIwasreadingabiographyofEinstein.The

booksaidthatasacollegestudent,EinsteinhadbeendazzledbysomethingcalledMaxwell’sequationsforelectricityandmagnetism,andIsaidIcouldn’twaituntilIknewenoughmathtolearnwhattheywere.Thisbeingaboardingschool,wewereeatingdinnertogetheratabigtable

withseveralotherstudents,hiswife,andhistwodaughters,andMr.DiCurciowasservingmashedpotatoes.AtthementionofMaxwell’sequations,hedroppedtheservingspoon,grabbedapapernapkin,andbeganwritinglinesofcrypticsymbols—dotsandcrosses,upside-downtriangles,EsandBswitharrowsoverthem—andsuddenlyheseemedtobespeakingintongues:“Thecurlofacurlisgraddivminusdelsquared...”Thatabracadabrahewasmumbling?Irealizenowhewasspeakinginvector

calculus,thebranchofmaththatdescribestheinvisiblefieldsallaroundus.Thinkofthemagneticfieldthattwistsacompassneedlenorthward,orthegravitationalfieldthatpullsyourchairtothefloor,orthemicrowavefieldthatnukesyourdinner.Thegreatestachievementsofvectorcalculuslieinthattwilightrealmwhere

mathmeetsreality.Indeed,thestoryofJamesClerkMaxwellandhisequationsoffersoneoftheeeriestinstancesoftheunreasonableeffectivenessofmathematics.Somehow,byshufflingafewsymbols,Maxwelldiscoveredwhatlightis.TogiveasenseofwhatMaxwellaccomplishedand,moregenerally,what

vectorcalculusisabout,let’sbeginwiththeword“vector.”ItcomesfromtheLatinrootvehere,“tocarry,”whichalsogivesuswordslike“vehicle”and“conveyorbelt.”Toanepidemiologist,avectoristhecarrierofapathogen,likethemosquitothatconveysmalariatoyourbloodstream.Toamathematician,avector(atleastinitssimplestform)isastepthatcarriesyoufromoneplacetoanother.Thinkaboutoneofthosediagramsforaspiringballroomdancerscoveredwith

arrowsindicatinghowtomovetherightfoot,thentheleftfoot,aswhendoingtherumba:

Thesearrowsarevectors.Theyshowtwokindsofinformation:adirection(whichwaytomovethatfoot)andamagnitude(howfartomoveit).Allvectorsdothatsamedoubleduty.Vectorscanbeaddedandsubtracted,justlikenumbers,excepttheir

directionalitymakesthingsalittletrickier.Still,therightwaytoaddvectorsbecomesclearifyouthinkofthemasdanceinstructions.Forexample,whatdoyougetwhenyoutakeonestepeastfollowedbyonestepnorth?Avectorthatpointsnortheast,naturally.

Remarkably,velocitiesandforcesworkthesameway—theytooaddjustlike

dancesteps.Thisshouldbefamiliartoanytennisplayerwho’severtriedtoimitatePeteSamprasandhitaforehanddownthelinewhilesprintingatfullspeedtowardthesideline.Ifyounaivelyaimyourshotwhereyouwantittogo,itwillsailwidebecauseyouforgottotakeyourownrunningintoaccount.Theball’svelocityrelativetothecourtisthesumoftwovectors:theball’svelocity

relativetoyou(avectorpointingdowntheline,asintended),andyourvelocityrelativetothecourt(avectorpointingsideways,sincethat’sthedirectionyou’rerunning).Tohittheballwhereyouwantittogo,youhavetoaimslightlycrosscourt,tocompensateforyoursidewaysmotion.

Beyondsuchvectoralgebraliesvectorcalculus,thekindofmathMr.DiCurciowasusing.Calculus,you’llrecall,isthemathematicsofchange.Andsowhatevervectorcalculusis,itmustinvolvevectorsthatchange,eitherfrommomenttomomentorfromplacetoplace.Inthelattercase,onespeaksofa“vectorfield.”Aclassicexampleistheforcefieldaroundamagnet.Tovisualizeit,puta

magnetonapieceofpaperandsprinkleironfilingseverywhere.Eachfilingactslikealittlecompassneedle—italignswiththedirectionoflocal“north,”determinedbythemagneticfieldatthatpoint.Viewedinaggregate,thesefilingsrevealaspectacularpatternofmagnetic-fieldlinesleadingfromonepoleofthemagnettotheother.

Thedirectionandmagnitudeofthevectorsinamagneticfieldvaryfrompointtopoint.Asinallofcalculus,thekeytoolforquantifyingsuchchangesisthederivative.Invectorcalculusthederivativeoperatorgoesbythenameofdel,whichhasafolksysouthernringtoit,thoughitactuallyalludestotheGreekletter∆(delta),commonlyusedtodenoteachangeinsomevariable.Asareminderofthatkinship,“del”iswrittenlikethis:∇.(Thatwasthemysteriousupside-downtriangleMr.DiCurciokeptwritingonthenapkin.)Itturnsouttherearetwodifferentbutequallynaturalwaystotakethe

derivativeofavectorfieldbyapplyingdeltoit.Thefirstgiveswhat’sknownasthefield’sdivergence(the“div”thatMr.DiCurciomuttered).Togetanintuitivefeelingforwhatthedivergencemeasures,takealookatthevectorfieldbelow,whichshowshowwaterwouldflowfromasourceonthelefttoasinkontheright.

Forthisexample,insteadofusingironfilingstotrackthevectorfield,

imaginelotsoftinycorksorbitsofleavesfloatingonthewatersurface.We’regoingtousethemasprobes.Theirmotionwilltellushowthewaterismovingateachpoint.Specifically,imaginewhatwouldhappenifweputasmallcircleofcorksaroundthesource.Obviously,thecorkswouldspreadapartandthecircle

wouldexpand,becausewaterflowsawayfromasource.Itdivergesthere.Andthestrongerthedivergence,thefastertheareaofourcork-circlewouldgrow.That’swhatthedivergenceofavectorfieldmeasures:howfasttheareaofasmallcircleofcorksgrows.Theimagebelowshowsthenumericalvalueofthedivergenceateachpointin

thefieldwe’vejustbeenlookingat,codedbyshadesofgray.Lightershadesshowpointswheretheflowhasapositivedivergence.Darkershadesshowplacesofnegativedivergence,meaningthattheflowwouldcompressatinycork-circlecenteredthere.

Theotherkindofderivativemeasuresthecurlofavectorfield.Roughly

speaking,itindicateshowstronglythefieldisswirlingaboutagivenpoint.(Thinkoftheweathermapsyou’veseenonthelocalnewsshowingtherotatingwindpatternsaroundhurricanesortropicalstorms.)Inthevectorfieldbelow,regionsthatlooklikehurricaneshavealargecurl.

Byembellishingthevectorfieldwithshading,wecannowshowwherethe

curlismostpositive(lightestregions)andmostnegative(darkestregions).Noticethatthisalsotellsuswhethertheflowisspinningcounterclockwiseorclockwise.

Thecurlisextremelyinformativeforscientistsworkinginfluidmechanics

andaerodynamics.AfewyearsagomycolleagueJaneWangusedacomputertosimulatethepatternofairflowaroundadragonflyasithoveredinplace.Bycalculatingthecurl,shefoundthatwhenadragonflyflapsitswings,itcreatespairsofcounter-rotatingvorticesthatactlikelittletornadoesbeneathitswings,producingenoughlifttokeeptheinsectaloft.Inthisway,vectorcalculusishelpingtoexplainhowdragonflies,bumblebees,andhummingbirdscanfly—somethingthathadlongbeenamysterytoconventionalfixed-wingaerodynamics.Withthenotionsofdivergenceandcurlinhand,we’renowreadytorevisit

Maxwell’sequations.Theyexpressfourfundamentallaws:oneforthedivergenceoftheelectricfield,anotherforitscurl,andtwomoreofthesametypebutnowforthemagneticfield.Thedivergenceequationsrelatetheelectricandmagneticfieldstotheirsources,thechargedparticlesandcurrentsthatproducetheminthefirstplace.Thecurlequationsdescribehowtheelectricandmagneticfieldsinteractandchangeovertime.Insodoing,theseequationsexpressabeautifulsymmetry:theylinkonefield’srateofchangeintimetothe

otherfield’srateofchangeinspace,asquantifiedbyitscurl.Usingmathematicalmaneuversequivalenttovectorcalculus—whichwasn’t

knowninhisday—Maxwellthenextractedthelogicalconsequencesofthosefourequations.Hissymbolshufflingledhimtotheconclusionthatelectricandmagneticfieldscouldpropagateasawave,somewhatlikearippleonapond,exceptthatthesetwofieldsweremorelikesymbioticorganisms.Eachsustainedtheother.Theelectricfield’sundulationsre-createdthemagneticfield,whichinturnre-createdtheelectricfield,andsoon,witheachpullingtheotherforward,somethingneithercoulddoonitsown.Thatwasthefirstbreakthrough—thetheoreticalpredictionofelectromagnetic

waves.Buttherealstunnercamenext.WhenMaxwellcalculatedthespeedofthesehypotheticalwaves,usingknownpropertiesofelectricityandmagnetism,hisequationstoldhimthattheytraveledatabout193,000milespersecond—thesamerateasthespeedoflightmeasuredbytheFrenchphysicistHippolyteFizeauadecadeearlier!HowIwishIcouldhavewitnessedthemomentwhenahumanbeingfirst

understoodthetruenatureoflight.Byidentifyingitwithanelectromagneticwave,Maxwellunifiedthreeancientandseeminglyunrelatedphenomena:electricity,magnetism,andlight.AlthoughexperimenterslikeFaradayandAmpèrehadpreviouslyfoundkeypiecesofthispuzzle,itwasonlyMaxwell,armedwithhismathematics,whoputthemalltogether.TodayweareawashinMaxwell’sonce-hypotheticalwaves:Radio.

Television.Cellphones.Wi-Fi.Thesearethelegacyofhisconjuringwithsymbols.

PartFiveDATA

22.TheNewNormal

STATISTICSHASSUDDENLYbecomecoolandtrendy.ThankstotheemergenceoftheInternet,e-commerce,socialnetworks,theHumanGenomeProject,anddigitalcultureingeneral,theworldisnowteemingwithdata.Marketersscrutinizeourhabitsandtastes.Intelligenceagenciescollectdataonourwhereabouts,e-mails,andphonecalls.Sportsstatisticianscrunchthenumberstodecidewhichplayerstotrade,whomtodraft,andwhethertogoforitonfourthdownwithtwoyardstogo.Everybodywantstoconnectthedots,tofindtheneedleofmeaninginthehaystackofdata.Soit’snotsurprisingthatstudentsarebeingadvisedaccordingly.“Learnsome

statistics,”exhortedGregMankiw,aneconomistatHarvard,ina2010columnintheNewYorkTimes.“HighschoolmathematicscurriculumsspendtoomuchtimeontraditionaltopicslikeEuclideangeometryandtrigonometry.Foratypicalperson,theseareusefulintellectualexercisesbuthavelittleapplicabilitytodailylife.Studentswouldbebetterservedbylearningmoreaboutprobabilityandstatistics.”DavidBrooksputitmorebluntly.Inacolumnaboutwhatcollegecourseseveryoneshouldtaketobeproperlyeducated,hewrote,“Takestatistics.Sorry,butyou’llfindlaterinlifethatit’shandytoknowwhatastandarddeviationis.”Yes,andevenhandiertoknowwhatadistributionis.That’sthefirstideaI’d

liketofocusonhere,becauseitembodiesoneofthecentrallessonsofstatistics—thingsthatseemhopelesslyrandomandunpredictablewhenviewedinisolationoftenturnouttobelawfulandpredictablewhenviewedinaggregate.Youmayhaveseenademonstrationofthisprincipleatasciencemuseum(if

not,youcanfindvideosonline).ThestandardexhibitinvolvesasetupcalledaGaltonboard,whichlooksabitlikeapinballmachineexceptithasnoflippersanditsbumpersconsistofaregulararrayofevenlyspacedpegsarrangedinrows.

ThedemobeginswhenhundredsofballsarepouredintothetopoftheGalton

board.Astheyraindown,theyrandomlybounceoffthepegs,sometimestotheleft,sometimestotheright,andultimatelydistributethemselvesintheevenlyspacedbinsatthebottom.Theheightofthestackedballsineachbinshowshowlikelyitwasforaballtolandthere.Mostballsendupsomewherenearthemiddle,withslightlyfewerballsflankingthemoneitherside,andfewerstillfaroffinthetailsateitherend.Overall,thepatternisutterlypredictable:italwaysformsabell-shapeddistribution—eventhoughit’simpossibletopredictwhereanygivenballwillendup.Howdoesindividualrandomnessturnintocollectiveregularity?Easy—the

oddsdemandit.Themiddlebinislikelytobethemostpopulatedspotbecause

mostballswillmakeaboutthesamenumberofleftwardandrightwardbouncesbeforetheyreachthebottom.Sothey’llprobablyendupsomewherenearthemiddle.Theonlyballsthatmakeitfarouttoeitherextreme,wayoffinthetailsofthedistributionwheretheoutlierslive,arethosethatjusthappentobounceinthesamedirectionalmosteverytimetheyhitapeg.That’sveryunlikely.Andthat’swhytherearesofewballsoutthere.Justastheultimatelocationofeachballisdeterminedbythesumofmany

chanceevents,lotsofphenomenainthisworldarethenetresultofmanytinyflukes,sotheytooaregovernedbyabell-shapedcurve.Insurancecompaniesbankonthis.Theyknow,withgreataccuracy,howmanyoftheircustomerswilldieeachyear.Theyjustdon’tknowwhotheunluckyoneswillbe.Orconsiderhowtallyouare.Yourheightdependsoncountlesslittleaccidents

ofgenetics,biochemistry,nutrition,andenvironment.Consequently,it’splausiblethatwhenviewedinaggregate,theheightsofadultmenandwomenshouldfallonabellcurve.Inablogposttitled“Thebigliespeopletellinonlinedating,”thestatistically

mindeddatingserviceOkCupidrecentlygraphedhowtalltheirmembersare—orrather,howtalltheysaytheyare—andfoundthattheheightsreportedbybothsexesfollowbellcurves,asexpected.What’ssurprising,however,isthatbothdistributionsareshiftedabouttwoinchestotherightofwheretheyshouldbe.

SoeitherthepeoplewhojoinOkCupidareunusuallytall,ortheyexaggeratetheirheightsbyacoupleofincheswhendescribingthemselvesonline.Anidealizedversionofthesebellcurvesiswhatmathematicianscallthe

normaldistribution.It’soneofthemostimportantconceptsinstatistics.Partof

itsappealistheoretical.Thenormaldistributioncanbeproventoarisewheneveralargenumberofmildlyrandomeffectsofsimilarsize,allactingindependently,areaddedtogether.Andmanythingsarelikethat.Butnoteverything.That’sthesecondpointI’dliketostress.Thenormal

distributionisnotnearlyasubiquitousasitonceseemed.Forabout100yearsnow,andespeciallyduringthepastfewdecades,statisticiansandscientistshavenoticedthatplentyofphenomenadeviatefromthispatternyetstillmanagetofollowapatternoftheirown.Curiously,thesetypesofdistributionsarebarelymentionedintheelementarystatisticstextbooks,andwhentheyare,they’reusuallytrottedoutaspathologicalspecimens.It’soutrageous.For,asI’lltrytoexplain,muchofmodernlifemakesalotmoresensewhenyouunderstandthesedistributions.They’rethenewnormal.TakethedistributionofcitysizesintheUnitedStates.Insteadofclustering

aroundsomeintermediatevalueinabell-shapedfashion,thevastmajorityoftownsandcitiesaretinyandthereforehuddletogetherontheleftofthegraph.

Andthelargerthepopulationofacity,themoreuncommonacityofthatsizeis.Sowhenviewedintheaggregate,thedistributionlooksmorelikeanL-curvethanabellcurve.There’snothingsurprisingaboutthis.Everybodyknowsthatbigcitiesare

rarerthansmallones.What’slessobvious,though,isthatcitysizesneverthelessfollowabeautifullysimpledistribution...aslongasyoulookatthemthroughlogarithmiclenses.Inotherwords,supposeweregardthesizedifferentialbetweenapairofcities

tobethesameiftheirpopulationsdifferbythesamefactor,ratherthanbythesameabsolutenumberofpeople(muchliketwopitchesanoctaveapartalwaysdifferbyaconstantfactorofdoublethefrequency).Andsupposewedolikewiseontheverticalaxis.

Thenthedatafallonacurvethat’salmostastraightline.Fromthepropertiesoflogarithms,wecanthendeducethattheoriginalL-curvewasapowerlaw,afunctionoftheform

wherexisacity’ssize,yishowmanycitieshavethatsize,Cisaconstant,andtheexponenta(thepowerinthepowerlaw)isthenegativeofthestraightline’sslope.Power-lawdistributionshavecounterintuitivepropertiesfromthestandpoint

ofconventionalstatistics.Forexample,unlikenormaldistributions’,theirmodes,medians,andmeansdonotagreebecauseoftheskewed,asymmetricalshapesoftheirL-curves.PresidentBushmadeuseofthispropertywhenhestatedthathis2003taxcutshadsavedfamiliesanaverageof$1,586each.Thoughthatistechnicallycorrect,hewasconvenientlyreferringtothemeanrebate,afigurethataveragedinthewhoppingrebatesofhundredsofthousandsofdollarsreceivedbytherichest0.1percentofthepopulation.Thetailonthefarrightoftheincomedistributionisknowntofollowapowerlaw,andinsituationslikethis,themeanisamisleadingstatistictousebecauseit’sfarfromtypical.Mostfamilies,infact,gotlessthan$650back.Themedianwasalotlessthanthemean.Thisexamplehighlightsthemostcrucialfeatureofpower-lawdistributions.

Theirtailsareheavy(alsoknownasfatorlong),atleastcomparedtothepunylittlewispofatailonthenormaldistribution.Soextremelylargeoutliers,thoughstillrare,aremuchmorecommonforthesedistributionsthantheywouldbefornormalbellcurves.OnOctober19,1987,nowknownasBlackMonday,theDowJonesindustrial

averagedroppedby22percentinasingleday.Comparedtotheusuallevelofvolatilityinthestockmarket,thiswasadropofmorethantwentystandarddeviations.Suchaneventisallbutimpossibleaccordingtotraditionalbell-curvestatistics;itsprobabilityislessthanonein100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000(that’s10raisedtothe50thpower).Yetithappened...becausefluctuationsinstockpricesdon’tfollownormaldistributions.They’rebetterdescribedbyheavy-taileddistributions.Soareearthquakes,wildfires,andfloods,whichcomplicatesthetaskofrisk

managementforinsuranceindustries.Thesamemathematicalpatternholdsforthenumbersofdeathscausedbywarsandterroristattacks,andevenformorebenignthingslikewordfrequenciesinnovelsandthenumberofsexualpartnerspeoplehave.Thoughtheadjectivesusedtodescribetheirprominenttailsweren’toriginally

meanttobeflattering,suchdistributionshavecometowearthemwithpride.Fat,heavy,andlong?Yeah,that’sright.Nowwho’snormal?

23.ChancesAre

HAVEYOUEVERhadthatanxietydreamwhereyousuddenlyrealizeyouhavetotakethefinalexaminsomecourseyou’veneverattended?Forprofessors,itworkstheotherwayaround—youdreamyou’regivingalectureinacourseyouknownothingabout.That’swhatit’slikeformewheneverIteachprobabilitytheory.Itwasnever

partofmyowneducation,sohavingtolectureaboutitnowisscaryandfun,inanamusement-park-thrill-housesortofway.Perhapsthemostpulse-quickeningtopicofallisconditionalprobability—the

probabilitythatsomeeventAhappens,given(orconditionalupon)theoccurrenceofsomeothereventB.It’saslipperyconcept,easilyconflatedwiththeprobabilityofBgivenA.They’renotthesame,butyouhavetoconcentratetoseewhy.Forexample,considerthefollowingwordproblem.Beforegoingonvacationforaweek,youaskyourspacyfriendtowateryour

ailingplant.Withoutwater,theplanthasa90percentchanceofdying.Evenwithproperwatering,ithasa20percentchanceofdying.Andtheprobabilitythatyourfriendwillforgettowateritis30percent.(a)What’sthechancethatyourplantwillsurvivetheweek?(b)Ifit’sdeadwhenyoureturn,what’sthechancethatyourfriendforgottowaterit?(c)Ifyourfriendforgottowaterit,what’sthechanceit’llbedeadwhenyoureturn?Althoughtheysoundalike,(b)and(c)arenotthesame.Infact,theproblemtellsusthattheanswerto(c)is90percent.Buthowdoyoucombinealltheprobabilitiestogettheanswerto(b)?Or(a)?Naturally,thefirstfewsemestersItaughtthistopic,Istucktothebook,

inchingalong,playingitsafe.ButgraduallyIbegantonoticesomething.AfewofmystudentswouldavoidusingBayes’stheorem,thelabyrinthineformulaIwasteachingthem.Insteadtheywouldsolvetheproblemsbyanequivalentmethodthatseemedeasier.Whattheseresourcefulstudentskeptdiscovering,yearafteryear,wasabetter

waytothinkaboutconditionalprobability.Theirwaycomportswithhumanintuitioninsteadofconfoundingit.Thetrickistothinkintermsofnaturalfrequencies—simplecountsofevents—ratherthaninmoreabstractnotionsofpercentages,odds,orprobabilities.Assoonasyoumakethismentalshift,thefoglifts.ThisisthecentrallessonofCalculatedRisks,afascinatingbookbyGerd

Gigerenzer,acognitivepsychologistattheMaxPlanckInstituteforHuman

DevelopmentinBerlin.InaseriesofstudiesaboutmedicalandlegalissuesrangingfromAIDScounselingtotheinterpretationofDNAfingerprinting,Gigerenzerexploreshowpeoplemiscalculateriskanduncertainty.Butratherthanscoldorbemoanhumanfrailty,hetellsushowtodobetter—howtoavoidcloudedthinkingbyrecastingconditional-probabilityproblemsintermsofnaturalfrequencies,muchasmystudentsdid.Inonestudy,GigerenzerandhiscolleaguesaskeddoctorsinGermanyandthe

UnitedStatestoestimatetheprobabilitythatawomanwhohasapositivemammogramactuallyhasbreastcancereventhoughshe’sinalow-riskgroup:fortytofiftyyearsold,withnosymptomsorfamilyhistoryofbreastcancer.Tomakethequestionspecific,thedoctorsweretoldtoassumethefollowingstatistics—couchedintermsofpercentagesandprobabilities—abouttheprevalenceofbreastcanceramongwomeninthiscohortandaboutthemammogram’ssensitivityandrateoffalsepositives:Theprobabilitythatoneofthesewomenhasbreastcanceris0.8percent.Ifawomanhasbreastcancer,theprobabilityis90percentthatshewillhaveapositivemammogram.Ifawomandoesnothavebreastcancer,theprobabilityis7percentthatshewillstillhaveapositivemammogram.Imagineawomanwhohasapositivemammogram.Whatistheprobabilitythatsheactuallyhasbreastcancer?Gigerenzerdescribesthereactionofthefirstdoctorhetested,adepartment

chiefatauniversityteachinghospitalwithmorethanthirtyyearsofprofessionalexperience:[He]wasvisiblynervouswhiletryingtofigureoutwhathewouldtellthewoman.Aftermullingthenumbersover,hefinallyestimatedthewoman’sprobabilityofhavingbreastcancer,giventhatshehasapositivemammogram,tobe90percent.Nervously,headded,“Oh,whatnonsense.Ican’tdothis.Youshouldtestmydaughter;sheisstudyingmedicine.”Heknewthathisestimatewaswrong,buthedidnotknowhowtoreasonbetter.Despitethefactthathehadspent10minuteswringinghismindforananswer,hecouldnotfigureouthowtodrawasoundinferencefromtheprobabilities.Gigerenzeraskedtwenty-fourotherGermandoctorsthesamequestion,and

theirestimateswhipsawedfrom1percentto90percent.Eightofthemthoughtthechanceswere10percentorless;eightotherssaid90percent;andthe

remainingeightguessedsomewherebetween50and80percent.Imaginehowupsettingitwouldbeasapatienttohearsuchdivergentopinions.AsfortheAmericandoctors,ninety-fiveoutofahundredestimatedthe

woman’sprobabilityofhavingbreastcancertobesomewherearound75percent.Therightansweris9percent.Howcanitbesolow?Gigerenzer’spointisthattheanalysisbecomesalmost

transparentifwetranslatetheoriginalinformationfrompercentagesandprobabilitiesintonaturalfrequencies:Eightoutofevery1,000womenhavebreastcancer.Ofthese8womenwithbreastcancer,7willhaveapositivemammogram.Oftheremaining992womenwhodon’thavebreastcancer,some70willstillhaveapositivemammogram.Imagineasampleofwomenwhohavepositivemammogramsinscreening.Howmanyofthesewomenactuallyhavebreastcancer?

Sinceatotalof7+70=77womenhavepositivemammograms,andonly7ofthemtrulyhavebreastcancer,theprobabilityofawoman’shavingbreastcancergivenapositivemammogramis7outof77,whichis1in11,orabout9percent.Noticetwosimplificationsinthecalculationabove.First,weroundedoff

decimalstowholenumbers.Thathappenedinafewplaces,likewhenwesaid,“Ofthese8womenwithbreastcancer,7willhaveapositivemammogram.”Reallyweshouldhavesaid90percentof8women,or7.2women,willhaveapositivemammogram.Sowesacrificedalittleprecisionforalotofclarity.Second,weassumedthateverythinghappensexactlyasfrequentlyasits

probabilitysuggests.Forinstance,sincetheprobabilityofbreastcanceris0.8percent,exactly8womenoutof1,000inourhypotheticalsamplewereassumedtohaveit.Inreality,thiswouldn’tnecessarilybetrue.Eventsdon’thavetofollowtheirprobabilities;acoinflipped1,000timesdoesn’talwayscomeupheads500times.Butpretendingthatitdoesgivestherightanswerinproblemslikethis.Admittedlythelogicisalittleshaky—that’swhythetextbookslookdown

theirnosesatthisapproach,comparedtothemorerigorousbuthard-to-useBayes’stheorem—butthegainsinclarityarejustificationenough.WhenGigerenzertestedanothersetoftwenty-fourdoctors,thistimeusingnaturalfrequencies,nearlyallofthemgotthecorrectanswer,orclosetoit.Althoughreformulatingthedataintermsofnaturalfrequenciesisahugehelp,

conditional-probabilityproblemscanstillbeperplexingforotherreasons.It’s

easytoaskthewrongquestionortocalculateaprobabilitythat’scorrectbutmisleading.BoththeprosecutionandthedefensewereguiltyofthisintheO.J.Simpson

trialof1994–95.Eachofthemaskedthecourttoconsiderthewrongconditionalprobability.Theprosecutionspentthefirsttendaysofthetrialintroducingevidencethat

O.J.hadahistoryofviolencetowardhisex-wifeNicoleBrown.Hehadallegedlybatteredher,thrownheragainstwalls,andgropedherinpublic,tellingonlookers,“Thisbelongstome.”Butwhatdidanyofthishavetodowithamurdertrial?Theprosecution’sargumentwasthatapatternofspousalabusereflectedamotivetokill.Asoneoftheprosecutorsputit,“Aslapisapreludetohomicide.”AlanDershowitzcounteredforthedefense,arguingthatevenifthe

allegationsofdomesticviolenceweretrue,theywereirrelevantandshouldthereforebeinadmissible.Helaterwrote,“Weknewwecouldprove,ifwehadto,thataninfinitesimalpercentage—certainlyfewerthan1of2,500—ofmenwhoslaporbeattheirdomesticpartnersgoontomurderthem.”Ineffect,bothsideswereaskingthecourttoconsidertheprobabilitythata

manmurderedhisex-wife,giventhathepreviouslybatteredher.ButasthestatisticianI.J.Goodpointedout,that’snottherightnumbertolookat.Therealquestionis:What’stheprobabilitythatamanmurderedhisex-wife,

giventhathepreviouslybatteredherandshewasmurderedbysomeone?Thatconditionalprobabilityturnsouttobeveryfarfrom1in2,500.Toseewhy,imagineasampleof100,000batteredwomen.Granting

Dershowitz’snumberof1in2,500,weexpectabout40ofthesewomentobemurderedbytheirabusersinagivenyear(since100,000dividedby2,500equals40).Wealsoexpect3moreofthesewomen,onaverage,tobekilledbysomeoneelse(thisestimateisbasedonstatisticsreportedbytheFBIforwomenmurderedin1992;seethenotesforfurtherdetails).Sooutofthe43murdervictimsaltogether,40ofthemwerekilledbytheirbatterers.Inotherwords,thebattererwasthemurdererabout93percentofthetime.Don’tconfusethisnumberwiththeprobabilitythatO.J.didit.That

probabilitywoulddependonalotofotherevidence,proandcon,suchasthedefense’sclaimthatthepoliceframedO.J.,andtheprosecution’sclaimthatthekillerandO.J.sharedthesamestyleofshoes,gloves,andDNA.Theprobabilitythatanyofthischangedyourmindabouttheverdict?Zero.

