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

Forinformationaboutpermissiontoreproduceselectionsfromthisbook,writetoPermissions,HoughtonMifflinHarcourtPublishingCompany,215Park

AvenueSouth,NewYork,NewYork10003.

www.hmhbooks.com

TheLibraryofCongresshascatalogedtheprinteditionasfollows:Strogatz,StevenH.(StevenHenry)

Thejoyofx:aguidedtourofmath,fromonetoinfinity/StevenStrogatz.p.cm.

ISBN978-0-547-51765-0(hardback)1.Mathematics—Popularworks.I.Title.

QA93.S772012510—dc232012017320

IllustrationsbyMargaretNelson

eISBN978-0-547-51766-7

v1.1012Chapters1–3,5,7,8,11,12,16–18,20,23,26,28,and30areadapted,with

permission,frompiecesoriginallypublishedintheNewYorkTimes.Gratefulacknowledgmentismadeforpermissiontoreprintanexcerptfromthefollowingcopyrightedwork:[>],fromp.111ofTheSolitudeofPrimeNumbers:ANovelbyPaoloGiordano,translatedbyShaunWhiteside,copyright©2008byArnoldoMondadoriEditoreS.p.A.,translationcopyright©2009byShaunWhiteside.UsedbypermissionofPamelaDormanBooks,animprintofViking

Penguin,adivisionofPenguinGroup(USA)Inc.

Creditsappearon[>].

Preface

Ihaveafriendwhogetsatremendouskickoutofscience,eventhoughhe’sanartist.Wheneverwegettogetherallhewantstodoischataboutthelatestthingin psychology or quantummechanics.Butwhen it comes tomath, he feels atsea,anditsaddenshim.Thestrangesymbolskeephimout.Hesayshedoesn’tevenknowhowtopronouncethem.Infact,hisalienationrunsalotdeeper.He’snotsurewhatmathematiciansdo

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

tourthroughtheelementsofmath,frompreschooltogradschool,foranyoneouttherewho’dliketohaveasecondchanceatthesubject—butthistimefromanadult perspective. It’s not intended to be remedial. The goal is to give you abetterfeelingforwhatmathisallaboutandwhyit’ssoenthrallingtothosewhogetit.We’lldiscoverhowMichaelJordan’sdunkscanhelpexplainthefundamentals

ofcalculus.I’llshowyouasimple—andmind-blowing—waytounderstandthatstapleofgeometry, thePythagorean theorem.We’ll try toget to thebottomofsomeoflife’smysteries,bigandsmall:DidO.J.doit?Howshouldyouflipyourmattresstogetthemaximumwearoutofit?Howmanypeopleshouldyoudatebeforesettlingdown?Andwe’llseewhysomeinfinitiesarebiggerthanothers.Math is everywhere, if you knowwhere to look.We’ll spot sine waves in

zebra stripes, hear echoes of Euclid in the Declaration of Independence, andrecognizesignsofnegativenumbersintherun-uptoWorldWarI.Andwe’llseehowourlivestodayarebeingtouchedbynewkindsofmath,aswesearchforrestaurants online and try to understand—not to mention survive—thefrighteningswingsinthestockmarket.Byacoincidence thatseemsonlyfittingforabookaboutnumbers, thisone

wasbornonthedayIturnedfifty.DavidShipley,whowasthentheeditoroftheop-ed page for theNew York Times, had invited me to lunch on the big day(unawareofitssemicentennialsignificance)andaskedifIwouldeverconsiderwriting a series aboutmath for his readers. I loved the thought of sharing thepleasuresofmathwithanaudiencebeyondmyinquisitiveartistfriend.“The Elements ofMath” appeared online in late January 2010 and ran for

fifteenweeks. In response, lettersandcommentspoured in fromreadersofallages.Manywhowrotewerestudentsandteachers.Otherswerecuriouspeoplewho, for whatever reason, had fallen off the track somewhere in their matheducationbutsensedtheyweremissingsomethingworthwhileandwantedtotryagain.EspeciallygratifyingwerethenotesIreceivedfromparentsthankingmeforhelping themexplainmath to theirkidsand, in theprocess, to themselves.Evenmycolleaguesandfellowmathaficionadosseemedtoenjoythepieces—whentheyweren’tsuggestingimprovements(orperhapsespeciallythen!).All in all, the experience convinced me that there’s a profound but little-

recognized hunger formath among the general public.Despite everythingwehear about math phobia, many peoplewant to understand the subject a littlebetter.Andoncetheydo,theyfinditaddictive.

TheJoyofxisanintroductiontomath’smostcompellingandfar-reachingideas.The chapters—some from the original Times series—are bite-size and largelyindependent,sofeelfreetosnackwhereveryoulike.Ifyouwanttowadedeeperinto anything, the notes at the end of the book provide additional details andsuggestionsforfurtherreading.For thebenefitof readerswhopreferastep-by-stepapproach, I’vearranged

thematerialintosixmainparts,followingthelinesofthetraditionalcurriculum.Part 1, “Numbers,” begins our journey with kindergarten and grade-school

arithmetic, stressinghowhelpful numbers canbe andhowuncannily effectivetheyareatdescribingtheworld.Part 2, “Relationships,” generalizes fromworkingwith numbers toworking

withrelationshipsbetweennumbers.Thesearetheideasattheheartofalgebra.Whatmakes them so crucial is that they provide the first tools for describinghow one thing affects another, through cause and effect, supply and demand,dose and response, and so on—thekinds of relationships thatmake theworldcomplicatedandrich.Part3,“Shapes,”turnsfromnumbersandsymbolstoshapesandspace—the

province of geometry and trigonometry. Along with characterizing all thingsvisual,thesesubjectsraisemathtonewlevelsofrigorthroughlogicandproof.In part 4, “Change,”we come to calculus, themost penetrating and fruitful

branchofmath.Calculusmadeitpossibletopredictthemotionsoftheplanets,therhythmofthetides,andvirtuallyeveryotherformofcontinuouschangeintheuniverseandourselves.Asupportingthemeinthispartistheroleofinfinity.Thedomesticationofinfinitywasthebreakthroughthatmadecalculuswork.Byharnessingtheawesomepoweroftheinfinite,calculuscouldfinallysolvemanylong-standingproblems that haddefied the ancients, and that ultimately led to

thescientificrevolutionandthemodernworld.Part5,“Data,”dealswithprobability,statistics,networks,anddatamining,all

relatively young subjects inspired by themessy side of life: chance and luck,uncertainty, risk, volatility, randomness, interconnectivity.With the right kindsofmath, and the right kinds of data,we’ll see how to pullmeaning from themaelstrom.Asweneartheendofourjourneyinpart6,“Frontiers,”weapproachtheedge

of mathematical knowledge, the borderland between what’s known and whatremainselusive.Thesequenceofchapters follows the familiar structurewe’veused throughout—numbers, relationships, shapes, change, and infinity—buteachofthesetopicsisnowrevisitedmoredeeply,andinitsmodernincarnation.I hope that all of the ideas aheadwill provide joy—and a good number of

Aha!moments.But any journeyneeds to begin at the beginning, so let’s startwiththesimple,magicalactofcounting.

PartOneNUMBERS

1.FromFishtoInfinity

THE BEST INTRODUCTION to numbers I’ve ever seen—the clearest and funniestexplanationofwhattheyareandwhyweneedthem—appearsinaSesameStreetvideo called123CountwithMe.Humphrey, an amiable but dimwitted fellowwith pink fur and a green nose, isworking the lunch shift at the FurryArmsHotel when he takes a call from a roomful of penguins. Humphrey listenscarefullyandthencallsouttheirordertothekitchen:“Fish,fish,fish,fish,fish,fish.”ThispromptsErnietoenlightenhimaboutthevirtuesofthenumbersix.

Children learn from this that numbers are wonderful shortcuts. Instead of

sayingtheword“fish”exactlyasmanytimesastherearepenguins,Humphreycouldusethemorepowerfulconceptofsix.Asadults,however,wemightnoticeapotentialdownside tonumbers.Sure,

theyaregreattimesavers,butataseriouscostinabstraction.Sixismoreetherealthansix fish,preciselybecause it’smoregeneral. Itapplies tosixofanything:sixplates,sixpenguins,sixutterancesoftheword“fish.”It’stheineffablething

theyallhaveincommon.Viewedinthislight,numbersstarttoseemabitmysterious.Theyapparently

existinsomesortofPlatonicrealm,alevelabovereality.Inthatrespecttheyaremorelikeotherloftyconcepts(e.g.,truthandjustice),andlessliketheordinaryobjects of daily life. Their philosophical status becomes even murkier uponfurtherreflection.Whereexactlydonumberscomefrom?Didhumanityinventthem?Ordiscoverthem?An additional subtlety is that numbers (and allmathematical ideas, for that

matter)havelivesoftheirown.Wecan’tcontrolthem.Eventhoughtheyexistinourminds,oncewedecidewhatwemeanbythemwehavenosayinhowtheybehave.Theyobey certain laws and have certain properties, personalities, andways of combining with one another, and there’s nothing we can do about itexceptwatchandtrytounderstand.Inthatsensetheyareeerilyreminiscentofatoms and stars, the things of this world, which are likewise subject to lawsbeyondourcontrol...exceptthatthosethingsexistoutsideourheads.This dual aspect of numbers—as part heaven, part earth—is perhaps their

mostparadoxical feature,and thefeature thatmakes themsouseful. It iswhatthephysicistEugeneWignerhad inmindwhenhewroteof“theunreasonableeffectivenessofmathematicsinthenaturalsciences.”In case it’s not clear what I mean about the lives of numbers and their

uncontrollable behavior, let’s go back to the FurryArms. Suppose that beforeHumphreyputs in thepenguins’order, he suddenlygets a call on another linefrom a room occupied by the same number of penguins, all of them alsoclamoringforfish.Aftertakingbothcalls,whatshouldHumphreyyellouttothekitchen? If he hasn’t learned anything, he could shout “fish” once for eachpenguin.Or,usinghisnumbers,hecouldtellthecookheneedssixordersoffishforthefirstroomandsixmoreforthesecondroom.Butwhathereallyneedsisanewconcept:addition.Oncehe’smastered it,he’llproudlysayheneedssixplussix(or,ifhe’sashowoff,twelve)fish.Thecreativeprocesshereisthesameastheonethatgaveusnumbersinthe

first place. Just as numbers are a shortcut for counting by ones, addition is ashortcutforcountingbyanyamount.Thisishowmathematicsgrows.Therightabstractionleadstonewinsight,andnewpower.Beforelong,evenHumphreymightrealizehecankeepcountingforever.Yet despite this infinite vista, there are always constraints on our creativity.

Wecandecidewhatwemeanbythingslike6and+,butoncewedo,theresultsofexpressionslike6+6arebeyondourcontrol.Logicleavesusnochoice.Inthat sense,math always involves both inventionand discovery:we invent theconceptsbutdiscovertheirconsequences.Aswe’llseeinthecomingchapters,

inmathematicsourfreedomliesinthequestionsweask—andinhowwepursuethem—butnotintheanswersawaitingus.

2.RockGroups

LIKEANYTHINGELSE,arithmetichasitsserioussideanditsplayfulside.Theserioussideiswhatwealllearnedinschool:howtoworkwithcolumns

of numbers, adding them, subtracting them, grinding them through thespreadsheetcalculationsneededfortaxreturnsandyear-endreports.Thissideofarithmeticisimportant,practical,and—formanypeople—joyless.Theplayfulsideofarithmeticisalotlessfamiliar,unlessyouweretrainedin

the ways of advanced mathematics. Yet there’s nothing inherently advancedaboutit.It’sasnaturalasachild’scuriosity.In his book A Mathematician’s Lament, Paul Lockhart advocates an

educationalapproachinwhichnumbersaretreatedmoreconcretelythanusual:heasksustoimaginethemasgroupsofrocks.Forexample,6correspondstoagroupofrockslikethis:

Youprobablydon’tseeanythingstrikinghere,andthat’sright—unlesswemakefurtherdemandsonnumbers,theyalllookprettymuchthesame.Ourchancetobecreativecomesinwhatweaskofthem.For instance, let’s focusongroupshavingbetween1 and10 rocks in them,

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

A less stringent challenge is to identify groups of rocks that can be neatly

organized into a rectangle with exactly two rows that come out even. That’spossibleaslongasthereare2,4,6,8,or10rocks;thenumberhastobeeven.Ifwetrytocoerceanyoftheothernumbersfrom1to10—theoddnumbers—intotworows,theyalwaysleaveanoddbitstickingout.

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

Ifweloosentherulesstillfurthertoadmitnumbersgreaterthan10andallow

arectangularpattern tohavemore than tworowsof rocks, someoddnumbersdisplayatalentformakingtheselargerrectangles.Forexample,thenumber15canforma3×5rectangle:

So 15, although undeniably odd, at least has the consolation of being acomposite number—it’s composedof three rowsof five rocks each.Similarly,every other entry in the multiplication table yields its own rectangular rockgroup.Yetsomenumbers, like2,3,5,and7, trulyarehopeless.Noneof themcan

formanysortofrectangleatall,otherthanasimplelineofrocks(asinglerow).Thesestrangelyinflexiblenumbersarethefamousprimenumbers.So we see that numbers have quirks of structure that endow them with

personalities.Buttoseethefullrangeoftheirbehavior,weneedtogobeyondindividualnumbersandwatchwhathappenswhentheyinteract.For example, instead of adding just two odd numbers together, supposewe

addalltheconsecutiveoddnumbers,startingfrom1:

Thesumsabove,remarkably,alwaysturnout tobeperfectsquares.(Wesaw4and9inthesquarepatternsdiscussedearlier,and16=4×4,and25=5×5.)Aquick check shows that this rule keeps working for larger and larger oddnumbers; it apparently holds all the way out to infinity. But what possibleconnection could there be between odd numbers, with their ungainlyappendages, and the classically symmetrical numbers that form squares? Byarranging our rocks in the right way, we can make this surprising link seemobvious—thehallmarkofanelegantproof.The key is to recognize that odd numbers can make L-shapes, with their

protuberancescastoffintothecorner.AndwhenyoustacksuccessiveL-shapestogether,yougetasquare!

This style of thinking appears in another recent book, though for altogether

different literary reasons. In Yoko Ogawa’s charming novel TheHousekeeperand theProfessor, anastutebutuneducatedyoungwomanwitha ten-year-oldson is hired to take care of an elderly mathematician who has suffered atraumatic brain injury that leaves himwith only eightyminutes of short-termmemory.Adriftinthepresent,andaloneinhisshabbycottagewithnothingbuthisnumbers, theProfessor tries toconnectwith theHousekeeper theonlywayhe knows how: by inquiring about her shoe size or birthday and makingmathematical small talk abouther statistics.TheProfessor also takes a speciallikingtotheHousekeeper’sson,whomhecallsRoot,becausetheflattopoftheboy’sheadremindshimofthesquarerootsymbol, .OnedaytheProfessorgivesRootalittlepuzzle:Canhefindthesumofallthe

numbersfrom1to10?AfterRootcarefullyaddsthenumbersandreturnswiththe answer (55), the Professor asks him to find a betterway.Can he find theanswerwithoutaddingthenumbers?Rootkicksthechairandshouts,“That’snotfair!”ButlittlebylittletheHousekeepergetsdrawnintotheworldofnumbers,and

shesecretlystartsexploringthepuzzleherself.“I’mnotsurewhyIbecamesoabsorbedinachild’smathproblemwithnopracticalvalue,”shesays.“Atfirst,Iwas conscious of wanting to please the Professor, but gradually that feelingfadedandIrealizedithadbecomeabattlebetweentheproblemandme.WhenI

wokeinthemorning,theequationwaswaiting:

and it followedmeall through theday, as though it hadburned itself intomyretinaandcouldnotbeignored.”ThereareseveralwaystosolvetheProfessor’sproblem(seehowmanyyou

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

By its very appearance this picture gives a clear sense of negative space. It

seems only half complete. And that suggests a creative leap. If you copy thetriangle, flip it upside down, and add it as themissing half towhat’s alreadythere, youget somethingmuch simpler: a rectanglewith ten rowsof 11 rockseach,foratotalof110.

Sincetheoriginaltriangleishalfofthisrectangle,thedesiredsummustbehalfof110,or55.Lookingatnumbersasgroupsofrocksmayseemunusual,butactuallyit’sas

oldasmathitself.Theword“calculate”reflectsthatlegacy—itcomesfromtheLatinwordcalculus,meaningapebbleusedforcounting.Toenjoyworkingwithnumbersyoudon’thave tobeEinstein (German for“onestone”),but itmighthelptohaverocksinyourhead.

3.TheEnemyofMyEnemy

IT’STRADITIONALTOteachkidssubtractionrightafteraddition.Thatmakessense—the same facts about numbers get used in both, though in reverse.And theblackartofborrowing,socrucialtosuccessfulsubtraction,isonlyalittlemorebaroquethanthatofcarrying,itscounterpartforaddition.Ifyoucancopewithcalculating23+9,you’llbereadyfor23–9soonenough.Atadeeper level,however,subtractionraisesamuchmoredisturbing issue,

onethatneverariseswithaddition.Subtractioncangeneratenegativenumbers.If I try to take6 cookies away fromyoubut youhaveonly2, I can’t do it—except in my mind, where you now have negative 4 cookies, whatever thatmeans.Subtractionforcesustoexpandourconceptionofwhatnumbersare.Negative

numbersarealotmoreabstractthanpositivenumbers—youcan’tseenegative4cookies and you certainly can’t eat them—but you can think about them, andyouhaveto,inallaspectsofdailylife,fromdebtsandoverdraftstocontendingwithfreezingtemperaturesandparkinggarages.Still, many of us haven’t quitemade peace with negative numbers. Asmy

colleagueAndyRuinahaspointedout,peoplehaveconcoctedallsortsoffunnylittle mental strategies to sidestep the dreaded negative sign. Onmutual fundstatements,losses(negativenumbers)areprintedinredornestledinparentheseswith nary a negative sign to be found. The history books tell us that JuliusCaesar was born in 100 B.C., not –100. The subterranean levels in a parkinggarageoftenhavedesignationslikeB1andB2.Temperaturesareoneofthefewexceptions:folksdosay,especiallyhereinIthaca,NewYork,thatit’s–5degreesoutside,thougheventhen,manyprefertosay5belowzero.There’ssomethingaboutthatnegativesignthatjustlookssounpleasant,so...negative.Perhaps the most unsettling thing is that a negative times a negative is a

positive.Soletmetrytoexplainthethinkingbehindthat.How shouldwedefine thevalueof an expression like–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, . . . At each step, we’re adding 1 to the number before it. Sowouldn’tyouagreethenextnumbershouldlogicallybe1?That’soneargumentforwhy(–1)×(–1)=1.Theappealofthisdefinitionis

that it preserves the rules of ordinary arithmetic; what works for positivenumbersalsoworksfornegativenumbers.But if you’re a hard-boiled pragmatist, you may be wondering if these

abstractions have any parallels in the real world. Admittedly, life sometimesseems to play by different rules. In conventional morality, two wrongs don’tmakearight.Likewise,doublenegativesdon’talwaysamounttopositives;theycanmakenegativesmore intense,as in“Ican’tgetnosatisfaction.” (Actually,languagescanbeverytrickyinthisrespect.TheeminentlinguisticphilosopherJ.L.AustinofOxfordoncegave a lecture inwhichhe asserted that there aremanylanguagesinwhichadoublenegativemakesapositivebutnoneinwhichadoublepositivemakesanegative—towhichtheColumbiaphilosopherSidneyMorgenbesser,sittingintheaudience,sarcasticallyreplied,“Yeah,yeah.”)Still, thereareplentyofcaseswhere the realworlddoesmirror the rulesof

negative numbers. One nerve cell’s firing can be inhibited by the firing of asecondnervecell.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,as balanced—there’s no reason for anyone to change how he feels, since it’sreasonabletolikeyourfriends’friends.Similarly,thetriangleontheright,withtwo negatives and a positive, is considered balanced because it causes nodissonance;eventhoughitallowsforhostility,nothingcementsafriendshiplikehatingthesameperson.Ofcourse,trianglescanalsobeunbalanced.Whenthreemutualenemiessize

up the situation, two of them—often the twowith the least animosity towardeachother—maybetemptedtojoinforcesandganguponthethird.Evenmore unbalanced is a trianglewith a single negative relationship. For

instance,supposeCarolisfriendlywithbothAliceandBob,butBobandAlicedespise each other. Perhaps they were once a couple but suffered a nastybreakup,andeachisnowbadmouthingtheothertoever-loyalCarol.Thiscausespsychologicalstressallaround.Torestorebalance,eitherAliceandBobhavetoreconcileorCarolhastochooseaside.