24.UntanglingtheWeb

INATIMElongago,inthedarkdaysbeforeGoogle,searchingtheWebwasanexerciseinfrustration.Thesitessuggestedbytheoldersearchenginesweretoooftenirrelevant,whiletheonesyoureallywantedwereeitherburiedwaydowninthelistofresultsormissingaltogether.Algorithmsbasedonlinkanalysissolvedtheproblemwithaninsightas

paradoxicalasaZenkoan:AWebsearchshouldreturnthebestpages.Andwhat,grasshopper,makesapagegood?Apageisgoodifothergoodpageslinktoit.Thatsoundslikecircularreasoning.Itis...whichiswhyit’ssodeep.By

grapplingwiththiscircleandturningittoadvantage,linkanalysisyieldsajujitsusolutiontosearchingtheWeb.Theapproachbuildsonideasfromlinearalgebra,thestudyofvectorsand

matrices.Whetheryouwanttodetectpatternsinlargedatasetsorperformgiganticcomputationsinvolvingmillionsofvariables,linearalgebrahasthetoolsyouneed.AlongwithunderpinningGoogle’sPageRankalgorithm,ithashelpedscientistsclassifyhumanfaces,analyzethevotingpatternsofSupremeCourtjustices,andwinthemillion-dollarNetflixPrize(awardedtothepersonorteamwhocouldimprovebymorethan10percentNetflix’ssystemforrecommendingmoviestoitscustomers).Foracasestudyoflinearalgebrainaction,let’slookathowPageRankworks.

Andtobringoutitsessencewithaminimumoffuss,let’simagineatoyWebthathasjustthreepages,allconnectedlikethis:

ThearrowsindicatethatpageXcontainsalinktopageY,butYdoesnotreturnthefavor.Instead,YlinkstoZ.MeanwhileXandZlinktoeachotherinafrenzyofdigitalback-scratching.InthislittleWeb,whichpageisthemostimportant,andwhichistheleast?

Youmightthinkthere’snotenoughinformationtosaybecausenothingisknownaboutthepages’content.Butthat’sold-schoolthinking.Worryingaboutcontentturnedouttobeanimpracticalwaytorankwebpages.Computersweren’tgoodatit,andhumanjudgescouldn’tkeepupwiththedelugeofthousandsofpagesaddedeachday.TheapproachtakenbyLarryPageandSergeyBrin,thegradstudentswho

cofoundedGoogle,wastoletwebpagesrankthemselvesbyvotingwiththeirfeet—or,rather,withtheirlinks.Intheexampleabove,pagesXandYbothlinktoZ,whichmakesZtheonlypagewithtwoincominglinks.Soit’sthemostpopularpageintheuniverse.Thatshouldcountforsomething.However,ifthoselinkscomefrompagesofdubiousquality,thatshouldcountagainstthem.Popularitymeansnothingonitsown.Whatmattersishavinglinksfromgoodpages.Whichbringsusbacktotheriddleofthecircle:Apageisgoodifgoodpages

linktoit,butwhodecideswhichpagesaregoodinthefirstplace?Thenetworkdoes.Andhere’show.(Actually,I’mskippingsomedetails;see

thenoteson[>]foramorecompletestory.)Google’salgorithmassignsafractionalscorebetween0and1toeachpage.

ThatscoreiscalleditsPageRank;itmeasureshowimportantthatpageisrelativetotheothersbycomputingtheproportionoftimethatahypotheticalWebsurferwouldspendthere.Wheneverthereismorethanoneoutgoinglinktochoose

from,thesurferselectsoneatrandom,withequalprobability.Underthisinterpretation,pagesareregardedasmoreimportantifthey’revisitedmorefrequently(bythisidealizedsurfer,notbyactualWebtraffic).AndbecausethePageRanksaredefinedasproportions,theyhavetoaddupto

1whensummedoverthewholenetwork.Thisconservationlawsuggestsanother,perhapsmorepalpable,waytovisualizePageRank.Pictureitasafluid,awaterysubstancethatflowsthroughthenetwork,drainingawayfrombadpagesandpoolingatgoodones.Thealgorithmseekstodeterminehowthisfluiddistributesitselfacrossthenetworkinthelongrun.Theansweremergesfromacleveriterativeprocess.Thealgorithmstartswith

aguess,thenupdatesallthePageRanksbyapportioningthefluidinequalsharestotheoutgoinglinks,anditkeepsdoingthatinaseriesofroundsuntileverythingsettlesdownandallthepagesgettheirrightfulshares.Initiallythealgorithmtakesanegalitarianstance.Itgiveseverypageanequal

portionofPageRank.Sincetherearethreepagesintheexamplewe’reconsidering,eachpagebeginswithascoreof1/3.

Next,thesescoresareupdatedtobetterreflecteachpage’strueimportance.

TheruleisthateachpagetakesitsPageRankfromthelastroundandparcelsit

outequallytoallthepagesitlinksto.Thus,afteroneround,theupdatedvalueofXwouldstillequal1/3,becausethat’showmuchPageRankitreceivesfromZ,theonlypagethatlinkstoit.ButY’sscoredropstoameasly1/6,sinceitgetsonlyhalfofX’sPageRankfromthepreviousround.TheotherhalfgoestoZ,whichmakesZthebigwinneratthisstage,sincealongwiththe1/6itreceivesfromX,italsogetsthefull1/3fromY,foratotalof1/2.Soafteroneround,thePageRankvaluesarethoseshownbelow:

Intheroundstocome,theupdaterulestaysthesame.Ifwewrite(x,y,z)for

thecurrentscoresofpagesX,Y,andZ,thentheupdatedscoreswillbe

wheretheprimesymbolinthesuperscriptsignifiesthatanupdatehasoccurred.

Thiskindofiterativecalculationiseasytodoinaspreadsheet(orevenbyhand,foranetworkassmallastheonewe’restudying).Afterteniterations,onefindsthatthenumbersdon’tchangemuchfromone

roundtothenext.Bythen,Xhasa40.6percentshareofthetotalPageRank,Yhas19.8percent,andZhas39.6percent.Thosenumberslooksuspiciouslycloseto40percent,20percent,and40percent,suggestingthatthealgorithmisconvergingtothosevalues.Infact,that’scorrect.ThoselimitingvaluesarewhatGoogle’salgorithm

woulddefineasthePageRanksforthenetwork.

TheimplicationisthatXandZareequallyimportantpages,eventhoughZhastwiceasmanylinkscomingin.Thatmakessense:XisjustasimportantasZbecauseitgetsthefullendorsementofZbutreciprocateswithonlyhalfitsownendorsement.TheotherhalfitsendstoY.ThisalsoexplainswhyYfaresonlyhalfaswellasXandZ.Remarkably,thesescorescanbeobtaineddirectly,withoutgoingthroughthe

iteration.Justthinkabouttheconditionsthatdefinethesteadystate.Ifnothingchangesafteranupdateisperformed,wemusthavex′=x,y′=y,andz′=z.Soreplacetheprimedvariablesintheupdateequationswiththeirunprimed

counterparts.Thenweget

andthissystemofequationscanbesolvedsimultaneouslytoobtainx=2y=z.Finally,sincethesescoresmustsumto1,weconcludex=2/5,y=1/5,andz=2/5,inagreementwiththepercentagesfoundabove.Let’sstepbackforamomenttolookathowallthisfitsintothelargercontext

oflinearalgebra.Thesteady-stateequationsabove,aswellastheearlierupdateequationswiththeprimesinthem,aretypicalexamplesoflinearequations.They’recalledlinearbecausethey’rerelatedtolines.Thevariablesx,y,zinthemappeartothefirstpoweronly,justastheydointhefamiliarequationforastraightline,y=mx+b,astapleofhigh-schoolalgebracourses.Linearequations,asopposedtothosecontainingnonlineartermslikex²oryz

orsinx,arecomparativelyeasytosolve.Thechallengecomeswhenthereareenormousnumbersofvariablesinvolved,asthereareintherealWeb.Oneofthecentraltasksoflinearalgebra,therefore,isthedevelopmentoffasterandfasteralgorithmsforsolvingsuchhugesetsofequations.Evenslightimprovementshaveramificationsforeverythingfromairlineschedulingtoimagecompression.Butthegreatesttriumphoflinearalgebra,fromthestandpointofreal-world

impact,issurelyitssolutiontotheZenriddleofrankingwebpages.“Apageisgoodinsofarasgoodpageslinktoit.”Translatedintosymbols,thatcriterionbecomesthePageRankequations.Googlegotwhereitistodaybysolvingthesameequationsaswedidhere—

justwithafewbillionmorevariables...andprofitstomatch.

PartSixFRONTIERS

25.TheLoneliestNumbers

ACCORDINGTOAmemorablesongfromthe1960s,oneistheloneliestnumber,andtwocanbeasbadasone.Maybeso,buttheprimenumbershaveitprettyroughtoo.PaoloGiordanoexplainswhyinhisbest-sellingnovelTheSolitudeofPrime

Numbers.It’sthemelancholylovestoryoftwomisfits,twoprimes,namedMattiaandAlice,bothscarredbychildhoodtragediesthatleftthemvirtuallyincapableofconnectingwithotherpeople,yetwhosenseineachotherakindreddamagedspirit.Giordanowrites,Primenumbersaredivisibleonlyby1andbythemselves.Theyholdtheirplaceintheinfiniteseriesofnaturalnumbers,squashed,likeallnumbers,betweentwoothers,butonestepfurtherthantherest.Theyaresuspicious,solitarynumbers,whichiswhyMattiathoughttheywerewonderful.Sometimeshethoughtthattheyhadendedupinthatsequencebymistake,thatthey’dbeentrapped,likepearlsstrungonanecklace.Othertimeshesuspectedthattheytoowouldhavepreferredtobelikealltheothers,justordinarynumbers,butforsomereasontheycouldn’tdoit.[...]Inhisfirstyearatuniversity,Mattiahadlearnedthat,amongprime

numbers,therearesomethatareevenmorespecial.Mathematicianscallthemtwinprimes:pairsofprimenumbersthatareclosetoeachother,almostneighbors,butbetweenthemthereisalwaysanevennumberthatpreventsthemfromtrulytouching.Numberslike11and13,like17and19,41and43.Ifyouhavethepatiencetogooncounting,youdiscoverthatthesepairsgraduallybecomerarer.Youencounterincreasinglyisolatedprimes,lostinthatsilent,measuredspacemadeonlyofciphers,andyoudevelopadistressingpresentimentthatthepairsencounteredupuntilthatpointwereaccidental,thatsolitudeisthetruedestiny.Then,justwhenyou’reabouttosurrender,whenyounolongerhavethedesiretogooncounting,youcomeacrossanotherpairoftwins,clutchingeachothertightly.Thereisacommonconvictionamongmathematiciansthathoweverfaryougo,therewillalwaysbeanothertwo,evenifnoonecansaywhereexactly,untiltheyarediscovered.MattiathoughtthatheandAlicewerelikethat,twinprimes,aloneand

lost,closebutnotcloseenoughtoreallytoucheachother.

HereI’dliketoexploresomeofthebeautifulideasinthepassageabove,particularlyastheyrelatetothesolitudeofprimenumbersandtwinprimes.Theseissuesarecentraltonumbertheory,thesubjectthatconcernsitselfwiththestudyofwholenumbersandtheirpropertiesandthatisoftendescribedasthepurestpartofmathematics.Beforeweascendtowheretheairisthin,letmedispensewithaquestionthat

oftenoccurstopractical-mindedpeople:Isnumbertheorygoodforanything?Yes.Almostinspiteofitself,numbertheoryprovidesthebasisfortheencryptionalgorithmsusedmillionsoftimeseachdaytosecurecreditcardtransactionsovertheInternetandtoencodemilitary-strengthsecretcommunications.Thosealgorithmsrelyonthedifficultyofdecomposinganenormousnumberintoitsprimefactors.Butthat’snotwhymathematiciansareobsessedwithprimenumbers.Thereal

reasonisthatthey’refundamental.They’retheatomsofarithmetic.JustastheGreekoriginoftheword“atom”suggests,theprimesare“a-tomic,”meaning“uncuttable,indivisible.”Andjustaseverythingiscomposedofatoms,everynumberiscomposedofprimes.Forexample,60equals2×2×3×5.Wesaythat60isacompositenumberwithprimefactorsof2(countedtwice),3,and5.Andwhatabout1?Isitprime?No,itisn’t,andwhenyouunderstandwhyit

isn’t,you’llbegintoappreciatewhy1trulyistheloneliestnumber—evenlonelierthantheprimes.Itdoesn’tdeservetobeleftout.Giventhat1isdivisibleonlyby1anditself,it

reallyshouldbeconsideredprime,andformanyyearsitwas.Butmodernmathematicianshavedecidedtoexcludeit,solelyforconvenience.If1wereallowedin,itwouldmessupatheoremthatwe’dliketobetrue.Inotherwords,we’veriggedthedefinitionofprimenumberstogiveusthetheoremwewant.Thedesiredtheoremsaysthatanynumbercanbefactoredintoprimesina

uniqueway.Butif1wereconsideredprime,theuniquenessofprimefactorizationwouldfail.Forexample,6wouldequal2×3,butitwouldalsoequal1×2×3and1×1×2×3andsoon,andthesewouldallhavetobeacceptedasdifferentprimefactorizations.Silly,ofcourse,butthat’swhatwe’dbestuckwithif1wereallowedin.Thissordidlittletaleisinstructive;itpullsbackthecurtainonhowmathis

donesometimes.Thenaiveviewisthatwemakeourdefinitions,settheminstone,thendeducewhatevertheoremshappentofollowfromthem.Notso.Thatwouldbemuchtoopassive.We’reinchargeandcanalterthedefinitionsasweplease—especiallyifaslighttweakleadstoatidiertheorem,asitdoeshere.Nowthat1hasbeenthrownunderthebus,let’slookateveryoneelse,thefull-

fledgedprimenumbers.Themainthingtoknowaboutthemishowmysterious

theyare,howalienandinscrutable.Noonehaseverfoundanexactformulafortheprimes.Unlikerealatoms,theydon’tfollowanysimplepattern,nothingakintotheperiodictableoftheelements.Youcanalreadyseethewarningsignsinthefirsttenprimes:2,3,5,7,11,13,

17,19,23,29.Rightoffthebat,thingsstartbadlywith2.It’safreak,amisfitamongmisfits—theonlyprimewiththeembarrassmentofbeinganevennumber.Nowonder“it’stheloneliestnumbersincethenumberone”(asthesongsays).Apartfrom2,therestoftheprimesareallodd...butstillquirky.Lookatthe

gapsbetweenthem.Sometimesthey’retwospacesapart(like5and7),sometimesfour(13and17),andsometimessix(23and29).Tofurtherunderscorehowdisorderlytheprimesare,comparethemtotheir

straight-arrowcousinstheoddnumbers:1,3,5,7,9,11,13,...Thegapsbetweenoddnumbersarealwaysconsistent:twospaces,steadyasadrumbeat.Sotheyobeyasimpleformula:thenthoddnumberis2n–1.Theprimes,bycontrast,marchtotheirowndrummer,toarhythmnooneelsecanperceive.Giventheirregularitiesinthespacingoftheprimes,numberstheoristshave

resortedtolookingatthemstatistically,asmembersofanensemble,ratherthandwellingontheiridiosyncrasies.Specifically,let’saskhowthey’redistributedamongtheordinarywholenumbers.Howmanyprimesarelessthanorequalto10?Or100?OranarbitrarynumberN?Thisconstructionisadirectparalleltothestatisticalconceptofacumulativedistribution.Soimaginecountingtheprimenumbersbywalkingamongthem,likean

anthropologisttakingacensus.Picturethemstandingthereonthex-axis.Youstartatthenumber1andbeginwalkingtotheright,tallyingprimesasyougo.Yourrunningtotallookslikethis:

Thevaluesonthey-axisshowhowmanyprimesyou’vecountedbythetimeyoureachagivenlocation,x.Forallx’slessthan2,thegraphofyremainsflatat0,sincenoprimeshavebeencountedyet.Thefirstprimeappearsatx=2.Sothegraphjumpsupthere.(Gotone!)Thenitremainsflatuntilx=3,afterwhichitjumpsupanotherstep.Thealternatingjumpsandplateausformastrange,irregularstaircase.Mathematicianscallitthecountingfunctionfortheprimes.Contrastthisimagewithitscounterpartfortheoddnumbers.

Nowthestaircasebecomesperfectlyregular,followingatrendlinewhoseslopeis1/2.That’sbecausethegapbetweenneighboringoddnumbersisalways2.Isthereanyhopeoffindingsomethingsimilarfortheprimenumbersdespite

theirerraticcharacter?Miraculously,yes.Thekeyistofocusonthetrend,notthedetailsofthestairsteps.Ifwezoomout,acurvebeginstoemergefromtheclutter.Here’sthegraphofthecountingfunctionforallprimesupto100.

Thestepsarelessdistractingnow.Thecurvelooksevensmootherifwecountalltheprimesouttoabillion:

Firstimpressionstothecontrary,thiscurveisnotactuallyastraightline.Itdroopsdownslightlyasitclimbs.Itsdroopinessmeansthattheprimesarebecomingrarer.Moreisolated.Morealone.That’swhatGiordanomeantbythe“solitudeofprimenumbers.”Thisthinningoutbecomesobviousifwelookatthecensusdatafromanother

angle.Rememberwetalliedtenprimesinthefirstthirtywholenumbers.Sonearthebeginningofthenumberline,aboutoneoutofeverythreenumbersisprime,makingthemarobust33percentofthepopulation.Butamongthefirsthundrednumbers,onlytwenty-fiveareprime.Theirrankshavedwindledtooneinfour,aworrisome25percent.Andamongthefirstbillionnumbers,amere5percentareprime.That’sthebleakmessageofthedroopycurve.Theprimesareadyingbreed.

Theyneverdieoutcompletely—we’veknownsinceEuclidtheygoonforever—buttheyfadeintonearoblivion.Byfindingfunctionsthatapproximatethedroopycurve,numbertheorists

havequantifiedhowdesolatetheprimenumberstrulyare,asexpressedbyaformulaforthetypicalspacingbetweenthem.IfNisalargenumber,theaveragegapbetweentheprimesnearNisapproximatelyequaltolnN,thenaturallogarithmofN.(Thenaturallogarithmbehavesliketheordinarylogarithm

encounteredinhighschool,exceptit’sbasedonthenumbereinsteadof10.It’snaturalinthesensethatitpopsupeverywhereinadvancedmath,thankstobeingpartofe’sentourage.Formoreontheubiquityofe,seechapter19.)AlthoughthelnNformulafortheaveragespacingbetweenprimesdoesn’t

worktoowellwhenNissmall,itimprovesinthesensethatitspercentageerrorgoestozeroasNapproachesinfinity.Togetafeelforthenumbersinvolved,supposeN=1,000.Itturnsoutthereare168primenumberslessthan1,000,sotheaveragegapbetweentheminthispartofthenumberlineis1,000/68,orabout5.9.Forcomparison,theformulapredictsanaveragegapofln(1,000)≈6.9,whichistoohighbyabout17percent.Butwhenwegomuchfartherout,saytoN=1,000,000,000,theactualandpredictedgapsbecome19.7and20.7,respectively,anoverestimateofonlyabout5percent.ThevalidityofthelnNformulaasNtendstoinfinityisnowknownasthe

primenumbertheorem.Itwasfirstnoticed(butnotpublished)byCarlFriedrichGaussin1792whenhewasfifteenyearsold.(SeewhatakidcandowhennotdistractedbyanXbox?)Asforthischapter’sotherteens,MattiaandAlice,Ihopeyoucanappreciate

howpoignantitisthattwinprimesapparentlycontinuetoexistinthefarthestreachesofthenumberline,“inthatsilent,measuredspacemadeonlyofciphers.”Theoddsarestackedagainstthem.Accordingtotheprimenumbertheorem,anyparticularprimenearNhasnorighttoexpectapotentialmatemuchcloserthanlnNaway,agulfmuchlargerthan2whenNislarge.Andyetsomecouplesdobeattheodds.Computershavefoundtwinprimesat

unbelievablyremotepartsofthenumberline.Thelargestknownpairconsistsoftwonumberswith100,355decimaldigitseach,snugglinginthedarkness.Thetwinprimeconjecturesayscoupleslikethiswillturnupforever.Butasforfindinganotherprimecouplenearbyforafriendlygameof

doubles?Goodluck.

26.GroupThink

MYWIFEANDIhavedifferentsleepingstyles—andourmattressshowsit.Shehoardsthepillows,thrashesaroundallnightlong,andbarelydentsthemattress,whileIlieonmyback,mummy-like,moldingacavernousdepressionintomysideofthebed.Bedmanufacturersrecommendflippingyourmattressperiodically,probably

withpeoplelikemeinmind.Butwhat’sthebestsystem?Howexactlyareyousupposedtoflipittogetthemostevenwearoutofit?BrianHayesexploresthisprobleminthetitleessayofhisbookGroupTheory

intheBedroom.Double-entendresaside,the“group”underdiscussionhereisacollectionofmathematicalactions—allthepossiblewaysyoucouldfliporrotatethemattresssothatitstillfitsneatlyonthebedframe.

Bylookingintomattressmathinsomedetail,Ihopetogiveyouafeelingfor

grouptheorymoregenerally.It’soneofthemostversatilepartsofmathematics.ItunderlieseverythingfromthechoreographyofsquaredancingandthefundamentallawsofparticlephysicstothemosaicsoftheAlhambraandtheirchaoticcounterparts,likethisimage:

Astheseexamplessuggest,grouptheorybridgestheartsandsciences.Itaddressessomethingthetwoculturesshare—anabidingfascinationwithsymmetry.Yetbecauseitencompassessuchawiderangeofphenomena,grouptheoryisnecessarilyabstract.Itdistillssymmetrytoitsessence.Normallywethinkofsymmetryasapropertyofashape.Butgrouptheorists

focusmoreonwhatyoucandotoashape—specifically,allthewaysyoucanchangeitwhilekeepingsomethingelseaboutitthesame.Moreprecisely,theylookforallthetransformationsthatleaveashapeunchanged,givencertainconstraints.Thesetransformationsarecalledthesymmetriesoftheshape.Takentogether,theyformagroup,acollectionoftransformationswhoserelationshipsdefinetheshape’smostbasicarchitecture.Inthecaseofamattress,thetransformationsalteritsorientationinspace

(that’swhatchanges)whilemaintainingitsrigidity(that’stheconstraint).Andafterthetransformationiscomplete,themattresshastofitsnuglyontherectangularbedframe(that’swhatstaysthesame).Withtheserulesinplace,let’sseewhattransformationsqualifyformembershipinthisexclusivelittle

group.Itturnsoutthereareonlyfourofthem.Thefirstisthedo-nothingtransformation,alazybutpopularchoicethat

leavesthemattressuntouched.Itcertainlysatisfiesalltherules,butit’snotmuchhelpinprolongingthelifeofyourmattress.Still,it’sveryimportanttoincludeinthegroup.Itplaysthesameroleforgrouptheorythat0doesforadditionofnumbers,orthat1doesformultiplication.Mathematicianscallittheidentityelement,soI’lldenoteitbythesymbolI.Nextcomethethreegenuinewaystoflipamattress.Todistinguishamong

them,ithelpstolabelthecornersofthemattressbynumberingthemlikeso:

Thefirstkindofflipisdepictednearthebeginningofthischapter.Thehandsomegentlemaninstripedpajamasistryingtoturnthemattressfromsidetosidebyrotatingit180degreesarounditslongaxis,inamoveI’llcallH,for“horizontalflip.”

Amorerecklesswaytoturnoverthemattressisaverticalflip,V.Thismaneuverswapsitsheadandfoot.Youstandthemattressuprightthelongway,sothatitalmostreachestheceiling,andthentoppleitendoverend.Theneteffect,besidestheenormousthud,isthatthemattressrotates180degreesaboutits

lateralaxis,shownbelow.

Thefinalpossibilityistospinthemattresshalfaturnwhilekeepingitflatonthebed.

UnliketheHandVflips,thisrotation,R,keepsthetopsurfaceontop.Thatdifferenceshowsupwhenwelookatthemattress—nowimaginedtobe

translucent—fromaboveandinspectthenumbersatthecornersaftereachofthepossibletransformations.Thehorizontalflipturnsthenumeralsintotheirmirrorimages.Italsopermutesthemsothat1and2tradeplaces,asdo3and4.

Theverticalflippermutesthenumbersinadifferentwayandstandsthemontheirheads,inadditiontomirroringthem.

Therotation,however,doesn’tgenerateanymirrorimages.Itmerelyturnsthenumbersupsidedown,thistimeexchanging1for4and2for3.

Thesedetailsarenotthemainpoint.Whatmattersishowthetransformations

relatetooneanother.Theirpatternsofinteractionencodethesymmetryofthemattress.Torevealthosepatternswithaminimumofeffort,ithelpstodrawthe

followingdiagram.(ImageslikethisaboundinaterrificbookcalledVisualGroupTheory,byNathanCarter.It’soneofthebestintroductionstogrouptheory—ortoanybranchofhighermath—I’veeverread.)

Thefourpossiblestatesofthemattressareshownatthecornersofthediagram.Theupperleftstateisthestartingpoint.Thearrowsindicatethemovesthattakethemattressfromonestatetoanother.Forexample,thearrowpointingfromtheupperlefttothelowerrightdepicts

theactionoftherotationR.Thearrowalsohasanarrowheadontheotherend,becauseifyoudoRtwice,it’stantamounttodoingnothing.Thatshouldn’tcomeasasurprise.Itjustmeansthatturningthemattresshead

tofootandthendoingthatagainreturnsthemattresstoitsoriginalstate.WecansummarizethispropertywiththeequationRR=I,whereRRmeans“doRtwice,”andIisthedo-nothingidentityelement.Thehorizontalandverticalfliptransformationsalsoundothemselves:HH=IandVV=I.Thediagramembodiesawealthofotherinformation.Forinstance,itshows

thatthedeath-defyingverticalflip,V,isequivalenttoHR,ahorizontalflipfollowedbyarotation—amuchsaferpathtothesameresult.Tocheckthis,beginatthestartingstateintheupperleft.HeaddueeastalongHtothenextstate,andfromtheregodiagonallysouthwestalongR.Becauseyouarriveatthesamestateyou’dreachifyoufollowedVtobeginwith,thediagramdemonstratesthatHR=V.Notice,too,thattheorderofthoseactionsisirrelevant:HR=RH,sinceboth

roadsleadtoV.Thisindifferencetoorderistrueforanyotherpairofactions.Youshouldthinkofthisasageneralizationofthecommutativelawforadditionofordinarynumbers,xandy,accordingtowhichx+y=y+x.Butbeware:Themattressgroupisspecial.Manyothergroupsviolatethecommutativelaw.Those

fortunateenoughtoobeyitareparticularlycleanandsimple.Nowforthepayoff.Thediagramshowshowtogetthemostevenwearoutof

amattress.Anystrategythatsamplesallfourstatesperiodicallywillwork.Forexample,alternatingRandHisconvenient—andsinceitbypassesV,it’snottoostrenuous.Tohelpyourememberit,somemanufacturerssuggestthemnemonic“spininthespring,flipinthefall.”Themattressgroupalsopopsupinsomeunexpectedplaces,fromthe

symmetryofwatermoleculestothelogicofapairofelectricalswitches.That’soneofthecharmsofgrouptheory.Itexposesthehiddenunityofthingsthatwouldotherwiseseemunrelated...likeinthisanecdoteabouthowthephysicistRichardFeynmangotadraftdeferment.ThearmypsychiatristquestioninghimaskedFeynmantoputouthishandsso

hecouldexaminethem.Feynmanstuckthemout,onepalmup,theotherdown.“No,theotherway,”saidthepsychiatrist.SoFeynmanreversedbothhands,leavingonepalmdownandtheotherup.Feynmanwasn’tmerelyplayingmindgames;hewasindulginginalittle

group-theoretichumor.Ifweconsiderallthepossiblewayshecouldhaveheldouthishands,alongwiththevarioustransitionsamongthem,thearrowsformthesamepatternasthemattressgroup!

Butifallthismakesmattressesseemwaytoocomplicated,maybethereal

lessonhereisoneyoualreadyknew—ifsomething’sbotheringyou,justsleeponit.

27.TwistandShout

OURLOCALELEMENTARYschoolinvitesparentstocomeandtalktotheirchildren’sclasses.Thisgivesthekidsachancetohearaboutdifferentjobsandtolearnaboutthingstheymightnotbeexposedtootherwise.Whenmyturncame,Ishowedupatmydaughter’sfirst-gradeclasswithabag

fullofMöbiusstrips.Thenightbefore,mywifeandIhadcutlongstripsofpaperandthengiveneachstripahalftwist,likethis,

beforetapingthetwoendstogethertomakeaMöbiusstrip:

Therearefunactivitiesthatasix-year-oldcandowiththeseshapesthatinvolvenothingmorethanscissors,crayons,tape,andalittlecuriosity.AsmywifeandIpassedouttheMöbiusstripsandartsupplies,theteacher

askedtheclasswhatsubjecttheythoughtweweredoingnow.Oneboyraisedhishandandsaid,“Well,I’mnotsure,butIknowit’snotlinguistics.”Ofcourse,theteacherhadbeenexpectingananswerof“art,”ormaybe,more

precociously,“math.”Thebestanswer,however,wouldhavebeen“topology.”(InIthacait’saltogetherpossiblethatafirst-gradercouldhavecomeupwiththat.Butthetopologist’skidhappenedtobeinanotherclassthatyear.)Sowhatistopology?It’savibrantbranchofmodernmath,anoffshootof

geometry,butmuchmoreloosey-goosey.Intopology,twoshapesareregardedasthesameifyoucanbend,twist,stretch,orotherwisedeformoneintotheothercontinuously—thatis,withoutanyrippingorpuncturing.Unliketherigidobjectsofgeometry,theobjectsoftopologybehaveasiftheywereinfinitelyelastic,asiftheyweremadeofanidealkindofrubberorSillyPutty.Topologyshinesaspotlightonashape’sdeepestproperties—theproperties

thatremainunchangedafteracontinuousdistortion.Forexample,arubberbandshapedlikeasquareandanothershapedlikeacirclearetopologicallyindistinguishable.Itdoesn’tmatterthatasquarehasfourcornersandfourstraightsides.Thosepropertiesareirrelevant.Acontinuousdeformationcangetridofthembyroundingoffthesquare’scornersandbendingitssidesintocirculararcs.

Buttheonethingsuchadeformationcan’tgetridofistheintrinsicloopinessofacircleandasquare.They’rebothclosedcurves.That’stheirsharedtopologicalessence.Likewise,theessenceofaMöbiusstripisthepeculiarhalftwistlockedintoit.

Thattwistendowstheshapewithitssignaturefeatures.Mostfamously,aMöbiusstriphasonlyonesideandonlyoneedge.Inotherwords,itsfrontandbacksurfacesareactuallythesame,andsoareitstopandbottomedges.(Tocheckthis,justrunyourfingeralongthemiddleofthestriporitsedgeuntilyoureturntoyourstartingpoint.)Whathappenedhereisthatthehalftwisthookedtheformertopandbottomedgesofthepaperintoonebig,longcontinuouscurve.Similarly,itfusedthefronttotheback.Oncethestriphasbeentapedshut,thesefeaturesarepermanent.YoucanstretchandtwistaMöbiusstripallyouwant,butnothingcanchangeitshalf-twistedness,itsone-sidedness,anditsone-edgedness.Byhavingthefirst-gradersexaminethestrangepropertiesofMöbiusstrips

thatfollowfromthesefeatures,Iwashopingtoshowthemhowmuchfunmathcouldbe—andalsohowamazing.FirstIaskedthemeachtotakeacrayonandcarefullydrawalinealltheway

aroundtheMöbiusstrip,rightdownthemiddleofthesurface.Withbrowsfurrowed,theybegantracingsomethinglikethedashedlineonthefollowing

page:

Afteronecircuit,manyofthestudentsstoppedandlookedpuzzled.Thentheybeganshoutingtooneanotherexcitedly,becausetheirlineshadnotclosedasthey’dexpected.Thecrayonhadnotcomebacktothestartingpoint;itwasnowonthe“other”sideofthesurface.Thatwassurprisenumberone:youhavetogotwicearoundaMöbiusstriptogetbacktowhereyoustarted.Suddenlyoneboybeganmeltingdown.Whenherealizedhiscrayonhadn’t

comebacktoitsstartingpoint,hethoughthe’ddonesomethingwrong.Nomatterhowmanytimeswereassuredhimthatthiswassupposedtohappen,thathewasdoingagreatjob,andthatheshouldjustgoaroundthestriponemoretime,itdidn’thelp.Itwastoolate.Hewasonthefloor,wailing,inconsolable.WithsometrepidationIaskedtheclasstotrythenextactivity.Whatdidthey

thinkwouldhappeniftheytooktheirscissorsandcutneatlydownthemidlineallthewayalongthelengthofthestrip?Itwillfallapart!Itwillmaketwopieces!theyguessed.Butaftertheytriedit

andsomethingincrediblehappened(thestripremainedinonepiecebutgrewtwiceaslong),therewereevenmoresquealsofsurpriseanddelight.Itwaslike

amagictrick.Afterthatitwashardtoholdthestudents’attention.Theyweretoobusy

tryingtheirownexperiments,makingnewkindsofMöbiusstripswithtwoorthreehalftwistsinthemandcuttingthemlengthwiseintohalves,thirds,orquarters,producingallsortsoftwistednecklaces,chains,andknots,allthewhileshoutingvariationsof“Hey,lookwhatIfound!”ButIstillcan’tgetoverthatonelittleboyItraumatized.Andapparentlymylessonwasn’tthefirsttohavedrivenastudenttotears.ViHartwassofrustratedbyherboringmathcoursesinhighschoolthatshe

begandoodlinginclass,sketchingsnakesandtreesandinfinitechainsofshrinkingelephants,insteadoflisteningtotheteacherdroningon.Vi,whocallsherselfa“full-timerecreationalmathemusician,”haspostedsomeofherdoodlesonYouTube.They’venowbeenwatchedhundredsofthousandsoftimes,andinthecaseoftheelephants,morethanamillion.She,andhervideos,arebreathtakinglyoriginal.TwoofmyfavoriteshighlightthefreakypropertiesofMöbiusstripsthrough

aninventiveuseofmusicandstories.Inthelessbafflingofthetwo,her“Möbiusmusicbox”playsathemefromapieceofmusicshecomposed,inspiredbytheHarryPotterbooks.

Themelodyisencodedasaseriesofholespunchedthroughatape,whichisthenfedthroughastandardmusicbox.HerinnovationwastotwisttheendsofthetapeandjointhemtogethertoformaMöbiusstrip.Bycrankingthecrankonthemusicbox,Vifeedsthetapethroughthedeviceandthemelodyplaysinthenormalfashion.Butaboutfiftysecondsintothevideo,theloopcompletesonecircuit,andbecauseofthehalftwistintheMöbiusstrip,themusicboxnowbeginsplayingwhatwasoriginallythebackofthepunchedtape,upsidedown.Hencethesamemelodybeginsagainbutnowwithallthenotesinverted.Highnotesbecomelownotes,andlownotesbecomehigh.They’restillplayedinthesameorder,butupsidedown,thankstothesomersaultsimposedbytheMöbiusstructure.Foranevenmorestrikingexampleofthetopsy-turvyimplicationsofMöbius

strips,in“MöbiusStory:WindandMr.Ug,”Vitellsabittersweetstoryofunattainablelove.AfriendlylittletrianglenamedWind,drawnwithanerasablemarker,unknowinglylivesinaflatworldmadeoutofclearacetateandshapedlikeaMöbiusstrip.She’slonesomebuteverhopeful,eagertomeettheworld’sonlyotherinhabitant,amysteriousgentnamedMr.Ugwholivesonedoordown.Althoughshe’snevermethim—healwaysseemstobeoutwhenshestopsbyhishouse—shelovesthemessagesheleavesforherandlongstomeethimsomeday.

Spoileralert:skipthenextparagraphifyoudon’twanttolearnthestory’ssecret.Mr.Ugdoesn’texist.WindisMr.Ug,viewedupsidedownandonthebackof

thetransparentMöbiusstrip.BecauseofthecleverwaythatViprintslettersandmakestheworldturnbyspinningtheacetate,whenWind’snameorherhouseorhermessagesgooncearoundtheMöbiusstrip,allthosethingsflipoverandlookliketheybelongtoMr.Ug.Myexplanationdoesn’tdothisvideojustice.You’vesimplygottowatchitto

seethetremendousingenuityatworkhere,combiningauniquelovestorywithvividillustrationsofthepropertiesofMöbiusstrips.Otherartistshavelikewisedrawninspirationfromtheperplexingfeaturesof

Möbiusstrips.Escherusedtheminhisdrawingsofantstrappedinaneternalloop.Sculptorsandstonecarvers,likeMaxBillandKeizoUshio,haveincorporatedMöbiusmotifsintheirmassiveworks.PerhapsthemostmonumentalofallMöbiusstructuresisthatbeingplanned

fortheNationalLibraryofKazakhstan.Itsdesign,bytheDanisharchitectural

firmBIG,callsforspiralingpublicpathsthatcoilupandthendownandinwhich“likeayurtthewallbecomestheroof,whichbecomesfloor,whichbecomesthewallagain.”

ThepropertiesofMöbiusstripsofferdesignadvantagestoengineersaswell.

Forexample,acontinuous-looprecordingtapeintheshapeofaMöbiusstripdoublestheplayingtime.TheB.F.GoodrichCompanypatentedaMöbiusstripconveyorbelt,whichlaststwiceaslongasaconventionalbeltsinceitwearsevenlyon“both”sidesofitssurface(youknowwhatImean).OtherMöbiuspatentsincludenoveldesignsforcapacitors,abdominalsurgicalretractors,andself-cleaningfiltersfordry-cleaningmachines.Butperhapstheniftiestapplicationoftopologyisonethatdoesn’tinvolve

Möbiusstripsatall.It’savariationonthethemeoftwistsandlinks,andyoumightfindithelpfulnexttimeyouhaveguestsoverforbrunchonaSundaymorning.It’stheworkofGeorgeHart,Vi’sdad.He’sageometerandasculptor,formerlyacomputerscienceprofessoratStonyBrookUniversityandthechiefofcontentatMoMath,theMuseumofMathematicsinNewYorkCity.Georgehasdevisedawaytosliceabagelinhalfsuchthatthetwopiecesarelockedtogetherlikethelinksofachain.

Theadvantage,besidesleavingyourguestsagog,isthatitcreatesmoresurfacearea—andhencemoreroomforthecreamcheese.

28.ThinkGlobally

THEMOSTFAMILIARideasofgeometrywereinspiredbyanancientvision—avisionoftheworldasflat.FromthePythagoreantheoremtoparallellinesthatnevermeet,theseareeternaltruthsaboutanimaginaryplace,thetwo-dimensionallandscapeofplanegeometry.ConceivedinIndia,China,Egypt,andBabyloniamorethan2,500yearsago

andcodifiedandrefinedbyEuclidandtheGreeks,thisflat-earthgeometryisthemainone(andoftentheonlyone)beingtaughtinhighschoolstoday.Butthingshavechangedinthepastfewmillennia.Inaneraofglobalization,GoogleEarth,andintercontinentalairtravel,allof

usshouldtrytolearnalittleaboutsphericalgeometryanditsmoderngeneralization,differentialgeometry.Thebasicideashereareonlyabout200yearsold.PioneeredbyCarlFriedrichGaussandBernhardRiemann,differentialgeometryunderpinssuchimposingintellectualedificesasEinstein’sgeneraltheoryofrelativity.Atitsheart,however,arebeautifulconceptsthatcanbegraspedbyanyonewho’severriddenabicycle,lookedataglobe,orstretchedarubberband.Andunderstandingthemwillhelpyoumakesenseofafewcuriositiesyoumayhavenoticedinyourtravels.Forexample,whenIwaslittle,mydadusedtoenjoyquizzingmeabout

geography.Whichisfarthernorth,he’dask,RomeorNewYorkCity?MostpeoplewouldguessNewYorkCity,butsurprisinglythey’reatalmostthesamelatitude,withRomebeingjustabitfarthernorth.Ontheusualmapoftheworld(themisleadingMercatorprojection,whereGreenlandappearsgigantic),itlookslikeyoucouldgostraightfromNewYorktoRomebyheadingdueeast.Yetairlinepilotsnevertakethatroute.TheyalwaysflynortheastoutofNew

York,huggingthecoastofCanada.Iusedtothinktheywerestayingclosetolandforsafety’ssake,butthat’snotthereason.It’ssimplythemostdirectroutewhenyoutaketheEarth’scurvatureintoaccount.TheshortestwaytogetfromNewYorktoRomeistogopastNovaScotiaandNewfoundland,headoutovertheAtlantic,andfinallypasssouthofIrelandandflyacrossFrancetoarriveinsunnyItaly.

Thiskindofpathontheglobeiscalledanarcofagreatcircle.Likestraight

linesinordinaryspace,greatcirclesonaspherecontaintheshortestpathsbetweenanytwopoints.They’recalledgreatbecausethey’rethelargestcirclesyoucanhaveonasphere.ConspicuousexamplesincludetheequatorandthelongitudinalcirclesthatpassthroughtheNorthandSouthPoles.Anotherpropertythatlinesandgreatcirclesshareisthatthey’rethe

straightestpathsbetweentwopoints.Thatmightsoundstrange—allpathsonaglobearecurved,sowhatdowemeanby“straightest”?Well,somepathsaremorecurvedthanothers.Thegreatcirclesdon’tdoanyadditionalcurvingaboveandbeyondwhatthey’reforcedtodobyfollowingthesurfaceofthesphere.Here’sawaytovisualizethis.Imagineyou’reridingatinybicycleonthe

surfaceofaglobe,andyou’retryingtostayonacertainpath.Ifit’spartofagreatcircle,youcankeepthefrontwheelpointedstraightaheadatalltimes.That’sthesenseinwhichgreatcirclesarestraight.Incontrast,ifyoutrytoridealongalineoflatitudenearoneofthepoles,you’llhavetokeepthehandlebarsturned.Ofcourse,assurfacesgo,theplaneandthesphereareabnormallysimple.The

surfaceofahumanbody,oratincan,orabagelwouldbemoretypical—theyallhavefarlesssymmetry,aswellasvariouskindsofholesandpassagewaysthat

makethemmoreconfusingtonavigate.Inthismoregeneralsetting,findingtheshortestpathbetweenanytwopointsbecomesalottrickier.Soratherthandelvingintotechnicalities,let’ssticktoanintuitiveapproach.Thisiswhererubberbandscomeinhandy.Specifically,imagineaslipperyelasticstringthatalwayscontractsasfarasit

canwhileremainingonanobject’ssurface.Withitshelp,wecaneasilydeterminetheshortestpathbetweenNewYorkandRomeor,forthatmatter,betweenanytwopointsonanysurface.Tietheendsofthestringtothepointsofdepartureandarrivalandletthestringpullitselftightwhileitcontinuesclingingtothesurface’scontours.Whenthestringisastautastheseconstraintsallow,voilà!Ittracestheshortestpath.Onsurfacesjustalittlemorecomplicatedthanplanesorspheres,something

strangeandnewcanhappen:manylocallyshortestpathscanexistbetweenthesametwopoints.Forexample,considerthesurfaceofasoupcan,withonepointlyingdirectlybelowtheother.

Thentheshortestpathbetweenthemisclearlyalinesegment,asshownabove,andourelasticstringwouldfindthatsolution.Sowhat’snewhere?Thecylindricalshapeofthecanopensupnewpossibilitiesforallkindsofcontortions.Supposewerequirethatthestringencirclesthecylinderoncebeforeconnectingtothesecondpoint.(ConstraintslikethisareimposedonDNAwhenitwrapsaroundcertainproteinsinchromosomes.)Nowwhenthestringpullsitselftaut,itformsahelix,likethecurvesonoldbarbershoppoles.

Thishelicalpathqualifiesasanothersolutiontotheshortest-pathproblem,in

thesensethatit’stheshortestofthecandidatepathsnearby.Ifyounudgethestringalittle,itwouldnecessarilygetlongerandthencontractbacktothehelix.Youcouldsayit’sthelocallyshortestpath—theregionalchampionofallthosethatwraponcearoundthecylinder.(Bytheway,thisiswhythesubjectiscalleddifferentialgeometry:itstudiestheeffectsofsmalllocaldifferencesonvariouskindsofshapes,suchasthedifferenceinlengthbetweenthehelicalpathanditsneighbors.)Butthat’snotall.There’sanotherchampthatwindsaroundtwice,andanother

thatgoesaroundthreetimes,andsoon.Thereareinfinitelymanylocallyshortestpathsonacylinder!Ofcourse,noneofthesehelicesisthegloballyshortestpath.Thestraight-linepathisshorterthanallofthem.Likewise,surfaceswithholesandhandlespermitmanylocallyshortestpaths,

distinguishedbytheirpatternofweavingaroundvariouspartsofthesurface.ThefollowingsnapshotfromavideobythemathematicianKonradPolthieroftheFreeUniversityofBerlinillustratesthenon-uniquenessoftheselocallyshortestpaths,orgeodesics,onthesurfaceofanimaginaryplanetshapedlikeafigureeight,asurfaceknowninthetradeasatwo-holedtorus:

Thethreegeodesicsshownherevisitverydifferentpartsoftheplanet,therebyexecutingdifferentlooppatterns.Butwhattheyallhaveincommonistheirsuperiordirectnesscomparedtothepathsnearby.Andjustlikelinesonaplaneorgreatcirclesonasphere,thesegeodesicsarethestraightestpossiblecurvesonthesurface.Theybendtoconformtothesurfacebutdon’tbendwithinit.Tomakethisclear,Polthierhasproducedanotherilluminatingvideo.

Here,anunmannedmotorcycleridesalongageodesichighwayonatwo-holedtorus,followingthelayoftheland.Theremarkablethingisthatitshandlebarsarelockedstraightahead;itdoesn’tneedtosteertostayontheroad.Thisunderscorestheearlierimpressionthatgeodesics,likegreatcircles,arethenaturalgeneralizationofstraightlines.Withalltheseflightsoffancy,youmaybewonderingifgeodesicshave

anythingtodowithreality.Ofcoursetheydo.Einsteinshowedthatlightbeamsfollowgeodesicsastheysailthroughtheuniverse.Thefamousbendingofstarlightaroundthesun,detectedintheeclipseobservationsof1919,confirmedthatlighttravelsongeodesicsthroughcurvedspace-time,withthewarpingbeingcausedbythesun’sgravity.Atamoredown-to-earthlevel,themathematicsoffindingshortestpathsis

criticaltotheroutingoftrafficontheInternet.Inthissituation,however,therelevantspaceisagargantuanmazeofaddressesandlinks,asopposedtothe

smoothsurfacesconsideredabove,andthemathematicalissueshavetodowiththespeedofalgorithms—what’sthemostefficientwaytofindtheshortestpaththroughanetwork?Giventhemyriadofpotentialroutes,theproblemwouldbeoverwhelmingwereitnotfortheingenuityofthemathematiciansandcomputerscientistswhocrackedit.Sometimeswhenpeoplesaytheshortestdistancebetweentwopointsisa

straightline,theymeanitfiguratively,asawayofridiculingnuanceandaffirmingcommonsense.Inotherwords,keepitsimple.Butbattlingobstaclescangiverisetogreatbeauty—somuchsothatinart,andinmath,it’softenmorefruitfultoimposeconstraintsonourselves.Thinkofhaiku,orsonnets,ortellingthestoryofyourlifeinsixwords.Thesameistrueofallthemaththat’sbeencreatedtohelpyoufindtheshortestwayfromheretotherewhenyoucan’ttaketheeasywayout.Twopoints.Manypaths.Mathematicalbliss.

29.AnalyzeThis!

MATHSWAGGERSWITHanintimidatingairofcertainty.LikeaMafiacapo,itcomesacrossasdecisive,unyielding,andstrong.It’llmakeyouanargumentyoucan’trefuse.Butinprivate,mathisoccasionallyinsecure.Ithasdoubts.Itquestionsitself

andisn’talwayssureit’sright.Especiallywhereinfinityisconcerned.Infinitycankeepmathupatnight,worrying,fidgeting,feelingexistentialdread.Fortherehavebeentimesinthehistoryofmathwhenunleashinginfinitywroughtsuchmayhem,therewerefearsitmightblowupthewholeenterprise.Andthatwouldbebadforbusiness.IntheHBOseriesTheSopranos,mobbossTonySopranoconsultsa

psychiatrist,seekingtreatmentforanxietyattacks,tryingtounderstandwhyhismotherwantstohavehimkilled,thatsortofthing.Beneathatoughexteriorofcertaintyliesaveryconfusedandfrightenedperson.

Inmuchthesameway,calculusputitselfonthecouchjustwhenitseemedto

beatitsmostlethal.Afterdecadesoftriumph,ofmowingdownalltheproblemsthatstoodinitsway,itstartedtobecomeawareofsomethingrottenatitscore.Theverythingsthathadmadeitmostsuccessful—itsbrutalskillandfearlessnessinmanipulatinginfiniteprocesses—werenowthreateningtodestroyit.Andthetherapythateventuallyhelpeditthroughthiscrisiscametobeknown,coincidentally,asanalysis.Here’sanexampleofthekindofproblemthatworriedthemathematiciansof

the1700s.Considertheinfiniteseries

It’sthenumericalequivalentofvacillatingforever,takingonestepforward,onestepback,onestepforward,onestepback,andsoon,adinfinitum.Doesthisseriesevenmakesense?Andifso,whatdoesitequal?Disorientedbyaninfinitelylongexpressionlikethis,anoptimistmighthope

thatsomeoftheoldrules—therulesforgedfromexperiencewithfinitesums—wouldstillapply.Forexample,weknowthat1+2=2+1;whenweaddtwoormorenumbersinafinitesum,wecanalwaysswitchtheirorderwithoutchangingtheresult:a+bequalsb+a(thecommutativelawofaddition).Andwhentherearemorethantwoterms,wecanalwaysinsertparentheseswithabandon,groupingthetermshoweverwelike,withoutaffectingtheultimateanswer.Forinstance,(1+2)+4=1+(2+4);adding1and2first,then4,givesthesameanswerasadding2and4first,then1.Thisiscalledtheassociativelawofaddition.Itworksevenifsomenumbersarebeingsubtracted,aslongaswerememberthatsubtractinganumberisthesameasaddingitsnegative.Forexample,considerathree-termversionoftheseriesabove,andask:Whatis1–1+1?Wecouldviewitaseither(1–1)+1or1+(–1+1),whereinthatsecondsetofparentheseswe’veaddednegative1insteadofsubtracting1.Eitherway,theanswercomesouttobe1.Butwhenwetrytogeneralizetheserulestoinfinitesums,afewunpleasant

surpriseslieinstoreforus.Lookatthecontradictionthatoccursifwetrotouttheassociativelawandtrustinglyapplyitto1–1+1–1+1–1+∙∙∙.Ontheonehand,itappearswecanannihilatethepositiveandnegative1sbypairingthemofflikeso:

Ontheotherhand,wecouldjustaswellinserttheparentheseslikethisandconcludethatthesumis1:

Neitherargumentseemsmoreconvincingthantheother,soperhapsthesumis

both0and1?Thatpropositionsoundsabsurdtoustoday,butatthetimesomemathematicianswerecomfortedbyitsreligiousovertones.ItremindedthemofthetheologicalassertionthatGodcreatedtheworldfromnothing.AsthemathematicianandpriestGuidoGrandiwrotein1703,“Byputtingparenthesesintotheexpression1–1+1–1+∙∙∙indifferentways,Ican,ifIwant,obtain0or1.Butthentheideaofcreationexnihiloisperfectlyplausible.”Nevertheless,itappearsthatGrandifavoredathirdvalueforthesum,

differentfromeither0or1.Canyouguesswhathethoughtitshouldbe?Thinkofwhatyou’dsayifyouwerekiddingbuttryingtosoundscholastic.Right—Grandibelievedthetruesumwas .Andfarsuperior

mathematicians,includingLeibnizandEuler,agreed.Therewereseverallinesofreasoningthatsupportedthiscompromise.Thesimplestwastonoticethat1–1+1–1+∙∙∙couldbeexpressedintermsofitself,asfollows.Let’susetheletterStodenotethesum.Thenbydefinition

Nowleavethefirst1ontheright-handsidealoneandlookatalltheotherterms.TheyharbortheirowncopyofS,positionedtotherightofthatfirst1andsubtractedfromit:

SoS=1–Sandtherefore .Thedebateovertheseries1–1+1–1+∙∙∙ragedforabout150years,until

anewbreedofanalystsputallofcalculusanditsinfiniteprocesses(limits,derivatives,integrals,infiniteseries)onafirmfoundation,onceandforall.Theyrebuiltthesubjectfromthegroundup,fashioningalogicalstructureassoundasEuclid’sgeometry.Twooftheirkeynotionsarepartialsumsandconvergence.Apartialsumisa

runningtotal.Yousimplyaddupafinitenumberoftermsandthenstop.Forexample,ifwesumthefirstthreetermsoftheseries1–1+1–1+∙∙∙,weget1–1+1=1.Let’scallthisS3.HeretheletterSstandsfor“sum”andthesubscript3indicatesthatweaddedonlythefirstthreeterms.Similarly,thefirstfewpartialsumsforthisseriesare

Thusweseethatthepartialsumsbobblebackandforthbetween0and1,withnotendencytosettledownto0or1,to ,ortoanythingelse.Forthisreason,mathematicianstodaywouldsaythattheseries1–1+1–1+∙∙∙doesnotconverge.Inotherwords,itspartialsumsdon’tapproachanylimitingvalueasmoreandmoretermsareincludedinthesum.Thereforethesumoftheinfiniteseriesismeaningless.Sosupposewekeeptothestraightandnarrow—nodallyingwiththedark

side—andrestrictourattentiontoonlythoseseriesthatconverge.Doesthatgetridoftheearlierparadoxes?Notyet.Thenightmarescontinue.Andit’sjustaswellthattheydo,because

byfacingdownthesenewdemons,theanalystsofthe1800sdiscovereddeepersecretsattheheartofcalculusandthenexposedthemtothelight.Thelessonslearnedhaveprovedinvaluable,notjustwithinmathbutformath’sapplications

toeverythingfrommusictomedicalimaging.Considerthisseries,knowninthetradeasthealternatingharmonicseries:

Insteadofonestepforward,onestepback,thestepsnowgetprogressivelysmaller.It’sonestepforward,butonlyhalfastepback,thenathirdofthestepforward,afourthofastepback,andsoon.Noticethepattern:thefractionswithodddenominatorshaveplussignsinfrontofthem,whiletheevenfractionshavenegativesigns.Thepartialsumsinthiscaseare

Andifyougofarenough,you’llfindthattheyhomeinonanumbercloseto0.69.Infact,theseriescanbeproventoconverge.Itslimitingvalueisthenaturallogarithmof2,denotedln2andapproximatelyequalto0.693147.Sowhat’snightmarishhere?Onthefaceofit,nothing.Thealternating

harmonicseriesseemslikeanice,well-behaved,convergentseries,thesortyourparentswouldapproveof.Andthat’swhatmakesitsodangerous.It’sachameleon,aconman,a

slipperysickothatwillbeanythingyouwant.Ifyouaddupitstermsinadifferentorder,youcanmakeitsumtoanything.Literally.Itcanberearrangedtoconvergetoanyrealnumber:297.126,or–42π,or0,orwhateveryourheartdesires.It’sasiftheserieshaduttercontemptforthecommutativelawofaddition.