Inallthesecases,thelogicofbalancematchesthelogicofmultiplication.Ina

balancedtriangle,thesignoftheproductofanytwosides,positiveornegative,alwaysagreeswiththesignofthethird.Inunbalancedtriangles,thispatternisbroken.Leavingasidetheverisimilitudeofthemodel, thereareinterestingquestions

hereofapurelymathematicalflavor.Forexample,inaclose-knitnetworkwhereeveryone knows everyone, what’s the most stable state? One possibility is anirvanaofgoodwill,whereallrelationshipsarepositiveandalltriangleswithinthenetworkarebalanced.Butsurprisingly,thereareotherstatesthatareequallystable. These are states of intractable conflict,with the network split into twohostilefactions(ofarbitrarysizesandcompositions).Allmembersofonefactionare friendly with one another but antagonistic toward everybody in the otherfaction.(Soundfamiliar?)Perhapsevenmoresurprisingly,thesepolarizedstatesaretheonlystatesasstableasnirvana.Inparticular,nothree-partysplitcanhaveallitstrianglesbalanced.Scholars have used these ideas to analyze the run-up toWorldWar I. The

diagramthat followsshows theshiftingalliancesamongGreatBritain,France,Russia,Italy,Germany,andAustria-Hungarybetween1872and1907.

The first five configurations were all unbalanced, in the sense that they eachcontained at least one unbalanced triangle. The resultant dissonance tended topushthesenationstorealignthemselves,triggeringreverberationselsewhereinthe network. In the final stage, Europe had split into two implacably opposedblocs—technicallybalanced,butonthebrinkofwar.The point is not that this theory is powerfully predictive. It isn’t. It’s too

simpletoaccountforallthesubtletiesofgeopoliticaldynamics.Thepointisthatsomepartofwhatweobserveisduetonothingmorethantheprimitivelogicof“the enemy of my enemy,” and this part is captured perfectly by themultiplicationofnegativenumbers.Bysortingthemeaningfulfromthegeneric,thearithmeticofnegativenumberscanhelpusseewheretherealpuzzleslie.

4.Commuting

EVERYDECADEORSOanewapproachtoteachingmathcomesalongandcreatesfreshopportunitiesforparentstofeelinadequate.Backinthe1960s,myparentswere flabbergasted by their inability to help me with my second-gradehomework.They’dneverheardofbase3orVenndiagrams.Now the tables have turned. “Dad, can you show me how to do these

multiplication problems?” Sure, I thought, until the headshaking began. “No,Dad,that’snothowwe’resupposedtodoit.That’stheold-schoolmethod.Don’tyouknowthelatticemethod?No?Well,whataboutpartialproducts?”These humbling sessions have prompted me to revisit multiplication from

scratch.Andit’sactuallyquitesubtle,onceyoustarttothinkaboutit.Take the terminology.Does“seven times three”mean“sevenadded to itself

threetimes”?Or“threeaddedtoitselfseventimes”?Insomeculturesthelanguageislessambiguous.AfriendofminefromBelize

used to recitehis times tables like this:“Sevenonesareseven,seven twosarefourteen,seventhreesaretwenty-one,”andsoon.Thisphrasingmakesitclearthat the first number is the multiplier; the second number is the thing beingmultiplied.It’sthesameconventionasinLionelRichie’simmortallyrics“She’sonce, twice, three timesa lady.” (“She’sa lady times three”wouldneverhavebeenahit.)Maybe all this semantic fuss strikes you as silly, since the order in which

numbersaremultiplieddoesn’tmatteranyway:7×3=3×7.Fairenough,butthatbegsthequestionI’dliketoexploreinsomedepthhere:Isthiscommutativelaw of multiplication, a × b = b × a, really so obvious? I remember beingsurprisedbyitasachild;maybeyouweretoo.Torecapture themagic, imaginenotknowingwhat7×3equals.Soyou try

countingbysevens:7,14,21.Nowturnitaroundandcountbythreesinstead:3,6,9,...Doyoufeelthesuspensebuilding?Sofar,noneofthenumbersmatchthoseinthesevenslist,butkeepgoing...12,15,18,andthen,bingo,21!My point is that if you regard multiplication as being synonymous with

repeated counting by a certain number (or, in other words, with repeatedaddition),thecommutativelawisn’ttransparent.Butitbecomesmoreintuitiveifyouconceiveofmultiplicationvisually.Think

of7×3asthenumberofdotsinarectangulararraywithsevenrowsandthreecolumns.

Ifyouturnthearrayonitsside,ittransformsintothreerowsandsevencolumns—andsince rotating thepicturedoesn’tchange thenumberofdots, itmustbetruethat7×3=3×7.

Yetstrangelyenough,inmanyreal-worldsituations,especiallywheremoney

is concerned, people seem to forget the commutative law, or don’t realize itapplies.Letmegiveyoutwoexamples.Suppose you’re shopping for a new pair of jeans. They’re on sale for 20

percent off the sticker price of $50,which sounds like a bargain, but keep inmindthatyoualsohave topay the8percentsales tax.After theclerkfinishes

complimentingyouon the flattering fit, she starts ringingup thepurchasebutthenpausesandwhispers,inaconspiratorialtone,“Hey,letmesaveyousomemoney. I’ll apply the tax first, and then take twenty percent off the total, soyou’llgetmoremoneyback.Okay?”Butsomethingaboutthatsoundsfishytoyou.“Nothanks,”yousay.“Could

youpleasetakethetwentypercentofffirst,thenapplythetaxtothesaleprice?Thatway,I’llpaylesstax.”Whichwayisabetterdealforyou?(Assumebotharelegal.)When confronted with a question like this, many people approach it

additively. Theywork out the tax and the discount under both scenarios, andthendowhateveradditionsorsubtractionsarenecessarytofindthefinalprice.Doingthingstheclerk’sway,youreason,wouldcostyou$4intax(8percentofthestickerpriceof$50).Thatwouldbringyourtotalto$54.Thenapplyingthe20percentdiscountto$54givesyou$10.80back,soyou’denduppaying$54minus $10.80, which equals $43.20. Whereas under your scenario, the 20percentdiscountwouldbeappliedfirst,savingyou$10offthe$50stickerprice.Then the8percent taxon that reducedpriceof$40wouldbe$3.20, soyou’dstillenduppaying$43.20.Amazing!But it’s merely the commutative law in action. To see why, think

multiplicatively, not additively. Applying an 8 percent tax followed by a 20percent discount amounts to multiplying the sticker price by 1.08 and thenmultiplyingthatresultby0.80.Switchingtheorderoftaxanddiscountreversesthemultiplication,butsince1.08×0.80=0.80×1.08,thefinalpricecomesoutthesame.Considerations like these also arise in larger financial decisions. Is a Roth

401(k)betterorworsethanatraditionalretirementplan?Moregenerally,ifyouhaveapileofmoneytoinvestandyouhavetopaytaxesonitatsomepoint,isitbetter to take the tax bite at the beginning of the investment period, or at theend?Once again, the commutative law shows it’s a wash, all other things being

equal(which,sadly,theyoftenaren’t).If,forbothscenarios,yourmoneygrowsbythesamefactorandgetstaxedatthesamerate,itdoesn’tmatterwhetheryoupaythetaxesupfrontorattheend.Please don’tmistake thismathematical remark for financial advice.Anyone

facingthesedecisionsinreallifeneedstobeawareofmanycomplicationsthatmuddy thewaters:Doyouexpect tobe inahigheror lower taxbracketwhenyou retire? Will you max out your contribution limits? Do you think thegovernmentwillchangeitspoliciesaboutthetax-exemptstatusofwithdrawalsbythetimeyou’rereadytotakethemoneyout?Leavingallthisaside(anddon’t

get me wrong, it’s all important; I’m just trying to focus here on a simplermathematical issue),mybasicpoint is that thecommutative law is relevant totheanalysisofsuchdecisions.You can find heated debates about this on personal finance sites on the

Internet.Evenaftertherelevanceofthecommutativelawhasbeenpointedout,somebloggersdon’tacceptit.It’sthatcounterintuitive.Maybe we’re wired to doubt the commutative law because in daily life, it

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

hewasworryingabouthis future.AsanundergraduateatYale,hedesperatelywanted to stay in the IvyLeague forgraduate school.UnfortunatelyPrincetonrejected his application. Harvard said yes but seemed to be dragging its feetabout providing the financial support he needed. His best option, though hefound it depressing, was MIT. In Gell-Mann’s eyes, MIT was a grubbytechnological institute,beneathhisrarefied taste.Nevertheless,heaccepted theoffer.Yearslaterhewouldexplainthathehadcontemplatedsuicideatthetimebut decided against it once he realized that attendingMIT and killing himselfdidn’tcommute.HecouldalwaysgotoMITandcommitsuicidelaterifhehadto,butnottheotherwayaround.Gell-Mann had probably been sensitized to the importance of non-

commutativity.Asaquantumphysicisthewouldhavebeenacutelyawarethatatthedeepest level,naturedisobeys thecommutative law.And it’sagood thing,too.Forthefailureofcommutativityiswhatmakestheworldthewayitis.It’swhymatterissolid,andwhyatomsdon’timplode.Specifically, early in the development of quantum mechanics, Werner

HeisenbergandPaulDirachaddiscoveredthatnaturefollowsacuriouskindoflogic in which p × q ≠ q × p, where p and q represent the momentum andpositionofaquantumparticle.Withoutthatbreakdownofthecommutativelaw,therewouldbenoHeisenberguncertaintyprinciple,atomswouldcollapse,andnothingwouldexist.That’swhyyou’dbettermindyourp’sandq’s.And tellyourkids todo the

same.

5.DivisionandItsDiscontents

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

count,add,andmultiply.Butonceweaskhowmuchremainswheneverythingistaken away, we are forced to create a new kind of number—zero—and sincedebts can be owed,we need negative numbers too. This enlarged universe ofnumbers,calledintegers,iseverybitasself-containedasthenaturalnumbersbutmuchmorepowerfulbecauseitembracessubtractionaswell.A new crisis comes when we try to work out the mathematics of sharing.

Dividingawholenumberevenlyisnotalwayspossible...unlessweexpandtheuniverseoncemore,nowby inventingfractions.Theseare ratiosof integers—hencetheirtechnicalname,rationalnumbers.Sadly,thisistheplacewheremanystudentshitthemathematicalwall.There are many confusing things about division and its consequences, but

perhapsthemostmaddeningisthattherearesomanydifferentwaystodescribeapartofawhole.Ifyoucutachocolatelayercakerightdownthemiddleintotwoequalpieces,

youcouldcertainlysaythateachpieceishalfthecake.Oryoumightexpressthesameideawiththefraction1/2,meaning“1of2equalpieces.”(Whenyouwriteitthisway,theslashbetweenthe1andthe2isavisualreminderthatsomethingisbeingsliced.)Athirdwayistosaythateachpieceis50percentofthewhole,meaning literally “50 parts out of 100.”As if thatweren’t enough, you couldalsoinvokedecimalnotationanddescribeeachpieceas0.5oftheentirecake.Thisprofusionofchoicesmaybepartlytoblameforthebewildermentmany

of us feelwhen confrontedwith fractions, percentages, and decimals.A vividexample appears in themovieMyLeftFoot, the true story of the Irishwriter,painter, and poet Christy Brown. Born into a large working-class family, hesufferedfromcerebralpalsythatmadeitalmostimpossibleforhimtospeakorcontrolanyofhislimbsexcepthisleftfoot.Asaboyhewasoftendismissedasmentally disabled, especially by his father,who resented him and treated himcruelly.A pivotal scene in the movie takes place around the kitchen table. One of

Christy’s older sisters is quietly doinghermathhomework, seatednext to her

father,whileChristy,asusual,isshuntedoffinthecorneroftheroom,twistedinhis chair. His sister breaks the silence: “What’s twenty-five percent of aquarter?”sheasks.Fathermullsitover.“Twenty-fivepercentofaquarter?Uhhh. . .That’sastupidquestion,eh?Imean, twenty-fivepercent isaquarter.Youcan’t have a quarter of a quarter.” Sister responds, “You can. Can’t you,Christy?”Father:“Ha!Whatwouldheknow?”Writhing, Christy struggles to pick up a piece of chalk with his left foot.

Positioningitoveraslateonthefloor,hemanages toscrawla1, thenaslash,then something unrecognizable. It’s the number 16, but the 6 comes outbackward.Frustrated,heerasesthe6withhisheelandtriesagain,butthistimethe chalk moves too far, crossing through the 6, rendering it indecipherable.“That’sonlyanervoussquiggle,”sneershisfather,turningaway.Christycloseshiseyesandslumpsback,exhausted.Aside from the dramatic power of the scene, what’s striking is the father’s

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

Since16of these thinslicesmake theoriginalwhole,eachslice is1/16of thewhole—theanswerChristywastryingtoscratchout.A version of the same kind of mental rigidity, updated for the digital age,

made the rounds on the Internet a few years ago when a frustrated customernamedGeorgeVaccaro recorded and posted his phone conversationwith twoservice representatives atVerizonWireless.Vaccaro’s complaintwas that he’d

beenquotedadatausagerateof.002centsperkilobyte,buthisbillshowedhe’dbeencharged.002dollarsperkilobyte,ahundredfoldhigherrate.TheensuingconversationclimbedtothetopfiftyinYouTube’scomedysection.Here’sahighlightthatoccursabouthalfwaythroughtherecording,duringan

exchangebetweenVaccaroandAndrea,theVerizonfloormanager:V:Doyou recognize that there’sadifferencebetweenonedollarandonecent?A:Definitely.V:Doyou recognize there’sadifferencebetweenhalfadollarandhalf acent?A:Definitely.V: Then, do you therefore recognize there’s a difference between .002dollarsand.002cents?A:No.V:No?A:Imeanthere’s...there’sno.002dollars.

A fewmoments laterAndrea says, “Obviously a dollar is ‘one, decimal, zero,zero,’ right?Sowhatwoulda ‘pointzerozero twodollars’ look like? . . . I’veneverheardof.002dollars...It’sjustnotafullcent.”The challenge of converting between dollars and cents is only part of the

problem for Andrea. The real barrier is her inability to envision a portion ofeither.From firsthand experience I can tell you what it’s like to be mystified by

decimals. In eighth grade, Ms. Stanton began teaching us how to convert afractionintoadecimal.Usinglongdivision,wefoundthatsomefractionsgivedecimalsthatterminateinallzeros.Forexample, =.2500...,whichcanberewritten as .25, since all those zeros amount to nothing.Other fractions givedecimalsthateventuallyrepeat,like

Myfavoritewas ,whosedecimalcounterpartrepeatseverysixdigits:

ThebafflementbeganwhenMs.Stantonpointedoutthatifyoutriplebothsidesofthesimpleequation

you’reforcedtoconcludethat1mustequal.9999...At the time Iprotested that theycouldn’tbeequal.Nomatterhowmany9s

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

thathad justbeenproven tome. Isawitbut refused tobelieve it. (Thismightremindyouofsomepeopleyouknow.)But itgetsworse—orbetter, ifyou like to feelyourneuronssizzle.Back in

Ms. Stanton’s class, what stopped us from looking at decimals that neitherterminate nor repeat periodically? It’s easy to cookup such stomach-churners.Here’sanexample:

By design, the blocks of 2 get progressively longer as wemove to the right.There’s no way to express this decimal as a fraction. Fractions always yielddecimalsthatterminateoreventuallyrepeatperiodically—thatcanbeproven—andsince thisdecimaldoesneither, itcan’tbeequal to the ratioofanywholenumbers.It’sirrational.Givenhowcontrivedthisdecimal is,youmightsupposeirrationality israre.

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

numbersand fractions, sobelovedand familiar,nowappear scarceandexotic.And that innocuous number line pinned to the molding of your grade-schoolclassroom?Nooneevertoldyou,butit’schaosupthere.

6.Location,Location,Location

I’DWALKEDPASTEzraCornell’sstatuehundredsof timeswithoutevenglancingathisgreenishlikeness.ButthenonedayIstoppedforacloserlook.

Ezraappearsoutdoorsyandruggedlydignifiedinhislongcoat,vest,andboots,hisrighthandrestingonawalkingstickandholdingarumpled,wide-brimmedhat. The monument comes across as unpretentious and disarmingly direct—muchlikethemanhimself,byallaccounts.Which iswhy it seems so discordant that Ezra’s dates are inscribed on the

pedestalinpompousRomannumerals:

EZRACORNELLMDCCCVII–MDCCCLXXIV

Whynotwrite simply1807–1874?Romannumeralsmay look impressive,butthey’rehardtoreadandcumbersometouse.Ezrawouldhavehadlittlepatienceforthat.Findingagoodwaytorepresentnumbershasalwaysbeenachallenge.Since

thedawnofcivilization,peoplehavetriedvarioussystemsforwritingnumbersandreckoningwiththem,whetherfortrading,measuringland,orkeepingtrackoftheherd.Whatnearlyall thesesystemshave incommonis thatourbiology isdeeply

embedded in them.Through thevagariesofevolution,wehappen tohavefivefingerson eachof twohands.That peculiar anatomical fact is reflected in theprimitivesystemoftallying;forexample,thenumber17iswrittenas

Here, each of the vertical strokes in each groupmust have originallymeant afinger. Maybe the diagonal slash was a thumb, folded across the other fourfingerstomakeafist?Roman numerals are only slightly more sophisticated than tallies. You can

spot the vestige of tallies in the way Romans wrote 2 and 3, as II and III.

Likewise,thediagonalslashisechoedintheshapeoftheRomansymbolfor5,V.But4isanambiguouscase.Sometimesit’swrittenasIIII,tallystyle(you’lloftensee thison fancyclocks), thoughmorecommonly it’swrittenas IV.Thepositioningofasmallernumber(I) to the leftofa largernumber(V) indicatesthat you’re supposed to subtract I, rather than add it, as youwould if itwerestationedontheright.ThusIVmeans4,whereasVImeans6.TheBabylonianswerenotnearlyasattached to their fingers.Theirnumeral

systemwas based on 60—a clear sign of their impeccable taste, for 60 is anexceptionallypleasantnumber.Itsbeautyisintrinsicandhasnothingtodowithhumanappendages.Sixtyisthesmallestnumberthatcanbedividedevenlyby1,2,3,4,5,and6.Andthat’sjustforstarters(there’salso10,12,15,20,and30).Becauseofitspromiscuousdivisibility,60ismuchmorecongenial than10forany sort of calculation ormeasurement that involves cutting things into equalparts.Whenwedivideanhourinto60minutes,oraminuteinto60seconds,orafullcircleinto360degrees,we’rechannelingthesagesofancientBabylon.But thegreatest legacyof theBabylonians isanideathat’ssocommonplace

todaythatfewofusappreciatehowsubtleandingeniousitis.To illustrate it, let’s consider our own Hindu-Arabic system, which

incorporatesthesameideainitsmodernform.Insteadof60,thissystemisbasedontensymbols:1,2,3,4,5,6,7,8,9,and,mostbrilliant,0.Thesearecalleddigits,naturally,fromtheLatinwordforafingeroratoe.The great innovation here is that even though this system is based on the

number 10, there is no single symbol reserved for 10. Ten is marked by aposition—the tens place—instead of a symbol. The same is true for 100, or1,000,oranyotherpowerof10.Theirdistinguishedstatusissignifiednotbyasymbolbutbyaparkingspot,areservedpieceofrealestate.Location,location,location.Contrast the elegance of this place-value system with the much cruder

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

ownsymbols,andexpressall theother, second-classnumbersascombinationsofthose.Unfortunately, Roman numerals creaked and groaned when faced with

anything larger than a few thousand. In a workaround solution that wouldnowadaysbecalledakludge,thescholarswhowerestillusingRomannumeralsin theMiddle Ages resorted to piling bars on top of the existing symbols to

indicatemultiplicationbyathousand.Forinstance, meanttenthousand,andmeantathousandthousandsor,inotherwords,amillion.Multiplyingbya

billion (a thousandmillion)was rarely necessary, but if you ever had to, youcouldalwaysputasecondbarontopofthe .Asyoucansee,thefunneverstopped.But in theHindu-Arabic system, it’s a snap towrite anynumber, nomatter

how big. All numbers can be expressed with the same ten digits, merely byslotting them into the right places. Furthermore, the notation is inherentlyconcise.Everynumberlessthanamillion,forexample,canbeexpressedinsixsymbolsorfewer.Trydoingthatwithwords,tallies,orRomannumerals.Best of all, with a place-value system, ordinary people can learn to do

arithmetic.Youjusthavetomasterafewfacts—themultiplicationtableanditscounterpart for addition.Onceyouget thosedown, that’s allyou’ll everneed.Any calculation involving any pair of numbers, no matter how big, can beperformedbyapplyingthesamesetsoffacts,overandoveragain,recursively.Ifitallsoundsprettymechanical,that’spreciselythepoint.Withplace-value

systems, you canprogramamachine todo arithmetic.From the earlydaysofmechanical calculators to the supercomputers of today, the automation ofarithmeticwasmadepossiblebythebeautifulideaofplacevalue.Buttheunsungherointhisstoryis0.Without0, thewholeapproachwould

collapse.It’stheplaceholderthatallowsustotell1,10,and100apart.All place-value systems are based on some number called, appropriately

enough,thebase.Oursystemisbase10,ordecimal(fromtheLatinrootdecem,meaning “ten”). After the ones place, the subsequent consecutive placesrepresenttens,hundreds,thousands,andsoon,eachofwhichisapowerof10:

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

systemusedincomputersandallthingsdigital,fromcellphonestocameras.Ofallthepossiblebases,itrequiresthefewestsymbols—justtwoofthem,0and1.