Merelybyaddingitstermsinadifferentorder,youcanchangetheanswer—somethingthatcouldneverhappenforafinitesum.Soeventhoughtheoriginalseriesconverges,it’sstillcapableofweirdnessunimaginableinordinaryarithmetic.Ratherthanprovethisastonishingfact(aresultknownastheRiemann

rearrangementtheorem),let’slookataparticularlysimplerearrangementwhosesumiseasytocalculate.Supposeweaddtwoofthenegativetermsinthe

alternatingharmonicseriesforeveryoneofitspositiveterms,asfollows:

Next,simplifyeachofthebracketedexpressionsbysubtractingthesecondtermfromthefirstwhileleavingthethirdtermuntouched.Thentheseriesreducesto

Afterfactoringout fromallthefractionsaboveandcollectingterms,thisbecomes

Lookwho’sback:thebeastinsidethebracketsisthealternatingharmonicseriesitself.Byrearrangingit,we’vesomehowmadeithalfasbigasitwasoriginally—eventhoughitcontainsallthesameterms!Arrangedinthisorder,theseriesnowconvergestoStrange,yes.Sick,yes.Andsurprisinglyenough,itmattersinreallifetoo.As

we’veseenthroughoutthisbook,eventhemostabstruseandfar-fetchedconceptsofmathoftenfindapplicationtopracticalthings.Thelinkinthepresentcaseisthatinmanypartsofscienceandtechnology,fromsignalprocessingandacousticstofinanceandmedicine,it’susefultorepresentvariouskindsofcurves,sounds,signals,orimagesassumsofsimplercurves,sounds,signals,orimages.Whenthebasicbuildingblocksaresinewaves,thetechniqueisknownasFourieranalysis,andthecorrespondingsumsarecalledFourierseries.Butwhentheseriesinquestionbearssomeofthesamepathologiesasthealternatingharmonicseriesanditsequallyderangedrelatives,theconvergencebehavioroftheFourierseriescanbeveryweirdindeed.Here,forexample,isaFourierseriesdirectlyinspiredbythealternating

harmonicseries:

Togetasenseforwhatthislookslike,let’sgraphthesumofitsfirsttenterms.

Thispartialsum(shownasasolidline)isclearlytryingtoapproximateamuchsimplercurve,awaveshapedliketheteethofasaw(shownbythedashedline).Notice,however,thatsomethinggoeswrongneartheedgesoftheteeth.Thesinewavesovershootthemarkthereandproduceastrangefingerthatisn’tinthesawtoothwaveitself.Toseethismoreclearly,here’sazoomnearoneofthoseedges,atx=π:

Supposewetrytogetridofthefingerbyincludingmoretermsinthesum.Noluck.Thefingerjustbecomesthinnerandmovesclosertotheedge,butitsheightstaysaboutthesame.

Theblamecanbelaidatthedoorstepofthealternatingharmonicseries.Its

pathologiesdiscussedearliernowcontaminatetheassociatedFourierseries.

They’reresponsibleforthatannoyingfingerthatjustwon’tgoaway.Thiseffect,commonlycalledtheGibbsphenomenon,ismorethana

mathematicalcuriosity.Knownsincethemid-1800s,itnowturnsupinourdigitalphotographsandonMRIscans.TheunwantedoscillationscausedbytheGibbsphenomenoncanproduceblurring,shimmering,andotherartifactsatsharpedgesintheimage.Inamedicalcontext,thesecanbemistakenfordamagedtissue,ortheycanobscurelesionsthatareactuallypresent.Fortunately,analystsacenturyagopinpointedwhatcausesGibbsartifacts(see

thenoteson[>]fordiscussion).Theirinsightshavetaughtushowtoovercomethem,oratleasthowtospotthemwhentheydooccur.Thetherapyhasbeenverysuccessful.Thecopayisduenow.

30.TheHilbertHotel

INFEBRUARY2010Ireceivedane-mailfromawomannamedKimForbes.Hersix-year-oldson,Ben,hadaskedheramathquestionshecouldn’tanswer,andshewashopingIcouldhelp:Todayisthe100thdayofschool.Hewasveryexcitedandtoldmeeverythingheknowsaboutthenumber100,includingthat100wasanevennumber.Hethentoldmethat101wasanoddnumberand1millionwasanevennumber,etc.Hethenpausedandasked:“Isinfinityevenorodd?”IexplainedtoKimthatinfinityisneitherevennorodd.It’snotanumberin

theusualsense,anditdoesn’tobeytherulesofarithmetic.Allsortsofcontradictionswouldfollowifitdid.Forinstance,Iwrote,“ifinfinitywereodd,2timesinfinitywouldbeeven.Butbothareinfinity!Sothewholeideaofoddandevendoesnotmakesenseforinfinity.”Kimreplied:Thankyou.Benwassatisfiedwiththatanswerandkindoflikestheideathatinfinityisbigenoughtobebothoddandeven.

Althoughsomethinggotgarbledintranslation(infinityisneitheroddnoreven,notboth),Ben’srenderinghintsatalargertruth.Infinitycanbemind-boggling.Someofitsstrangestaspectsfirstcametolightinthelate1800s,withGeorg

Cantor’sgroundbreakingworkonsettheory.Cantorwasparticularlyinterestedininfinitesetsofnumbersandpoints,liketheset{1,2,3,4,...}ofnaturalnumbersandthesetofpointsonaline.Hedefinedarigorouswaytocomparedifferentinfinitesetsanddiscovered,shockingly,thatsomeinfinitiesarebiggerthanothers.Atthetime,Cantor’stheoryprovokednotjustresistance,butoutrage.Henri

Poincaré,oneoftheleadingmathematiciansoftheday,calledita“disease.”Butanothergiantoftheera,DavidHilbert,sawitasalastingcontributionandlaterproclaimed,“NooneshallexpelusfromtheParadisethatCantorhascreated.”Mygoalhereistogiveyouaglimpseofthisparadise.Butratherthan

workingdirectlywithsetsofnumbersorpoints,letmefollowanapproachintroducedbyHilberthimself.HevividlyconveyedthestrangenessandwonderofCantor’stheorybytellingaparableaboutagrandhotel,nowknownasthe

HilbertHotel.It’salwaysbookedsolid,yetthere’salwaysavacancy.FortheHilbertHoteldoesn’thavemerelyhundredsofrooms—ithasan

infinitenumberofthem.Wheneveranewguestarrives,themanagershiftstheoccupantofroom1toroom2,room2toroom3,andsoon.Thatfreesuproom1forthenewcomerandaccommodateseveryoneelseaswell(thoughinconveniencingthembythemove).Nowsupposeinfinitelymanynewguestsarrive,sweatyandshort-tempered.

Noproblem.Theunflappablemanagermovestheoccupantofroom1toroom2,room2toroom4,room3toroom6,andsoon.Thisdoublingtrickopensupalltheodd-numberedrooms—infinitelymanyofthem—forthenewguests.Laterthatnight,anendlessconvoyofbusesrumblesuptoreception.There

areinfinitelymanybuses,andworsestill,eachoneisloadedwithaninfinityofcrabbypeopledemandingthatthehotelliveuptoitsmotto,“There’salwaysroomattheHilbertHotel.”Themanagerhasfacedthischallengebeforeandtakesitinstride.Firsthedoesthedoublingtrick.Thatreassignsthecurrentgueststotheeven-

numberedroomsandclearsoutalltheodd-numberedones—agoodstart,becausehenowhasaninfinitenumberofroomsavailable.Butisthatenough?Aretherereallyenoughodd-numberedroomsto

accommodatetheteeminghordeofnewguests?Itseemsunlikely,sincetherearesomethinglikeinfinitysquaredpeopleclamoringfortheserooms.(Whyinfinitysquared?Becausetherewasaninfinitenumberofpeopleoneachofaninfinitenumberofbuses,andthatamountstoinfinitytimesinfinity,whateverthatmeans.)Thisiswherethelogicofinfinitygetsveryweird.Tounderstandhowthemanagerisgoingtosolvehislatestproblem,ithelpsto

visualizeallthepeoplehehastoserve.

Ofcourse,wecan’tshowliterallyallofthemhere,sincethediagramwouldneedtobeinfiniteinbothdirections.Butafiniteversionofthepictureisadequate.Thepointisthatanyspecificbuspassenger(yourauntInez,say,onvacationfromLouisville)issuretoappearonthediagramsomewhere,aslongasweincludeenoughrowsandcolumns.Inthatsense,everybodyoneverybusisaccountedfor.Younamethepassenger,andheorsheiscertaintobedepictedatsomefinitenumberofstepseastandsouthofthediagram’scorner.Themanager’schallengeistofindawaytoworkthroughthispicture

systematically.Heneedstodeviseaschemeforassigningroomssothateverybodygetsoneeventually,afteronlyafinitenumberofotherpeoplehavebeenserved.Sadly,thepreviousmanagerhadn’tunderstoodthis,andmayhemensued.

Whenasimilarconvoyshoweduponhiswatch,hebecamesoflusteredtryingtoprocessallthepeopleonbus1thathenevergotaroundtoanyotherbus,leavingallthoseneglectedpassengersscreamingandfurious.Illustratedonthediagrambelow,thismyopicstrategywouldcorrespondtoapaththatmarchedeastwardalongrow1forever.

Thenewmanager,however,haseverythingundercontrol.Insteadoftending

tojustonebus,hezigsandzagsthroughthediagram,fanningoutfromthecorner,asshownbelow.

Hestartswithpassenger1onbus1andgivesherthefirstemptyroom.Thesecondandthirdemptyroomsgotopassenger2onbus1andpassenger1onbus2,bothofwhomaredepictedontheseconddiagonalfromthecornerofthediagram.Afterservingthem,themanagerproceedstothethirddiagonalandhandsoutasetofroomkeystopassenger1onbus3,passenger2onbus2,andpassenger3onbus1.Ihopethemanager’sprocedure—progressingfromonediagonaltoanother—

isclearfromthepictureabove,andyou’reconvincedthatanyparticularpersonwillbereachedinafinitenumberofsteps.So,asadvertised,there’salwaysroomattheHilbertHotel.TheargumentI’vejustpresentedisafamousoneinthetheoryofinfinitesets.

Cantorusedittoprovethatthereareexactlyasmanypositivefractions(ratiosp/qofpositivewholenumberspandq)astherearenaturalnumbers(1,2,3,4,...).That’samuchstrongerstatementthansayingbothsetsareinfinite.Itsaystheyareinfinitetopreciselythesameextent,inthesensethataone-to-onecorrespondencecanbeestablishedbetweenthem.Youcouldthinkofthiscorrespondenceasabuddysysteminwhicheach

naturalnumberispairedwithsomepositivefraction,andviceversa.Theexistenceofsuchabuddysystemseemsutterlyatoddswithcommonsense—it’s

thesortofsophistrythatmadePoincarérecoil.Foritimplieswecouldmakeanexhaustivelistofallpositivefractions,eventhoughthere’snosmallestone!Andyetthereissuchalist.We’vealreadyfoundit.Thefractionp/q

correspondstopassengerponbusq,andtheargumentaboveshowsthateachofthesefractionscanbepairedoffwithacertainnaturalnumber1,2,3,...,givenbythepassenger’sroomnumberattheHilbertHotel.ThecoupdegrâceisCantor’sproofthatsomeinfinitesetsarebiggerthan

this.Specifically,thesetofrealnumbersbetween0and1isuncountable—itcan’tbeputinone-to-onecorrespondencewiththenaturalnumbers.Forthehospitalityindustry,thismeansthatifalltheserealnumbersshowupatthereceptiondeskandbangonthebell,therewon’tbeenoughroomsforallofthem,evenattheHilbertHotel.Theproofisbycontradiction.Supposeeachrealnumbercouldbegivenits

ownroom.Thentherosterofoccupants,identifiedbytheirdecimalexpansionsandlistedbyroomnumber,wouldlooksomethinglikethis:

Remember,thisissupposedtobeacompletelist.Everyrealnumberbetween0and1issupposedtoappearsomewhere,atsomefiniteplaceontheroster.Cantorshowedthatalotofnumbersaremissingfromanysuchlist;that’sthe

contradiction.Forinstance,toconstructonethatappearsnowhereonthelistshownabove,godownthediagonalandbuildanewnumberfromtheunderlineddigits:

Thedecimalsogeneratedis.6975...

Butwe’renotdoneyet.Thenextstepistotakethisdecimalandchangeallitsdigits,replacingeachofthemwithanyotherdigitbetween1and8.Forexample,wecouldchangethe6toa3,the9toa2,the7toa5,andsoon.Thisnewdecimal.325...isthekiller.It’scertainlynotinroom1,sinceithas

adifferentfirstdigitfromthenumberthere.It’salsonotinroom2,sinceitsseconddigitdisagrees.Ingeneral,itdiffersfromthenthnumberinthenthdecimalplace.Soitdoesn’tappearanywhereonthelist!TheconclusionisthattheHilbertHotelcan’taccommodateallthereal

numbers.Therearesimplytoomanyofthem,aninfinitybeyondinfinity.Andwiththathumblingthought,wecometotheendofthisbook,which

beganwithasceneinanotherimaginaryhotel.ASesameStreetcharacternamedHumphrey,workingthelunchshiftattheFurryArms,tookanorderfromaroomfulofhungrypenguins—“Fish,fish,fish,fish,fish,fish”—andsoonlearnedaboutthepowerofnumbers.It’sbeenalongjourneyfromfishtoinfinity.Thanksforjoiningme.

Acknowledgments

Manyfriendsandcolleagueshelpedimprovethisbookbygenerouslyofferingtheirsageadvice—mathematical,stylistic,historical,andotherwise.ThankstoDougArnold,SheldonAxler,LarryBraden,DanCallahan,BobConnelly,TomGilovich,GeorgeHart,ViHart,DianeHopkins,HerbertHui,CindyKlauss,MichaelLewis,MichaelMauboussin,BarryMazur,EriNoguchi,CharliePeskin,StevePinker,RaviRamakrishna,DavidRand,RichardRand,PeterRenz,DouglasRogers,JohnSmillie,GrantWiggins,StephenYeung,andCarlZimmer.Othercolleaguescreatedimagesforthisbookorallowedmetoincludetheir

visualwork.ThankstoRickAllmendinger,PaulBourke,MikeField,BrianMadsen,NikDayman(Teamfresh),MarkNewman,KonradPolthier,ChristianRudderatOkCupid,SimonTatham,andJaneWang.IamimmenselygratefultoDavidShipleyforinvitingmetowritetheNew

YorkTimesseriesthatledtothisbook,andespeciallyforhisvisionofhowtheseriesshouldbestructured.Simplicity,simplicity,simplicity,urgedThoreau—andbothheandShipleywereright.GeorgeKalogerakis,myeditorattheTimes,wieldedhispenlightly,movingcommas,butonlywhennecessary,whileprotectingmefrommoreseriousinfelicities.Hisconfidencewasenormouslyreassuring.KatieO’Brienontheproductionteammadesurethemathalwayslookedrightandputupwiththerequisitetypographicalfussingwithgraceandgoodhumor.IfeelsofortunatetohaveKatinkaMatsoninmycornerasmyliteraryagent.

Shechampionedthisbookfromthebeginningwithanenthusiasmthatwasinspirational.PaulGinsparg,JonKleinberg,TimNovikoff,andAndyRuinareaddraftsof

nearlyeverychapter,theironlycompensationbeingthepleasureofcatchingclunkersandusingtheirbrilliantmindsforgoodinsteadofevil.Normallyit’sadragtobearoundsuchknow-it-alls,butthefactis,theydoknowitall.Andthisbookisthebetterforit.I’mtrulygratefulfortheireffortandencouragement.ThankstoMargyNelson,theillustrator,forherplayfulnessandscientific

sensibility.Sheoftenfelttomelikeapartnerinthisproject,withherknackforfindingoriginalwaystoconveytheessenceofamathematicalconcept.AnywriterwouldbeblessedtohaveAmandaCookasaneditor.Howcan

anyonebesogentleandwiseanddecisive,allatthesametime?Thankyou,Amanda,forbelievinginthisbookandforhelpingmeshapeeverybitofit.EamonDolan,anotheroftheworld’sgreateditors,guidedthisproject(andme)

towardthefinishlinewithasurehandandinfectiousexcitement.EditorialassistantsAshleyGilliamandBenHymanweremeticulousandfuntoworkwithandtookgoodcareofthebookateverystageofitsdevelopment.CopyeditorTracyRoetaughtmeaboutappositives,apostrophes,andwordsaswords.Butmoreimportant(not“importantly”!),shesharpenedthewritingandthinkinginthesepages.ThanksaswelltopublicistMichelleBonanno,marketingmanagerAyeshaMirza,productioneditorRebeccaSpringer,productionmanagerDavidFutato,andtheentireteamatHoughtonMifflinHarcourt.Finally,letmeaddmymostheartfeltthankstomyfamily.LeahandJo,

you’vebeenhearingaboutthebookforalongtimenow,andbelieveitornot,itreallyhascometoanend.Yournextchore,naturally,istolearnallthemathinit.Andasformyfantasticallypatientwife,Carole,whosloggedthroughthefirstndraftsofeachchapterandtherebylearnedthetruemeaningoftheexpression“asntendstoinfinity,”letmesaysimply,Iloveyou.FindingyouwasthebestproblemIeversolved.

Notes

1.FromFishtoInfinity

[>]SesameStreet:ThevideoSesameStreet:123CountwithMe(1997)isavailableforpurchaseonlineineitherVHSorDVDformat.[>]numbers...havelivesoftheirown:Forapassionatepresentationoftheideathatnumbershavelivesoftheirownandthenotionthatmathematicscanbeviewedasaformofart,seeP.Lockhart,AMathematician’sLament(BellevueLiteraryPress,2009).[>]“theunreasonableeffectivenessofmathematics”:Theessaythatintroducedthisnow-famousphraseisE.Wigner,“Theunreasonableeffectivenessofmathematicsinthenaturalsciences,”CommunicationsinPureandAppliedMathematics,Vol.13,No.1(February1960),[>].Anonlineversionisavailableathttp://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html.Forfurtherreflectionsontheseideasandontherelatedquestionofwhether

mathwasinventedordiscovered,seeM.Livio,IsGodaMathematician?(SimonandSchuster,2009),andR.W.Hamming,“Theunreasonableeffectivenessofmathematics,”AmericanMathematicalMonthly,Vol.87,No.2(February1980),availableonlineathttp://www-lmmb.ncifcrf.gov/~toms/Hamming.unreasonable.html.

2.RockGroups

[>]Theplayfulsideofarithmetic:AsIhopetomakeclear,thischapterowesmuchtotwobooks—oneapolemic,theotheranovel,bothofthembrilliant:P.Lockhart,AMathematician’sLament(BellevueLiteraryPress,2009),whichinspiredtherockmetaphorandsomeoftheexamplesusedhere;andY.Ogawa,TheHousekeeperandtheProfessor(Picador,2009).[>]achild’scuriosity:Foryoungreaderswholikeexploringnumbersandthepatternstheymake,seeH.M.Enzensberger,TheNumberDevil(HoltPaperbacks,2000).[>]hallmarkofanelegantproof:DelightfulbutmoreadvancedexamplesofvisualizationinmathematicsarepresentedinR.B.Nelsen,ProofswithoutWords(MathematicalAssociationofAmerica,1997).

3.TheEnemyofMyEnemy

[>]“Yeah,yeah”:FormoreofSidneyMorgenbesser’switticismsandacademicone-liners,seethesamplingatLanguageLog(August5,2004),“IfP,sowhynotQ?”onlineathttp://itre.cis.upenn.edu/%7Emyl/languagelog/archives/001314.html.[>]relationshiptriangles:BalancetheorywasfirstproposedbythesocialpsychologistFritzHeiderandhassincebeendevelopedandappliedbysocialnetworktheorists,politicalscientists,anthropologists,mathematicians,andphysicists.Fortheoriginalformulation,seeF.Heider,“Attitudesandcognitiveorganization,”JournalofPsychology,Vol.21(1946),[>],andF.Heider,ThePsychologyofInterpersonalRelations(JohnWileyandSons,1958).Forareviewofbalancetheoryfromasocialnetworkperspective,seeS.WassermanandK.Faust,SocialNetworkAnalysis(CambridgeUniversityPress,1994),chapter6.[>]polarizedstatesaretheonlystatesasstableasnirvana:ThetheoremthatabalancedstateinafullyconnectednetworkmustbeeitherasinglenirvanaofallfriendsortwomutuallyantagonisticfactionswasfirstproveninD.CartwrightandF.Harary,“Structuralbalance:AgeneralizationofHeider’stheory,”PsychologicalReview,Vol.63(1956),[>].Averyreadableversionofthatproof,andagentleintroductiontothemathematicsofbalancetheory,hasbeengivenbytwoofmycolleaguesatCornell:D.EasleyandJ.Kleinberg,Networks,Crowds,andMarkets(CambridgeUniversityPress,2010).Inmuchoftheearlyworkonbalancetheory,atriangleofthreemutual

enemies(andhencethreenegativesides)wasconsideredunbalanced.Iassumedthisimplicitlywhenquotingtheresultsaboutnirvanaandthetwo-blocstatebeingtheonlyconfigurationsofafullyconnectednetworkinwhichalltrianglesarebalanced.However,someresearchershavechallengedthisassumptionandhaveexploredtheimplicationsoftreatingatriangleofthreenegativesasbalanced.Formoreonthisandothergeneralizationsofbalancetheory,seethebooksbyWassermanandFaustandbyEasleyandKleinbergcitedabove.[>]WorldWarI:TheexampleandgraphicaldepictionoftheshiftingalliancesbeforeWorldWarIarefromT.Antal,P.L.Krapivsky,andS.Redner,“Socialbalanceonnetworks:Thedynamicsoffriendshipandenmity,”PhysicaD,Vol.224(2006),[>],availableonlineathttp://arxiv.org/abs/physics/0605183.Thispaper,writtenbythreestatisticalphysicists,isnotableforrecastingbalancetheoryinadynamicframework,thusextendingitbeyondtheearlierstaticapproaches.ForthehistoricaldetailsoftheEuropeanalliances,seeW.L.Langer,EuropeanAlliancesandAlignments,1871–1890,2ndedition(Knopf,

1956),andB.E.Schmitt,TripleAllianceandTripleEntente(HenryHoltandCompany,1934).

4.Commuting

[>]revisitmultiplicationfromscratch:KeithDevlinhaswrittenaprovocativeseriesofessaysaboutthenatureofmultiplication:whatitis,whatitisnot,andwhycertainwaysofthinkingaboutitaremorevaluableandreliablethanothers.Hearguesinfavorofthinkingofmultiplicationasscaling,notrepeatedaddition,andshowsthatthetwoconceptsareverydifferentinreal-worldsettingswhereunitsareinvolved.SeehisJanuary2011blogpost“Whatexactlyismultiplication?”athttp://www.maa.org/devlin/devlin_01_11.html,aswellasthreeearlierpostsfrom2008:“Itain’tnorepeatedaddition”(http://www.maa.org/devlin/devlin_06_08.html);“It’sstillnotrepeatedaddition”(http://www.maa.org/devlin/devlin_0708_08.html);and“MultiplicationandthosepeskyBritishspellings”(http://www.maa.org/devlin/devlin_09_08.html).Theseessaysgeneratedalotofdiscussionintheblogosphere,especiallyamongschoolteachers.Ifyou’reshortontime,I’drecommendreadingtheonefrom2011first.[>]shoppingforanewpairofjeans:Forthejeansexample,theorderinwhichthetaxanddiscountareappliedmaynotmattertoyou—inbothscenariosyouenduppaying$43.20—butitmakesabigdifferencetothegovernmentandthestore!Intheclerk’sscenario(whereyoupaytaxbasedontheoriginalprice),youwouldpay$4intax;inyourscenario,only$3.20.Sohowcanthefinalpricecomeoutthesame?It’sbecauseintheclerk’sscenariothestoregetstokeep$39.20,whereasinyoursitwouldkeep$40.I’mnotsurewhatthelawrequires,anditmayvaryfromplacetoplace,buttherationalthingwouldbeforthegovernmenttochargesalestaxbasedontheactualpaymentthestorereceives.Onlyyourscenariosatisfiesthiscriterion.Forfurtherdiscussion,seehttp://www.facebook.com/TeachersofMathematics/posts/166897663338316.[>]financialdecisions:ForheatedonlineargumentsabouttherelativemeritsofaRoth401(k)versusatraditionalone,andwhetherthecommutativelawhasanythingtodowiththeseissues,seetheFinanceBuff,“Commutativelawofmultiplication”(http://thefinancebuff.com/commutative-law-of-multiplication.html),andtheSimpleDollar,“ThenewRoth401(k)versusthetraditional401(k):Whichisthebetterroute?”(http://www.thesimpledollar.com/2007/06/20/the-new-roth-401k-versus-the-traditional-401k-which-is-the-better-route/).[>]attendingMITandkillinghimselfdidn’tcommute:ThisstoryaboutMurray

Gell-MannisrecountedinG.Johnson,StrangeBeauty(Knopf,1999),[>].InGell-Mann’sownwords,hewasofferedadmissiontothe“dreaded”MassachusettsInstituteofTechnologyatthesametimeashewas“contemplatingsuicide,asbefitssomeonerejectedfromtheIvyLeague.Itoccurredtomehowever(anditisaninterestingexampleofnon-commutationofoperators)thatIcouldtryM.I.T.firstandkillmyselflater,whilethereverseorderofeventswasimpossible.”ThisexcerptappearsinH.Fritzsch,MurrayGell-Mann:SelectedPapers(WorldScientific,2009),[>].[>]developmentofquantummechanics:ForanaccountofhowHeisenbergandDiracdiscoveredtheroleofnon-commutingvariablesinquantummechanics,seeG.Farmelo,TheStrangestMan(BasicBooks,2009),[>].