As such, itmeshes perfectlywith the logic of electronic switches or anythingelsethatcantogglebetweentwostates—onoroff,openorclosed.Binarytakessomegettingusedto.Insteadofpowersof10,itusespowersof

2.Itstillhasaonesplacelikethedecimalsystem,butthesubsequentplacesnowstandfortwos,fours,andeights,because

Of course,wewouldn’twrite the symbol2, because it doesn’t exist inbinary,justasthere’snosinglenumeralfor10indecimal.Inbinary,2iswrittenas10,meaningone2andzero1s.Similarly,4wouldbewrittenas100(one4,zero2s,andzero1s),and8wouldbe1000.Theimplicationsreachfarbeyondmath.Ourworldhasbeenchangedbythe

powerof2.Inthepastfewdecadeswe’vecometorealizethatallinformation—not just numbers, but also language, images, and sound—can be encoded instreamsofzerosandones.WhichbringsusbacktoEzraCornell.Tucked at the rear of hismonument, and almost completely obscured, is a

telegraphmachine—amodest reminder of his role in the creation ofWesternUnionandthetyingtogetheroftheNorthAmericancontinent.

As a carpenter turned entrepreneur,Cornellworked for SamuelMorse,whosenamelivesoninthecodeofdotsanddashesthroughwhichtheEnglishlanguagewas reduced to the clicks of a telegraph key. Those two little symbols weretechnologicalforerunnersoftoday’szerosandones.MorseentrustedCornelltobuildthenation’sfirsttelegraphline,alinkfrom

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

PartTwoRELATIONSHIPS

7.TheJoyofx

NOW IT’S TIME to move on from grade-school arithmetic to high-school math.Overthenexttenchapterswe’llberevisitingalgebra,geometry,andtrig.Don’tworry ifyou’ve forgotten themall—therewon’tbeany tests this timearound.Instead ofworrying about the details of these subjects,we have the luxury ofconcentratingontheirmostbeautiful,important,andfar-reachingideas.Algebra, for example, may have struck you as a dizzyingmix of symbols,

definitions, and procedures, but in the end they all boil down to just twoactivities—solvingforxandworkingwithformulas.Solvingforxisdetectivework.You’researchingforanunknownnumber,x.

You’vebeenhandedafewcluesaboutit,eitherintheformofanequationlike2x+3=7or, lessconveniently, inaconvolutedverbaldescriptionof it (as inthose scaryword problems). In either case, the goal is to identify x from theinformationgiven.Workingwithformulas,bycontrast, isablendofartandscience.Insteadof

dwellingonaparticularx,you’remanipulatingandmassagingrelationshipsthatcontinuetoholdevenasthenumbersinthemchange.Thesechangingnumbersare called variables, and they are what truly distinguishes algebra fromarithmetic. The formulas in question might express elegant patterns aboutnumbers for their own sake. This is where algebra meets art. Or they mightexpressrelationshipsbetweennumbersintherealworld,astheydointhelawsof nature for falling objects or planetary orbits or genetic frequencies in apopulation.Thisiswherealgebrameetsscience.Thisdivisionofalgebraintotwograndactivitiesisnotstandard(infact,Ijust

madeitup),butitseemstoworkprettywell.InthenextchapterI’llhavemoretosayaboutsolvingforx,sofornowlet’sfocusonformulas,startingwithsomeeasyexamplestoclarifytheideas.A few years ago, my daughter Jo realized something about her big sister,

Leah. “Dad, there’s always a number betweenmy age andLeah’s.Right nowI’msixandLeah’seight,andsevenisinthemiddle.Andevenwhenwe’reold,likewhenI’mtwentyandshe’s twenty-two, therewillstillbeanumber in themiddle!”Jo’sobservationqualifiesasalgebra(thoughnoonebutaproudfatherwould

see it that way) because she was noticing a relationship between two ever-changing variables: her age, x, andLeah’s age, y. Nomatter how old both ofthemare,Leahwillalwaysbetwoyearsolder:y=x+2.

Algebraisthelanguageinwhichsuchpatternsaremostnaturallyphrased.Ittakessomepractice tobecomefluent inalgebra,because it’s loadedwithwhattheFrenchcallfauxamis,“falsefriends”:apairofwords,eachfromadifferentlanguage(inthiscase,Englishandalgebra),thatsoundrelatedandseemtomeanthe same thing but that actuallymean something horribly different from eachotherwhentranslated.For example, suppose the lengthof ahallway isywhenmeasured inyards,

andfwhenmeasuredinfeet.Writeanequationthatrelatesytof.My friend Grant Wiggins, an education consultant, has been posing this

problem to students and faculty for years. He says that in his experience,studentsget itwrongmorethanhalf the time,evenif theyhaverecently takenandpassedanalgebracourse.Ifyouthinktheanswerisy=3f,welcometotheclub.It seems like such a straightforward translation of the sentence “One yard

equals three feet.”But as soon as you try a few numbers, you’ll see that thisformulagetseverythingbackward.Say thehallway is10yards long;everyoneknowsthat’s30feet.Yetwhenyoupluginy=10andf=30,theformuladoesn’twork!Thecorrect formula is f=3y.Here3 reallymeans“3 feetperyard.”When

youmultiplyitbyyinyards,theunitsofyardscanceloutandyou’releftwithunitsoffeet,asyoushouldbe.Checkingthattheunitscancelproperlyhelpsavoidthiskindofblunder.For

example, it could have saved theVerizon customer service reps (discussed inchapter5)fromconfusingdollarsandcents.Another kind of formula is known as an identity. When you factored or

multipliedpolynomials inalgebraclass,youwereworkingwith identities.Youcanuse themnow to impressyour friendswithnumericalparlor tricks.Here’sonethatimpressedthephysicistRichardFeynman,noslouchhimselfatmentalmath:When I was at Los Alamos I found out that Hans Bethe was absolutelytopnotch at calculating. For example, one time we were putting somenumbers into a formula, and got to 48 squared. I reach for theMarchantcalculator,andhesays,“That’s2,300.”Ibegintopushthebuttons,andhesays,“Ifyouwantitexactly,it’s2,304.”Themachinesays2,304.“Gee!That’sprettyremarkable!”Isay.“Don’tyouknowhowtosquarenumbersnear50?”hesays.“Yousquare

50—that’s 2,500—and subtract 100 times the difference of your numberfrom50(inthiscaseit’s2),soyouhave2,300.Ifyouwantthecorrection,

squarethedifferenceandadditon.Thatmakes2,304.”

Bethe’strickisbasedontheidentity

Hehadmemorizedthatequationandwasapplyingitforthecasewherexis–2,correspondingtothenumber48=50–2.Foran intuitiveproofof this formula, imaginea squarepatchof carpet that

measures50+xoneachside.

Then its area is (50 + x) squared, which is what we’re looking for. But thediagram above shows that this area is made of a 50-by-50 square (thiscontributesthe2,500totheformula),tworectanglesofdimensions50byx(eachcontributesanareaof50x,foracombinedtotalof100x),andfinallythelittlex-by-xsquaregivesanareaofxsquared,thefinalterminBethe’sformula.Relationships like these are not just for theoretical physicists. An identity

similar toBethe’s is relevant to anyonewho hasmoney invested in the stockmarket. Suppose your portfolio drops catastrophically by 50 percent one yearandthengains50percentthenext.Evenafterthatdramaticrecovery,you’dstillbedown25percent.Toseewhy,observethata50percentlossmultipliesyourmoneyby0.50,anda50percentgainmultipliesitby1.50.Whenthosehappenbacktoback,yourmoneymultipliesby0.50times1.50,whichequals0.75—inotherwords,a25percentloss.

In fact, you never get back to even when you lose and gain by the samepercentageinconsecutiveyears.Withalgebrawecanunderstandwhy.Itfollowsfromtheidentity

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

andaccordingtotheformulaabove,thisequals

Thepointisthatthisexpressionisalwayslessthan1foranyxotherthan0.Soyounevercompletelyrecoupyourlosses.Needlesstosay,noteveryrelationshipbetweenvariablesisasstraightforward

as those above.Yet the allure of algebra is seductive, and in gullible hands itspawnssuchsillinessasaformulaforthesociallyacceptableagedifferenceinaromance.AccordingtosomesitesontheInternet,ifyourageisx,politesocietywilldisapproveifyoudatesomeoneyoungerthanx/2+7.In other words, it would be creepy for anyone over eighty-two to eye my

forty-eight-year-old wife, even if she were available. But eighty-one? Noproblem.Ick.Ick.Ick...

8.FindingYourRoots

FORMORE THAN 2,500 years,mathematicians have been obsessedwith solvingfor x. The story of their struggle to find the roots —the solutions—ofincreasingly complicated equations is one of the great epics in the history ofhumanthought.OneoftheearliestsuchproblemsperplexedthecitizensofDelosaround430

B.C.Desperate to stave off a plague, they consulted the oracle ofDelphi,whoadvised them to double the volume of their cube-shaped altar to Apollo.Unfortunately, it turns out that doubling a cube’s volume required them toconstruct thecube rootof2,a task that isnowknown tobe impossible,giventheir restriction to use nothing but a straightedge and compass, the only toolsallowedinGreekgeometry.Later studies of similar problems revealed another irritant, a nagging little

thing that wouldn’t go away: even when solutions were possible, they ofteninvolvedsquarerootsofnegativenumbers.Suchsolutionswerelongderidedassophisticorfictitiousbecausetheyseemednonsensicalontheirface.Until the1700sorso,mathematiciansbelieved thatsquarerootsofnegative

numberssimplycouldn’texist.Theycouldn’tbepositivenumbers,afterall,sinceapositivetimesapositive

isalwayspositive,andwe’relookingfornumberswhosesquareisnegative.Norcould negative numbers work, since a negative times a negative is, again,positive.Thereseemed tobenohopeof findingnumbers thatwhenmultipliedbythemselveswouldgivenegativeanswers.We’veseencriseslikethisbefore.Theyoccurwheneveranexistingoperation

is pushed too far, into a domain where it no longer seems sensible. Just assubtracting bigger numbers from smaller ones gave rise to negative numbers(chapter 3) and division spawned fractions and decimals (chapter 5), thefreewheelinguseof square roots eventually forced theuniverseof numbers toexpand...again.Historically, thisstepwasthemostpainfulofall.Thesquarerootof–1still

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

number)isdefinedbythepropertythat

It’s true that ican’tbe foundanywhereon thenumber line. In that respect it’smuch stranger than zero, negative numbers, fractions, or even irrationalnumbers,allofwhich—weirdastheyare—stillhavetheirplacesinline.Butwithenoughimagination,ourmindscanmakeroomforiaswell.Itlives

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

Thesearethecomplexnumbers.Here“complex”doesn’tmean“complicated”;itmeans that two typesofnumbers, realand imaginary,havebonded together toformacomplex,ahybridnumberlike2+3i.Complex numbers are magnificent, the pinnacle of number systems. They

enjoyallthesamepropertiesasrealnumbers—youcanaddandsubtractthem,multiplyanddivide them—but theyarebetter than realnumbersbecause theyalwayshave roots.Youcan take the square root or cube root or any root of acomplexnumber,andtheresultwillstillbeacomplexnumber.

Betteryet,agrandstatementcalledthefundamentaltheoremofalgebrasaysthat the roots of any polynomial are always complex numbers. In that sensethey’retheendofthequest,theholygrail.Theuniverseofnumbersneedneverexpandagain.Complexnumbersare theculminationof the journey thatbeganwith1.Youcanappreciatetheutilityofcomplexnumbers(orfinditmoreplausible)

ifyouknowhowtovisualizethem.Thekeyis tounderstandwhatmultiplyingby i looks like.Supposewemultiplyanarbitrarypositivenumber,say3,by i.Theresultistheimaginarynumber3i.

Somultiplying by i produces a rotation counterclockwise by a quarter turn. Ittakesanarrowoflength3pointingeastandchangesitintoanewarrowofthesamelengthbutnowpointingnorth.Electrical engineers love complex numbers for exactly this reason. Having

such a compact way to represent 90-degree rotations is very useful whenworking with alternating currents and voltages, or with electric andmagneticfields,becausetheseofteninvolveoscillationsorwavesthatareaquartercycle(i.e.,90degrees)outofphase.In fact, complex numbers are indispensable to all engineers. In aerospace

engineeringtheyeasedthefirstcalculationsoftheliftonanairplanewing.Civiland mechanical engineers use them routinely to analyze the vibrations offootbridges,skyscrapers,andcarsdrivingonbumpyroads.

The90-degreerotationpropertyalsoshedslightonwhati²=–1reallymeans.If we multiply a positive number by i², the corresponding arrow rotates 180degrees,flippingfromeasttowest,becausethetwo90-degreerotations(oneforeachfactorofi)combinetomakea180-degreerotation.

Butmultiplyingby–1producestheverysame180-degreeflip.That’sthesenseinwhichi²=–1.Computers have breathed new life into complex numbers and the age-old

problemofrootfinding.Whenthey’renotbeingusedforWebsurfingore-mail,themachinesonourdeskscanrevealthingstheancientsneverdreamedof.In1976,myCornellcolleagueJohnHubbardbeganlookingat thedynamics

ofNewton’smethod,apowerfulalgorithmforfindingrootsofequationsinthecomplexplane.Themethodtakesastartingpoint(anapproximationtotheroot)anddoesacertaincomputationthatimprovesit.Bydoingthisrepeatedly,alwaysusingthepreviouspointtogenerateabetterone,themethodbootstrapsitswayforwardandrapidlyhomesinonaroot.Hubbardwas interested inproblemswithmultiple roots. In that case,which

root would themethod find?He proved that if therewere just two roots, thecloser one would always win. But if there were three or more roots, he wasbaffled.Hisearlierproofnolongerapplied.SoHubbarddidanexperiment.Anumericalexperiment.HeprogrammedacomputertorunNewton’smethod.Thenhetoldittocolor-

code millions of different starting points according to which root theyapproachedandtoshadethemaccordingtohowfasttheygotthere.Before he peeked at the results, he anticipated that the roots would most

quicklyattractthepointsnearbyandthusshouldappearasbrightspotsinasolidpatch of color.Butwhat about the boundaries between the patches?Those hecouldn’tpicture,atleastnotinhismind’seye.Thecomputer’sanswerwasastonishing.

Theborderlandslookedlikepsychedelichallucinations.Thecolorsintermingled

there in an almost impossibly promiscuous manner, touching each other atinfinitelymanypointsandalwaysinathree-way.Inotherwords,wherevertwocolorsmet,thethirdwouldalwaysinsertitselfandjointhem.Magnifyingtheboundariesrevealedpatternswithinpatterns.

Thestructurewasafractal—anintricateshapewhoseinnerstructurerepeatedatfinerandfinerscales.Furthermore, chaos reigned near the boundary. Two pointsmight start very

close together, bounce side by side for awhile, and then veer off to different

roots.Thewinningrootwasasunpredictableasthewinningnumberinagameofroulette.Littlethings—tiny,imperceptiblechangesintheinitialconditions—couldmakeallthedifference.Hubbard’sworkwasanearlyforayintowhat’snowcalledcomplexdynamics,

avibrantblendofchaostheory,complexanalysis,andfractalgeometry.Inawayitbroughtgeometrybacktoitsroots.In600B.C.amanualwritteninSanskritfortemple builders in India gave detailed geometric instructions for computingsquareroots,neededinthedesignofritualaltars.Morethan2,500yearslater,in1976, mathematicians were still searching for roots, but now the instructionswerewritteninbinarycode.Someimaginaryfriendsyouneveroutgrow.

9.MyTubRunnethOver

UNCLEIRVWASmydad’sbrotheraswellashispartnerinashoestoretheyownedin our town. He handled the business end of things andmostly stayed in theoffice upstairs, because hewas goodwith numbers and not so goodwith thecustomers.WhenIwasabouttenoreleven,UncleIrvgavememyfirstwordproblem.It

stickswithmetothisday,probablybecauseIgotitwrongandfeltembarrassed.Ithadtodowithfillingabathtub.Ifthecold-waterfaucetcanfillthetubina

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

I’mprettysure Iguessed forty-fiveminutes,asmanypeoplewould.Uncle Irvshookhisheadandgrinned.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.Over the years since then, I’ve thought about this bathtub problem many

times,alwayswithaffectionforbothUncleIrvandthequestionitself.Therearebroader lessons to be learned here—lessons about how to solve problemsapproximately when you can’t solve them exactly, and how to solve themintuitively,forthepleasureoftheAha!moment.Considermyinitialguessofforty-fiveminutes.Bylookingatanextreme,or

limiting, case,we can see that that answer can’t possiblybe right. In fact, it’sabsurd.Tounderstandwhy, suppose the hotwaterwasn’t turnedon.Then thecoldwater—on itsown—would fill the tub in thirtyminutes.Sowhatever theanswertoUncleIrv’squestionis,ithastobelessthanthis.Afterall,runningthehotwateralongwiththecoldcanonlyhelp.Admittedly,thisconclusionisnotasinformativeastheexactansweroftwenty

minutes we found by Uncle Irv’s method, but it has the advantage of notrequiringanycalculation.Adifferentwaytosimplifytheproblemistopretendthetwofaucetsrunatthe

samerate.Sayeachcanfillthetubinthirtyminutes(meaningthatthehotwaterrunsjustasfastasthecold).Thentheanswerwouldbeobvious.Becauseofthesymmetryofthenewsituation,thetwoperfectlymatchedfaucetswouldtogetherfillthetubinfifteenminutes,sinceeachdoeshalfthework.This instantly tellsus thatUncleIrv’sscenariomust take longer thanfifteen

minutes.Why? Because fast plus fast beats slow plus fast. Ourmake-believe

symmetricalproblemhastwofastfaucets,whereasUncleIrv’shasoneslow,onefast.Andsincefifteenminutesistheanswerwhenthey’rebothfast,UncleIrv’stubcanonlytakelonger.Theupshot is that by considering twohypothetical cases—onewith the hot

water off, and anotherwith itmatched to the coldwater—we learned that theanswer lies somewhere between fifteen and thirty minutes. In much harderproblemswhereitmaybeimpossibletofindanexactanswer—notjustinmathbut in other domains as well—this sort of partial information can be veryvaluable.Even ifwe’re luckyenough tocomeupwithanexactanswer, that’sstillno

causeforcomplacency.Theremaybeeasierorclearerwaystofindthesolution.Thisisoneplacewheremathallowsforcreativity.For example, insteadofUncle Irv’s textbookmethod,with its fractions and

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

singlebathtub,imaginetwoassemblylinesofthem,twoseparateconveyorbeltsshuttlingbathtubspastahot-waterfaucetononesideandacold-waterfaucetontheother.

Eachfaucetstands inplaceandfills itsowntubs—nomixingallowed.Andassoonasatubfillsup,itmovesondowntheline,makingwayforthenextone.Noweverythingbecomeseasy.Inonehour,thehot-waterfaucetfillsonetub,

whilethecold-waterfaucetfillstwo(sinceeachtakesahalf-hour).Thatamountstothreetubsperhour,oronetubeverytwentyminutes.Eureka!So why do so many people, including my childhood self, blunder into

guessing forty-five minutes? Why is it so tempting to split the difference ofthirtyandsixtyminutes?I’mnotsure,butitseemstobeacaseoffaultypatternrecognition.Maybe the bathtub problem is being conflatedwith others wheresplitting the difference would make sense. My wife explained it to me byanalogy. Imagineyou’rehelpinga littleold ladycross thestreet.Withoutyourhelp, itwould takehersixtyseconds,whileyou’dzipacross in thirtyseconds.

How long, then, would it take the two of you, walking arm in arm? Acompromisearoundforty-fivesecondsseemsplausiblebecausewhengrannyisclingingtoyourelbow,sheslowsyoudownandyouspeedherup.Thedifferencehereisthatyouandgrannyaffecteachother’sspeeds,butthe

faucets don’t. They’re independent. Apparently our subconsciousminds don’tspotthisdistinction,atleastnotwhenthey’releapingtothewrongconclusion.Thesilverliningisthatevenwronganswerscanbeeducational...aslongas

you realize they’rewrong.Theyexposemisguidedanalogiesandotherwoollythinking,andbringthecruxoftheproblemintosharperrelief.Otherclassicwordproblemsareexpresslydesigned to trick theirvictimsby

misdirection,likeamagician’ssleightofhand.Thephrasingofthequestionsetsatrap.Ifyouanswerbyinstinct,you’llprobablyfallforit.Try thisone.Suppose threemencanpaint three fences in threehours.How

longwouldittakeonemantopaintonefence?It’s tempting toblurt out “onehour.”Thewords themselvesnudgeyou that

way.Thedrumbeatinthefirstsentence—threemen,threefences,threehours—catches your attention by establishing a rhythm, so when the next sentencerepeatsthepatternwithoneman,onefence,____hours,it’shardtoresistfillingin the blank with “one.” The parallel construction suggests an answer that’slinguisticallyrightbutmathematicallywrong.Thecorrectansweristhreehours.If you visualize the problem—mentally picture three men painting three

fences and all finishing after threehours, just as theproblemstates—the rightanswerbecomesclear.Forallthreefencestobedoneafterthreehours,eachmanmusthavespentthreehoursonhis.

The undistracted reasoning that this problem requires is one of the mostvaluablethingsaboutwordproblems.Theyforceustopauseandthink,ofteninunfamiliarways.Theygiveuspracticeinbeingmindful.Perhapsevenmoreimportant,wordproblemsgiveuspracticeinthinkingnot

just about numbers, but about relationships between numbers—how the flowratesofthefaucetsaffectthetimerequiredtofillthetub,forexample.Andthatistheessentialnextstepinanyone’smatheducation.Understandably,alotofushave trouble with it; relationships are muchmore abstract than numbers. Butthey’re also much more powerful. They express the inner logic of the worldaround us. Cause and effect, supply and demand, input and output, dose andresponse—all involve pairs of numbers and the relationships between them.Wordproblemsinitiateusintothiswayofthinking.However, Keith Devlin raises an interesting criticism in his essay “The

problemwithwordproblems.”Hispointisthattheseproblemstypicallyassumeyouunderstand the rulesof thegameandagree toplayby them, even thoughthey’re often artificial, sometimes absurdly so. For example, in our problemaboutthreemenpaintingthreefencesinthreehours,itwasimplicitthat(1)allthreemenpaint at the same rate and (2) they all paint steadily, never slowingdown or speeding up. Both assumptions are unrealistic. You’re supposed toknownottoworryaboutthatandgoalongwiththegag,becauseotherwisetheproblemwouldbe toocomplicatedandyouwouldn’thaveenoughinformationtosolveit.You’dneedtoknowexactlyhowmucheachpainterslowsdownashegetstiredinthethirdhour,howofteneachonestopsforasnack,andsoon.Those of uswho teachmath should try to turn this bug into a feature.We

should be up front about the fact that word problems force us to makesimplifying assumptions. That’s a valuable skill—it’s called mathematicalmodeling.Scientistsdo itall the timewhen theyapplymath to therealworld.Butthey,unliketheauthorsofmostwordproblems,areusuallycarefultostatetheirassumptionsexplicitly.So thanks,Uncle Irv, for that first lesson.Humiliating?Yes.Unforgettable?

Yes,thattoo...butinagoodway.

10.WorkingYourQuads

THEQUADRATICFORMULAistheRodneyDangerfieldofalgebra.Eventhoughit’soneoftheall-timegreats,itdon’tgetnorespect.Professionals certainly aren’t enamored of it. When mathematicians and

physicistsareaskedto list thetoptenmostbeautifulor importantequationsofalltime,thequadraticformulanevermakesthecut.Ohsure,everybodyswoonsover1+1=2,andE=mc²,and thepert littlePythagorean theorem,struttinglike it’s all that just because a² + b² = c². But the quadratic formula? Not achance.Admittedly,it’sunsightly.Somestudentsprefertosounditout,treatingitasa

ritual incantation: “x equals negative b, plus or minus the square root of bsquaredminusfourac, allover twoa.”Othersmadeof sterner stuff look theformula straight in the face, confronting a hodgepodge of letters and symbolsmoreformidablethananythingthey’veencountereduptothatpoint:

It’sonlywhenyouunderstandwhatthequadraticformulaistryingtodothat

youcanbegintoappreciateitsinnerbeauty.InthischapterIhopetogiveyouafeeling for the clevernesspacked into thatporcupineof symbols, alongwithabettersenseofwhattheformulameansandwhereitarises.Therearemanysituationsinwhichwe’dliketofigureoutthevalueofsome

unknownnumber.What dose of radiation therapy should be given to shrink athyroidtumor?Howmuchmoneywouldyouhavetopayeachmonthtocoverathirty-yearmortgage of $200,000 at a fixed annual interest rate of 5 percent?HowfastdoesarockethavetogotoescapetheEarth’sgravity?Algebraistheplacewherewecutourteethonthesimplestproblemsofthis

type. The subject was developed by Islamic mathematicians around A.D. 800,buildingonearlierworkbyEgyptian,Babylonian,Greek,and Indianscholars.OnepracticalimpetusatthattimewasthechallengeofcalculatinginheritancesaccordingtoIslamiclaw.For example, suppose a widower dies and leaves his entire estate of 10

dirhams to his daughter and two sons. Islamic law requires that both the sonsmustreceiveequalshares.Moreover,eachsonmustreceivetwiceasmuchasthedaughter.Howmanydirhamswilleachheirreceive?Let’s use the letter x to denote the daughter’s inheritance. Even thoughwe

don’t know what x is yet, we can reason about it as if it were an ordinarynumber.Specifically,weknowthateachsongetstwiceasmuchasthedaughterdoes, so they each receive 2x. Thus, taken together, the amount that the threeheirsinheritisx+2x+2x,foratotalof5x,andthismustequalthetotalvalueoftheestate,10dirhams.Hence5x=10dirhams.Finally,bydividingbothsidesoftheequationby5,weseethatx=2dirhamsisthedaughter’sshare.Andsinceeachofthesonsinherits2x,theybothget4dirhams.Noticethattwotypesofnumbersappearedinthisproblem:knownnumbers,

like2,5,and10,andunknownnumbers, likex.Oncewemanaged toderivearelationship between the unknown and the known (as encapsulated in theequation5x=10),wewereabletochipawayattheequation,dividingbothsidesby5to isolate theunknownx. Itwasabit likeasculptorworking themarble,tryingtoreleasethestatuefromthestone.A slightly different tacticwould have beenneeded ifwehad encountered a

knownnumberbeingsubtractedfromanunknown,asinanequationlikex–2=5.Tofreexinthiscase,wewouldpareawaythe2byaddingittobothsidesoftheequation.Thisyields anunencumberedx on the left and5+2=7on theright.Thusx=7,whichyoumayhavealreadyrealizedbycommonsense.Although this tactic isnow familiar toall studentsof algebra, theymaynot

realizetheentiresubjectisnamedafterit.Intheearlypartoftheninthcentury,Muhammad ibn Musa al-Khwarizmi, a mathematician working in Baghdad,wroteaseminal textbook inwhichhehighlighted theusefulnessof restoringaquantity being subtracted (like 2, above) by adding it to the other side of anequation. He called this process al-jabr (Arabic for “restoring”), which latermorphed into “algebra.” Then, long after his death, he hit the etymologicaljackpot again. His own name, al-Khwarizmi, lives on today in the word“algorithm.”Inhistextbook,beforewadingintotheintricaciesofcalculatinginheritances,

al-Khwarizmi considered a more complicated class of equations that embodyrelationshipsamongthreekindsofnumbers,notthemeretwoconsideredabove.Alongwithknownnumbersandanunknown(x), theseequationsalsoincludedthesquareof theunknown(x²).Theyarenowcalledquadraticequations,fromtheLatinquadratus,for“square.”AncientscholarsinBabylonia,Egypt,Greece,China, and India had already tackled such brainteasers, which often arose inarchitectural or geometrical problems involving areas or proportions, and had

shownhowtosolvesomeofthem.Anexamplediscussedbyal-Khwarizmiis

Inhisday,however,suchproblemswereposedinwords,notsymbols.Heasked:“What must be the square which, when increased by ten of its own roots,amountstothirty-nine?”(Here,theterm“root”referstotheunknownx.)Thisproblemismuchtougherthanthetwoweconsideredabove.Howcanwe

isolatex now?The tricks used earlier are insufficient, because the x² and 10xtermstendtosteponeachother’stoes.Evenifyoumanagetoisolatexinoneofthem,theotherremainstroublesome.Forinstance,ifwedividebothsidesoftheequationby10,the10xsimplifiestox,whichiswhatwewant,but thenthex²becomesx²/10,whichbringsusnoclosertofindingxitself.Thebasicobstacle,in a nutshell, is thatwe have to do two things at once, and they seem almostincompatible.Thesolutionthatal-Khwarizmipresentsisworthdelvingintoinsomedetail,

firstbecause it’ssoslick,andsecondbecause it’ssopowerful—itallowsus tosolveallquadraticequationsinasinglestroke.BythatImeanthatiftheknownnumbers10and39abovewerechangedtoanyothernumbers,themethodwouldstillwork.Theideaistointerpreteachofthetermsintheequationgeometrically.Think

ofthefirstterm,x²,astheareaofasquarewithdimensionsxbyx.

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

Attach the two new rectangles onto the square to produce a notched shape ofareax²+10x:

Viewedinthislight,al-Khwarizmi’spuzzleamountstoasking:Ifthenotched

shapeoccupies39squareunitsofarea,howlargewouldxhavetobe?

Thepictureitselfsuggestsanalmostirresistiblenextstep.Lookatthatmissingcorner. If only it were filled in, the notched shape would turn into a perfectsquare.Solet’stakethehintandcompletethesquare.

Supplyingthemissing5×5squareadds25squareunitstotheexistingareaofx²+ 10x, for a total of x² + 10x + 25. Equivalently, that combined area can beexpressedmoreneatlyas(x+5)²,sincethecompletedsquareisx+5unitslongoneachside.Theupshot is thatx²and10xarenowmovinggracefullyasacouple, rather

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

x² + 10x = 39, we must also add 25 to the right side, to keep the equation

balanced.Since39+25=64,ourequationthenbecomes

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

(giving9)andthenadd10times3(giving30),thesumis39,asdesired.There’s only one snag. If al-Khwarizmi were taking algebra today, he

wouldn’treceivefullcreditfor thisanswer.Hefails tomentionthatanegativenumber,x=–13,alsoworks.Squaringitgives169;addingittentimesgives–130;andtheytooaddupto39.Butthisnegativesolutionwasignoredinancienttimes,sinceasquarewithasideofnegativelengthisgeometricallymeaningless.Today, algebra is less beholden to geometry and we regard the positive andnegativesolutionsasequallyvalid.Inthecenturiesafteral-Khwarizmi,scholarscametorealizethatallquadratic

equationscouldbesolvedinthesameway,bycompletingthesquare—aslongasonewaswillingtoallowthenegativenumbers(andtheirbewilderingsquareroots)thatoftencameupintheanswers.Thislineofargumentrevealedthatthesolutionstoanyquadraticequation

(wherea,b,andcareknownbutarbitrarynumbers,andxisunknown)couldbeexpressedbythequadraticformula,

What’s so remarkable about this formula is how brutally explicit andcomprehensiveitis.There’stheanswer,rightthere,nomatterwhata,b,andchappen to be. Considering that there are infinitely many possible choices foreachofthem,that’salotforasingleformulatomanage.Inourowntime, thequadratic formulahasbecomean irreplaceable tool for

practicalapplications.Engineersandscientistsuse it toanalyzethetuningofa

radio, the swaying of a footbridge or a skyscraper, the arc of a baseball or acannonball,theupsanddownsofanimalpopulations,andcountlessotherreal-worldphenomena.Foraformulabornofthemathematicsofinheritance,that’squitealegacy.

11.PowerTools

IFYOUWEREanavidtelevisionwatcherinthe1980s,youmayrememberaclevershow calledMoonlighting. Known for its snappy dialogue and the romanticchemistrybetween its costars, it featuredCybillShepherdandBruceWillis asMaddieHayesandDavidAddison,acoupleofwisecrackingprivatedetectives.

While investigating one particularly tough case, David asks a coroner’s

assistant for his best guess about possible suspects. “Beats me,” says the

assistant. “But you know what I don’t understand?” David replies,“Logarithms?”Then,reactingtoMaddie’slook:“What?Youunderstoodthose?”That pretty well sums up how many people feel about logarithms. Their

peculiar name is just part of their image problem.Most folks never use themagain after high school, at least not consciously, and are oblivious to thelogarithmshidingbehindthescenesoftheirdailylives.Thesame is trueofmanyof theother functionsdiscussed in algebra II and

precalculus.Powerfunctions,exponential functions—whatwas thepointofallthat?Mygoalinthischapteristohelpyouappreciatethefunctionofall thosefunctions, even if you never have occasion to press their buttons on yourcalculator.A mathematician needs functions for the same reason that a builder needs

hammers and drills. Tools transform things. So do functions. In fact,mathematicians often refer to them as transformations because of this. Butinstead of wood and steel, the materials that functions pound away on arenumbersandshapesand,sometimes,evenotherfunctions.ToshowyouwhatImean,let’splotthegraphoftheequationy=4–x².You

mayrememberhowthissortofactivitygoes:Youdrawapictureofthexyplanewith thex-axis running horizontally and the y-axis vertically. Then for each xyoucomputethecorrespondingyandplotthemtogetherasasinglepointinthexyplane.Forexample,whenxis1,theequationsaysy=4–1²,whichis4–1,or3.So(x,y)=(1,3)isapointonthegraph.Afteryoucalculateandplotafewmorepoints,thefollowingpictureemerges.

Thebowedshapeofthecurveisduetotheactionofmathematicalpliers.In

theequationfory, the function that transformsx intox² behaves a lot like thecommontoolforbendingandpullingthings.Whenit’sappliedtoeverypointonapieceofthex-axis(whichyoucouldvisualizeasastraightpieceofwire),thepliers bend and elongate that piece into the downward-curving arch shownabove.Andwhatroledoesthe4playintheequationy=4–x²?Itactslikeanailfor

hanging a picture on awall. It lifts up the bentwire arch by 4 units. Since itraisesallpointsbythesameamount,it’sknownasaconstantfunction.Thisexampleillustratesthedualnatureoffunctions.Ontheonehand,they’re

tools:thex²bendsthepieceofthex-axis,andthe4lifts it.Ontheotherhand,they’rebuildingblocks:the4andthe–x²canberegardedascomponentpartsofamorecomplicatedfunction,4–x², justaswires,batteries,andtransistorsarecomponentpartsofaradio.Onceyoustarttolookatthingsthisway,you’llnoticefunctionseverywhere.

Thearchingcurveabove—technicallyknownasaparabola—isthesignatureofthe squaring functionx² operating behind the scenes.Look for itwhen you’retakingasipfromawaterfountainorwatchingabasketballarctowardthehoop.And if you ever have a few minutes to spare on a layover in Detroit’sinternationalairport,besuretostopbythewaterfeatureintheDeltaterminaltoenjoytheworld’smostbreathtakingparabolasatplay.

Parabolasandconstantsareassociatedwithawiderclassoffunctions—power

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

firstpower(n=1)givesafunction thatworks likearamp,asteadyinclineofgrowthordecay.It’scalledalinearfunctionbecauseitsxygraphisaline.Ifyouleave a bucket out in a steady rain, the water collecting at the bottom riseslinearlyintime.Anotheruseful toolis theinversesquarefunction,1/x²,corresponding to the

casen=–2.(Thepowerbecomes–2becausethefunctionisaninversesquare;the x² appears in the denominator.) This function is good for describing howwavesandforcesattenuateastheyspreadoutinthreedimensions—forinstance,howasoundsoftensasitmovesawayfromitssource.Powerfunctionslikethesearethebuildingblocksthatscientistsandengineers

usetodescribegrowthanddecayintheirmildestforms.But when you need mathematical dynamite, it’s time to unpack the

exponentialfunctions.Theydescribeallsortsofexplosivegrowth,fromnuclearchainreactionstotheproliferationofbacteriainapetridish.Themostfamiliarexampleisthefunction10ˣ,inwhich10israisedtothepowerx.Makesurenottoconfusethiswiththeearlierpowerfunctions.Heretheexponent(thepowerx)is avariable, and thebase (thenumber10) is a constant—whereas inapower

functionlikex²,it’stheotherwayaround.Thisswitchmakesahugedifference:asxgetslargerandlarger,anexponentialfunctionofxeventuallygrowsfasterthananypowerfunction,nomatterhowlargethepower.Exponentialgrowthisalmostunimaginablyrapid.That’swhy it’s so hard to fold a piece of paper in halfmore than seven or

eight times. Each folding approximately doubles the thickness of the wad,causing it to grow exponentially.Meanwhile, thewad’s length shrinks in halfevery time, and thus decreases exponentially fast. For a standard sheet ofnotebookpaper,aftersevenfoldsthewadbecomesthickerthanitislong,soitcan’tbefoldedagain.Itdoesn’tmatterhowstrongthepersondoingthefoldingis.Forasheettobeconsideredlegitimatelyfoldedntimes,theresultingwadisrequiredtohave2ⁿ layersinastraight line,andthiscan’thappenif thewadisthickerthanitislong.The challenge was thought to be impossible until Britney Gallivan, then a

juniorinhighschool,solveditin2002.Shebeganbyderivingaformula

thatpredictedthemaximumnumberoftimes,n,thatpaperofagiventhicknessTandlengthLcouldbefoldedinonedirection.Noticetheforbiddingpresenceoftheexponentialfunction2ⁿintwoplaces—oncetoaccountforthedoublingofthewad’sthicknessateachfold,andanothertimetoaccountforthehalvingofitslength.Usingherformula,Britneyconcludedthatshewouldneedtouseaspecialroll

oftoiletpapernearlythree-quartersofamilelong.Sheboughtthepaper,andinJanuary 2002, she went to a shopping mall in her hometown of Pomona,California, andunrolled thepaper.Sevenhours later, andwith thehelpof herparents,shesmashedtheworldrecordbyfoldingthepaperinhalftwelvetimes!Intheory,exponentialgrowthisalsosupposedtograceyourbankaccount.If

yourmoneygrowsatanannualinterestrateofr,afteroneyearitwillbeworth(1+r)timesyouroriginaldeposit;aftertwoyears,(1+r)²;andafterxyears,(1+ r)ˣ times your initial deposit. Thus themiracle of compounding that we sooftenhearaboutiscausedbyexponentialgrowthinaction.Whichbringsusbacktologarithms.Weneedthembecauseit’susefultohave

toolsthatcanundotheactionsofothertools.Justaseveryofficeworkerneedsboth a stapler and a staple remover, every mathematician needs exponential

functions and logarithms. They’re inverses. This means that if you type anumberxintoyourcalculatorandthenpunchthe10ˣbuttonfollowedbythelogxbutton,you’llgetbacktothenumberyoustartedwith.Forexample,ifx=2,then10ˣwouldbe10²,whichequals100.Takingthelogofthatthenbringstheresult back to 2; the log button undoes the action of the 10ˣ button. Hencelog(100)equals2.Likewise,log(1,000)=3andlog(10,000)=4,because1,000=103and10,000=104.Notice somethingmagical here: as the numbers inside the logarithms grew

multiplicatively,increasingtenfoldeachtimefrom100to1,000to10,000,theirlogarithmsgrewadditively, increasing from2 to 3 to 4.Our brains perform asimilartrickwhenwelistentomusic.Thefrequenciesofthenotesinascale—do,re,mi,fa,sol, la, ti,do—soundtouslikethey’rerisinginequalsteps.Butobjectively their vibrational frequencies are rising by equal multiples. Weperceivepitchlogarithmically.In every place where they arise, from the Richter scale for earthquake

magnitudestopHmeasuresofacidity,logarithmsmakewonderfulcompressors.They’re ideal for taking quantities that vary over awide range and squeezingthem together so they become more manageable. For instance, 100 and 100million differ amillionfold, a gulf thatmost of us find incomprehensible.Buttheir logarithmsdiffer only fourfold (they are2 and8, because100=10² and100million=108). In conversation,we all use a crudeversionof logarithmicshorthandwhenwerefertoanysalarybetween$100,000and$999,999asbeingsix figures.That“six” is roughly the logarithmof thesesalaries,which in factspantherangefromfivetosix.As impressive as all these functionsmaybe, amathematician’s toolbox can

onlydosomuch—whichiswhyIstillhaven’tassembledmyIkeabookcases.