5.DivisionandItsDiscontents

[>]MyLeftFoot:AclipofthescenewhereyoungChristystrugglesvaliantlytoanswerthequestion“What’stwenty-fivepercentofaquarter?”isavailableonlineathttp://www.tcm.com/mediaroom/video/223343/My-Left-Foot-Movie-Clip-25-Percent-of-a-Quarter.html.[>]VerizonWireless:GeorgeVaccaro’sblog(http://verizonmath.blogspot.com/)providestheexasperatingdetailsofhisencounterswithVerizon.Thetranscriptofhisconversationwithcustomerserviceisavailableathttp://verizonmath.blogspot.com/2006/12/transcription-jt.html.Theaudiorecordingisathttp://imgs.xkcd.com/verizon_billing.mp3.[>]you’reforcedtoconcludethat1mustequal.9999...:Forreaderswhomaystillfindithardtoacceptthat1=.9999...,theargumentthateventuallyconvincedmewasthis:theymustbeequal,becausethere’snoroomforanyotherdecimaltofitbetweenthem.(Whereasiftwodecimalsareunequal,theiraverageisbetweenthem,asareinfinitelymanyotherdecimals.)[>]almostalldecimalsareirrational:TheamazingpropertiesofirrationalnumbersarediscussedatahighermathematicallevelontheMathWorldpage“IrrationalNumber,”http://mathworld.wolfram.com/IrrationalNumber.html.Thesenseinwhichthedigitsofirrationalnumbersarerandomisclarifiedathttp://mathworld.wolfram.com/NormalNumber.html.

6.Location,Location,Location

[>]EzraCornell’sstatue:FormoreaboutCornell,includinghisroleinWesternUnionandtheearlydaysofthetelegraph,seeP.Dorf,TheBuilder:ABiographyofEzraCornell(Macmillan,1952);W.P.Marshall,EzraCornell(Kessinger

Publishing,2006);andhttp://rmc.library.cornell.edu/ezra/index.html,anonlineexhibitioninhonorofCornell’s200thbirthday.[>]systemsforwritingnumbers:Ancientnumbersystemsandtheoriginsofthedecimalplace-valuesystemarediscussedinV.J.Katz,AHistoryofMathematics,2ndedition(AddisonWesleyLongman,1998),andinC.B.BoyerandU.C.Merzbach,AHistoryofMathematics,3rdedition(Wiley,2011).Forachattieraccount,seeC.Seife,Zero(Viking,2000),chapter1.[>]Romannumerals:MarkChu-CarrollclarifiessomeofthepeculiarfeaturesofRomannumeralsandarithmeticinthisblogpost:http://scienceblogs.com/goodmath/2006/08/roman_numerals_and_arithmetic.php[>]Babylonians:AfascinatingexhibitionofBabylonianmathisdescribedbyN.Wade,“Anexhibitionthatgetstothe(square)rootofSumerianmath,”NewYorkTimes(November22,2010),onlineathttp://www.nytimes.com/2010/11/23/science/23babylon.html,accompaniedbyaslideshowathttp://www.nytimes.com/slideshow/2010/11/18/science/20101123-babylon.html.[>]nothingtodowithhumanappendages:Thismaywellbeanoverstatement.Youcancounttotwelveononehandbyusingyourthumbtoindicateeachofthethreelittlefingerbones(phalanges)ontheotherfourfingers.Thenyoucanuseallfivefingersonyourotherhandtokeeptrackofhowmanysetsoftwelveyou’vecounted.Thebase60systemusedbytheSumeriansmayhaveoriginatedinthisway.Formoreonthishypothesisandotherspeculationsabouttheoriginsofthebase60system,seeG.Ifrah,TheUniversalHistoryofNumbers(Wiley,2000),chapter9.

7.TheJoyofx

[>]Jorealizedsomethingaboutherbigsister,Leah:Forsticklers,Leahisactuallytwenty-onemonthsolderthanJo.HenceJo’sformulaisonlyanapproximation.Obviously![>]“WhenIwasatLosAlamos”:FeynmantellsthestoryofBethe’strickforsquaringnumberscloseto50inR.P.Feynman,“SurelyYou’reJoking,Mr.Feynman!”(W.W.NortonandCompany,1985),[>].[>]moneyinvestedinthestockmarket:Theidentityabouttheeffectofequalup-and-downpercentageswingsinthestockmarketcanbeprovensymbolically,bymultiplying1+xby1–x,orgeometrically,bydrawingadiagramsimilartothatusedtoexplainBethe’strick.Ifyou’reinthemood,trybothapproachesasanexercise.[>]sociallyacceptableagedifferenceinaromance:The“halfyourageplus

seven”ruleabouttheacceptableagegapinaromanticrelationshipiscalledthestandardcreepinessruleinthisxkcdcomic:http://xkcd.com/314/.

8.FindingYourRoots

[>]theirstruggletofindtheroots:Thequestforsolutionstoincreasinglycomplicatedequations,fromquadratictoquintic,isrecountedinvividdetailinM.Livio,TheEquationThatCouldn’tBeSolved(SimonandSchuster,2005).[>]doublingacube’svolume:Formoreabouttheclassicproblemofdoublingthecube,seehttp://www-history.mcs.st-and.ac.uk/HistTopics/Doubling_the_cube.html.[>]squarerootsofnegativenumbers:Tolearnmoreaboutimaginaryandcomplexnumbers,theirapplications,andtheircheckeredhistory,seeP.J.Nahin,AnImaginaryTale(PrincetonUniversityPress,1998),andB.Mazur,ImaginingNumbers(Farrar,StrausandGiroux,2003).[>]dynamicsofNewton’smethod:ForasuperbjournalisticaccountofJohnHubbard’swork,seeJ.Gleick,Chaos(Viking,1987),[>].Hubbard’sowntakeonNewton’smethodappearsinsection2.8ofJ.HubbardandB.B.Hubbard,VectorCalculus,LinearAlgebra,andDifferentialForms,4thedition(MatrixEditions,2009).ForreaderswhowanttodelveintothemathematicsofNewton’smethod,a

moresophisticatedbutstillreadableintroductionisgiveninH.-O.PeitgenandP.H.Richter,TheBeautyofFractals(Springer,1986),chapter6,andalsoseethearticlebyA.Douady(Hubbard’scollaborator)entitled“JuliasetsandtheMandelbrotset,”startingon[>]ofthesamebook.[>]Theborderlandslookedlikepsychedelichallucinations::HubbardwasnotthefirstmathematiciantoaskquestionsaboutNewton’smethodinthecomplexplane;ArthurCayleyhadwonderedaboutthesamethingsin1879.Hetoolookedatbothquadraticandcubicpolynomialsandrealizedthatthefirstcasewaseasyandthesecondwashard.Althoughhecouldn’thaveknownaboutthefractalsthatwouldbediscoveredacenturylater,heclearlyunderstoodthatsomethingnastycouldhappenwhenthereweremorethantworoots.Inhisone-pagearticle“Desiderataandsuggestions:No.3—theNewton-Fourierimaginaryproblem,”AmericanJournalofMathematics,Vol.2,No.1(March1879),[>],availableonlineathttp://www.jstor.org/pss/2369201,Cayley’sfinalsentenceisamarvelofunderstatement:“Thesolutioniseasyandelegantinthecaseofaquadricequation,butthenextsucceedingcaseofthecubicequationappearstopresentconsiderabledifficulty.”[>]Thestructurewasafractal:Thesnapshotsshowninthischapterwere

computedusingNewton’smethodappliedtothepolynomialz3–1.Therootsarethethreecuberootsof1.Forthiscase,Newton’salgorithmtakesapointzinthecomplexplaneandmapsittoanewpoint

Thatpointthenbecomesthenextz.Thisprocessisrepeateduntilzcomessufficientlyclosetoarootor,equivalently,untilz3–1comessufficientlyclosetozero,where“sufficientlyclose”isaverysmalldistance,arbitrarilychosenbythepersonwhoprogrammedthecomputer.Allinitialpointsthatleadtoaparticularrootarethenassignedthesamecolor.Thusredlabelsallthepointsthatconvergetooneroot,greenlabelsanother,andbluelabelsthethird.ThesnapshotsoftheresultingNewtonfractalwerekindlyprovidedbySimon

Tatham.Formoreonhiswork,seehiswebpage“FractalsderivedfromNewton-Raphsoniteration”(http://www.chiark.greenend.org.uk/~sgtatham/newton/).VideoanimationsoftheNewtonfractalhavebeencreatedbyTeamfresh.

Stunninglydeepzoomsintootherfractals,includingthefamousMandelbrotset,areavailableattheTeamfreshwebsite,http://www.hd-fractals.com.[>]templebuildersinIndia:ForanintroductiontotheancientIndianmethodsforfindingsquareroots,seeD.W.HendersonandD.Taimina,ExperiencingGeometry,3rdexpandedandrevisededition(PearsonPrenticeHall,2005).

9.MyTubRunnethOver

[>]myfirstwordproblem:Alargecollectionofclassicwordproblemsisavailableathttp://MathNEXUS.wwu.edu/Archive/oldie/list.asp.[>]fillingabathtub:Amoredifficultbathtubproblemappearsinthe1941movieHowGreenWasMyValley.Foraclip,seehttp://www.math.harvard.edu/~knill/mathmovies/index.html.Andwhileyou’rethere,youshouldalsocheckoutthisclipfromthebaseballcomedyLittleBigLeague:http://www.math.harvard.edu/~knill/mathmovies/m4v/league.m4v.Itcontainsawordproblemaboutpaintinghouses:“IfIcanpaintahouseinthreehours,andyoucanpaintitinfive,howlongwillittakeustopaintittogether?”Thesceneshowsthebaseballplayersgivingvarioussillyanswers.“It’ssimple,fivetimesthree,sothat’sfifteen.”“No,no,no,lookit.Ittakeseighthours:fiveplusthree,that’seight.”Afterafewmoreblunders,oneplayerfinallygetsitright: hours.

10.WorkingYourQuads

[>]toptenmostbeautifulorimportantequations:Forbooksaboutgreatequations,seeM.Guillen,FiveEquationsThatChangedtheWorld(Hyperion,1995);G.Farmelo,ItMustBeBeautiful(Granta,2002);andR.P.Crease,TheGreatEquations(W.W.NortonandCompany,2009).Thereareseverallistspostedonlinetoo.I’dsuggeststartingwithK.Chang,“Whatmakesanequationbeautiful?,”NewYorkTimes(October24,2004),http://www.nytimes.com/2004/10/24/weekinreview/24chan.html.Foroneofthefewliststhatincludethequadraticequation,seehttp://www4.ncsu.edu/~kaltofen/top10eqs/top10eqs.html.[>]calculatinginheritances:ManyexamplesarediscussedinS.Gandz,“Thealgebraofinheritance:Arehabilitationofal-Khuwarizmi,”Osiris,Vol.5(1938),pp.319–391.[>]al-Khwarizmi:Al-Khwarizmi’sapproachtothequadraticequationisexplainedinV.J.Katz,AHistoryofMathematics,2ndedition(AddisonWesleyLongman,1998),[>].

11.PowerTools

[>]Moonlighting:Thebanteraboutlogarithmsisfromtheepisode“InGodWeStronglySuspect.”ItoriginallyairedonFebruary11,1986,duringtheshow’ssecondseason.Avideoclipisavailableathttp://opinionator.blogs.nytimes.com/2010/03/28/power-tools/.[>]functions:Forsimplicity,I’vereferredtoexpressionslikex²asfunctions,thoughtobemorepreciseIshouldspeakof“thefunctionthatmapsxintox².”Ihopethissortofabbreviationwon’tcauseconfusion,sincewe’veallseenitoncalculatorbuttons.[>]breathtakingparabolas:ForapromotionalvideoofthewaterfeatureattheDetroitairport,createdbyWETDesign,seehttp://www.youtube.com/watch?v=VSUKNxVXE4E.SeveralhomevideosofitarealsoavailableatYouTube.Oneofthemostvividis“DetroitAirportWaterFeature”byPassTravelFool(http://www.youtube.com/watch?v=or8i_EvIRdE).WillHoffmanandDerekPaulBoylehavefilmedanintriguingvideoofthe

parabolas(alongwiththeirexponentialcousins,curvescalledcatenaries,sonamedfortheshapeofhangingchains)thatareallaroundusintheeverydayworld.See“WNYC/NPR’sRadioLabpresentsParabolas(etc.)”onlineathttp://www.youtube.com/watch?v=rdSgqHuI-mw.Fulldisclosure:thefilmmakerssaythisvideowasinspiredbyastoryItoldonanepisodeof

Radiolab(“Yellowfluffandothercuriousencounters,”availableathttp://www.radiolab.org/2009/jan/12/).[>]foldapieceofpaperinhalfmorethansevenoreighttimes:ForthestoryofBritneyGallivan’sadventuresinpaperfolding,seeB.Gallivan,“Howtofoldapaperinhalftwelvetimes:An‘impossiblechallenge’solvedandexplained,”Pomona,CA:HistoricalSocietyofPomonaValley,2002,onlineathttp://pomonahistorical.org/12times.htm.Forajournalist’saccountaimedatchildren,seeI.Peterson,“Championpaper-folder,”Muse(July/August2004),[>],availableonlineathttp://musemath.blogspot.com/2007/06/champion-paper-folder.html.TheMythBustershaveattemptedtoreplicateBritney’sexperimentontheirtelevisionshow(http://kwc.org/mythbusters/2007/01/episode_72_underwater_car_and.html).[>]Weperceivepitchlogarithmically:Forreferencesandfurtherdiscussionofmusicalscalesandour(approximately)logarithmicperceptionofpitch,seeJ.H.McDermottandA.J.Oxenham,“Musicperception,pitch,andtheauditorysystem,”CurrentOpinioninNeurobiology,Vol.18(2008),[>];http://en.wikipedia.org/wiki/Pitch_(music);http://en.wikipedia.org/wiki/Musical_scale;andhttp://en.wikipedia.org/wiki/Piano_key_frequencies.Forevidencethatourinnatenumbersenseisalsologarithmic,seeS.Dehaene,

V.Izard,E.Spelke,andP.Pica,“Logorlinear?DistinctintuitionsofthenumberscaleinWesternandAmazonianindigenecultures,”Science,Vol.320(2008),pp.1217–1220.PopularaccountsofthisstudyareavailableatScienceDaily(http://www.sciencedaily.com/releases/2008/05/080529141344.htm)andinanepisodeofRadiolabcalled“Numbers”(http://www.radiolab.org/2009/nov/30/).

12.SquareDancing

[>]Pythagoreantheorem:TheancientBabylonians,Indians,andChineseappeartohavebeenawareofthecontentofthePythagoreantheoremseveralcenturiesbeforePythagorasandtheGreeks.Formoreaboutthehistoryandsignificanceofthetheorem,aswellasasurveyofthemanyingeniouswaystoproveit,seeE.Maor,ThePythagoreanTheorem(PrincetonUniversityPress,2007).[>]hypotenuse:Onp.xiiiofhisbook,Maorexplainsthattheword“hypotenuse”means“stretchedbeneath”andpointsoutthatthismakessenseiftherighttriangleisviewedwithitshypotenuseatthebottom,asdepictedinEuclid’sElements.HealsonotesthatthisinterpretationfitswellwiththeChinesewordforhypotenuse:“hsien,astringstretchedbetweentwopoints(as

inalute).”[>]buildingthesquaresoutofmanylittlecrackers:ChildrenandtheirparentswillenjoytheedibleillustrationsofthePythagoreantheoremsuggestedbyGeorgeHartinhispostfortheMuseumofMathematics“Pythagoreancrackers,”http://momath.org/home/pythagorean-crackers/.[>]anotherproof:Ifyouenjoyseeingdifferentproofs,anicelyannotatedcollectionofdozensofthem—withcreatorsrangingfromEuclidtoLeonardodaVincitoPresidentJamesGarfield—isavailableatAlexanderBogomolny’sblogCuttheKnot.Seehttp://www.cut-the-knot.org/pythagoras/index.shtml.[>]missingsteps:Withanyluck,thefirstproofinthechaptershouldhavegivenyouanAha!sensation.Buttomaketheargumentcompletelyairtight,wealsoneedtoprovethatthepicturesaren’tdeceivingus—inotherwords,theytrulyhavethepropertiestheyappeartohave.Amorerigorousproofwouldestablish,forexample,thattheouterframeistrulyasquare,andthatthemediumandsmallsquaresmeetatasinglepoint,asshown.Checkingthesedetailsisfunandnottoodifficult.

Herearethemissingstepsinthesecondproof.Takethisequation:

andrearrangeittoget

Similarlymassaginganotheroftheequationsyields

Finally,substitutingtheexpressionsabovefordandeintotheequationc=d+eyields

Thenmultiplyingbothsidesbycgivesthedesiredformula:

13.SomethingfromNothing

[>]Euclid:ForallthirteenbooksoftheElementsinaconvenientone-volumeformat,withplentifuldiagrams,seeEuclid’sElements,editedbyD.Densmore,translatedbyT.L.Heath(GreenLionPress,2002).AnotherexcellentoptionisafreedownloadablePDFdocumentbyRichardFitzpatrickgivinghisownmoderntranslationofEuclid’sElements,availableathttp://farside.ph.utexas.edu/euclid.html.

[>]ThomasJefferson:FormoreaboutThomasJefferson’sreverenceforEuclidandNewtonandhisuseoftheiraxiomaticapproachintheDeclarationofIndependence,seeI.B.Cohen,ScienceandtheFoundingFathers(W.W.NortonandCompany,1995),[>],aswellasJ.Fauvel,“Jeffersonandmathematics,”http://www.math.virginia.edu/Jefferson/jefferson.htm,especiallythepageabouttheDeclarationofIndependence:http://www.math.virginia.edu/Jefferson/jeff_r(4).htm.[>]equilateraltriangle:ForEuclid’sversionoftheequilateraltriangleproof,inGreek,seehttp://en.wikipedia.org/wiki/File:Euclid-proof.jpg.[>]logicalandcreative:Ihaveglossedoveranumberofsubtletiesinthetwoproofspresentedinthischapter.Forexample,intheequilateraltriangleproof,weimplicitlyassumed(asEucliddid)thatthetwocirclesintersectsomewhere—inparticular,atthepointwelabeledC.ButtheexistenceofthatintersectionisnotguaranteedbyanyofEuclid’saxioms;oneneedsanadditionalaxiomaboutthecontinuityofthecircles.BertrandRussell,amongothers,notedthislacuna:B.Russell,“TheTeachingofEuclid,”MathematicalGazette,Vol.2,No.33(1902),[>],availableonlineathttp://www-history.mcs.st-and.ac.uk/history/Extras/Russell_Euclid.html.Anothersubtletyinvolvestheimplicituseoftheparallelpostulateintheproof

thattheanglesofatrianglesumto180degrees.Thatpostulateiswhatgaveuspermissiontoconstructthelineparalleltothetriangle’sbase.Inothertypesofgeometry(knownasnon-Euclideangeometries),theremightexistnolineparalleltothebase,orinfinitelymanysuchlines.Inthosekindsofgeometries,whichareeverybitaslogicallyconsistentasEuclid’s,theanglesofatriangledon’talwaysaddupto180degrees.Thus,thePythagoreanproofgivenhereismorethanjuststunninglyelegant;itrevealssomethingdeepaboutthenatureofspaceitself.Formorecommentaryontheseissues,seetheblogpostbyA.Bogomolny,“Anglesintriangleaddto180°,”http://www.cut-the-knot.org/triangle/pythpar/AnglesInTriangle.shtml,andthearticlebyT.Beardon,“Whentheanglesofatriangledon’taddupto180degrees,”http://nrich.maths.org/1434.

14.TheConicConspiracy

[>]parabolasandellipses:Forbackgroundonconicsectionsandreferencestothevastliteratureonthem,seehttp://mathworld.wolfram.com/ConicSection.htmlandhttp://en.wikipedia.org/wiki/Conic_section.Forreaderswithsomemathematicaltraining,alargeamountofinterestingandunusualinformationhasbeen

collectedbyJamesB.Calvertonhiswebsite;see“Ellipse”(http://mysite.du.edu/~jcalvert/math/ellipse.htm),“Parabola”(http://mysite.du.edu/~jcalvert/math/parabola.htm),and“Hyperbola”(http://mysite.du.edu/~jcalvert/math/hyperb.htm).[>]parabolicmirror:You’llgainalotofintuitionbywatchingtheonlineanimationscreatedbyLouTalmananddiscussedonhispage“Thegeometryoftheconicsections,”http://rowdy.mscd.edu/~talmanl/HTML/GeometryOfConicSections.html.Inparticular,watchhttp://clem.mscd.edu/~talmanl/HTML/ParabolicReflector.htmlandfixonasinglephotonasitapproachesandthenbouncesofftheparabolicreflector.Thenlookatallthephotonsmovingtogether.You’llneverwanttotanyourfacewithasunreflectoragain.Theanalogousanimationforanellipseisshownathttp://rowdy.mscd.edu/~talmanl/HTML/EllipticReflector.html.

15.SineQuaNon

[>]sunrisesandsunsets:ThechartsshowninthetextareforJupiter,Florida,usingdatafrom2011.Forconvenience,thetimesofsunriseandsunsethavebeenexpressedrelativetoEasternStandardTime(theUTC-05:00timezone)allyearlongtoavoidtheartificialdiscontinuitiescausedbydaylight-savingtime.Youcancreatesimilarchartsofsunriseandsunsetforyourownlocationatwebsitessuchashttp://ptaff.ca/soleil/?lang=en_CAorhttp://www.gaisma.com/en/.Studentsseemtofindthesechartssurprising(forinstance,someofthem

expectthecurvestobetriangularinappearanceinsteadofroundedandsmooth),whichcanmakeforinstructiveclassroomactivitiesatthehigh-schoolormiddle-schoollevel.Forapedagogicalcasestudy,seeA.FriedlanderandT.Resnick,“Sunrise,sunset,”MontanaMathematicsEnthusiast,Vol.3,No.2(2006),[>],availableathttp://www.math.umt.edu/tmme/vol3no2/TMMEvol3no2_Israel_pp249_255.pdf.Derivingformulasforthetimesofsunriseandsunsetiscomplicated,both

mathematicallyandintermsofthephysicsinvolved.See,forexample,T.L.Watts’swebpage“Variationinthetimeofsunrise”athttp://www.physics.rutgers.edu/~twatts/sunrise/sunrise.html.Watts’sanalysisclarifieswhythetimesofsunriseandsunsetdonotvaryassimplesinewavesthroughouttheyear.Theyalsoincludeasecondharmonic(asinewavewithaperiodofsixmonths),mainlyduetoasubtleeffectoftheEarth’stiltthatcausesasemiannualvariationinlocalnoon,thetimeofdaywhenthesunishighestinthesky.Happily,thistermisthesameintheformulasforbothsunriseandsunset

times.Sowhenyousubtractonefromtheothertocomputethelengthoftheday(thenumberofhoursbetweensunriseandsunset),thesecondharmoniccancelsout.What’sleftisverynearlyaperfectsinewave.MoreinformationaboutallthiscanbefoundontheWebbysearchingfor“the

EquationofTime.”(Seriously—that’swhatit’scalled!)AgoodstartingpointisK.Taylor’swebpage“Theequationoftime:Whysundialtimediffersfromclocktimedependingontimeofyear,”http://myweb.tiscali.co.uk/moonkmft/Articles/EquationOfTime.html,ortheWikipediapagehttp://en.wikipedia.org/wiki/Equation_of_time.[>]trigonometry:ThesubjectislovinglysurveyedinE.Maor,TrigonometricDelights(PrincetonUniversityPress,1998).[>]patternformation:Forabroadoverviewofpatternsinnature,seeP.Ball,TheSelf-MadeTapestry(OxfordUniversityPress,1999).ThemathematicalmethodsinthisfieldarepresentedatagraduatelevelinR.Hoyle,PatternFormation(CambridgeUniversityPress,2006).Formathematicalanalysesofzebrastripes,butterfly-wingpatterns,andotherbiologicalexamplesofpatternformation,seeJ.D.Murray,MathematicalBiology:II.SpatialModelsandBiomedicalApplications,3rdedition(Springer,2003).[>]cosmicmicrowavebackground:TheconnectionsbetweenbiologicalpatternformationandcosmologyareoneofthemanydelightstobefoundinJannaLevin’sbookHowtheUniverseGotItsSpots(PrincetonUniversityPress,2002).It’sstructuredasaseriesofunsentletterstohermotherandrangesgracefullyoverthehistoryandideasofmathematicsandphysics,interwovenwithanintimatediaryofayoungscientistassheembarksonhercareer.[>]inflationarycosmology:Forabriefintroductiontocosmologyandinflation,seetwoarticlesbyStephenBattersby:“Introduction:Cosmology,”NewScientist(September4,2006),onlineathttp://www.newscientist.com/article/dn9988-introduction-cosmology.html,and“Bestevermapoftheearlyuniverserevealed,”NewScientist(March17,2006),onlineathttp://www.newscientist.com/article/dn8862-best-ever-map-of-the-early-universe-revealed.html.Thecaseforinflationremainscontroversial,however.ItsstrengthsandweaknessesareexplainedinP.J.Steinhardt,“Theinflationdebate:Isthetheoryattheheartofmoderncosmologydeeplyflawed?”ScientificAmerican(April2011),[>].

16.TakeIttotheLimit

[>]Zeno:ThehistoryandintellectuallegacyofZeno’sparadoxesarediscussedinJ.Mazur,Zeno’sParadox(Plume,2008).