PartThreeSHAPES

12.SquareDancing

IBETICANguessyourfavoritemathsubjectinhighschool.Itwasgeometry.So many people I’ve met over the years have expressed affection for that

subject. Is it because geometry draws on the right side of the brain, and thatappealstovisualthinkerswhomightotherwisecringeatitscoldlogic?Maybe.Butsomepeopletellmetheylovedgeometrypreciselybecauseitwassological.The step-by-step reasoning, with each new theorem resting firmly on thosealreadyestablished—that’sthesourceofsatisfactionformany.Butmybest hunch (and, full disclosure, I personally lovegeometry) is that

peopleenjoyitbecauseitmarries logicandintuition.Itfeelsgoodtousebothhalvesofthebrain.Toillustratethepleasuresofgeometry,let’srevisitthePythagoreantheorem,

whichyouprobablyrememberasa²+b²=c².Partofthegoalhereistoseewhyit’s trueandappreciatewhyitmatters.Beyondthat,byprovingthetheoremintwodifferentways,we’llcometoseehowoneproofcanbemoreelegantthananother,eventhoughbotharecorrect.The Pythagorean theorem is concernedwith right triangles—meaning those

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

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

youmightwanttoknowhowfaritisfromonecornertothediagonallyoppositecorner.(Bytheway,thisiswheregeometrystarted,historically—inproblemsofland measurement, or measuring the earth: geo = “earth,” and metry =“measurement.”)ThePythagoreantheoremtellsyouhowlongthediagonaliscomparedtothe

sidesof the rectangle. If one sidehas lengtha and theotherhas lengthb, thetheoremsaysthediagonalhaslengthc,where

For some reason, thediagonal is traditionally called thehypotenuse, though

I’venevermetanyonewhoknowswhy.(AnyLatinorGreekscholars?)Itmusthavesomethingtodowith thediagonalsubtendingarightangle,butas jargongoes,“subtending”isaboutasopaqueas“hypotenuse.”Anyway,here’showthetheoremworks.Tokeepthenumberssimple,let’ssay

a=3yardsandb=4yards.Thentofigureouttheunknownlengthc,wedonourblackhoodsandintonethatc²isthesumof3²plus4²,whichis9plus16.(Keep inmind that all of these quantities are nowmeasured in square yards,sincewesquaredtheyardsaswellasthenumbersthemselves.)Finally,since9+16=25,wegetc²=25squareyards,andthentakingsquarerootsofbothsidesyieldsc=5yardsasthelengthofthehypotenuse.This way of looking at the Pythagorean theorem makes it seem like a

statement about lengths. But traditionally it was viewed as a statement aboutareas.Thatbecomesclearerwhenyouhearhowtheyusedtosayit:The squareon thehypotenuse is the sumof the squareson theother twosides.

Notice the word “on.”We’re not speaking of the square of the hypotenuse—that’sanewfangledalgebraicconceptaboutmultiplyinganumber(thelengthofthehypotenuse)byitself.No,we’rereferringheretoasquareliterallysittingonthehypotenuse,likethis:

Let’s call this the large square, to distinguish it from the small and mediumsquareswecanbuildontheothertwosides:

Thenthetheoremsaysthatthelargesquarehasthesameareaasthesmallandmediumsquarescombined.Forthousandsofyears,thismarvelousfacthasbeenexpressedinadiagram,

aniconicmnemonicofdancingsquares:

Viewingthetheoremintermsofareasmakesitalotmorefuntothinkabout.

For example, you can test it—and then eat it—by building the squares out ofmany little crackers. Or you can treat the theorem like a child’s puzzle, withpiecesofdifferentshapesandsizes.Byrearrangingthesepuzzlepieces,wecanprovethetheoremverysimply,asfollows.Let’sgobacktothetiltedsquaresittingonthehypotenuse.

Ataninstinctivelevel,youshouldfeelabitdisquietedbythisimage.Thesquarelooks potentially unstable, like it might topple or slide down the ramp. Andthere’s also an unpleasant arbitrariness about which of the four sides of thesquaregetstotouchthetriangle.Guidedbytheseintuitivefeelings,let’saddthreemorecopiesofthetriangle

aroundthesquaretomakeamoresolidandsymmetricalpicture:

Nowrecallwhatwe’retryingtoprove:thatthetiltedwhitesquareinthepictureabove(whichis justourearlier largesquare—it’sstillsittingright thereonthehypotenuse) has the same area as the small andmedium squares put together.Butwherearethoseothersquares?Well,wehavetoshiftsometrianglesaroundtofindthem.Thinkof thepictureaboveasdepictingapuzzle,withfour triangularpieces

wedgedintothecornersofarigidpuzzleframe.

In this interpretation, the tilted square is the empty space in themiddle of thepuzzle.Therestoftheareainsidetheframeisoccupiedbythepuzzlepieces.Nowlet’s trymoving thepiecesaround invariousways.Ofcourse,nothing

wedocaneverchange the totalamountofemptyspace inside the frame—it’salwayswhateverarealiesoutsidethepieces.Thebrainstorm,then,istorearrangethepieceslikethis:

Allofasuddentheemptyspacehaschangedintothetwoshapeswe’relookingfor—thesmallsquareandthemediumsquare.Andsincethetotalareaofemptyspacealwaysstaysthesame,we’vejustproventhePythagoreantheorem!Thisproofdoesfarmore thanconvince; it illuminates.That’swhatmakes it

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

simplestproofthatavoidsusingareas.As before, consider a right triangle with sides of length a and b and

hypotenuseoflengthc,asshownbelowontheleft.

Now,bydivine inspirationor a strokeofgenius, something tells us todrawalinesegmentperpendicular to thehypotenuseanddownto theoppositecorner,asshowninthetriangleontheright.Thiscleverlittleconstructioncreatestwosmallertrianglesinsidetheoriginal

one. It’s easy to prove that all these triangles are similar—whichmeans theyhaveidenticalshapesbutdifferentsizes.Thatinturnimpliesthatthelengthsoftheir corresponding parts have the same proportions,which translates into thefollowingsetofequations:

Wealsoknowthat

because our constructionmerely split the original hypotenuse of length c intotwosmallersidesoflengthsdande.Atthispointyoumightbefeelingabitlost,oratleastunsureofwhattodo

next.There’samorassoffiveequationsabove,andwe’retryingtowhittlethemdowntodeducethat

Tryitforafewminutes.You’lldiscoverthattwooftheequationsareirrelevant.That’s ugly; an elegant proof should involve nothing superfluous. Withhindsight,ofcourse,youwouldn’thavelistedthoseequationstobeginwith.Butthatwouldjustbeputtinglipstickonap...(themissingwordhereis“proof”).Nevertheless, by manipulating the right three equations, you can get the

theoremtopopout.Seethenoteson[>]forthemissingsteps.Wouldyouagreewithmethat,onaestheticgrounds, thisproofis inferiorto

the first one? For one thing, it drags near the end. And who invited all thatalgebratotheparty?Thisissupposedtobeageometryevent.Butamoreseriousdefect is theproof’smurkiness.Bythetimeyou’redone

slogging through it, youmight believe the theorem (grudgingly), but you stillmightnotseewhyit’strue.Leaving proofs aside, why does the Pythagorean theorem even matter?

Becauseitrevealsafundamentaltruthaboutthenatureofspace.Itimpliesthatspace is flat, as opposed to curved. For the surface of a globe or a bagel, forexample,thetheoremneedstobemodified.Einsteinconfrontedthischallengeinhisgeneraltheoryofrelativity(wheregravityisnolongerviewedasaforce,but

rather as a manifestation of the curvature of space), and so did BernhardRiemannandothersbeforehimwhen laying the foundationsofnon-Euclideangeometry.It’salongroadfromPythagorastoEinstein.Butatleastit’sastraightline...

formostoftheway.

13.SomethingfromNothing

EVERY MATH COURSE contains at least one notoriously difficult topic. Inarithmetic, it’s longdivision. Inalgebra, it’swordproblems.And ingeometry,it’sproofs.Most students who take geometry have never seen a proof before. The

experiencecancomeasashock,soperhapsawarninglabelwouldbeinorder,something like this:Proofs can cause dizziness or excessive drowsiness. Sideeffects of prolonged exposuremay includenight sweats, panicattacks, and, inrarecases,euphoria.Askyourdoctorifproofsarerightforyou.Disorienting as proofs can be, learning to do them has long been thought

essential to a liberal education—more essential than the subject matter itself,somewouldsay.Accordingtothisview,geometryisgoodforthemind;ittrainsyou to think clearly and logically. It’s not the study of triangles, circles, andparallellinespersethatmatters.What’simportantistheaxiomaticmethod,theprocessofbuildingarigorousargument,stepbystep,untiladesiredconclusionhasbeenestablished.Euclid laid down this deductive approach in the Elements (now the most

reprinted textbook of all time) about 2,300 years ago. Ever since, Euclideangeometry has been a model for logical reasoning in all walks of life, fromscience and law to philosophy and politics. For example, Isaac Newtonchanneled Euclid in the structure of his masterwork The MathematicalPrinciples of Natural Philosophy. Using geometrical proofs, he deducedGalileo’sandKepler’s lawsaboutprojectilesandplanets fromhisowndeeperlawsofmotionandgravity.Spinoza’sEthics follows the samepattern. Its fulltitle is Ethica Ordine Geometrico Demonstrata (Ethics Demonstrated inGeometricalOrder).YoucanevenhearechoesofEuclid in theDeclarationofIndependence.WhenThomasJeffersonwrote,“Weholdthesetruthstobeself-evident,”hewasmimickingthestyleoftheElements.Euclidhadbegunwiththedefinitions, postulates, and self-evident truths of geometry—the axioms—andfrom them erected an edifice of propositions and demonstrations, one truthlinked to thenextbyunassailable logic. Jeffersonorganized theDeclaration inthe sameway so that its radical conclusion, that the colonies had the right togovernthemselves,wouldseemasinevitableasafactofgeometry.If that intellectual legacy seems far-fetched, keep in mind that Jefferson

reveredEuclid.A fewyearsafterhe finishedhis second termaspresidentand

steppedoutofpubliclife,hewrotetohisoldfriendJohnAdamsonJanuary12,1812, about the pleasures of leaving politics behind: “I have given upnewspapersinexchangeforTacitusandThucydides,forNewtonandEuclid;andIfindmyselfmuchthehappier.”Still, what’s missing in all this worship of Euclidean rationality is an

appreciation of geometry’smore intuitive aspects.Without inspiration, there’dbenoproofs—ortheoremstoproveinthefirstplace.Likecomposingmusicorwritingpoetry,geometryrequiresmakingsomethingfromnothing.Howdoesapoetfindtherightwords,oracomposerahauntingmelody?Thisisthemysteryofthemuse,andit’snolessmysteriousinmaththanintheothercreativearts.Asanillustration,considertheproblemofconstructinganequilateraltriangle

—atrianglewithallthreesidesthesamelength.Therulesofthegamearethatyou’regivenonesideofthetriangle,thelinesegmentshownhere:

Yourtaskistousethatsegment,somehow,toconstructtheothertwosides,andtoprove that theyeachhave thesame lengthas theoriginal.Theonly toolsatyour disposal are a straightedge and a compass.A straightedge allows you todraw a straight line of any length, or to draw a straight line between any twopoints.A compass allows you to draw a circle of any radius, centered at anypoint.Keepinmind,however,thatastraightedgeisnotaruler.Ithasnomarkings

onitandcan’tbeusedtomeasurelengths.(Specifically,youcan’tuseittocopyormeasuretheoriginalsegment.)Norcanacompassserveasaprotractor;allitcandoisdrawcircles,notmeasureangles.Ready?Begin.Thisisthemomentofparalysis.Wheretostart?Logic won’t help you here. Skilled problem solvers know that a better

approach is to relax and playwith the puzzle, hoping to get a feel for it. Forinstance,maybeitwouldhelptousethestraightedgetodrawtiltedlinesthroughtheendsofthesegment,likethis:

No luck. Although the lines form a triangle, there’s no guarantee it’s anequilateraltriangle.Anotherstabinthedarkwouldbetodrawsomecircleswiththecompass.But

where?Aroundoneoftheendpoints?

Oraroundapointinsidethesegment?

That second choice looks unpromising, since there’s no reason to pick oneinteriorpointoveranother.Solet’sgobacktodrawingcirclesaroundendpoints.

Unfortunatelythere’sstillalotofarbitrarinesshere.Howbigshouldthecirclesbe?Nothing’spoppingoutatusyet.After a few more minutes of flailing around like this, frustration (and an

impendingheadache)may tempt us to giveup.But ifwedon’t,wemight getluckyandrealizethere’sonlyonenaturalcircletodraw.Let’sseewhathappensifweputthesharppointofthecompassatoneendofthesegmentandthepencilattheother,andthentwirlthecompassthroughafullcircle.We’dgetthis:

Ofcourse,ifwe’dusedtheotherendpointasthecenter,we’dgetthis:

How about drawing both circles at the same time—for no good reason, justnoodling?

Diditjusthityou?Ashiverofinsight?Lookatthediagramagain.There’sa

curvyversionofanequilateral trianglestaringatus. Its topcorner is thepointwherethecirclesintersect.

Sonowlet’s turn that intoa real triangle,withstraightsides,bydrawing linesfrom the intersection point to the endpoints of the original segment. Theresultingtrianglesurelooksequilateral.

Havingallowedintuitiontoguideusthisfar,nowandonlynowisittimefor

logic to take over and finish the proof. For clarity, let’s pan back to the fulldiagramandlabelthepointsofinterestA,B,andC.

Theproofalmostprovesitself.ThesidesACandBChavethesamelengthastheoriginalsegmentAB,sincethat’showweconstructedthecircles;weusedABastheradiusforboth.SinceACandBCarealsoradii,theytoohavethislength,soallthreelengthsareequal,andthetriangleisequilateral.QED.This argument has been around for centuries. In fact, it’s Euclid’s opening

shot—the first proposition in Book I of the Elements. But the tendency hasalwaysbeentopresentthefinaldiagramwiththeartfulcirclesalreadyinplace,which robs the student of the joy of discovering them. That’s a pedagogical

mistake.Thisisaproofthatanyonecanfind.Itcanbenewwitheachgeneration,ifweteachitright.Thekeytothisproof,clearly,wastheinspiredconstructionofthetwocircles.

A more famous result in geometry can be proven by a similarly deftconstruction.It’sthetheoremthattheanglesofatriangleaddupto180degrees.Inthiscase,thebestproofisnotEuclid’sbutanearlieroneattributedtothe

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

Drawalineparalleltothebase,goingthroughthetopcorner.

Nowweneedtodigressforasecondtorecallapropertyofparallel lines: if

anotherlinecutsacrosstwoparallellineslikeso,

theangleslabeledhereasa(knowninthetradeasalternateinteriorangles)areequal.Let’sapplythisfacttotheconstructionaboveinwhichwedrewalinethrough

thetopcornerofatriangleparalleltoitsbase.

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

aboltoflightning,theconstructionoftheparallelline.Oncethatlinehasbeendrawn, the proof practically rises off the table and walks by itself, like Dr.Frankenstein’screation.And who knows? If we highlight this other side of geometry—its playful,

intuitiveside,whereasparkofimaginationcanbequicklyfannedintoaproof—maybe someday all studentswill remember geometry as the classwhere theylearnedtobelogicalandcreative.

14.TheConicConspiracy

WHISPERING GALLERIES ARE remarkable acoustic spaces found beneath certaindomes,vaults,orcurvedceilings.AfamousoneislocatedoutsidetheOysterBarrestaurant inNewYorkCity’sGrandCentralStation. It’sa funplace to takeadate:thetwoofyoucanexchangesweetnothingswhileyou’refortyfeetapartandseparatedbyabustlingpassageway.You’llheareachotherclearly,but thepassersbywon’thearawordyou’resaying.

To produce this effect, the two of you should stand at diagonally opposite

cornersofthespace,facingthewall.Thatputsyoueachnearafocus,aspecialpoint at which the sound of your voice gets focused as it reflects off thepassageway’scurvedwallsandceiling.Ordinarily,thesoundwavesyouproducetravel in all directions and bounce off thewalls at disparate times and places,scramblingthemsomuchthattheyareinaudiblewhentheyarriveattheearofa

listener forty feet away (which is why the passersby can’t hear what you’resaying).Butwhenyouwhisperata focus, the reflectedwavesallarriveat thesame time at the other focus, thus reinforcing one another and allowing yourwordstobeheard.Ellipsesdisplayasimilarflairforfocusing,thoughinamuchsimplerform.If

wefashiona reflector in theshapeofanellipse, twoparticularpoints inside it(markedasF1andF2inthefigurebelow)willactasfociinthefollowingsense:alltheraysemanatingfromalightsourceatoneofthosepointswillbereflectedtotheother.

Letmetrytoconveyhowamazingthatisbyrestatingitinafewways.SupposethatDarthandLukeenjoyplayinglasertaginanellipticalarenawith

mirroredwalls.Bothhaveagreednottoaimdirectlyattheother—theyhavetozap each other with bank shots. Darth, not being particularly astute aboutgeometryoroptics,suggeststheyeachstandatafocalpoint.“Fine,”saysLuke,“as long as I get to take the first shot.”Well, itwouldn’t bemuch of a duel,becauseLukecan’tmiss!Nomatterhowfoolishlyheaimshislaser,he’llalwaystagDarth.Everyshot’sawinner.Orifpoolisyourgame,imagineplayingbilliardsonanellipticaltablewitha

pocket at one focus.To set up a trick shot that is guaranteed to scratch everytime,placethecueballattheotherfocus.Nomatterhowyoustriketheballandnomatterwhereitcaromsoffthecushion,it’llalwaysgointothepocket.

Parabolic curves and surfaces have an impressive focusing power of their

own: each can takeparallel incomingwaves and focus themat a singlepoint.This feature of their geometry has been very useful in settings where lightwaves, sound waves, or other signals need to be amplified. For instance,parabolic microphones can be used to pick up hushed conversations and arethereforeofinterestinsurveillance,espionage,andlawenforcement.Theyalsocomeinhandyinnaturerecording,tocapturebirdsongsoranimalgrunts,andintelevisedsports,tolisteninonacoachcursingatanofficial.Parabolicantennascanamplifyradiowavesinthesameway,whichiswhysatellitedishesforTVreceptionandgiant radio telescopes forastronomyhave theircharacteristicallycurvedshapes.Thisfocusingpropertyofparabolasisjustasusefulwhendeployedinreverse.

Supposeyouwanttomakeastronglydirectionalbeamoflight,likethatneededfor spotlights or car headlights. On its own, a bulb—even a powerful one—typicallywouldn’tbegoodenough;itwastestoomuchlightbyshiningitinalldirections. But now place that bulb at the focus of a parabolic reflector, andvoilà!Theparabolacreatesadirectionalbeamautomatically.Ittakesthebulb’srays and, by reflecting them off the parabola’s silvered inner surface, makesthemallparallel.

Onceyoubegintoappreciatethefocusingabilitiesofparabolasandellipses,

youcan’thelpbutwonderifsomethingdeeperisatworkhere.Arethesecurvesrelatedinsomemorefundamentalway?Mathematicians and conspiracy theorists have thismuch in common:we’re

suspiciousofcoincidences—especiallyconvenientones.Therearenoaccidents.Things happen for a reason.While thismindsetmay be just a touch paranoidwhenapplied to real life, it’s aperfectly saneway to thinkaboutmath. In theidealworldofnumbersandshapes,strangecoincidencesusuallyareclues thatwe’remissingsomething.Theysuggestthepresenceofhiddenforcesatwork.So let’s look more deeply into the possible links between parabolas and

ellipses.Atfirstglancetheyseemanunlikelycouple.Parabolasarearch-shapedand expansive, stretching out at both ends. Ellipses are oval-shaped, likesquashedcircles,closedandcompact.

But as soon as you move beyond their appearances and probe their inneranatomy,youstart tonoticehowsimilar theyare.Theybothbelong toa royal

familyofcurves, agenetic tie thatbecomesobvious—onceyouknowwhat tolookfor.Toexplainhowthey’rerelated,wehavetorecallwhat,exactly,thesecurves

are.A parabola is commonly defined as the set of all points equidistant from a

given point and a given line not containing that point. That’s amouthful of adefinition, but it’s actually pretty easy to understandonceyou translate it intopictures.CallthegivenpointF,for“focus,”andcallthelineL.

Now,accordingtothedefinition,aparabolaconsistsofallthepointsthatliejustasfarfromFas theydofromL.Forexample, thepointP lyingstraightdownfromF,halfwaytoL,wouldcertainlyqualify:

InfinitelymanyotherpointsP1,P2,...worktoo.Theylieofftoeithersidelikethis:

Here,thepointP1liesatthesamedistance,d1,fromthelineasitdoesfromthefocus.ThesameistrueforthepointP2,exceptnowtheshareddistanceissomeother number,d2. Taken together, all the pointsPwith this property form theparabola.ThereasonforcallingFthefocusbecomesclearifwethinkoftheparabolaas

acurvedmirror.Itturnsout(thoughIwon’tproveit)thatifyoushineabeamoflight straight at a parabolic mirror, all the reflected rays will intersectsimultaneouslyatthepointF,producinganintenselyfocusedspotoflight.

Itworkssomethinglikethoseoldsuntanningreflectorsthatfriedagenerationoffaces,backinthedaysbeforeanybodyworriedaboutskincancer.Nowlet’sturntothecorrespondingstoryforellipses.Anellipseisdefinedas

thesetofpointsthesumofwhosedistancesfromtwogivenpointsisaconstant.When rephrased in more down-to-earth language, this description provides arecipe fordrawing an ellipse.Get apencil, a sheet ofpaper, a corkboard, twopushpins,andapieceofstring.Laythepaperontheboard.Pintheendsofthestringdownthroughthepaper,beingcarefultoleavesomeslack.Thenpullthestring tautwith the pencil to form an angle, as shown below. Begin drawing,keeping thestring taut.After thepencilhasgoneall thewayaroundbothpinsandreturnedtoitsstartingpoint,theresultingcurveisanellipse.

Noticehowthisrecipeimplementsthedefinitionabove,wordforword.The

pinsplay the roleof the twogivenpoints.And the sumof thedistances fromthemtoapointonthecurvealwaysremainsconstant,nomatterwherethepencilis—becausethosedistancesalwaysadduptothelengthofthestring.Sowherearetheellipse’sfociinthisconstruction?Atthepins.Again,Iwon’t

proveit,butthosearethepointsthatallowLukeandDarthtoplaytheirgameofcan’t-misslasertagandthatgiverisetoascratch-every-timepooltable.Parabolasandellipses:Whyisitthatthey,andonlythey,havesuchfantastic

powersoffocusing?What’sthesecrettheyshare?They’rebothcross-sectionsofthesurfaceofacone.A cone? If you feel like that just came out of nowhere, that’s precisely the

point.Thecone’sroleinallthishasbeenhiddensofar.To see how it’s implicated, imagine slicing through a cone with a meat

cleaver,somewhatlikecuttingthroughasalami,atprogressivelysteeperangles.Ifthecutislevel,thecurveofintersectionisacircle.

If instead the cone is sliced on a gentle bias, the resulting curve becomes anellipse.

As the cuts become steeper, the ellipse gets longer and slimmer in itsproportions.Atacriticalangle,when thebiasgetssosteep that itmatches theslopeoftheconeitself,theellipseturnsintoaparabola.

So that’s the secret: a parabola is an ellipse in disguise, in a certain limitingsense. No wonder it shares the ellipse’s marvelous focusing ability. It’s beenpasseddownthroughthebloodline.Infact,circles,ellipses,andparabolasareallmembersofa larger, tight-knit

family. They’re collectively known as conic sections—curves obtained bycuttingthesurfaceofaconewithaplane.Plusthere’soneadditionalsibling:iftheconeisslicedverysteeply,onabiasgreaterthanitsownslope,theresultingincisionformsacurvecalledahyperbola.Unlikealltheothers,itcomesintwopieces,notone.

Thesefourtypesofcurvesappearevenmoreintimatelyrelatedwhenviewed

from other mathematical perspectives. In algebra, they arise as the graphs ofsecond-degreeequations

wheretheconstantsA,B,C,...determinewhetherthegraphisacircle,ellipse,parabola,orhyperbola.Incalculus,theyariseastrajectoriesofobjectstuggedbytheforceofgravity.So it’s no accident that planetsmove in elliptical orbitswith the sun at one

focus; or that comets sail through the solar system on elliptic, parabolic, orhyperbolic trajectories; or that a child’s ball tossed to a parent follows aparabolicarc.They’reallmanifestationsoftheconicconspiracy.Focusonthatthenexttimeyouplaycatch.

15.SineQuaNon

MYDADHADafriendnamedDavewhoretiredtoJupiter,Florida.WevisitedhimforafamilyvacationwhenIwasabouttwelveorso,andsomethingheshowedusmadeanindelibleimpressiononme.Dave liked to chart the times of the glorious sunrises and sunsets he could

watchfromhisdeckallyearlong.Everydayhemarkedtwodotsonhischart,and aftermanymonths he noticed something curious. The two curves lookedlike opposingwaves.One usually rosewhile the other fell;when sunrisewasearlier,sunsetwaslater.

But there were exceptions. For the last three weeks of June and for most ofDecemberandearlyJanuary,sunriseandsunsetbothcamelatereachday,givingthewavesaslightlylopsidedappearance.Still,themessageofthecurveswasunmistakable:theoscillatinggapbetween

them showed the days growing longer and shorter with the changing of theseasons.Bysubtractingthelowercurvefromtheupperone,Davealsofigured

out how the number of hours of daylight varied throughout the year. To hisamazement,thiscurvewasn’tlopsidedatall.Itwasbeautifullysymmetrical.

Whathewaslookingatwasanearlyperfectsinewave.Youmayremember

havingheardaboutsinewavesifyoutooktrigonometryinhighschool,althoughyour teacherprobably talkedmore about the sine function, a fundamental toolforquantifyinghowthesidesandanglesofatrianglerelatetooneanother.Thatwastrigonometry’soriginalkillerapp,ofgreatutilitytoancientastronomersandsurveyors.Yettrigonometry,belyingitsmuchtoomodestname,nowgoesfarbeyondthe

measurementof triangles.Byquantifyingcirclesaswell, ithaspaved thewayfortheanalysisofanythingthatrepeats,fromoceanwavestobrainwaves.It’sthekeytothemathematicsofcycles.To see how trig connects circles, triangles, and waves, imagine a little girl

goingroundandroundonaFerriswheel.

Sheandhermom,whobothhappentobemathematicallyinclined,havedecidedthisisaperfectopportunityforanexperiment.ThegirltakesaGPSgadgetwithherontheridetorecordheraltitude,momentbymoment,asthewheelcarriesher up, then thrillingly over the top and back toward the ground, then up andaroundagain,andsoon.Theresultslooklikethis:

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

supposeweexamineasnapshotofthegirl.Atthemomentshown,she’satsome

angle,callita,relativetothedashedlineinthediagram.

For convenience, let’s suppose the hypotenuse of the right triangle shown—whichisalsothewheel’sradius—is1unitlong.Thensina(pronounced“sineofa”) tells us howhigh the girl is.More precisely, sina is defined as the girl’saltitudemeasuredfromthecenterofthewheelwhenshe’slocatedattheanglea.As she goes round and round, her angle a will progressively increase.

Eventuallyitexceeds90degrees,atwhichpointwecannolongerregardaasanangleinarighttriangle.Doesthatmeantrignolongerapplies?No.Undeterredasusual,mathematicianssimplyenlargethedefinitionofthe

sinefunctiontoallowforanyangle,notjustthoselessthan90degrees,andthendefine“sina”as thegirl’sheightaboveorbelow thecenterof thecircle.Thecorrespondinggraphofsina,asakeepsincreasing(orevengoesnegative,ifthewheelreversesdirection),iswhatwemeanbyasinewave.Itrepeatsitselfeverytimeachangesby360degrees,correspondingtoafullrevolution.Thissamesortofconversionofcircularmotionintosinewavesisapervasive,

thoughoftenunnoticed,partofourdailyexperience. It creates thehumof thefluorescent lights overhead in our offices, a reminder that somewhere in thepowergrid,generatorsarespinningatsixtycyclespersecond,convertingtheirrotarymotionintoalternatingcurrent,theelectricalsinewavesonwhichmodernlifedepends.WhenyouspeakandIhear,bothourbodiesareusingsinewaves—yours in thevibrationsofyourvocalcords toproducethesounds,andmineintheswayingofthehaircellsinmyearstoreceivethem.Ifweopenourheartsto

thesesinewavesandtunein to theirsilent thrumming, theyhavethepowertomoveus.There’ssomethingalmostspiritualaboutthem.Whenaguitarstringispluckedorwhenchildrenjiggleajumprope,theshape

thatappearsisasinewave.Theripplesonapond,theridgesofsanddunes,thestripesofa zebra—all aremanifestationsofnature’smostbasicmechanismofpatternformation: theemergenceof sinusoidal structure fromabackgroundofblanduniformity.

There are deepmathematical reasons for this.Whenever a state of featurelessequilibrium loses stability—for whatever reason, and by whatever physical,biological,orchemicalprocess—thepatternthatappearsfirstisasinewave,oracombinationofthem.Sine waves are the atoms of structure. They’re nature’s building blocks.

Without them there’dbenothing,givingnewmeaning to thephrase“sinequanon.”Infact,thewordsareliterallytrue.Quantummechanicsdescribesrealatoms,

andhenceallofmatter,aspacketsofsinewaves.Evenatthecosmologicalscale,sine waves form the seeds of all that exists. Astronomers have probed thespectrum(thepatternofsinewaves)of thecosmicmicrowavebackgroundandfoundthattheirmeasurementsmatchthepredictionsofinflationarycosmology,theleadingtheoryforthebirthandgrowthoftheuniverse.Soitseemsthatoutof a featureless Big Bang, primordial sine waves—ripples in the density ofmatter and energy—emerged spontaneously and spawned the stuff of the

cosmos.Stars.Galaxies.And,ultimately,littlekidsridingFerriswheels.

16.TakeIttotheLimit

INMIDDLESCHOOLmyfriendsandIenjoyedchewingontheclassicconundrums.What happenswhen an irresistible forcemeets an immovable object? Easy—theybothexplode.Philosophy’strivialwhenyou’rethirteen.Butonepuzzlebotheredus:Ifyoukeepmovinghalfwaytothewall,willyou

evergetthere?Somethingaboutthisonewasdeeplyfrustrating,thethoughtofgetting closer and closer and yet never quite making it. (There’s probably ametaphorforteenageangstintheresomewhere.)Anotherconcernwasthethinlyveiled presence of infinity. To reach the wall you’d need to take an infinitenumberofsteps,andbytheendthey’dbecomeinfinitesimallysmall.Whoa.Questionslikethishavealwayscausedheadaches.Around500B.C.,Zenoof

Elea posed four paradoxes about infinity that puzzled his contemporaries andthatmayhavebeenpartlytoblameforinfinity’sbanishmentfrommathematicsfor centuries thereafter. In Euclidean geometry, for example, the onlyconstructions allowed were those that involved a finite number of steps. Theinfinitewas considered too ineffable, too unfathomable, and too hard tomakelogicallyrigorous.ButArchimedes, thegreatestmathematicianof antiquity, realized thepower

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

But for now I’d like to beginwith the first beautiful hints of them, visible inancientcalculationsaboutcirclesandpi.Let’srecallwhatwemeanby“pi.”It’saratiooftwodistances.Oneofthemis

thediameter, thedistanceacross the circle through its center.Theother is thecircumference,thedistancearound thecircle.“Pi” isdefinedas their ratio, thecircumferencedividedbythediameter.

If you’re a careful thinker, youmight be worried about something already.

Howdoweknowthatpiisthesamenumberforallcircles?Coulditbedifferentfor big circles and little circles? The answer is no, but the proof isn’t trivial.Here’sanintuitiveargument.Imagineusingaphotocopiertoreduceanimageofacircleby,say,50percent.

Thenalldistancesinthepicture—includingthecircumferenceandthediameter—would shrink in proportion by 50 percent. So when you divide the newcircumference by the new diameter, that 50 percent changewould cancel out,leavingtheratiobetweenthemunaltered.Thatratioispi.Ofcourse,thisdoesn’ttellushowbigpiis.Simpleexperimentswithstrings

anddishesaregoodenoughtoyieldavaluenear3,orifyou’remoremeticulous,. But supposewewant to find pi exactly or at least approximate it to any

desired accuracy. What then? This was the problem that confounded theancients.Before turning to Archimedes’s brilliant solution, we should mention one

otherplacewherepiappearsinconnectionwithcircles.Theareaofacircle(theamountofspaceinsideit)isgivenbytheformula

HereA is the area, π is the Greek letter pi, and r is the radius of the circle,definedashalfthediameter.

Allofusmemorizedthisformulainhighschool,butwheredoesitcomefrom?It’snotusuallyproven ingeometryclass. Ifyouwenton to takecalculus,youprobably saw a proof of it there, but is it really necessary to use calculus toobtainsomethingsobasic?Yes,itis.Whatmakestheproblemdifficultisthatcirclesareround.Iftheyweremade

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

madeupoflotsoflittlestraightpieces.That’snotreallytrue,butitworks...aslong as you take it to the limit and imagine infinitely many pieces, eachinfinitesimallysmall.That’sthecrucialideabehindallofcalculus.Here’soneway touse it to find theareaofacircle.Beginbychopping the

areaintofourequalquarters,andrearrangethemlikeso.

The strange scalloped shape on the bottom has the same area as the circle,thoughthatmightseemprettyuninformativesincewedon’tknowitsareaeither.Butat leastweknowtwoimportant factsabout it.First, the twoarcsalong itsbottomhaveacombined lengthequal tohalf thecircumferenceof theoriginalcircle (because theotherhalfof thecircumference isaccountedforby the twoarcsontop).Sincethewholecircumferenceispitimesthediameter,halfofitispitimeshalfthediameteror,equivalently,pitimestheradiusr.That’swhythediagramaboveshowsπrasthecombinedlengthofthearcsatthebottomofthescallopedshape.Second,thestraightsidesofthesliceshavealengthofr,sinceeachofthemwasoriginallyaradiusofthecircle.Next,repeattheprocess,butthistimewitheightslices,stackedalternatelyas

before.

The scalloped shape looks a bit less bizarre now.The arcs on the top and thebottomarestillthere,butthey’renotaspronounced.Anotherimprovementistheleft and right sides of the scalloped shape don’t tilt asmuch as they used to.Despite these changes, the two facts above continue to hold: the arcs on thebottomstillhaveanetlengthofπrandeachsidestillhasalengthofr.Andofcourse the scalloped shape still has the same area as before—the area of thecirclewe’reseeking—sinceit’sjustarearrangementofthecircle’seightslices.As we take more and more slices, something marvelous happens: the

scalloped shape approaches a rectangle. The arcs become flatter and the sidesbecomealmostvertical.

Inthelimitofinfinitelymanyslices,theshapeisarectangle.Justasbefore,thetwofactsstillhold,whichmeansthisrectanglehasabottomofwidthπrandasideofheightr.

Butnowtheproblemiseasy.Theareaofarectangleequals itswidthtimesitsheight, somultiplying πr times r yields an area of πr² for the rectangle. Andsince the rearranged shape always has the same area as the circle, that’s theanswerforthecircletoo!What’s so charming about this calculation is the way infinity comes to the

rescue.Ateveryfinitestage,thescallopedshapelooksweirdandunpromising.Butwhenyoutakeittothelimit—whenyoufinallygettothewall—itbecomessimpleandbeautiful,andeverythingbecomesclear.That’showcalculusworks

atitsbest.Archimedes used a similar strategy to approximate pi. He replaced a circle

withapolygonwithmanystraightsidesandthenkeptdoublingthenumberofsides to get closer to perfect roundness. But rather than settling for anapproximation of uncertain accuracy, he methodically bounded pi bysandwichingthecirclebetweeninscribedandcircumscribedpolygons,asshownbelowfor6-,12-,and24-sidedfigures.

ThenheusedthePythagoreantheoremtoworkouttheperimetersoftheseinnerandouterpolygons,startingwith thehexagonandbootstrappinghiswayup to12,24,48,and,ultimately,96sides.Theresultsforthe96-gonsenabledhimtoprovethat

Indecimalnotation (whichArchimedesdidn’thave), thismeanspi isbetween3.1408and3.1429.This approach is known as themethod of exhaustion because of theway it

trapstheunknownnumberpibetweentwoknownnumbersthatsqueezeitfromeitherside.Theboundstightenwitheachdoubling, thusexhaustingthewiggleroomforpi.Inthelimitofinfinitelymanysides,boththeupperandlowerboundswould

convergetopi.Unfortunately,thislimitisn’tassimpleastheearlierone,wherethescallopedshapemorphedintoarectangle.Sopiremainsaselusiveasever.We can discover more and more of its digits—the current record is over 2.7

trilliondecimalplaces—butwe’llneverknowitcompletely.Aside from laying the groundwork for calculus, Archimedes taught us the

power of approximation and iteration.He bootstrapped a good estimate into abetterone,usingmoreandmorestraightpiecestoapproximateacurvedobjectwithincreasingaccuracy.Morethantwomillennialater,thisstrategymaturedintothemodernfieldof

numerical analysis. When engineers use computers to design cars that areoptimallystreamlined,orwhenbiophysicistssimulatehowanewchemotherapydrug latches onto a cancer cell, they are using numerical analysis. Themathematicians and computer scientistswho pioneered this field have createdhighlyefficient,repetitivealgorithms,runningbillionsoftimespersecond,thatenablecomputerstosolveproblemsineveryaspectofmodernlife,frombiotechtoWall Street to the Internet. In each case, the strategy is to find a series ofapproximationsthatconvergetothecorrectanswerasalimit.Andthere’snolimittowherethat’lltakeus.

PartFourCHANGE

17.ChangeWeCanBelieveIn

LONGBEFORE I knewwhat calculuswas, I sensed therewas something specialaboutit.Mydadhadspokenaboutitinreverentialtones.Hehadn’tbeenabletogo to college, being a child of theDepression, but somewhere along the line,maybeduringhistimeintheSouthPacificrepairingB-24bomberengines,he’dgottenafeelforwhatcalculuscoulddo.Imagineamechanicallycontrolledbankofantiaircraftgunsautomaticallyfiringatan incomingfighterplane.Calculus,hesupposed,couldbeusedtotellthegunswheretoaim.Every year about a million American students take calculus. But far fewer

really understand what the subject is about or could tell you why they werelearning it. It’s not their fault.There are somany techniques tomaster and somanynewideastoabsorbthattheoverallframeworkiseasytomiss.Calculusisthemathematicsofchange.Itdescribeseverythingfromthespread

of epidemics to the zigs and zags of a well-thrown curveball. The subject isgargantuan—andsoareitstextbooks.Manyexceedathousandpagesandworknicelyasdoorstops.But within that bulk you’ll find two ideas shining through. All the rest, as

RabbiHillelsaidofthegoldenrule,isjustcommentary.Thosetwoideasarethederivativeand the integral.Eachdominates itsownhalfof thesubject,named,respectively,differentialandintegralcalculus.Roughly speaking, the derivative tells you how fast something is changing;

the integral tellsyouhowmuch it’s accumulating.Theywereborn in separatetimesandplaces:integralsinGreecearound250B.C.;derivativesinEnglandandGermany in the mid-1600s. Yet in a twist straight out of a Dickens novel,they’veturnedouttobebloodrelatives—thoughittookalmosttwomillenniaforanyonetoseethefamilyresemblance.The next chapter will explore that astonishing connection, as well as the

meaningofintegrals.Butfirst,tolaythegroundwork,let’slookatderivatives.Derivatives are all aroundus, even ifwedon’t recognize themas such.For

example,theslopeofarampisaderivative.Likeallderivatives,itmeasuresarateofchange—inthiscase,howfaryou’regoingupordownforeverystepyoutake.Asteepramphasalargederivative.Awheelchair-accessibleramp,withitsgentlegradient,hasasmallderivative.Everyfieldhas itsownversionofaderivative.Whether itgoesbymarginal

returnorgrowth rateorvelocityor slope, aderivativebyanyothernamestill

smellsassweet.Unfortunately,manystudentsseemtocomeawayfromcalculuswith a much narrower interpretation, regarding the derivative as synonymouswiththeslopeofacurve.Their confusion is understandable. It’s caused by our reliance on graphs to

express quantitative relationships.Byplottingy versusx to visualize howonevariableaffectsanother,all scientists translate theirproblems into thecommonlanguageofmathematics.Therateofchangethatreallyconcernsthem—aviralgrowth rate, a jet’s velocity, orwhatever—then gets converted into somethingmuchmoreabstractbuteasiertopicture:aslopeonagraph.Likeslopes,derivativescanbepositive,negative,orzero,indicatingwhether

something is rising, falling, or leveling off. Consider Michael Jordan flyingthroughtheairbeforemakingoneofhisthunderousdunks.