[>]circlesandpi:Foradelightfullyopinionatedandwittyhistoryofpi,seeP.Beckmann,AHistoryofPi(St.Martin’sPress,1976).[>]Archimedesusedasimilarstrategy:ThePBStelevisionseriesNovaranawonderfulepisodeaboutArchimedes,infinity,andlimits,called“InfiniteSecrets.”ItoriginallyairedonSeptember30,2003.Theprogramwebsite(http://www.pbs.org/wgbh/nova/archimedes/)offersmanyonlineresources,includingtheprogramtranscriptandinteractivedemonstrations.[>]methodofexhaustion:ForreaderswishingtoseethemathematicaldetailsofArchimedes’smethodofexhaustion,NealCarothershasusedtrigonometry(equivalenttothePythagoreangymnasticsthatArchimedesreliedon)toderivetheperimetersoftheinscribedandcircumscribedpolygonsbetweenwhichthecircleistrapped;seehttp://personal.bgsu.edu/~carother/pi/Pi3a.html.PeterAlfeld’swebpage“Archimedesandthecomputationofpi”featuresaninteractiveJavaappletthatletsyouchangethenumberofsidesinthepolygons;seehttp://www.math.utah.edu/~alfeld/Archimedes/Archimedes.html.TheindividualstepsinArchimedes’soriginalargumentareofhistoricalinterestbutyoumightfindthemdisappointinglyobscure;seehttp://itech.fgcu.edu/faculty/clindsey/mhf4404/archimedes/archimedes.html.[>]piremainsaselusiveasever:AnyonecuriousabouttheheroiccomputationsofpitoimmensenumbersofdigitsshouldenjoyRichardPreston’sprofileoftheChudnovskybrothers.Entitled“Themountainsofpi,”thisaffectionateandsurprisinglycomicalpieceappearedintheMarch2,1992,issueoftheNewYorker,andmorerecentlyasachapterinR.Preston,PanicinLevelFour(RandomHouse,2008).[>]numericalanalysis:Foratextbookintroductiontothebasicsofnumericalanalysis,seeW.H.Press,S.A.Teukolsky,W.T.Vetterling,andB.P.Flannery,NumericalRecipes,3rdedition(CambridgeUniversityPress,2007).

17.ChangeWeCanBelieveIn

[>]EveryyearaboutamillionAmericanstudentstakecalculus:D.M.Bressoud,“Thecrisisofcalculus,”MathematicalAssociationofAmerica(April2007),availableathttp://www.maa.org/columns/launchings/launchings_04_07.html.[>]MichaelJordanflyingthroughtheair:ForvideoclipsofMichaelJordan’smostspectaculardunks,seehttp://www.youtube.com/watch?v=H8M2NgjvicA.[>]Myhigh-schoolcalculusteacher:ForacollectionofMr.Joffray’scalculusproblems,bothclassicandoriginal,seeS.Strogatz,TheCalculusofFriendship(PrincetonUniversityPress,2009).

[>]Snell’slaw:Severalarticles,videos,andwebsitespresentthedetailsofSnell’slawanditsderivationfromFermat’sprinciple(whichstatesthatlighttakesthepathofleasttime).Forexample,seeM.Golomb,“ElementaryproofsfortheequivalenceofFermat’sprincipleandSnell’slaw,”AmericanMathematicalMonthly,Vol.71,No.5(May1964),pp.541–543,andhttp://en.wikibooks.org/wiki/Optics/Fermat%27s_Principle.Othersprovidehistoricalaccounts;seehttp://en.wikipedia.org/wiki/Snell%27s_law.Fermat’sprinciplewasanearlyforerunnertothemoregeneralprincipleof

leastaction.Forentertaininganddeeplyenlighteningdiscussionsofthisprinciple,includingitsbasisinquantummechanics,seeR.P.Feynman,R.B.Leighton,andM.Sands,“Theprincipleofleastaction,”TheFeynmanLecturesonPhysics,Vol.2,chapter19(Addison-Wesley,1964),andR.Feynman,QED(PrincetonUniversityPress,1988).[>]allpossiblepaths:Inanutshell,Feynman’sastonishingpropositionisthatnatureactuallydoestryallpaths.Butnearlyallofthepathscanceloneanotheroutthroughaquantumanalogofdestructiveinterference—exceptforthoseveryclosetotheclassicalpathwheretheactionisminimized(or,moreprecisely,madestationary).Therethequantuminterferencebecomesconstructive,renderingthosepathsexceedinglymorelikelytobeobserved.This,inFeynman’saccount,iswhynatureobeysminimumprinciples.ThekeyisthatweliveinthemacroscopicworldofeverydayexperiencewheretheactionsarecolossalcomparedtoPlanck’sconstant.Inthatclassicallimit,quantumdestructiveinterferencebecomesextremelystrongandobliteratesnearlyeverythingelsethatcouldotherwisehappen.

18.ItSlices,ItDices

[>]oncology:Formoreaboutthewaysthatintegralcalculushasbeenusedtohelpcancerresearchers,seeD.Mackenzie,“Mathematicalmodelingofcancer,”SIAMNews,Vol.37(January/February2004),andH.P.Greenspan,“Modelsforthegrowthofasolidtumorbydiffusion,”StudiesinAppliedMathematics(December1972),pp.317–340.[>]solidcommontotwoidenticalcylinders:TheregioncommontotwoidenticalcircularcylinderswhoseaxesintersectatrightanglesisknownvariouslyasaSteinmetzsolidorabicylinder.Forbackground,seehttp://mathworld.wolfram.com/SteinmetzSolid.htmlandhttp://en.wikipedia.org/wiki/Steinmetz_solid.TheWikipediapagealsoincludesaveryhelpfulcomputeranimationthatshowstheSteinmetzsolidemerging,ghostlike,fromtheintersectingcylinders.Itsvolumecanbecalculated

straightforwardlybutopaquelybymoderntechniques.AnancientandmuchsimplersolutionwasknowntobothArchimedesand

TsuCh’ung-Chih.Itusesnothingmorethanthemethodofslicingandacomparisonbetweentheareasofasquareandacircle.Foramarvelouslyclearexposition,seeMartinGardner’scolumn“Mathematicalgames:Somepuzzlesbasedoncheckerboards,”ScientificAmerican,Vol.207(November1962),[>].AndforArchimedesandTsuCh’ung-Chih,see:Archimedes,TheMethod,EnglishtranslationbyT.L.Heath(1912),reprintedbyDover(1953);andT.Kiang,“AnoldChinesewayoffindingthevolumeofasphere,”MathematicalGazette,Vol.56(May1972),[>].MoretonMoorepointsoutthatthebicylinderalsohasapplicationsin

architecture:“TheRomansandNormans,inusingthebarrelvaulttospantheirbuildings,werefamiliarwiththegeometryofintersectingcylinderswheretwosuchvaultscrossedoneanothertoformacrossvault.”Forthis,aswellasapplicationstocrystallography,seeM.Moore,“Symmetricalintersectionsofrightcircularcylinders,”MathematicalGazette,Vol.58(October1974),[>].[>]Computergraphics:InteractivedemonstrationsofthebicylinderandotherproblemsinintegralcalculusareavailableonlineattheWolframDemonstrationsProject(http://demonstrations.wolfram.com/).Toplaythem,you’llneedtodownloadthefreeMathematicaPlayer(http://www.wolfram.com/products/player/),whichwillthenallowyoutoexplorehundredsofotherinteractivedemonstrationsinallpartsofmathematics.Thebicylinderdemoisathttp://demonstrations.wolfram.com/IntersectingCylinders/.MamikonMnatsakanianatCaltechhasproducedaseriesofanimationsthatillustratetheArchimedeanspiritandthepowerofslicing.Myfavoriteishttp://www.its.caltech.edu/~mamikon/Sphere.html,whichdepictsabeautifulrelationshipamongthevolumesofasphereandacertaindouble-coneandcylinderwhoseheightandradiusmatchthoseofthesphere.Healsoshowsthesamethingmorephysicallybydraininganimaginaryvolumeofliquidfromthecylinderandpouringitintotheothertwoshapes;seehttp://www.its.caltech.edu/~mamikon/SphereWater.html.SimilarlyelegantmechanicalargumentsintheserviceofmatharegiveninM.Levi,TheMathematicalMechanic(PrincetonUniversityPress,2009).[>]Archimedesmanagedtodothis:ForArchimedes’sapplicationofhismechanicalmethodtotheproblemoffindingthevolumeofthebicylinder,seeT.L.Heath,ed.,Proposition15,TheMethodofArchimedes,RecentlyDiscoveredbyHeiberg(CosimoClassics,2007),[>].On[>]ofthesamevolume,Archimedesconfessesthatheviewshis

mechanicalmethodasameansfordiscoveringtheoremsratherthanprovingthem:“certainthingsfirstbecamecleartomebyamechanicalmethod,althoughtheyhadtobedemonstratedbygeometryafterwardsbecausetheirinvestigationbythesaidmethoddidnotfurnishanactualdemonstration.Butitisofcourseeasier,whenwehavepreviouslyacquired,bythemethod,someknowledgeofthequestions,tosupplytheproofthanitistofinditwithoutanypreviousknowledge.”ForapopularaccountofArchimedes’swork,seeR.NetzandW.Noel,The

ArchimedesCodex(DaCapoPress,2009).

19.Allaboute

[>]whatise,exactly:Foranintroductiontoallthingseandexponential,seeE.Maor,e:TheStoryofaNumber(PrincetonUniversityPress,1994).ReaderswithabackgroundincalculuswillenjoytheChauvenetPrize–winningarticlebyB.J.McCartin,“e:Themasterofall,”MathematicalIntelligencer,Vol.28,No.2(2006),[>].APDFversionisavailableathttp://mathdl.maa.org/images/upload_library/22/Chauvenet/mccartin.pdf.[>]about13.5percentoftheseatsgotowaste:Theexpectedpackingfractionforcouplessittinginatheateratrandomhasbeenstudiedinotherguisesinthescientificliterature.Itfirstaroseinorganicchemistry—seeP.J.Flory,“Intramolecularreactionbetweenneighboringsubstituentsofvinylpolymers,”JournaloftheAmericanChemicalSociety,Vol.61(1939),pp.1518–1521.Themagicnumber1/e²appearsonthetoprightcolumnofp.1519.Amorerecenttreatmentrelatesthisquestiontotherandomparkingproblem,aclassicpuzzleinprobabilitytheoryandstatisticalphysics;seeW.H.Olson,“AMarkovchainmodelforthekineticsofreactantisolation,”JournalofAppliedProbability,Vol.15,No.4(1978),pp.835–841.Computerscientistshavetackledsimilarquestionsintheirstudiesof“randomgreedymatching”algorithmsforpairingneighboringnodesofnetworks;seeM.DyerandA.Frieze,“Randomizedgreedymatching,”RandomStructuresandAlgorithms,Vol.2(1991),[>].[>]howmanypeopleyoushoulddate:Thequestionofwhentostopdatingandchooseamatehasalsobeenstudiedinvariousforms,leadingtosuchdesignationsasthefiancéeproblem,themarriageproblem,thefussy-suitorproblem,andthesultan’s-dowryproblem.Butthemostcommontermforitnowadaysisthesecretaryproblem(theimaginedscenariobeingthatyou’retryingtohirethebestsecretaryfromagivenpoolofcandidates;youinterviewthecandidatesoneatatimeandhavetodecideonthespotwhethertohirethepersonorsaygoodbyeforever).Foranintroductiontothemathematicsand

historyofthiswonderfulpuzzle,seehttp://mathworld.wolfram.com/SultansDowryProblem.htmlandhttp://en.wikipedia.org/wiki/Secretary_problem.Formoredetails,seeT.S.Ferguson,“Whosolvedthesecretaryproblem?”StatisticalScience,Vol.4,No.3(1989),[>].Aclearexpositionofhowtosolvetheproblemisgivenathttp://www.math.uah.edu/stat/urn/Secretary.html.Foranintroductiontothelargersubjectofoptimalstoppingtheory,seeT.P.Hill,“Knowingwhentostop:Howtogambleifyoumust—themathematicsofoptimalstopping,”AmericanScientist,Vol.97(2009),[>].

20.LovesMe,LovesMeNot

[>]supposeRomeoisinlovewithJuliet:Formodelsofloveaffairsbasedondifferentialequations,seesection5.3inS.H.Strogatz,NonlinearDynamicsandChaos(Perseus,1994).[>]“Itisusefultosolvedifferentialequations”:ForNewton’sanagram,seep.viiinV.I.Arnold,GeometricalMethodsintheTheoryofOrdinaryDifferentialEquations(Springer,1994).[>]chaos:Chaosinthethree-bodyproblemisdiscussedinI.Peterson,Newton’sClock(W.H.Freeman,1993).[>]“madehisheadache”:Forthequoteabouthowthethree-bodyproblemmadeNewton’sheadache,seeD.Brewster,MemoirsoftheLife,Writings,andDiscoveriesofSirIsaacNewton(ThomasConstableandCompany,1855),Vol.2,[>].

21.StepIntotheLight

[>]vectorcalculus:AgreatintroductiontovectorcalculusandMaxwell’sequations,andperhapsthebesttextbookI’veeverread,isE.M.Purcell,ElectricityandMagnetism,2ndedition(CambridgeUniversityPress,2011).AnotherclassicisH.M.Schey,Div,Grad,Curl,andAllThat,4thedition(W.W.NortonandCompany,2005).[>]Maxwelldiscoveredwhatlightis:Thesewordsarebeingwrittenduringthe150thanniversaryofMaxwell’s1861paper“Onphysicallinesofforce.”See,specifically,“PartIII.Thetheoryofmolecularvorticesappliedtostaticalelectricity,”PhilosophicalMagazine(AprilandMay,1861),[>],availableathttp://en.wikisource.org/wiki/On_Physical_Lines_of_Forceandscannedfromtheoriginalathttp://www.vacuum-physics.com/Maxwell/maxwell_oplf.pdf.Theoriginalpaperiswellworthalook.Ahighpointoccursjustbelow

equation137,whereMaxwell—asobermannotpronetotheatrics—couldn’tresistitalicizingthemostrevolutionaryimplicationofhiswork:“Thevelocityoftransverseundulationsinourhypotheticalmedium,calculatedfromtheelectro-magneticexperimentsofM.M.KohlrauschandWeber,agreessoexactlywiththevelocityoflightcalculatedfromtheopticalexperimentsofM.Fizeau,thatwecanscarcelyavoidtheinferencethatlightconsistsinthetransverseundulationsofthesamemediumwhichisthecauseofelectricandmagneticphenomena.”[>]airflowaroundadragonflyasithovered:ForJaneWang’sworkondragonflyflight,seeZ.J.Wang,“Twodimensionalmechanismforinsecthovering,”PhysicalReviewLetters,Vol.85,No.10(September2000),pp.2216–2219,andZ.J.Wang,“Dragonflyflight,”PhysicsToday,Vol.61,No.10(October2008),[>].Herpapersarealsodownloadablefromhttp://dragonfly.tam.cornell.edu/insect_flight.html.Avideoofdragonflyflightisatthebottomofhttp://ptonline.aip.org/journals/doc/PHTOAD-ft/vol_61/iss_10/74_1.shtml.[>]HowIwishIcouldhavewitnessedthemoment:EinsteinalsoseemstohavewishedhewereaflyonthewallinMaxwell’sstudy.Ashewrotein1940,“Imagine[Maxwell’s]feelingswhenthedifferentialequationshehadformulatedprovedtohimthatelectromagneticfieldsspreadintheformofpolarizedwavesandatthespeedoflight!Tofewmenintheworldhassuchanexperiencebeenvouchsafed.”Seep.489inA.Einstein,“Considerationsconcerningthefundamentsoftheoreticalphysics,”Science,Vol.91(May24,1940),pp.487–492(availableonlineathttp://www.scribd.com/doc/30217690/Albert-Einstein-Considerations-Concerning-the-Fundaments-of-Theoretical-Physics).[>]thelegacyofhisconjuringwithsymbols:Maxwell’sequationsareoftenportrayedasatriumphofpurereason,butSimonSchaffer,ahistorianofscienceatCambridge,hasarguedtheyweredrivenasmuchbythetechnologicalchallengeoftheday:theproblemoftransmittingsignalsalongunderseatelegraphcables.SeeS.Schaffer,“Thelairdofphysics,”Nature,Vol.471(2011),[>].

22.TheNewNormal

[>]theworldisnowteemingwithdata:Forthenewworldofdatamining,seeS.Baker,TheNumerati(HoughtonMifflinHarcourt,2008),andI.Ayres,SuperCrunchers(Bantam,2007).[>]Sportsstatisticianscrunchthenumbers:M.Lewis,Moneyball(W.W.NortonandCompany,2003).

[>]“Learnsomestatistics”:N.G.Mankiw,“Acourseloadforthegameoflife,”NewYorkTimes(September4,2010).[>]“Takestatistics”:D.Brooks,“Harvard-bound?Chinup,”NewYorkTimes(March2,2006).[>]centrallessonsofstatistics:Forinformativeintroductionstostatisticsleavenedbyfinestorytelling,seeD.Salsburg,TheLadyTastingTea(W.H.Freeman,2001),andL.Mlodinow,TheDrunkard’sWalk(Pantheon,2008).[>]Galtonboard:Ifyou’veneverseenaGaltonboardinaction,checkoutthedemonstrationsavailableonYouTube.Oneofthemostdramaticofthesevideosmakesuseofsandratherthanballs;seehttp://www.youtube.com/watch?v=xDIyAOBa_yU.[>]heightsofadultmenandwomen:Youcanfindyourplaceontheheightdistributionbyusingtheonlineanalyzerathttp://www.shortsupport.org/Research/analyzer.html.Basedondatafrom1994,itshowswhatfractionoftheU.S.populationisshorterortallerthananygivenheight.Formorerecentdata,seeM.A.McDowelletal.,“Anthropometricreferencedataforchildrenandadults:UnitedStates,2003–2006,”NationalHealthStatisticsReports,No.10(October22,2008),availableonlineathttp://www.cdc.gov/nchs/data/nhsr/nhsr010.pdf.[>]OkCupid:OkCupidisthelargestfreedatingsiteintheUnitedStates,with7millionactivemembersasofsummer2011.TheirstatisticiansperformoriginalanalysisontheanonymizedandaggregateddatatheycollectfromtheirmembersandthenposttheirresultsandinsightsontheirblogOkTrends(http://blog.okcupid.com/index.php/about/).TheheightdistributionsarepresentedinC.Rudder,“Thebigliespeopletellinonlinedating,”http://blog.okcupid.com/index.php/the-biggest-lies-in-online-dating/.ThankstoChristianRudderforgenerouslyallowingmetoadapttheplotsfromhispost.[>]Power-lawdistributions:MarkNewmangivesasuperbintroductiontothistopicinM.E.J.Newman,“Powerlaws,ParetodistributionsandZipf’slaw,”ContemporaryPhysics,Vol.46,No.5(2005),pp.323–351(availableonlineathttp://www-personal.umich.edu/~mejn/courses/2006/cmplxsys899/powerlaws.pdf).ThisarticleincludesplotsofwordfrequenciesinMoby-Dick,themagnitudesofearthquakesinCaliforniafrom1910to1992,thenetworthofthe400richestpeopleintheUnitedStatesin2003,andmanyoftheotherheavy-taileddistributionsmentionedinthischapter.AnearlierbutstillexcellenttreatmentofpowerlawsisM.Schroder,Fractals,Chaos,PowerLaws(W.H.Freeman,1991).[>]2003taxcuts:I’veborrowedthisexamplefromC.Seife,Proofiness

(Viking,2010).ThetranscriptofPresidentBush’sspeechisavailableathttp://georgewbush-whitehouse.archives.gov/news/releases/2004/02/print/20040219-4.html.ThefiguresusedinthetextarebasedontheanalysisbyFactCheck.org(anonpartisanprojectoftheAnnenbergPublicPolicyCenteroftheUniversityofPennsylvania),availableonlineathttp://www.factcheck.org/here_we_go_again_bush_exaggerates_tax.html,andthisanalysispublishedbythenonpartisanTaxPolicyCenter:W.G.Gale,P.Orszag,andI.Shapiro,“Distributionaleffectsofthe2001and2003taxcutsandtheirfinancing,”http://www.taxpolicycenter.org/publications/url.cfm?ID=411018.[>]fluctuationsinstockprices:B.MandelbrotandR.L.Hudson,The(Mis)BehaviorofMarkets(BasicBooks,2004);N.N.Taleb,TheBlackSwan(RandomHouse,2007).[>]Fat,heavy,andlong:Thesethreewordsaren’talwaysusedsynonymously.Whenstatisticiansspeakofalongtail,theymeansomethingdifferentthanwhenbusinessandtechnologypeoplediscussit.Forexample,inChrisAnderson’sWiredarticle“Thelongtail”fromOctober2004(http://www.wired.com/wired/archive/12.10/tail.html)andinhisbookofthesamename,he’sreferringtothehugenumbersoffilms,books,songs,andotherworksthatareobscuretomostofthepopulationbutthatnonethelesshavenicheappealandsosurviveonline.Inotherwords,forhim,thelongtailisthemillionsoflittleguys;forstatisticians,thelongtailistheveryfewbigguys:thesuperwealthy,orthelargeearthquakes.ThedifferenceisthatAndersonswitchestheaxesonhisplots,whichis

somewhatlikelookingthroughtheotherendofthetelescope.Hisconventionisoppositethatusedbystatisticiansintheirplotsofcumulativedistributions,butithasalongtraditiongoingbacktoVilfredoPareto,anengineerandeconomistwhostudiedtheincomedistributionsofEuropeancountriesinthelate1800s.Inanutshell,AndersonandParetoplotfrequencyasafunctionofrank,whereasZipfandthestatisticiansplotrankasafunctionoffrequency.Thesameinformationisshowneitherway,butwiththeaxesflipped.Thisleadstomuchconfusioninthescientificliterature.See

http://www.hpl.hp.com/research/idl/papers/ranking/ranking.htmlforLadaAdamic’stutorialsortingthisout.MarkNewmanalsoclarifiesthispointinhispaperonpowerlaws,mentionedabove.

23.ChancesAre

[>]probabilitytheory:ForagoodtextbooktreatmentofconditionalprobabilityandBayes’stheorem,seeS.M.Ross,IntroductiontoProbabilityandStatisticsforEngineersandScientists,4thedition(AcademicPress,2009).ForahistoryofReverendBayesandthecontroversysurroundinghisapproachtoprobabilisticinference,seeS.B.McGrayne,TheTheoryThatWouldNotDie(YaleUniversityPress,2011).[>]ailingplant:Theanswertopart(a)oftheailing-plantproblemis59percent.Theanswertopart(b)is27/41,orapproximately65.85percent.Toderivetheseresults,imagine100ailingplantsandfigureouthowmanyofthem(onaverage)getwateredornot,andthenhowmanyofthosegoontodieornot,basedontheinformationgiven.Thisquestionappears,thoughwithslightlydifferentnumbersandwording,asproblem29onp.84ofRoss’stext.[>]mammogram:ThestudyofhowdoctorsinterpretmammogramresultsisdescribedinG.Gigerenzer,CalculatedRisks(SimonandSchuster,2002),chapter4.[>]conditional-probabilityproblemscanstillbeperplexing:Formanyentertaininganecdotesandinsightsaboutconditionalprobabilityanditsreal-worldapplications,aswellashowit’smisperceived,seeJ.A.Paulos,Innumeracy(Vintage,1990),andL.Mlodinow,TheDrunkard’sWalk(Vintage,2009).[>]O.J.Simpsontrial:FormoreontheO.J.Simpsoncaseandadiscussionofwifebatteringinalargercontext,seechapter8ofGigerenzer,CalculatedRisks.ThequotespertainingtotheO.J.SimpsontrialandAlanDershowitz’sestimateoftherateatwhichbatteredwomenaremurderedbytheirpartnersappearedinA.Dershowitz,ReasonableDoubts(Touchstone,1997),[>].ProbabilitytheorywasfirstcorrectlyappliedtotheSimpsontrialin1995.The

analysisgiveninthischapterisbasedonthatproposedbythelateI.J.Goodin“Whenbattererturnsmurderer,”Nature,Vol.375(1995),p.541,andrefinedin“Whenbattererbecomesmurderer,”Nature,Vol.381(1996),p.481.GoodphrasedhisanalysisintermsofoddsratiosandBayes’stheoremratherthaninthemoreintuitivenaturalfrequencyapproachusedhereandinGigerenzer’sbook.(Incidentally,Goodhadaninterestingcareer.InadditiontohismanycontributionstoprobabilitytheoryandBayesianstatistics,hehelpedbreaktheNaziEnigmacodeduringWorldWarIIandintroducedthefuturisticconceptnowknownasthetechnologicalsingularity.)Foranindependentanalysisthatreachesessentiallythesameconclusionand

thatwasalsopublishedin1995,seeJ.F.MerzandJ.P.Caulkins,“Propensitytoabuse—propensitytomurder?”Chance,Vol.8,No.2(1995),p.14.TheslightdifferencesbetweenthetwoapproachesarediscussedinJ.B.GarfieldandL.

Snell,“Teachingbits:Aresourceforteachersofstatistics,”JournalofStatisticsEducation,Vol.3,No.2(1995),onlineathttp://www.amstat.org/publications/jse/v3n2/resource.html.[>]Dershowitzcounteredforthedefense:HereishowDershowitzseemstohavecalculatedthatfewerthan1in2,500batterersperyeargoontomurdertheirpartners.On[>]ofhisbookReasonableDoubts,hecitesanestimatethatin1992,somewherebetween2.5and4millionwomenintheUnitedStateswerebatteredbytheirhusbands,boyfriends,andex-boyfriends.Inthatsameyear,accordingtotheFBIUniformCrimeReports(http://www.fbi.gov/about-us/cjis/ucr/ucr),913womenweremurderedbytheirhusbands,and519werekilledbytheirboyfriendsorex-boyfriends.Dividingthetotalof1,432homicidesby2.5millionbeatingsyields1murderper1,746beatings,whereasusingthehigherestimateof4millionbeatingsperyearyields1murderper2,793beatings.Dershowitzapparentlychose2,500asaroundnumberinbetweentheseextremes.What’suncleariswhatproportionofthemurderedwomenhadpreviously

beenbeatenbythesemen.ItseemsthatDershowitzwasassumingthatnearlyallthehomicidevictimshadearlierbeenbeaten,presumablytomakethepointthatevenwhentherateisoverestimatedinthisway,it’sstill“infinitesimal.”AfewyearsaftertheverdictwashandeddownintheSimpsoncase,

DershowitzandthemathematicianJohnAllenPaulosengagedinaheatedexchangevialetterstotheeditoroftheNewYorkTimes.Theissuewaswhetherevidenceofahistoryofspousalabuseshouldberegardedasrelevanttoamurdertrialinlightofprobabilisticargumentssimilartothosediscussedhere.SeeA.Dershowitz,“Thenumbersgame,”NewYorkTimes(May30,1999),archivedathttp://www.nytimes.com/1999/05/30/books/l-the-numbers-game-789356.html,andJ.A.Paulos,“Onceuponanumber,”NewYorkTimes(June27,1999),http://www.nytimes.com/1999/06/27/books/l-once-upon-a-number-224537.html.[>]expect3moreofthesewomen,onaverage,tobekilledbysomeoneelse:AccordingtotheFBIUniformCrimeReports,4,936womenweremurderedin1992.Ofthesemurdervictims,1,432(about29percent)werekilledbytheirhusbandsorboyfriends.Theremaining3,504werekilledbysomebodyelse.Therefore,consideringthatthetotalpopulationofwomenintheUnitedStatesatthattimewasabout125million,therateatwhichwomenweremurderedbysomeoneotherthantheirpartnerswas3,504dividedby125,000,000,or1murderper35,673womenperyear.Let’sassumethatthisrateofmurderbynonpartnerswasthesameforallwomen,batteredornot.Theninourhypotheticalsampleof100,000batteredwomen,we’dexpectabout100,000dividedby35,673,or2.8,womentobe

killedbysomeoneotherthanapartner.Rounding2.8to3,weobtaintheestimategiveninthetext.

24.UntanglingtheWeb

[>]searchingtheWeb:ForanintroductiontoWebsearchandlinkanalysis,seeD.EasleyandJ.Kleinberg,Networks,Crowds,andMarkets(CambridgeUniversityPress,2010),chapter14.Theirelegantexpositionhasinspiredmytreatmenthere.ForapopularaccountofthehistoryofInternetsearch,includingstoriesaboutthekeycharactersandcompanies,seeJ.Battelle,TheSearch(PortfolioHardcover,2005).Theearlydevelopmentoflinkanalysis,forreaderscomfortablewithlinearalgebra,issummarizedinS.Robinson,“TheongoingsearchforefficientWebsearchalgorithms,”SIAMNews,Vol.37,No.9(2004).[>]grasshopper:Foranyoneconfusedbymyuseoftheword“grasshopper,”itisanaffectionatenicknameforastudentwhohasmuchtolearnfromaZenmaster.InthetelevisionseriesKungFu,onmanyoftheoccasionswhentheblindmonkPoimpartswisdomtohisstudentCaine,hecallshimgrasshopper,harkingbacktotheirfirstlesson,ascenedepictedinthe1972pilotfilm(andonlineathttp://www.youtube.com/watch?v=WCyJRXvPNRo):MasterPo:Closeyoureyes.Whatdoyouhear?YoungCaine:Ihearthewater.Ihearthebirds.Po:Doyouhearyourownheartbeat?Caine:No.Po:Doyouhearthegrasshopperwhichisatyourfeet?Caine:Oldman,howisitthatyouhearthesethings?Po:Youngman,howisitthatyoudonot?

[>]circularreasoning:Therecognitionofthecircularityproblemforrankingwebpagesanditssolutionvialinearalgebragrewoutoftwolinesofresearchpublishedin1998.OnewasbymyCornellcolleagueJonKleinberg,thenworkingasavisitingscientistatIBMAlmadenResearchCenter.Forhisseminalpaperonthe“hubsandauthorities”algorithm(analternativeformoflinkanalysisthatappearedslightlyearlierthanGoogle’sPageRankalgorithm),seeJ.Kleinberg,“Authoritativesourcesinahyperlinkedenvironment,”ProceedingsoftheNinthAnnualACM-SIAMSymposiumonDiscreteAlgorithms(1998).TheotherlineofresearchwasbyGooglecofoundersLarryPageandSergey

Brin.TheirPageRankmethodwasoriginallymotivatedbythinkingabouttheproportionoftimearandomsurferwouldspendateachpageontheWeb—a

processwithadifferentdescriptionbutonethatleadstothesamewayofresolvingthecirculardefinition.ThefoundationalpaperonPageRankisS.BrinandL.Page,“Theanatomyofalarge-scalehypertextualWebsearchengine,”ProceedingsoftheSeventhInternationalWorldWideWebConference(1998),[>].Assooftenhappensinscience,strikinglysimilarprecursorsoftheseideashad

alreadybeendiscoveredinotherfields.ForthisprehistoryofPageRankinbibliometrics,psychology,sociology,andeconometrics,seeM.Franceschet,“PageRank:Standingontheshouldersofgiants,”CommunicationsoftheACM,Vol.54,No.6(2011),availableathttp://arxiv.org/abs/1002.2858;andS.Vigna,“Spectralranking,”http://arxiv.org/abs/0912.0238.[>]linearalgebra:Foranyoneseekinganintroductiontolinearalgebraanditsapplications,GilStrang’sbooksandonlinevideolectureswouldbeafineplacetostart:G.Strang,IntroductiontoLinearAlgebra,4thedition(Wellesley-CambridgePress,2009),andhttp://web.mit.edu/18.06/www/videos.html.[>]linearalgebrahasthetoolsyouneed:Someofthemostimpressiveapplicationsoflinearalgebrarelyonthetechniquesofsingularvaluedecompositionandprincipalcomponentanalysis.SeeD.James,M.Lachance,andJ.Remski,“Singularvectors’subtlesecrets,”CollegeMathematicsJournal,Vol.42,No.2(March2011),[>].[>]PageRankalgorithm:AccordingtoGoogle,theterm“PageRank”referstoLarryPage,notwebpage.Seehttp://web.archive.org/web/20090424093934/http://www.google.com/press/funfacts.html[>]classifyhumanfaces:Theideahereisthatanyhumanfacecanbeexpressedasacombinationofasmallnumberoffundamentalfaceingredients,oreigenfaces.ThisapplicationoflinearalgebratofacerecognitionandclassificationwaspioneeredbyL.SirovichandM.Kirby,“Low-dimensionalprocedureforthecharacterizationofhumanfaces,”JournaloftheOpticalSocietyofAmericaA,Vol.4(1987),pp.519–524,andfurtherdevelopedbyM.TurkandA.Pentland,“Eigenfacesforrecognition,”JournalofCognitiveNeuroscience,Vol.3(1991),[>],whichisalsoavailableonlineathttp://cse.seu.edu.cn/people/xgeng/files/under/turk91eigenfaceForRecognition.pdfForacomprehensivelistofscholarlypapersinthisarea,seetheFace

RecognitionHomepage(http://www.face-rec.org/interesting-papers/).[>]votingpatternsofSupremeCourtjustices:L.Sirovich,“ApatternanalysisofthesecondRehnquistU.S.SupremeCourt,”ProceedingsoftheNationalAcademyofSciences,Vol.100,No.13(2003),pp.7432–7437.Forajournalisticaccountofthiswork,seeN.Wade,“AmathematiciancrunchestheSupremeCourt’snumbers,”NewYorkTimes(June24,2003).Foradiscussionaimedat

legalscholarsbyamathematicianandnowlawprofessor,seeP.H.Edelman,“ThedimensionoftheSupremeCourt,”ConstitutionalCommentary,Vol.20,No.3(2003),pp.557–570.[>]NetflixPrize:ForthestoryoftheNetflixPrize,withamusingdetailsaboutitsearlycontestantsandtheimportanceofthemovieNapoleonDynamite,seeC.Thompson,“Ifyoulikedthis,you’resuretolovethat—WinningtheNetflixprize,”NewYorkTimesMagazine(November23,2008).TheprizewaswoninSeptember2009,threeyearsafterthecontestbegan;seeS.Lohr,“A$1millionresearchbargainforNetflix,andmaybeamodelforothers,”NewYorkTimes(September22,2009).TheapplicationofthesingularvaluedecompositiontotheNetflixPrizeisdiscussedinB.Cipra,“Blockbusteralgorithm,”SIAMNews,Vol.42,No.4(2009).[>]skippingsomedetails:Forsimplicity,IhavepresentedonlythemostbasicversionofthePageRankalgorithm.Tohandlenetworkswithcertaincommonstructuralfeatures,PageRankneedstobemodified.Suppose,forexample,thatthenetworkhassomepagesthatpointtoothersbutthathavenonepointingbacktothem.Duringtheupdateprocess,thosepageswilllosetheirPageRank,asifleakingorhemorrhagingit.Theygiveittoothersbutit’sneverreplenished.Sothey’llallendupwithPageRanksofzeroandwillthereforebeindistinguishableinthatrespect.Attheotherextreme,considernetworksinwhichsomepages,orgroupsof

pages,hoardPageRankbybeingclubby,byneverlinkingbackouttoanyoneelse.SuchpagestendtoactassinksforPageRank.Toovercometheseandothereffects,BrinandPagemodifiedtheiralgorithm

asfollows:Aftereachstepintheupdateprocess,allthecurrentPageRanksarescaleddownbyaconstantfactor,sothetotalislessthan1.Whateverisleftoveristhenevenlydistributedtoallthenodesinthenetwork,asifbeingraineddownoneveryone.It’stheultimateegalitarianact,spreadingthePageRanktotheneediestnodes.JoethePlumberwouldnotbehappy.ForadeeperlookatthemathematicsofPageRank,withinteractive

explorations,seeE.Aghapour,T.P.Chartier,A.N.Langville,andK.E.Pedings,“GooglePageRank:ThemathematicsofGoogle”(http://www.whydomath.org/node/google/index.html).Acomprehensiveyetaccessiblebook-lengthtreatmentisA.N.LangvilleandC.D.Meyer,Google’sPageRankandBeyond(PrincetonUniversityPress,2006).

25.TheLoneliestNumbers

[>]oneistheloneliestnumber:HarryNilssonwrotethesong“One.”Three

DogNight’scoverofitbecameahit,reachingnumber5ontheBillboardHot100,andAimeeMannhasaterrificversionofitthatcanbeheardinthemovieMagnolia.[>]TheSolitudeofPrimeNumbers:P.Giordano,TheSolitudeofPrimeNumbers(PamelaDormanBooks/VikingPenguin,2010).Thepassageexcerptedhereappearson[>].[>]numbertheory:Forpopularintroductionstonumbertheory,andthemysteriesofprimenumbersinparticular,themostdifficultchoiceiswheretobegin.Thereareatleastthreeexcellentbookstochoosefrom.Allappearedataroundthesametime,andallcenteredontheRiemannhypothesis,widelyregardedasthegreatestunsolvedprobleminmathematics.Forsomeofthemathematicaldetails,alongwiththeearlyhistoryoftheRiemannhypothesis,I’drecommendJ.Derbyshire,PrimeObsession(JosephHenryPress,2003).Formoreemphasisonthelatestdevelopmentsbutstillataveryaccessiblelevel,seeD.Rockmore,StalkingtheRiemannHypothesis(Pantheon,2005),andM.duSautoy,TheMusicofthePrimes(HarperCollins,2003).[>]encryptionalgorithms:Fortheuseofnumbertheoryincryptography,seeM.Gardner,PenroseTilestoTrapdoorCiphers(MathematicalAssociationofAmerica,1997),chapters13and14.ThefirstofthesechaptersreprintsGardner’sfamouscolumnfromtheAugust1977issueofScientificAmericaninwhichhetoldthepublicaboutthevirtuallyunbreakableRSAcryptosystem.Thesecondchapterdescribesthe“intensefuror”itarousedwithintheNationalSecurityAgency.Formorerecentdevelopments,seechapter10ofduSautoy,TheMusicofthePrimes.[>]primenumbertheorem:AlongwiththebooksbyDerbyshire,Rockmore,andduSautoymentionedabove,therearemanyonlinesourcesofinformationabouttheprimenumbertheorem,suchasChrisK.Caldwell’spage“Howmanyprimesarethere?”(http://primes.utm.edu/howmany.shtml),theMathWorldpage“Primenumbertheorem”(http://mathworld.wolfram.com/PrimeNumberTheorem.html),andtheWikipediapage“Primenumbertheorem”(http://en.wikipedia.org/wiki/Prime_number_theorem).[>]CarlFriedrichGauss:ThestoryofhowGaussnoticedtheprimenumbertheorematagefifteenistoldon[>]ofDerbyshire,PrimeObsession,andingreaterdetailbyL.J.Goldstein,“Ahistoryoftheprimenumbertheorem,”AmericanMathematicalMonthly,Vol.80,No.6(1973),pp.599–615.Gaussdidnotprovethetheorembutguesseditbyporingovertablesofprimenumbersthathehadcomputed—byhand—forhisownamusement.Thefirstproofswerepublishedin1896,aboutacenturylater,byJacquesHadamardandCharlesdela

ValléePoussin,eachofwhomhadbeenworkingindependentlyontheproblem.[>]twinprimesapparentlycontinuetoexist:HowcantwinprimesexistatlargeN,inlightoftheprimenumbertheorem?ThetheoremsaysonlythattheaveragegapislnN.Buttherearefluctuationsaboutthisaverage,andsincethereareinfinitelymanyprimes,someofthemareboundtogetluckyandbeattheodds.Inotherwords,eventhoughmostwon’tfindaneighboringprimemuchcloserthanlnNaway,someofthemwill.Forreaderswhowanttoseesomeofthemathgoverning“verysmallgaps

betweenprimes”beautifullyexplainedinaconciseway,seeAndrewGranville’sarticleonanalyticnumbertheoryinT.Gowers,ThePrincetonCompaniontoMathematics(PrincetonUniversityPress,2008),pp.332–348,especiallyp.343.Also,there’saniceonlinearticlebyTerryTaothatgivesalotofinsightinto

twinprimes—specifically,howthey’redistributedandwhymathematiciansbelievethereareinfinitelymanyofthem—andthenwadesintodeeperwaterstoexplaintheproofofhiscelebratedtheorem(withBenGreen)thattheprimescontainarbitrarilylongarithmeticprogressions.SeeT.Tao,“Structureandrandomnessintheprimenumbers,”http://terrytao.wordpress.com/2008/01/07/ams-lecture-structure-and-randomness-in-the-prime-numbers/.Forfurtherdetailsandbackgroundinformationabouttwinprimes,see

http://en.wikipedia.org/wiki/Twin_primeandhttp://mathworld.wolfram.com/TwinPrimeConjecture.html.[>]anotherprimecouplenearby:I’mjustkiddingaroundhereandnottryingtomakeaseriouspointaboutthespacingbetweenconsecutivepairsoftwinprimes.Maybesomewhereoutthere,fardownthenumberline,twosetsoftwinshappentobeextremelyclosetogether.Foranintroductiontosuchquestions,seeI.Peterson,“Primetwins”(June4,2001),http://www.maa.org/mathland/mathtrek_6_4_01.html.Inanycase,themetaphorofawkwardcouplesastwinprimeshasnotbeen

lostonHollywood.Forlightentertainment,youmightwanttorentamoviecalledTheMirrorHasTwoFaces,aBarbraStreisandvehiclecostarringJeffBridges.He’sahandsomebutsociallycluelessmathprofessor.She’saprofessorintheEnglishliteraturedepartment,aplucky,energetic,buthomelywoman(oratleastthat’showwe’resupposedtoseeher)wholiveswithhermotherandgorgeoussister.Eventually,thetwoprofessorsmanagetogettogetherfortheirfirstdate.Whentheirdinnerconversationdriftstothetopicofdancing(whichembarrasseshim),hechangesthesubjectabruptlytotwinprimes.Shegetstheideaimmediatelyandasks,“Whatwouldhappenifyoucountedpastamillion?Wouldtherestillbepairslikethat?”Hepracticallyfallsoffhischairandsays,“I

can’tbelieveyouthoughtofthat!Thatisexactlywhatisyettobeproveninthetwinprimeconjecture.”Laterinthemovie,whentheystartfallinginlove,shegiveshimabirthdaypresentofcufflinkswithprimenumbersonthem.

26.GroupThink

[>]mattressmath:ThemattressgroupistechnicallyknownastheKleinfour-group.It’soneofthesimplestinagiganticzooofpossibilities.Mathematicianshavebeenanalyzinggroupsandclassifyingtheirstructuresforabout200years.Foranengagingaccountofgrouptheoryandthemorerecentquesttoclassifyallthefinitesimplegroups,seeM.duSautoy,Symmetry(Harper,2008).[>]grouptheory:Tworecentbooksinspiredthischapter:N.Carter,VisualGroupTheory(MathematicalAssociationofAmerica,2009);andB.Hayes,GroupTheoryintheBedroom(HillandWang,2008).Carterintroducesthebasicsofgrouptheorygentlyandpictorially.HealsotouchesonitsconnectionstoRubik’scube,contradancingandsquaredancing,crystals,chemistry,art,andarchitecture.AnearlierversionofHayes’smattress-flippingarticleappearedinAmericanScientist,Vol.93,No.5(September/October2005),p.395,availableonlineathttp://www.americanscientist.org/issues/pub/group-theory-in-the-bedroom.Readersinterestedinseeingadefinitionofwhata“group”isshouldconsult

anyoftheauthoritativeonlinereferencesorstandardtextbooksonthesubject.AgoodplacetostartistheMathWorldpagehttp://mathworld.wolfram.com/topics/GroupTheory.htmlortheWikipediapagehttp://en.wikipedia.org/wiki/Group_(mathematics).ThetreatmentI’vegivenhereemphasizessymmetrygroupsratherthangroupsinthemostgeneralsense.[>]chaoticcounterparts:MichaelFieldandMartinGolubitskyhavestudiedtheinterplaybetweengrouptheoryandnonlineardynamics.Inthecourseoftheirinvestigations,they’vegeneratedstunningcomputergraphicsofsymmetricchaos,manyofwhichcanbefoundonMikeField’swebpage(http://www.math.uh.edu/%7Emike/ag/art.html).Fortheart,science,andmathematicsofthistopic,seeM.FieldandM.Golubitsky,SymmetryinChaos,2ndedition(SocietyforIndustrialandAppliedMathematics,2009).[>]thediagramdemonstratesthatHR=V:Awordaboutsomepotentiallyconfusingnotationusedthroughoutthischapter:InequationslikeHR=V,theHwaswrittenonthelefttoindicatethatit’sthetransformationbeingperformedfirst.Carterusesthisnotationforfunctionalcompositioninhisbook,butthereadershouldbeawarethatmanymathematiciansusetheoppositeconvention,placingtheHontheright.

[>]Feynmangotadraftdeferment:FortheanecdoteaboutFeynmanandthepsychiatrist,seeR.P.Feynman,“SurelyYou’reJoking,Mr.Feynman!”(W.W.NortonandCompany,1985),[>],andJ.Gleick,Genius(RandomHouse,1993),[>].

27.TwistandShout

[>]Möbiusstrips:Art,limericks,patents,parlortricks,andseriousmath—younameit,andifithasanythingtodowithMöbiusstrips,it’sprobablyinCliffPickover’sjovialbookTheMöbiusStrip(BasicBooks,2006).AnearliergenerationfirstlearnedaboutsomeofthesewondersinM.Gardner,“TheworldoftheMöbiusstrip:Endless,edgeless,andone-sided,”ScientificAmerican,Vol.219,No.6(December1968).[>]funactivitiesthatasix-year-oldcando:Forstep-by-stepinstructions,withphotographs,ofsomeoftheactivitiesdescribedinthischapter,see“HowtoexploreaMöbiusstrip”athttp://www.wiki-how.com/Explore-a-Möbius-Strip.JulianFlerongivesmanyotherideas—Möbiusgarlands,hearts,paperclipstars—in“RecyclingMöbius,”http://artofmathematics.wsc.ma.edu/sculpture/workinprogress/Mobius1206.pdf.Forfurtherfunwithpapermodels,seeS.Barr’sclassicbookExperimentsin

Topology(Crowell,1964).[>]topology:ThebasicsoftopologyareexpertlyexplainedinR.CourantandH.Robbins(revisedbyI.Stewart),WhatIsMathematics?2ndedition(OxfordUniversityPress,1996),chapter5.Foraplayfulsurvey,seeM.Gardner,TheColossalBookofMathematics(W.W.NortonandCompany,2001).HediscussesKleinbottles,knots,linkeddoughnuts,andotherdelightsofrecreationaltopologyinpart5,chapters18–20.AverygoodcontemporarytreatmentisD.S.Richeson,Euler’sGem(PrincetonUniversityPress,2008).Richesonpresentsahistoryandcelebrationoftopologyandanintroductiontoitsmainconcepts,usingEuler’spolyhedronformulaasacenterpiece.Atamuchhigherlevel,butstillwithincomfortablereachofpeoplewithacollegemathbackground,seethechaptersonalgebraictopologyanddifferentialtopologyinT.Gowers,ThePrincetonCompaniontoMathematics(PrincetonUniversityPress,2008),pp.383–408.[>]theintrinsicloopinessofacircleandasquare:Giventhatacircleandsquarearetopologicallyequivalentcurves,youmightbewonderingwhatkindsofcurveswouldbetopologicallydifferent.Thesimplestexampleisalinesegment.Toprovethis,observethatifyoutravelinonedirectionaroundacircle,asquare,oranyotherkindofloop,you’llalwaysreturntoyourstarting

point,butthat’snottruefortravelingonalinesegment.Sincethispropertyisleftunchangedbyalltransformationsthatpreserveanobject’stopology(namely,continuousdeformationswhoseinverseisalsocontinuous),andsincethispropertydiffersbetweenloopsandsegments,wecanconcludethatloopsandsegmentsaretopologicallydifferent.[>]ViHart:Vi’svideosdiscussedinthischapter,“Möbiusmusicbox”and“Möbiusstory:WindandMr.Ug,”canbefoundonYouTubeandalsoathttp://vihart.com/musicbox/andhttp://vihart.com/blog/mobius-story/.Formoreofheringeniousandfun-lovingexcursionsintomathematicalfood,doodling,balloons,beadwork,andmusicboxes,seeherwebsiteathttp://vihart.com/everything/.ShewasprofiledinK.Chang,“Bendingandstretchingclassroomlessonstomakemathinspire,”NewYorkTimes(January17,2011),availableonlineathttp://www.nytimes.com/2011/01/18/science/18prof.html.[>]artistshavelikewisedrawninspiration:ToseeimagesoftheMöbiusartworkbyMauritsEscher,MaxBill,andKeizoUshio,searchtheWebusingtheartist’snameand“Möbius”assearchterms.IvarsPetersonhaswrittenabouttheuseofMöbiusstripsinliterature,art,architecture,andsculpture,withphotographsandexplanations,athisMathematicalTouristblog:http://mathtourist.blogspot.com/search/label/Moebius%20Strips.[>]NationalLibraryofKazakhstan:Thelibraryiscurrentlyunderconstruction.Foritsdesignconceptandintriguingimagesofwhatitwilllooklike,gotothewebsiteforthearchitecturalfirmBIG(BjarkeIngelsGroup),http://www.big.dk/.ClickontheiconforANL,AstanaNationalLibrary;itappearsinthe2009column(thefourthcolumnfromtheright)whentheprojectsarearrangedintheirdefaultchronologicalorder.Thesitecontainsforty-oneslidesofthelibrary’sinternalandexternalstructure,museumcirculation,thermalexposure,andsoon,allofwhichareunusualbecauseofthebuilding’sMöbiuslayout.ForaprofileofBjarkeIngelsandhispractice,seeG.Williams,“Opensourcearchitect:Meetthemaestroof‘hedonisticsustainability,’”http://www.wired.co.uk/magazine/archive/2011/07/features/open-source-architect.[>]Möbiuspatents:SomeofthesearediscussedinPickover,TheMöbiusStrip.Youcanfindhundredsofothersbysearchingfor“Möbiusstrip”inGooglePatents.[>]bagel:Ifyouwanttotrycuttingabagelthisway,GeorgeHartexplainshistechniqueathiswebsitehttp://www.georgehart.com/bagel/bagel.html.OryoucanseeacomputeranimationbyBillGileshere:http://www.youtube.com/watch?v=hYXnZ8-ux80.Ifyouprefertowatchit

happeninginrealtime,checkoutavideobyUltraNurdcalled“MöbiusBagel”(http://www.youtube.com/watch?v=Zu5z1BCC70s).Butstrictlyspeaking,thisshouldnotbecalledaMöbiusbagel—apointofconfusionamongmanypeoplewhohavewrittenaboutorcopiedGeorge’swork.ThesurfaceonwhichthecreamcheeseisspreadisnotequivalenttoaMöbiusstripbecauseithastwohalftwistsinit,notone,andtheresultingsurfaceistwo-sided,notone-sided.Furthermore,atrueMöbiusbagelwouldremaininonepiece,nottwo,afterbeingcutinhalf.ForademonstrationofhowtocutabagelinthisgenuineMöbiusfashion,seehttp://www.youtube.com/watch?v=l6Vuh16r8o8.

28.ThinkGlobally

[>]avisionoftheworldasflat:Byreferringtoplanegeometryasflat-earthgeometry,Imightseemtobedisparagingthesubject,butthat’snotmyintent.Thetacticoflocallyapproximatingacurvedshapebyaflatonehasoftenturnedouttobeausefulsimplificationinmanypartsofmathematicsandphysics,fromcalculustorelativitytheory.Planegeometryisthefirstinstanceofthisgreatidea.NordoImeantosuggestthatalltheancientsthoughttheworldwasflat.For

anengagingaccountofEratosthenes’smeasurementofthedistancearoundtheglobe,seeN.Nicastro,Circumference(St.Martin’sPress,2008).Foramorecontemporaryapproachthatyoumightliketotryonyourown,RobertVanderbeiatPrincetonUniversityrecentlygaveapresentationtohisdaughter’shigh-schoolgeometryclassinwhichheusedaphotographofasunsettoshowthattheEarthisnotflatandtoestimateitsdiameter.Hisslidesarepostedathttp://orfe.princeton.edu/~rvdb/tex/sunset/34-39.OPN.1108twoup.pdf.[>]differentialgeometry:AsuperbintroductiontomoderngeometrywascoauthoredbyDavidHilbert,oneofthegreatestmathematiciansofthetwentiethcentury.Thisclassic,originallypublishedin1952,hasbeenreissuedasD.HilbertandS.Cohn-Vossen,GeometryandtheImagination(AmericanMathematicalSociety,1999).SeveralgoodtextbooksandonlinecoursesindifferentialgeometryarelistedontheWikipediapagehttp://en.wikipedia.org/wiki/Differential_geometry.[>]mostdirectroute:ForaninteractiveonlinedemonstrationthatletsyouplottheshortestroutebetweenanytwopointsonthesurfaceoftheEarth,seehttp://demonstrations.wolfram.com/GreatCirclesOnMercatorsChart/.(You’llneedtodownloadthefreeMathematicaPlayer,whichwillthenallowyoutoexplorehundredsofotherinteractivedemonstrationsinallpartsofmathematics.)

[>]KonradPolthier:ExcerptsfromanumberofPolthier’sfascinatingeducationalvideosaboutmathematicaltopicscanbefoundonlineathttp://page.mi.fu-berlin.de/polthier/video/Geodesics/Scenes.html.Award-winningvideosbyPolthierandhiscolleaguesappearintheVideoMathFestivalcollection(http://page.mi.fu-berlin.de/polthier/Events/VideoMath/index.html),availableasaDVDfromSpringer-Verlag.Formoredetails,seeG.GlaeserandK.Polthier,AMathematicalPictureBook(Springer,2012).TheimagesshowninthetextarefromtheDVDTouchingSoapFilms(Springer,1995),byAndreasArnez,KonradPolthier,MartinSteffens,andChristianTeitzel.[>]shortestpaththroughanetwork:Theclassicalgorithmforshortest-pathproblemsonnetworkswascreatedbyEdsgerDijkstra.Foranintroduction,seehttp://en.wikipedia.org/wiki/Dijkstra’s_algorithm.StevenSkienahaspostedaninstructiveanimationofDijkstra’salgorithmathttp://www.cs.sunysb.edu/~skiena/combinatorica/animations/dijkstra.html.Naturecansolvecertainshortest-pathproblemsbydecentralizedprocesses

akintoanalogcomputation.Forchemicalwavesthatsolvemazes,seeO.Steinbock,A.Toth,andK.Showalter,“Navigatingcomplexlabyrinths:Optimalpathsfromchemicalwaves,”Science,Vol.267(1995),p.868.Nottobeoutdone,slimemoldscansolvethemtoo:T.Nakagaki,H.Yamada,andA.Toth,“Maze-solvingbyanamoeboidorganism,”Nature,Vol.407(2000),p.470.ThisslimycreaturecanevenmakenetworksasefficientastheTokyorailsystem:A.Teroetal.,“Rulesforbiologicallyinspiredadaptivenetworkdesign,”Science,Vol.327(2010),p.439.[>]tellingthestoryofyourlifeinsixwords:Delightfulexamplesofsix-wordmemoirsaregivenathttp://www.smithmag.net/sixwords/andhttp://en.wikipedia.org/wiki/Six-Word_Memoirs.