Justafterliftoff,hisverticalvelocity(therateatwhichhiselevationchangesintime and, thus, another derivative) is positive, because he’s going up. Hiselevationisincreasing.Onthewaydown,thisderivativeisnegative.Andatthehighest point of his jump,where he seems to hang in the air, his elevation ismomentarily unchanging and his derivative is zero. In that sense he truly ishanging.There’samoregeneralprincipleatworkhere—thingsalwayschangeslowest

at the top or the bottom. It’s especially noticeable here in Ithaca. During thedarkest depths ofwinter, the days are not just unmercifully short; they barely

improve from one to the next.Whereas once spring starts popping, the dayslengthen rapidly. All of this makes sense. Change is most sluggish at theextremes precisely because the derivative is zero there. Things stand still,momentarily.Thiszero-derivativepropertyofpeaksandtroughsunderliessomeofthemost

practical applications of calculus. It allows us to use derivatives to figure outwhere a function reaches its maximum or minimum, an issue that ariseswheneverwe’relookingforthebestorcheapestorfastestwaytodosomething.Myhigh-school calculus teacher,Mr. Joffray, had a knack formaking such

“max-min” questions come alive. One day he came bounding into class andbegan telling us about his hike through a snow-covered field. The wind hadapparentlyblownalotofsnowacrosspartofthefield,blanketingitheavilyandforcing him to walkmuchmore slowly there, while the rest of the field wasclear, allowing him to stride through it easily. In a situation like that, hewonderedwhatpathahikershouldtaketogetfrompointAtopointBasquicklyaspossible.

Onethoughtwouldbetotrudgestraightacrossthedeepsnow,tocutdownon

theslowestpartofthehike.Thedownside,though,istherestofthetripwilltakelongerthanitwouldotherwise.

Another strategy is to head straight fromA toB. That’s certainly the shortestdistance,butitdoescostextratimeinthemostarduouspartofthetrip.

With differential calculus you can find the best path. It’s a certain specificcompromisebetweenthetwopathsconsideredabove.

Theanalysisinvolvesfourmainsteps.(Forthosewho’dliketoseethedetails,referencesaregiveninthenoteson[>].)First, notice that the total time of travel—which is what we’re trying to

minimize—dependsonwherethehikeremergesfromthesnow.Hecouldchoosetoemergeanywhere,so let’sconsiderallhispossibleexitpointsasavariable.Each of these locations can be characterized succinctly by specifying a singlenumber:thedistancexwherethehikeremergesfromthesnow.

(Implicitly,thetraveltimealsodependsonthelocationsofAandBandonthehiker’sspeedsinbothpartsofthefield,butallthoseparametersaregiven.Theonlythingunderthehiker’scontrolisx.)Second,givenachoiceofxand theknownlocationsof thestartingpointA

andthedestinationB,wecancalculatehowmuchtimethehikerspendswalkingthrough the fast and slow parts of the field. For each leg of the trip, thiscalculation requires the Pythagorean theorem and the old algebra mantra“distanceequalsrate times time.”Adding the timesforboth legs together thenyieldsaformulaforthetotaltraveltime,T,asafunctionofx.Third,wegraphTversusx.Thebottomofthecurveisthepointwe’reseeking

—itcorrespondstotheleasttimeoftravelandhencethefastesttrip.

Fourth, to find this lowest point, we invoke the zero-derivative principle

mentionedabove.WecalculatethederivativeofT,setitequaltozero,andsolveforx.These four steps require a command of geometry, algebra, and various

derivative formulas from calculus—skills equivalent to fluency in a foreignlanguageand,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

MATHEMATICALSIGNSAND symbolsareoftencryptic,but thebestof themoffervisualcluestotheirownmeaning.Thesymbolsforzero,one,andinfinityaptlyresemble an emptyhole, a singlemark, andan endless loop:0, 1,∞.And theequal sign,=, is formedby twoparallel linesbecause, as its originator,Welshmathematician Robert Recorde, wrote in 1557, “no two things can be moreequal.”Incalculusthemostrecognizableiconistheintegralsign:

Its graceful lines are evocativeof amusical clef or a violin’s f-hole—a fittingcoincidence,giventhatsomeofthemostenchantingharmoniesinmathematicsareexpressedbyintegrals.ButtherealreasonthatthemathematicianGottfriedLeibnizchosethissymbolismuchlesspoetic.It’ssimplyalong-neckedS, for“summation.”As for what’s being summed, that depends on context. In astronomy, the

gravitationalpullofthesunontheEarthisdescribedbyanintegral.ItrepresentsthecollectiveeffectofalltheminusculeforcesgeneratedbyeachsolaratomattheirvaryingdistancesfromtheEarth.Inoncology,thegrowingmassofasolidtumor can bemodeled by an integral. So can the cumulative amount of drugadministeredduringthecourseofachemotherapyregimen.To understand why sums like these require integral calculus and not the

ordinary kind of addition we learned in grade school, let’s consider whatchallengeswe’dfaceifweactuallytriedtocalculatethesun’sgravitationalpullontheEarth.Thefirstdifficultyisthatthesunisnotapoint...andneitheristheEarth.Both of them are gigantic ballsmade up of stupendous numbers ofatoms. Every atom in the sun exerts a gravitational tug on every atom in theEarth. Of course, since atoms are tiny, their mutual attractions are almostinfinitesimally small, yet because there are almost infinitelymany of them, inaggregatetheycanstillamounttosomething.Somehowwehavetoaddthemallup.

Butthere’sasecondandmoreseriousdifficulty:Thosepullsaredifferentfordifferent pairs of atoms. Some are stronger than others. Why? Because thestrengthofgravitychangeswithdistance—thecloser two thingsare, themorestronglytheyattract.TheatomsonthefarsidesofthesunandtheEarthfeeltheleastattraction;thoseonthenearsidesfeelthestrongest;andthoseinbetweenfeel forces of middling strength. Integral calculus is needed to sum all thosechangingforces.Amazingly,itcanbedone—atleastintheidealizedlimitwherewe treat the Earth and the sun as solid spheres composed of infinitely manypoints of continuous matter, each exerting an infinitesimal attraction on theothers.Asinallofcalculus:infinityandlimitstotherescue!Historically,integralsarosefirstingeometry,inconnectionwiththeproblem

of finding the areas of curved shapes.Aswe saw in chapter 16, the area of acirclecanbeviewedasthesumofmanythinpieslices.Inthelimitofinfinitelymany slices, each of which is infinitesimally thin, those slices could then becunninglyrearrangedintoarectanglewhoseareawasmucheasiertofind.Thatwasa typicaluseof integrals.They’reall about takingsomethingcomplicatedandslicinganddicingittomakeiteasiertoaddup.Ina3-Dgeneralizationofthismethod,Archimedes(andbeforehim,Eudoxus,

around400B.C.)calculatedthevolumesofvarioussolidshapesbyreimaginingthemasstacksofmanywafersordisks,likeasalamislicedthin.Bycomputingthe changing volumes of the varying slices and then ingeniously integratingthem—addingthembacktogether—theywereabletodeducethevolumeoftheoriginalwhole.Today we still ask budding mathematicians and scientists to sharpen their

skills at integration by applying them to these classic geometry problems.They’resomeofthehardestexercisesweassign,andalotofstudentshatethem,butthere’snosurerwaytohonethefacilitywithintegralsneededforadvancedworkineveryquantitativedisciplinefromphysicstofinance.One such mind-bender concerns the volume of the solid common to two

identicalcylinderscrossingatrightangles,likestovepipesinakitchen.

Ittakesanunusualgiftofimaginationtovisualizethisthree-dimensionalshape.Sothere’snoshameinadmittingdefeatandlookingforawaytomakeitmorepalpable. To do so, you can resort to a trickmy high-school calculus teacherused—takeatincanandcutthetopoffwithmetalshearstoformacylindricalcoring tool.Then core a large Idahopotatoor a pieceofStyrofoam from twomutuallyperpendiculardirections.Inspecttheresultingshapeatyourleisure.Computergraphicsnowmakeitpossibletovisualizethisshapemoreeasily.

Remarkably,ithassquarecross-sections,eventhoughitwascreatedfromroundcylinders.

It’sastackofinfinitelymanylayers,eachawafer-thinsquare, taperingfromalarge square in themiddle to progressively smaller ones and finally to singlepointsatthetopandbottom.Still,picturingtheshapeismerelythefirststep.Whatremainsistodetermine

its volume, by tallying the volumes of all the separate slices. Archimedesmanaged todo this, but onlybyvirtueof his astounding ingenuity.Heused amechanicalmethodbasedonleversandcentersofgravity,ineffectweighingtheshape in his mind by balancing it against others he already understood. Thedownsideofhisapproach,besidestheprohibitivebrillianceitrequired,wasthatitappliedtoonlyahandfulofshapes.Conceptualroadblockslikethisstumpedtheworld’sfinestmathematiciansfor

thenextnineteencenturies...untilthemid-1600s,whenJamesGregory,IsaacBarrow,IsaacNewton,andGottfriedLeibnizestablishedwhat’snowknownasthefundamentaltheoremofcalculus.Itforgedapowerfullinkbetweenthetwotypesofchangebeingstudiedincalculus:thecumulativechangerepresentedbyintegrals,andthelocalrateofchangerepresentedbyderivatives(thesubjectofchapter 17). By exposing this connection, the fundamental theorem greatlyexpanded the universe of integrals that could be solved, and it reduced theircalculationtogruntwork.Nowadayscomputerscanbeprogrammedtouseit—andsocanstudents.Withitshelp,eventhestovepipeproblemthatwasonceaworld-class challenge becomes an exercise within common reach. (For thedetails of Archimedes’s approach as well as the modern one, consult thereferencesinthenoteson[>].)Beforecalculusand the fundamental theoremcamealong,only the simplest

kindsofnetchangecouldbeanalyzed.Whensomethingchangessteadily,ataconstantrate,algebraworksbeautifully.This is thedomainof“distanceequalsratetimestime.”Forexample,acarmovingatanunchangingspeedof60milesperhourwillsurelytravel60milesinthefirsthour,and120milesbytheendofthesecondhour.

But what about change that proceeds at a changing rate? Such changingchangeisallaroundus—intheacceleratingdescentofapennydroppedfromatall building, in the ebb and flow of the tides, in the elliptical orbits of theplanets, in the circadian rhythms within us. Only calculus can cope with thecumulativeeffectsofchangesasnon-uniformasthese.For nearly two millennia after Archimedes, just one method existed for

predictingtheneteffectofchangingchange:addupthevaryingslices,bitbybit.Youweresupposedtotreattherateofchangeasconstantwithineachslice,theninvoketheanalogof“distanceequalsratetimestime”toinchforwardtotheendof thatslice,andrepeatuntilall theslicesweredealtwith.Mostof the timeitcouldn’tbedone.Theinfinitesumsweretoohard.Thefundamental theoremenableda lotof theseproblems tobesolved—not

all of them, but many more than before. It often gave a shortcut for solvingintegrals, at least for the elementary functions (sums and products of powers,exponentials, logarithms, and trig functions) that describe so many of thephenomenainthenaturalworld.Here’sananalogythat Ihopewillshedsomelightonwhat thefundamental

theoremsaysandwhyit’ssohelpful.(MycolleagueCharliePeskinatNewYorkUniversitysuggestedit.)Imagineastaircase.Thetotalchangeinheightfromthetoptothebottomisthesumoftherisesofallthestepsinbetween.That’strueevenifsomeof thestepsrisemore thanothersandnomatterhowmanystepsthereare.Thefundamentaltheoremofcalculussayssomethingsimilarforfunctions—if

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

asoccurswheneveryou’recalculatinganintegralbyslices.Ifyoucansomehowmanagetofindthecorrespondingstaircase—inotherwords,ifyoucanfindanelevation function forwhich those numbers are the rises—then computing theintegralisasnap.It’sjustthetopminusthebottom.That’sthegreatspeedupmadepossiblebythefundamentaltheorem.Andit’s

whywetortureallbeginningcalculusstudentswithmonthsoflearninghowtofindelevationfunctions,technicallycalledantiderivativesorindefiniteintegrals.This advance allowed mathematicians to forecast events in a changing worldwithmuchgreaterprecisionthanhadeverbeenpossible.Fromthisperspective,thelastinglegacyofintegralcalculusisaVeg-O-Matic

view of the universe. Newton and his successors discovered that nature itselfunfolds in slices.Virtually all the lawsof physics found in thepast 300yearsturnedouttohavethischaracter,whethertheydescribethemotionsofparticlesor theflowofheat,electricity,air,orwater.Togetherwith thegoverning laws,the conditions in each slice of time or space determine what will happen inadjacentslices.The implications were profound. For the first time in history, rational

predictionbecamepossible...notjustonesliceatatimebut,withthehelpofthefundamentaltheorem,byleapsandbounds.Sowe’re longoverdue toupdateoursloganfor integrals—from“Itslices, it

dices”to“Recalculating.Abetterrouteisavailable.”

19.Allaboute

AFEWNUMBERSARESUCHCELEBRITIESthattheygobysingle-letterstagenames,somethingnot evenMadonnaorPrince canmatch.Themost famous isπ, thenumberformerlyknownas3.14159...Closebehindisi,theit-numberofalgebra,theimaginarynumbersoradicalit

changedwhatitmeanttobeanumber.NextontheAlist?Say hello to e.Nicknamed for its breakout role in exponential growth, e is

now the Zelig of advanced mathematics. It pops up everywhere, peeking outfromthecornersofthestage, teasingusbyitspresenceinincongruousplaces.For example, along with the insights it offers about chain reactions andpopulation booms, e has a thing or two to say about how many people youshoulddatebeforesettlingdown.Butbeforeweget to that,what ise,exactly?Itsnumericalvalue is2.71828

...butthat’snotterriblyenlightening.Icouldtellyouthateequalsthelimitingnumberapproachedbythesum

as we take more and more terms. But that’s not particularly helpful either.Instead,let’slookateinaction.Imaginethatyou’vedeposited$1,000inasavingsaccountatabankthatpays

anincrediblygenerousinterestrateof100percent,compoundedannually.Ayearlater,youraccountwouldbeworth$2,000—theinitialdepositof$1,000plusthe100percentinterestonit,equaltoanother$1,000.Knowing a sucker when you see one, you ask the bank for even more

favorable terms: How would they feel about compounding the interestsemiannually,meaningthatthey’dbepayingonly50percentinterestforthefirstsixmonths, followedbyanother50percent for the second sixmonths?You’dclearlydobetterthanbefore—sinceyou’dgaininterestontheinterest—buthowmuchbetter?Theansweristhatyourinitial$1,000wouldgrowbyafactorof1.50overthe

first half of the year, and thenby another factor of 1.50over the secondhalf.Andsince1.50times1.50is2.25,yourmoneywouldamounttoacool$2,250afteroneyear,substantiallymorethanthe$2,000yougotfromtheoriginaldeal.

Whatifyoupushedevenharderandpersuadedthebanktodividetheyearintomore and more periods—daily, by the second, or even by the nanosecond?Wouldyoumakeasmallfortune?Tomakethenumberscomeoutnicely,here’stheresultforayeardividedinto

100equalperiods,aftereachofwhichyou’dbepaid1percentinterest(the100percent annual rate divided evenly into 100 installments): yourmoney wouldgrowby a factor of 1.01 raised to the 100th power, and that comes out to beabout 2.70481. In other words, instead of $2,000 or $2,250, you’d have$2,704.81.Finally, theultimate: if the interestwascompounded infinitelyoften—this is

calledcontinuouscompounding—yourtotalafteroneyearwouldbebiggerstill,but not bymuch: $2,718.28. The exact answer is $1,000 times e, where e isdefinedasthelimitingnumberarisingfromthisprocess:

This is a quintessential calculus argument.Aswe discussed in the last few

chapters when we calculated the area of a circle or pondered the sun’sgravitationalpullontheEarth,whatdistinguishescalculusfromtheearlierpartsof math is its willingness to confront—and harness—the awesome power ofinfinity.Whether we’re looking at limits, derivatives, or integrals, we alwayshavetosidleuptoinfinityinonewayoranother.In the limiting process that led to e above,we imagined slicing a year into

moreandmorecompoundingperiods,windowsoftimethatbecamethinnerandthinner,ultimatelyapproachingwhat canonlybedescribedas infinitelymany,infinitesimally thinwindows. (Thismight soundparadoxical, but it’s reallynoworsethantreatingacircleasthelimitofaregularpolygonwithmoreandmoresides,eachofwhichgetsshorterandshorter.)The fascinating thing is that themoreoftentheinterestiscompounded,thelessyourmoneygrowsduringeachperiod. Yet it still amounts to something substantial after a year, because it’sbeenmultipliedoversomanyperiods!This is a clue to the ubiquity of e. It often ariseswhen something changes

throughthecumulativeeffectofmanytinyevents.Consider a lump of uranium undergoing radioactive decay. Moment by

moment,everyatomhasacertainsmallchanceofdisintegrating.Whetherandwhen each one does is completely unpredictable, and each such event has an

infinitesimal effect on the whole. Nevertheless, in ensemble these trillions ofevents produce a smooth, predictable, exponentially decaying level ofradioactivity.Or think about the world’s population, which grows approximately

exponentially.Allaroundtheworld,childrenarebeingbornatrandomtimesandplaces, while other people are dying, also at random times and places. Eacheventhasaminusculeimpact,percentagewise,ontheworld’soverallpopulation—yetinaggregatethatpopulationgrowsexponentiallyataverypredictablerate.Another recipe for e combines randomness with enormous numbers of

choices. Let me give you two examples inspired by everyday life, though inhighlystylizedform.Imaginethere’saverypopularnewmovieshowingatthelocaltheater.It’sa

romantic comedy, and hundreds of couples (many more than the theater canaccommodate)are linedupat theboxoffice,desperate toget in.Oncea luckycouplegettheirtickets,theyscrambleinsideandchoosetwoseatsrightnexttoeach other. To keep things simple, let’s suppose they choose these seats atrandom,whereverthere’sroom.Inotherwords,theydon’tcarewhethertheysitclosetothescreenorfaraway,ontheaisleorinthemiddleofarow.Aslongasthey’retogether,sidebyside,they’rehappy.Also, let’sassumenocouplewill ever slideover tomake roomforanother.

Once a couple sits down, that’s it.No courtesywhatsoever.Knowing this, thebox office stops selling tickets as soon as there are only single seats left.Otherwisebrawlswouldensue.At first,when the theater is pretty empty, there’s no problem.Every couple

can find twoadjacent seats.Butafterawhile, theonlyseats leftare singles—solitary, uninhabitable dead spaces that a couple can’t use. In real life, peopleoftencreate thesebuffersdeliberately,eitherfor theircoatsor toavoidsharinganarmrestwitharepulsivestranger.Inthismodel,however,thesedeadspacesjusthappenbychance.The question is: When there’s no room left for any more couples, what

fractionofthetheater’sseatsareunoccupied?The answer, in the case of a theater withmany seats per row, turns out to

approach

soabout13.5percentoftheseatsgotowaste.Although the details of the calculation are too intricate to present here, it’s

easy to see that13.5percent is in the rightballparkbycomparing itwith twoextreme cases. If all couples sat next to each other, packed in with perfectefficiencylikesardines,there’dbenowastedseats.

However, if they’d positioned themselves as inefficiently as possible, alwayswithanemptyseatbetweenthem(andleavinganemptyaisleseatononeendortheotherofeachrow,asinthediagrambelow),one-thirdoftheseatswouldbewasted,becauseeverycoupleusesthreeseats: twoforthemselves,andoneforthedeadspace.

Guessingthattherandomcaseshouldliesomewherebetweenperfectefficiencyand perfect inefficiency, and taking the average of 0 and , we’d expect thatabout ,or16.7percent,of the seatswouldbewasted,not far from theexactanswerof13.5percent.Here the large number of choices came about because of all the ways that

couples could be arranged in a huge theater. Our final example is also aboutarrangingcouples, exceptnow in time,not space.What I’m referring to is thevexingproblemofhowmanypeopleyoushoulddatebeforechoosingamate.The real-life version of this problem is too hard formath, so let’s consider asimplifiedmodel.Despiteitsunrealisticassumptions,itstillcapturessomeoftheheartbreakinguncertaintiesofromance.Let’s suppose you know how many potential mates you’re going to meet

duringyourlifetime.(Theactualnumberisnotimportantaslongasit’sknownaheadoftimeandit’snottoosmall.)Also assume you could rank these people unambiguously if you could see

themallatonce.Thetragedy,ofcourse,isthatyoucan’t.Youmeetthemoneata time, in randomorder.Soyoucanneverbe sure ifDreamboat—who’d ranknumber1onyourlist—isjustaroundthecorner,orwhetheryou’vealreadymetandparted.Andthewaythisgameworksis,onceyouletsomeonego,heorsheisgone.