29.AnalyzeThis!

[>]analysis:Analysisgrewoutoftheneedtoshoreupthelogicalfoundationsofcalculus.WilliamDunhamtracesthisstorythroughtheworksofelevenmasters,fromNewtontoLebesgue,inW.Dunham,TheCalculusGallery(PrincetonUniversityPress,2005).Thebookcontainsexplicitmathematicspresentedaccessiblyforreadershavingacollege-levelbackground.AtextbookinasimilarspiritisD.Bressoud,ARadicalApproachtoRealAnalysis,2ndedition(MathematicalAssociationofAmerica,2006).Foramorecomprehensivehistoricalaccount,seeC.B.Boyer,TheHistoryoftheCalculusandItsConceptualDevelopment(Dover,1959).[>]vacillatingforever:Grandi’sseries1–1+1–1+1–1+∙∙∙isdiscussed

inameticulouslysourcedWikipediaarticleaboutitshistory,withlinkstofurtherthreadsaboutitsmathematicalstatusanditsroleinmatheducation.Allthesemaybereachedfromthemainpage“Grandi’sseries”:http://en.wikipedia.org/wiki/Grandi’s_series.[>]Riemannrearrangementtheorem:ForaclearexpositionoftheRiemannrearrangementtheorem,seeDunham,TheCalculusGallery,pp.112–115.[>]Strange,yes.Sick,yes:Thealternatingharmonicseriesisconditionallyconvergent,meaningit’sconvergentbutnotabsolutelyconvergent(thesumoftheabsolutevaluesofitstermsdonotconverge).Foraserieslikethat,youcanreorderthesumtogetanyrealnumber.That’stheshockingimplicationoftheRiemannrearrangementtheorem.Itshowsthataconvergentsumcanviolateourintuitiveexpectationsifitdoesnotconvergeabsolutely.Inthemuch-better-behavedcaseofanabsolutelyconvergentseries,all

rearrangementsoftheseriesconvergetothesamevalue.That’swonderfullyconvenient.Itmeansthatanabsolutelyconvergentseriesbehaveslikeafinitesum.Inparticular,itobeysthecommutativelawofaddition.Youcanrearrangethetermsanywayyouwantwithoutchangingtheanswer.Formoreonabsoluteconvergence,seehttp://mathworld.wolfram.com/AbsoluteConvergence.htmlandhttp://en.wikipedia.org/wiki/Absolute_convergence.[>]Fourieranalysis:TomKörner’sextraordinarybookFourierAnalysis(CambridgeUniversityPress,1989)isaself-described“shopwindow”oftheideas,techniques,applications,andhistoryofFourieranalysis.Thelevelofmathematicalrigorishigh,yetthebookiswitty,elegant,andpleasantlyquirky.ForanintroductiontoFourier’sworkanditsconnectiontomusic,seeM.Kline,MathematicsinWesternCulture(OxfordUniversityPress,1974),chapter19.[>]Gibbsphenomenon:TheGibbsphenomenonanditstortuoushistoryisreviewedbyE.HewittandR.E.Hewitt,“TheGibbs-Wilbrahamphenomenon:AnepisodeinFourieranalysis,”ArchivefortheHistoryofExactSciences,Vol.21(1979),[>].[>]digitalphotographsandonMRIscans:TheGibbsphenomenoncanaffectMPEGandJPEGcompressionofdigitalvideo:http://www.doc.ic.ac.uk/~nd/surprise_96/journal/vol4/sab/report.html.WhenitappearsinMRIscans,theGibbsphenomenonisknownastruncationorGibbsringing;seehttp://www.mr-tip.com/serv1.php?type=art&sub=Gibbs%20Artifact.Formethodstohandletheseartifacts,seeT.B.SmithandK.S.Nayak,“MRIartifactsandcorrectionstrategies,”ImagingMedicine,Vol.2,No.4(2010),pp.445–457,onlineathttp://mrel.usc.edu/pdf/Smith_IM_2010.pdf.[>]pinpointedwhatcausesGibbsartifacts:Theanalystsofthe1800sidentifiedtheunderlyingmathematicalcauseoftheGibbsphenomenon.Forfunctions(or,

nowadays,images)displayingsharpedgesorothermildtypesofjumpdiscontinuities,thepartialsumsofthesinewaveswereproventoconvergepointwisebutnotuniformlytotheoriginalfunction.Pointwiseconvergencemeansthatatanyparticularpointx,thepartialsumsgetarbitrarilyclosetotheoriginalfunctionasmoretermsareadded.Sointhatsense,theseriesdoesconverge,asonewouldhope.Thecatchisthatsomepointsaremuchmorefinickythanothers.TheGibbsphenomenonoccursneartheworstofthosepoints—theedgesintheoriginalfunction.Forexample,considerthesawtoothwavediscussedinthischapter.Asxgets

closertotheedgeofasawtooth,ittakesmoreandmoretermsintheFourierseriestoreachagivenlevelofapproximation.That’swhatwemeanbysayingtheconvergenceisnotuniform.Itoccursatdifferentratesfordifferentx.Inthiscase,thenon-uniformityoftheconvergenceisattributabletothe

pathologiesofthealternatingharmonicseries,whosetermsappearasFouriercoefficientsforthesawtoothwave.Asdiscussedabove,thealternatingharmonicseriesconverges,butonlybecauseofthemassivecancellationcausedbyitsalternatingmixofpositiveandnegativeterms.Ifallitstermsweremadepositivebytakingtheirabsolutevalues,theserieswoulddiverge—thesumwouldapproachinfinity.That’swhythealternatingharmonicseriesissaidtoconvergeconditionally,notabsolutely.ThisprecariousformofconvergencetheninfectstheassociatedFourierseriesandrendersitnon-uniformlyconvergent,leadingtotheGibbsphenomenonanditsmockingupraisedfingerneartheedge.Incontrast,inthemuchbettercasewheretheseriesofFouriercoefficientsis

absolutelyconvergent,theassociatedFourierseriesconvergesuniformlytotheoriginalfunction.ThentheGibbsphenomenondoesn’toccur.Formoredetails,seehttp://mathworld.wolfram.com/GibbsPhenomenon.htmlandhttp://en.wikipedia.org/wiki/Gibbs_phenomenon.Thebottomlineisthattheanalyststaughtustobewaryofconditionally

convergentseries.Convergenceisgood,butnotgoodenough.Foraninfiniteseriestobehavelikeafinitesuminallrespects,itneedsmuchtighterconstraintsthanconditionalconvergencecanprovide.Insistingonabsoluteconvergenceyieldsthebehaviorwe’dexpectintuitively,bothfortheseriesitselfandforitsassociatedFourierseries.

30.TheHilbertHotel

[>]GeorgCantor’s:FormoreaboutCantor,includingthemathematical,philosophical,andtheologicalcontroversiessurroundinghiswork,seeJ.W.Dauben,GeorgCantor(PrincetonUniversityPress,1990).

[>]settheory:Ifyouhaven’treadityet,IrecommendthesurprisebestsellerLogicomix,abrilliantlycreativegraphicnovelaboutsettheory,logic,infinity,madness,andthequestformathematicaltruth:A.DoxiadisandC.H.Papadimitriou,Logicomix(Bloomsbury,2009).ItstarsBertrandRussell,butCantor,Hilbert,Poincaré,andmanyothersmakememorableappearances.[>]DavidHilbert:TheclassicbiographyofDavidHilbertisamovingandnontechnicalaccountofhislife,hiswork,andhistimes:C.Reid,Hilbert(Springer,1996).Hilbert’scontributionstomathematicsaretoonumeroustolisthere,butperhapshisgreatestishiscollectionoftwenty-threeproblems—allofwhichwereunsolvedwhenheproposedthem—thathethoughtwouldshapethecourseofmathematicsinthetwentiethcentury.FortheongoingstoryandsignificanceoftheseHilbertproblemsandthepeoplewhosolvedsomeofthem,seeB.H.Yandell,TheHonorsClass(AKPeters,2002).Severaloftheproblemsstillremainopen.[>]HilbertHotel:Hilbert’sparableoftheinfinitehotelismentionedinGeorgeGamow’severgreenmasterpieceOneTwoThree...Infinity(Dover,1988),[>].Gamowalsodoesagoodjobofexplainingcountableanduncountablesetsandrelatedideasaboutinfinity.ThecomedicanddramaticpossibilitiesoftheHilbertHotelhaveoftenbeen

exploredbywritersofmathematicalfiction.Forexample,seeS.Lem,“TheextraordinaryhotelorthethousandandfirstjourneyofIontheQuiet,”reprintedinImaginaryNumbers,editedbyW.Frucht(Wiley,1999),andI.Stewart,ProfessorStewart’sCabinetofMathematicalCuriosities(BasicBooks,2009).Achildren’sbookonthesamethemeisI.Ekeland,TheCatinNumberland(CricketBooks,2006).[>]anyotherdigitbetween1and8:AtinyfinesseoccurredintheargumentfortheuncountabilityoftherealnumberswhenIrequiredthatthediagonaldigitsweretobereplacedbydigitsbetween1and8.Thiswasn’tessential.ButIwantedtoavoidusing0and9tosidestepanyfussinesscausedbythefactthatsomerealnumbershavetwodecimalrepresentations.Forexample,.200000...equals.199999...Thus,ifwehadn’texcludedtheuseof0sand9sasreplacementdigits,it’sconceivablethediagonalargumentcouldhaveinadvertentlyproducedanumberalreadyonthelist(andthatwouldhaveruinedtheproof).Bymyforbiddingtheuseof0and9,wedidn’thavetoworryaboutthisannoyance.[>]infinitybeyondinfinity:Foramoremathematicalbutstillveryreadablediscussionofinfinity(andmanyoftheotherideasdiscussedinthisbook),seeJ.C.Stillwell,YearningfortheImpossible(AKPeters,2006).ReaderswishingtogodeeperintoinfinitymightenjoyTerryTao’sblogpostaboutself-defeating

objects,http://terrytao.wordpress.com/2009/11/05/the-no-self-defeating-object-argument/.Inaveryaccessibleway,hepresentsandelucidatesalotoffundamentalargumentsaboutinfinitythatariseinsettheory,philosophy,physics,computerscience,gametheory,andlogic.Forasurveyofthefoundationalissuesraisedbythesesortsofideas,seealsoJ.C.Stillwell,RoadstoInfinity(AKPeters,2010).

Credits

[>]:“SesameWorkshop”®,“SesameStreet”®andassociatedcharacters,trademarks,anddesignelementsareownedandlicensedbySesameWorkshop.©2011SesameWorkshop.AllRightsReserved.[>]:MarkH.Anbinder[>],[>]:SimonTatham[>]:ABCPhotoArchives/ABCviaGettyImages[>]:Photograph©2011WET.AllRightsReserved.[>]:GarryJenkins[>]:L.Clarke/Corbis[>]:Corbis[>]:Photodisc/GettyImages[>]:MannyMillan/GettyImages[>],[>]:PaulBourke[>]:J.R.Eyerman/GettyImages[>]:Alchemy/Alamy[>]:ChristianRudder/OkCupid[>],[>]:M.E.J.Newman[>]:StudyforAlhambraStars(2000)/MikeField[>],[>]:ViHartp.[>]BIG—BjarkeIngelsGroup[>]:©GeorgeW.Hart[>],[>]:AndreasArnez,KonradPolthier,MartinSteffens,ChristianTeitzel.

ImagesfromDVDTouchingSoapFilms,Springer,1995.[>]:AFarchive/Alamy

Index

Adams,John,[>]addition,[>],[>],[>],[>],[>],[>]aerodynamics,[>]algebra,[>]–[>],[>],[>]–[>],[>]differentialcalculusand,[>],[>]fundamentaltheoremof,[>]imaginarynumbersin,[>]linear,[>]–[>],[>],[>]quadraticequations,[>]–[>]quadraticformula,[>]–[>],[>]relationshipsand,[>]vector,[>]

algorithms,[>],[>],[>]al-Khwarizmiand,[>]encryption,[>]–[>]numericalanalysisand,[>]“randomgreedymatching,”[>]SeealsoPageRank

al-Khwarizmi,MuhammadibnMusa,[>]–[>],[>],[>]alternatingharmonicseries,[>]–[>],[>],[>]–[>]analysiscalculusand,[>]–[>],[>]–[>]complex,[>]link,[>]numerical,[>]–[>]

Anderson,Chris,[>]Archimedes(Greekmathematician),[>]–[>],[>],[>],[>],[>],[>],[>],[>],[>]arithmetic,[>],[>].Seealsoaddition;division;multiplication;negativenumbers;primenumbers;subtractionassociativelawofaddition,[>]Austin,J.L.,[>]axiomaticmethod,[>]axioms,[>]Babylonians,[>],[>]balancetheory,[>]–[>]

Barrow,Isaac,[>]Bayes’stheorem,[>],[>],[>]bell-shapeddistributions,[>]–[>],[>]Bethe,Hans,[>]–[>],[>]B.F.GoodrichCompany,[>]bicylinder,[>],[>]BIG(Danisharchitecturalfirm),[>]Bill,Max,[>]binarysystem,ofwritingnumbers,[>],[>]BlackMonday,[>]Boyle,DerekPaul,[>]–[>]breastcancer,[>]–[>]Bridges,Jeff,[>]Brin,Sergey,[>],[>],[>]Brooks,David,[>]–[>]Brown,Christy,[>]–[>],[>]Brown,Nicole,[>]–[>]Bush,GeorgeW.,[>]calculate,originsofword,[>]CalculatedRisks(Gigerenzer),[>]calculus,[>]–[>],[>],[>],[>]analysisand,[>]–[>],[>]–[>]Archimedesand,[>],[>]defined,[>]fundamentaltheoremof,[>],[>]–[>],[>]vector,[>]–[>]Seealsodifferentialcalculus;differentialequations;integralcalculus

CalculusGallery,The(Dunham),[>]–[>]Cantor,Georg,[>],[>]–[>]Carothers,Neal,[>]Carter,Nathan,[>],[>]catenaries,[>]Cayley,Arthur,[>]change,[>]–[>],[>].Seealsocalculuschaostheory,[>]Christopher,Peter,[>]circles,[>],[>],[>],[>]–[>],[>],[>]Cohn-Vossen,S.,[>]coincidences,[>]

commutativelawsofaddition,[>],[>],[>],[>]ofmultiplication,[>]–[>],[>]

complexdynamics,[>]complexnumbers,[>]–[>]compounding,[>]computersalgorithmsand,[>]binarysystemusedin,[>],[>]numericalanalysisand,[>]–[>],[>]statisticsand,[>]twinprimesand,[>],[>],[>],[>]SeealsoInternet

conditionalprobability,[>]–[>]conicsections,[>]–[>]constantfunctions,[>]continuouscompounding,[>]convergence,[>]–[>],[>],[>]–[>]Cornell,Ezra,[>]–[>],[>]–[>]cylinders,[>]–[>],[>]–[>],[>],[>]datamining,[>]datingstrategies,eand,[>]–[>],[>]decimals,[>],[>]–[>],[>],[>]DeclarationofIndependence,[>]derivatives.SeedifferentialcalculusDershowitz,Alan,[>],[>]–[>]Detroit’sinternationalairport,[>]Devlin,Keith,[>],[>]–[>]differentialcalculus,[>]–[>],[>],[>]–[>]differentialequations,[>]–[>]differentialgeometry,[>]–[>],[>]Dijkstra,Edsger,[>]Dirac,Paul,[>],[>]distributions,[>]–[>],[>]division,[>].SeealsofractionsDowJonesindustrialaverage,[>]Dunham,William,[>]–[>]e,[>]–[>]Einstein,Albert,[>],[>],[>],[>],[>],[>]–[>]

electricfields,[>]–[>]electromagneticwaves,[>],[>]Elements(Euclid),[>]–[>],[>],[>]“ElementsofMath,The”(Strogatz),[>],[>]ellipses,[>]–[>],[>]–[>],[>]–[>]encryption,[>]–[>]engineers,[>],[>],[>],[>],[>]equilateraltriangles,[>]–[>]Escher,M.C.,[>]Ethics(Spinoza),[>]Euclid(Greekmathematician),[>]–[>],[>],[>],[>],[>],[>],[>]Euclideangeometry,[>]–[>],[>],[>]Eudoxus(Greekmathematician),[>]Euler,Leonhard,[>],[>]exponentialfunctions,[>]–[>],[>]exponentialgrowth,[>],[>]facerecognition/classification,[>]Feynman,Richard,[>]–[>],[>]–[>],[>]–[>]Field,Michael,[>]Fizeau,Hippolyte,[>]fluidmechanics,[>]Forbes,Kim,[>]–[>]Fourieranalysis,[>],[>]FourierAnalysis(Körner),[>]Fourierseries,[>]–[>],[>]–[>]fractalgeometry,[>],[>]–[>]fractals,[>]fractions,[>]–[>],[>]FreeUniversityofBerlin,[>]functions,[>]–[>],[>]–[>],[>]fundamentaltheoremofalgebra,[>]fundamentaltheoremofcalculus,[>],[>]–[>],[>]GalileoGalilei,[>]Gallivan,Britney,[>],[>]Galtonboards,[>]–[>],[>]–[>]Gardner,M.,[>]–[>]Gauss,CarlFriedrich,[>],[>]–[>],[>]Gell-Mann,Murray,[>],[>]–[>]generaltheoryofrelativity,[>],[>]

geodesics,[>]–[>]geometry,[>],[>],[>],[>]differential,[>]–[>],[>]differentialcalculusand,[>]Euclidean,[>]–[>],[>],[>]fractal,[>],[>]–[>]integralcalculusand,[>]non-Euclidean,[>],[>]plane,[>],[>]proofs,[>]–[>],[>]–[>],[>]–[>]shapesand,[>]spherical,[>]topologyand,[>]SeealsoPythagoreantheorem

GeometryandtheImagination(HilbertandCohn-Vossen),[>]Gibbsphenomenon,[>],[>]–[>]Gigerenzer,Gerd,[>]–[>],[>]Giordano,Paolo,[>]–[>],[>]Golubitsky,Martin,[>]Good,I.J.,[>],[>]Google,[>]–[>],[>].SeealsoPageRankGoogleEarth,[>]GrandCentralStation,[>]Grandi,Guido,[>]grasshopper,[>]Gregory,James,[>]grouptheory,[>]–[>],[>]GroupTheoryintheBedroom(Hayes),[>]Hadamard,Jacques,[>]Halley,Edmund,[>]Hart,George,[>],[>],[>]Hart,Vi,[>]–[>],[>]Hayes,Brian,[>]Heider,Fritz,[>]Heisenberg,Werner,[>],[>]Heisenberguncertaintyprinciple,[>]Hilbert,David,[>],[>],[>]–[>]HilbertHotelparable,[>]–[>],[>]Hindu-Arabicsystem,ofwritingnumbers,[>]–[>]

Hoffman,Will,[>]–[>]HousekeeperandtheProfessor,The(Ogawa),[>]–[>]HowGreenWasMyValley(movie),[>]Hubbard,John,[>]–[>],[>],[>]hyperbola,[>]–[>]hypotenuse,[>]–[>],[>],[>]IBMAlmadenResearchCenter,[>]identities,[>]–[>]imaginarynumbers,[>]–[>],[>]“InfiniteSecrets”(PBSseries),[>]infinity,[>]–[>],[>]–[>],[>]–[>],[>]–[>],[>]–[>]calculusand,[>],[>]–[>],[>],[>],[>]–[>]HilbertHotelparable/settheoryand,[>]–[>],[>]symbolfor,[>]

integers,[>]integralcalculus,[>],[>]–[>],[>]–[>]Internet,[>],[>]–[>],[>],[>]–[>].SeealsoGoogleinverses,[>]inversesquarefunction,[>]–[>]Islamicmathematicians,[>]Jefferson,Thomas,[>]Jordan,Michael,[>]Kepler,Johannes,[>]Kleinberg,Jon,[>]Kleinfour-group,[>]Körner,Tom,[>]Leibniz,GottfriedWilhelmvon,[>],[>],[>],[>]light,[>],[>]linearalgebra,[>]–[>],[>],[>]linearfunctions,[>]linkanalysis,[>]LittleBigLeague(movie),[>]Lockhart,Paul,[>]logarithms,[>],[>]–[>],[>],[>],[>],[>]longtail,[>]magneticfields,[>]–[>],[>]Magnolia(movie),[>]mammograms,[>]–[>]Mandelbrotset,[>]

Mankiw,Greg,[>]Mann,Aimee,[>]mathematicalmodeling,[>]MathematicalPrinciplesofNaturalPhilosophy,The(Newton),[>]Mathematician’sLament,A(Lockhart),[>]mathemusician,[>]mattresses,[>]–[>],[>]MaxPlanckInstituteforHumanDevelopment(Berlin),[>]Maxwell,JamesClerk,[>]–[>],[>]–[>],[>]–[>]means,[>]medians,[>]methodofexhaustion,[>]MirrorHasTwoFaces,The(movie),[>]–[>]“Möbiusmusicbox”(videobyHart),[>]–[>]“MöbiusStory:WindandMr.Ug”(videobyHart),[>]–[>]Möbiusstrips,[>]–[>],[>],[>]MoMath(MuseumofMathematics),[>]Moonlighting(televisionshow),[>]–[>]Moore,Moreton,[>]Morgenbesser,Sidney,[>]Morse,Samuel,[>]–[>]movietheaterseating,[>]–[>],[>]multiplication,[>]–[>],[>]–[>]MyLeftFoot(movie),[>]–[>],[>]NationalLibraryofKazakhstan,[>],[>]NationalSecurityAgency,[>]naturalfrequencies,[>]negativenumbers,[>]–[>],[>],[>]–[>],[>]Netflix,[>]–[>],[>]networks,[>]Newton,SirIsaac,[>],[>],[>],[>],[>],[>],[>]–[>],[>]–[>]NewYorkTimes,[>],[>],[>],[>]NewYorkUniversity,[>]Nilsson,Harry,[>]normaldistributions,[>]–[>],[>]numbers,[>]complex,[>]–[>]asgroupsofrocks,[>]–[>],[>]–[>]TheHousekeeperandtheProfessorproblemwith,[>]–[>]

imaginary,[>]–[>],[>]integers,[>]introductionto,[>]–[>]natural,[>]negative,[>]–[>],[>],[>]–[>],[>]thenumber“1,”[>]–[>]prime,[>]–[>],[>]rational,[>]systemsforwriting,[>]–[>],[>]SeealsoRomannumerals

numbertheory,[>]–[>],[>]numericalanalysis,[>]–[>]Ogawa,Yoko,[>]OkCupid(datingservice),[>],[>]“One”(Nilsson),[>]one,thenumber,[>]–[>]123CountwithMe(video),[>]–[>],[>]“Onphysicallinesofforce”(Maxwell),[>]OysterBarrestaurant(NewYorkCity),[>]Page,Larry,[>],[>],[>]PageRank,[>]–[>],[>]–[>],[>]–[>]parabolas,[>]–[>],[>]–[>],[>]–[>]Pareto,Vilfredo,[>]partialsums,[>]–[>]Paulos,JohnAllen,[>]percentages,[>]perfectsquares,[>]personalfinance,[>]–[>],[>]–[>],[>],[>]–[>],[>],[>]Peskin,Charlie,[>]physics,lawsof,[>],[>]–[>]pi,[>]–[>],[>],[>]place-valuesystem,ofwritingnumbers,[>]–[>]planegeometry,[>],[>]Poincaré,Henri,[>],[>]Polthier,Konrad,[>]–[>]powerfunctions,[>]–[>]power-lawdistributions,[>]–[>]primenumbers,[>]–[>],[>].Seealsotwinprimesprimenumbertheorem,[>],[>]

PrincetonUniversity,[>]probability,[>],[>]–[>],[>]–[>],[>],[>]–[>]“problemwithwordproblems,The”(Devlin),[>]proofs,ingeometry,[>]–[>],[>]–[>],[>]–[>]Pythagoreantheorem,[>]–[>],[>],[>],[>]quadraticequations,[>]–[>]quadraticformula,[>]–[>],[>]quantummechanics,[>]–[>],[>],[>],[>],[>]rationalnumbers,[>]Recorde,Robert,[>]relationships,[>],[>]–[>],[>],[>]–[>],[>].SeealsoalgebraRichie,Lionel,[>]–[>]Riemann,Bernhard,[>],[>]–[>]Riemannhypothesis,[>]Riemannrearrangementtheorem,[>],[>]Romannumerals,[>]–[>],[>],[>]Ruina,Andy,[>]Russell,Bertrand,[>]Schaffer,Simon,[>]ScientificAmerican,[>]SesameStreet,[>]–[>],[>]settheory,[>]–[>],[>]shapes,[>]Shepherd,Cybill,[>]Shipley,David,[>]Simpson,O.J.,[>]–[>],[>]–[>]sinefunction,[>]sinewaves,[>]–[>],[>],[>],[>]–[>],[>]Snell’slaw,[>],[>]SolitudeofPrimeNumbers,The(Giordano),[>]–[>]Sopranos,The(televisionshow),[>]–[>]sphericalgeometry,[>]Spinoza,Baruch,[>]squareroots,[>]–[>]statistics,[>],[>]–[>]Steinmetzsolid,[>]StonyBrookUniversity,[>]Streisand,Barbra,[>]subtraction,[>],[>]

sunrises/sunsets,[>]–[>],[>]–[>]symmetriesoftheshape,[>]symmetry,[>]–[>],[>],[>]Tatham,Simon,[>]technologicalsingularity,[>]telegraphmachines,[>]–[>]tennis,[>]–[>]three-bodyproblems,[>]–[>]ThreeDogNight,[>]topology,[>]–[>],[>]–[>].SeealsoMöbiusstripstorus,two-holed,[>]–[>]triangles,[>]–[>],[>]–[>].SeealsoPythagoreantheoremtrigonometry,[>],[>],[>]–[>],[>],[>],[>]twinprimes,[>],[>],[>]–[>]two-bodyproblems,[>]Ushio,Keizo,[>]Vaccaro,George,[>],[>]ValléePoussin,Charlesdela,[>]Vanderbei,Robert,[>]variables.Seealgebravectorcalculus,[>]–[>]VerizonWireless,[>],[>],[>]VisualGroupTheory(Carter),[>]Wang,Jane,[>]WesternUnion,[>]whisperinggalleries,[>]–[>]Wiggins,Grant,[>]Wigner,Eugene,[>]–[>],[>]Willis,Bruce,[>]WorcesterPolytechnicInstitute,[>]wordproblems,[>]–[>],[>]WorldWarI,[>]–[>],[>]YouTube,[>],[>],[>]–[>]ZenoofElea(Greekphilosopher),[>]zero,[>]

AbouttheAuthor

StevenStrogatzistheSchurmanProfessorofAppliedMathematicsatCornellUniversity.Arenownedteacherandoneoftheworld’smosthighlycitedmathematicians,hehasbeenafrequentguestonNationalPublicRadio’sRadiolab.HeistheauthorofSyncandTheCalculusofFriendship,andtherecipientofalifetimeachievementawardformathcommunication.HealsowroteapopularNewYorkTimesonlinecolumn,“TheElementsofMath,”whichformedthebasisforthisbook.