Nosecondchances.Finally,assumeyoudon’twanttosettle.IfyouendupwithSecondBest,or

anyoneelsewho, inretrospect,wouldn’thavemadethetopofyour list,you’llconsideryourlovelifeafailure.Is thereanyhopeofchoosingyouronetrue love?Ifso,whatcanyoudoto

giveyourselfthebestodds?Agoodstrategy,thoughnotthebestone,istodivideyourdatinglifeintotwo

equalhalves.Inthefirsthalf,you’rejustplayingthefield;inthesecond,you’reready togetserious,andyou’regoing tograb the firstpersonyoumeetwho’sbetterthaneveryoneelseyou’vedatedsofar.Withthisstrategy,there’satleasta25percentchanceofsnaggingDreamboat.

Here’swhy:Youhavea50-50chanceofmeetingDreamboatinthesecondhalfof your dating life, your “get serious” phase, and a 50-50 chance of meetingSecond Best in the first half, while you’re playing the field. If both of thosethingshappen—andthereisa25percentchancethattheywill—thenyou’llendupwithyouronetruelove.That’s because Second Best raised the bar so high. No one youmeet after

you’re ready toget seriouswill temptyouexceptDreamboat.So even thoughyoucan’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, and so painful. To explain these swings, many lovelorn souls havesoughtanswersindrink;othershaveturnedtopoetry.We’llconsultcalculus.The analysis below is offered tongue-in-cheek, but it touches on a serious

point:While the lawsof theheartmayeludeus forever, the lawsof inanimatethings are nowwell understood. They take the form of differential equations,which describe how interlinked variables change from moment to moment,dependingon theircurrentvalues.Asforwhatsuchequationshave todowithromance—well, at the very least theymight shed a little light onwhy, in thewordsofanotherpoet,“thecourseoftrueloveneverdidrunsmooth.”To illustrate theapproach, supposeRomeo is in lovewith Juliet but that, in

ourversionofthestory,Julietisaficklelover.ThemoreRomeolovesher,themoreshewantstorunawayandhide.Butwhenhetakesthehintandbacksoff,shebeginstofindhimstrangelyattractive.He,however,tendstomirrorher:hewarmsupwhensheloveshimandcoolsdownwhenshehateshim.Whathappens toourstar-crossed lovers?Howdoes their loveebbandflow

overtime?That’swherecalculuscomesin.BywritingequationsthatsummarizehowRomeoandJulietrespondtoeachother’saffectionsandthensolvingthoseequationswithcalculus,wecanpredict thecourseof theiraffair.Theresultingforecastfor thiscouple is, tragically,anever-endingcycleof loveandhate.Atleasttheymanagetoachievesimultaneousloveaquarterofthetime.

Toreachthisconclusion,I’veassumedthatRomeo’sbehaviorcanbemodeled

bythedifferentialequation

which describes how his love (represented by R) changes in the next instant(representedbydt).Accordingtothisequation,theamountofchange(dR)isjustamultiple(a)ofJuliet’scurrentlove(J)forhim.Thisreflectswhatwealreadyknow—that Romeo’s love goes up when Juliet loves him—but it assumessomethingmuch stronger. It says that Romeo’s love increases in direct linearproportion to howmuch Juliet loves him. This assumption of linearity is notemotionallyrealistic,butitmakestheequationsmucheasiertosolve.Juliet’sbehavior,bycontrast,canbemodeledbytheequation

The negative sign in front of the constant b reflects her tendency to cool offwhenRomeoishotforher.Theonlyremainingthingweneedtoknowishowtheloversfeltabouteach

other initially (R and J at time t = 0). Then everything about their affair ispredetermined. We can use a computer to inch R and J forward, instant byinstant, changing their values as prescribed by the differential equations.Actually,withthehelpofthefundamentaltheoremofcalculus,wecandomuchbetterthanthat.Becausethemodelissosimple,wedon’thavetotrudgeforwardonemomentatatime.CalculusyieldsapairofcomprehensiveformulasthattellushowmuchRomeoandJulietwilllove(orhate)eachotheratanyfuturetime.The differential equations above should be recognizable to students of

physics:RomeoandJulietbehavelikesimpleharmonicoscillators.Sothemodelpredicts thatR(t)andJ(t)—the functions that describe the time courseof theirrelationship—will be sine waves, each waxing and waning but peaking atdifferenttimes.Themodelcanbemademorerealisticinvariousways.Forinstance,Romeo

mightreacttohisownfeelingsaswellastoJuliet’s.Hemightbethetypeofguywhoissoworriedaboutthrowinghimselfatherthatheslowshimselfdownashisloveforhergrows.Orhemightbetheothertype,onewholovesfeelinginlovesomuchthathelovesherallthemoreforit.Add to those possibilities the two ways Romeo could react to Juliet’s

affections—eitherincreasingordecreasinghisown—andyouseethattherearefour personality types, each corresponding to a different romantic style. MystudentsandthoseinPeterChristopher’sclassatWorcesterPolytechnicInstitutehavesuggestedsuchdescriptivenamesasHermitandMalevolentMisanthropefortheparticularkindofRomeowhodampsdownhisownloveandalsorecoilsfromJuliet’s.WhereasthesortofRomeowhogetspumpedbyhisownardorbutturned off by Juliet’s has been called Narcissistic Nerd, Better Latent ThanNever,andaFlirtingFink.(Feelfreetocomeupwithyourownnamesforthesetwotypesandtheothertwopossibilities.)Although these examples arewhimsical, thekindsof equations that arise in

them are profound. They represent themost powerful tool humanity has evercreated for making sense of the material world. Sir Isaac Newton useddifferential equations to solve the ancient mystery of planetary motion. In so

doing,heunifiedtheearthlyandcelestialspheres,showingthatthesamelawsofmotionappliedtoboth.In thenearly350years sinceNewton,mankindhascome to realize that the

lawsofphysicsarealwaysexpressed in the languageofdifferentialequations.Thisistruefortheequationsgoverningtheflowofheat,air,andwater;forthelaws of electricity and magnetism; even for the unfamiliar and oftencounterintuitiveatomicrealm,wherequantummechanicsreigns.Inallcases,thebusinessoftheoreticalphysicsboilsdowntofindingtheright

differentialequationsandsolvingthem.WhenNewtondiscoveredthiskeytothesecretsoftheuniverse,hefeltitwassopreciousthathepublisheditonlyasananagraminLatin.Loosely translated, it reads:“It isuseful tosolvedifferentialequations.”The silly idea that love affairs might likewise be described by differential

equations occurred to me when I was in love for the first time, trying tounderstandmy girlfriend’s baffling behavior. Itwas a summer romance at theendofmysophomoreyearincollege.IwasalotlikethefirstRomeoabove,andshewasevenmorelikethefirstJuliet.Thecyclingofourrelationshipdrovemecrazyuntil I realized thatwewereboth actingmechanically, following simplerules of push andpull.But by the endof the summermy equations started tobreakdown,andIwasmoremystifiedthanever.Asitturnedout,therewasanimportantvariable that I’d leftoutof theequations—heroldboyfriendwantedherback.Inmathematicswecallthisathree-bodyproblem.It’snotoriouslyintractable,

especiallyintheastronomicalcontextwhereitfirstarose.AfterNewtonsolvedthe differential equations for the two-body problem (thus explaining why theplanetsmove inellipticalorbits around the sun),he turnedhis attention to thethree-body problem for the sun, Earth, and moon. He couldn’t solve it, andneither could anyone else. It later turned out that the three-body problemcontained the seeds of chaos, rendering its behavior unpredictable in the longrun.Newton knew nothing about chaotic dynamics, but according to his friend

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

21.StepIntotheLight

MR. DICURCIO WAS my mentor in high school. He was disagreeable anddemanding,withnerdyblack-rimmedglassesandapenchantforsarcasm,sohischarmswereeasytomiss.ButIfoundhispassionforphysicsirresistible.OnedayImentionedtohimthatIwasreadingabiographyofEinstein.The

book said that as a college student, Einstein had been dazzled by somethingcalledMaxwell’sequationsforelectricityandmagnetism,andIsaidIcouldn’twaituntilIknewenoughmathtolearnwhattheywere.Thisbeing aboarding school,wewere eatingdinner together at a big table

withseveralotherstudents,hiswife,andhis twodaughters,andMr.DiCurciowas serving mashed potatoes. At the mention of Maxwell’s equations, hedroppedtheservingspoon,grabbedapapernapkin,andbeganwritinglinesofcryptic symbols—dots and crosses, upside-down triangles, Es and Bs witharrowsoverthem—andsuddenlyheseemedtobespeakingintongues:“Thecurlofacurlisgraddivminusdelsquared...”Thatabracadabrahewasmumbling?Irealizenowhewasspeakinginvector

calculus, the branch of math that describes the invisible fields all around us.Think of the magnetic field that twists a compass needle northward, or thegravitationalfieldthatpullsyourchairtothefloor,orthemicrowavefieldthatnukesyourdinner.Thegreatestachievementsofvectorcalculuslieinthattwilightrealmwhere

mathmeetsreality.Indeed,thestoryofJamesClerkMaxwellandhisequationsoffers one of the eeriest instances of the unreasonable effectiveness ofmathematics.Somehow,byshufflingafewsymbols,Maxwelldiscoveredwhatlightis.To give a sense of whatMaxwell accomplished and, more generally, what

vectorcalculus is about, let’sbeginwith theword“vector.” It comes from theLatin root vehere, “to carry,” which also gives us words like “vehicle” and“conveyorbelt.”Toanepidemiologist,avectoristhecarrierofapathogen,likethemosquito thatconveysmalaria toyourbloodstream.Toamathematician,avector(atleastinitssimplestform)isastepthatcarriesyoufromoneplacetoanother.Thinkaboutoneofthosediagramsforaspiringballroomdancerscoveredwith

arrowsindicatinghowtomovetherightfoot,thentheleftfoot,aswhendoingtherumba:

These arrows are vectors. They show two kinds of information: a direction(whichwaytomovethatfoot)andamagnitude(howfartomoveit).Allvectorsdothatsamedoubleduty.Vectors can be added and subtracted, just like numbers, except their

directionality makes things a little trickier. Still, the right way to add vectorsbecomesclearifyouthinkofthemasdanceinstructions.Forexample,whatdoyougetwhenyoutakeonestepeastfollowedbyonestepnorth?Avectorthatpointsnortheast,naturally.

Remarkably,velocitiesandforcesworkthesameway—theytooaddjustlike

dance steps. This should be familiar to any tennis player who’s ever tried toimitate Pete Sampras and hit a forehand down the linewhile sprinting at fullspeedtowardthesideline.Ifyounaivelyaimyourshotwhereyouwantittogo,itwillsailwidebecauseyouforgottotakeyourownrunningintoaccount.Theball’svelocityrelativetothecourtisthesumoftwovectors:theball’svelocityrelativetoyou(avectorpointingdowntheline,asintended),andyourvelocityrelativetothecourt(avectorpointingsideways,sincethat’sthedirectionyou’rerunning). To hit the ball where you want it to go, you have to aim slightlycrosscourt,tocompensateforyoursidewaysmotion.

Beyond such vector algebra lies vector calculus, the kind of math Mr.DiCurciowasusing.Calculus,you’llrecall,isthemathematicsofchange.Andsowhatevervectorcalculusis, itmustinvolvevectorsthatchange,eitherfrommoment tomoment or fromplace to place. In the latter case, one speaks of a“vectorfield.”A classic example is the force field around amagnet. To visualize it, put a

magnetonapieceofpaperandsprinkleironfilingseverywhere.Eachfilingactslike a little compass needle—it aligns with the direction of local “north,”determinedbythemagneticfieldatthatpoint.Viewedinaggregate,thesefilingsrevealaspectacularpatternofmagnetic-fieldlinesleadingfromonepoleofthemagnettotheother.

Thedirectionandmagnitudeofthevectorsinamagneticfieldvaryfrompoint

topoint.As inallofcalculus, thekey tool forquantifyingsuchchanges is thederivative. In vector calculus the derivative operator goes by the nameof del,which has a folksy southern ring to it, though it actually alludes to theGreekletter ∆ (delta), commonly used to denote a change in some variable. As areminderofthatkinship,“del”iswrittenlikethis:∇.(Thatwasthemysteriousupside-downtriangleMr.DiCurciokeptwritingonthenapkin.)It turns out there are two different but equally natural ways to take the

derivativeofavectorfieldbyapplyingdeltoit.Thefirstgiveswhat’sknownasthefield’sdivergence(the“div”thatMr.DiCurciomuttered).Togetanintuitivefeelingforwhatthedivergencemeasures,takealookatthevectorfieldbelow,whichshowshowwaterwouldflowfromasourceon the left toasinkon theright.

For this example, instead of using iron filings to track the vector field,

imaginelotsoftinycorksorbitsofleavesfloatingonthewatersurface.We’regoingtousethemasprobes.Theirmotionwilltellushowthewaterismovingateachpoint.Specifically,imaginewhatwouldhappenifweputasmallcircleofcorksaroundthesource.Obviously,thecorkswouldspreadapartandthecirclewouldexpand,becausewaterflowsawayfromasource.Itdivergesthere.Andthe stronger the divergence, the faster the area of our cork-circlewould grow.That’swhat the divergence of a vector fieldmeasures: how fast the area of asmallcircleofcorksgrows.Theimagebelowshowsthenumericalvalueofthedivergenceateachpointin

the fieldwe’ve just been looking at, coded by shades of gray. Lighter shadesshow points where the flow has a positive divergence. Darker shades showplaces of negative divergence, meaning that the flow would compress a tinycork-circlecenteredthere.

The other kind of derivative measures the curl of a vector field. Roughlyspeaking, it indicates how strongly the field is swirling about a given point.(Thinkoftheweathermapsyou’veseenonthelocalnewsshowingtherotatingwind patterns around hurricanes or tropical storms.) In the vector field below,regionsthatlooklikehurricaneshavealargecurl.

Byembellishing thevector fieldwith shading,wecannowshowwhere the

curl is most positive (lightest regions) and most negative (darkest regions).Notice that this also tells uswhether the flow is spinning counterclockwiseorclockwise.

The curl is extremely informative for scientistsworking in fluidmechanics

andaerodynamics.AfewyearsagomycolleagueJaneWangusedacomputertosimulate the pattern of airflow around a dragonfly as it hovered in place. Bycalculating thecurl, shefound thatwhenadragonflyflaps itswings, itcreatespairsofcounter-rotatingvorticesthatactlikelittletornadoesbeneathitswings,producing enough lift to keep the insect aloft. In this way, vector calculus ishelping to explain how dragonflies, bumblebees, and hummingbirds can fly—something that had long been a mystery to conventional fixed-wingaerodynamics.With thenotionsofdivergenceandcurl inhand,we’renowready to revisit

Maxwell’s equations. They express four fundamental laws: one for thedivergenceof theelectric field,another for itscurl,and twomoreof thesametypebutnowforthemagneticfield.Thedivergenceequationsrelatetheelectricand magnetic fields to their sources, the charged particles and currents that

producetheminthefirstplace.Thecurlequationsdescribehowtheelectricandmagnetic fields interact and change over time. In so doing, these equationsexpressabeautifulsymmetry:theylinkonefield’srateofchangeintimetotheotherfield’srateofchangeinspace,asquantifiedbyitscurl.Usingmathematicalmaneuversequivalent tovector calculus—whichwasn’t

known in his day—Maxwell then extracted the logical consequences of thosefourequations.Hissymbolshufflingledhimtotheconclusionthatelectricandmagnetic fields couldpropagateas awave, somewhat likea rippleonapond,exceptthatthesetwofieldsweremorelikesymbioticorganisms.Eachsustainedtheother.Theelectricfield’sundulationsrecreatedthemagneticfield,whichinturnrecreatedtheelectricfield,andsoon,witheachpullingtheotherforward,somethingneithercoulddoonitsown.Thatwasthefirstbreakthrough—thetheoreticalpredictionofelectromagnetic

waves.But therealstunnercamenext.WhenMaxwellcalculated thespeedofthesehypotheticalwaves,usingknownpropertiesofelectricityandmagnetism,hisequationstoldhimthattheytraveledatabout193,000milespersecond—thesame rate as the speed of light measured by the French physicist HippolyteFizeauadecadeearlier!How I wish I could havewitnessed themomentwhen a human being first

understood the true nature of light. By identifying it with an electromagneticwave, Maxwell unified three ancient and seemingly unrelated phenomena:electricity, magnetism, and light. Although experimenters like Faraday andAmpèrehadpreviously foundkeypieces of this puzzle, itwasonlyMaxwell,armedwithhismathematics,whoputthemalltogether.Today we are awash in Maxwell’s once-hypothetical waves: Radio.

Television. Cell phones. Wi-Fi. These are the legacy of his conjuring withsymbols.

PartFiveDATA

22.TheNewNormal

STATISTICSHASSUDDENLYbecomecooland trendy.Thanks to theemergenceofthe Internet, e-commerce, social networks, the Human Genome Project, anddigital culture in general, the world is now teeming with data. Marketersscrutinize our habits and tastes. Intelligence agencies collect data on ourwhereabouts,e-mails,andphonecalls.Sportsstatisticianscrunchthenumberstodecidewhichplayerstotrade,whomtodraft,andwhethertogoforitonfourthdownwith two yards to go. Everybodywants to connect the dots, to find theneedleofmeaninginthehaystackofdata.Soit’snotsurprisingthatstudentsarebeingadvisedaccordingly.“Learnsome

statistics,”exhortedGregMankiw,aneconomistatHarvard,ina2010columnintheNew York Times. “High school mathematics curriculums spend too muchtime on traditional topics like Euclidean geometry and trigonometry. For atypicalperson,theseareusefulintellectualexercisesbuthavelittleapplicabilitytodailylife.Studentswouldbebetterservedbylearningmoreaboutprobabilityandstatistics.”DavidBrooksputitmorebluntly.Inacolumnaboutwhatcollegecourseseveryoneshouldtaketobeproperlyeducated,hewrote,“Takestatistics.Sorry, but you’ll find later in life that it’s handy to know what a standarddeviationis.”Yes,andevenhandiertoknowwhatadistributionis.That’sthefirstideaI’d

liketofocusonhere,becauseitembodiesoneofthecentrallessonsofstatistics—things that seem hopelessly random and unpredictable when viewed inisolationoftenturnouttobelawfulandpredictablewhenviewedinaggregate.Youmayhaveseenademonstrationofthisprincipleatasciencemuseum(if

not,youcanfindvideosonline).ThestandardexhibitinvolvesasetupcalledaGaltonboard,whichlooksabitlikeapinballmachineexceptithasnoflippersand its bumpers consist of a regular array of evenly spaced pegs arranged inrows.

ThedemobeginswhenhundredsofballsarepouredintothetopoftheGalton

board.Astheyraindown,theyrandomlybounceoffthepegs,sometimestotheleft, sometimes to the right, andultimatelydistribute themselves in the evenlyspacedbinsatthebottom.Theheightofthestackedballsineachbinshowshowlikely it was for a ball to land there.Most balls end up somewhere near themiddle,withslightlyfewerballsflankingthemoneitherside,andfewerstillfaroffinthetailsateitherend.Overall,thepatternisutterlypredictable:italwaysformsabell-shapeddistribution—even though it’s impossible topredictwhere

anygivenballwillendup.How does individual randomness turn into collective regularity? Easy—the

oddsdemandit.Themiddlebinislikelytobethemostpopulatedspotbecausemostballswillmakeaboutthesamenumberofleftwardandrightwardbouncesbefore they reach the bottom.So they’ll probably end up somewhere near themiddle.Theonlyballsthatmakeitfarouttoeitherextreme,wayoffinthetailsofthedistributionwheretheoutlierslive,arethosethatjusthappentobounceinthesamedirectionalmostevery time theyhitapeg.That’sveryunlikely.Andthat’swhytherearesofewballsoutthere.Just as theultimate locationof eachball isdeterminedby the sumofmany

chanceevents, lotsofphenomena in thisworldare thenet resultofmany tinyflukes, so they too are governed by a bell-shaped curve. Insurance companiesbankonthis.Theyknow,withgreataccuracy,howmanyoftheircustomerswilldieeachyear.Theyjustdon’tknowwhotheunluckyoneswillbe.Orconsiderhowtallyouare.Yourheightdependsoncountlesslittleaccidents

of genetics, biochemistry, nutrition, and environment. Consequently, it’splausible thatwhenviewed inaggregate, theheightsofadultmenandwomenshouldfallonabellcurve.Inablogposttitled“Thebigliespeopletellinonlinedating,”thestatistically

mindeddatingserviceOkCupidrecentlygraphedhowtalltheirmembersare—orrather,howtall theysay theyare—andfoundthat theheightsreportedbybothsexes followbell curves, as expected.What’s surprising,however, is thatbothdistributionsareshiftedabouttwoinchestotherightofwheretheyshouldbe.

So either the peoplewho joinOkCupid are unusually tall, or they exaggeratetheirheightsbyacoupleofincheswhendescribingthemselvesonline.An idealized version of these bell curves is what mathematicians call the

normaldistribution.It’soneofthemostimportantconceptsinstatistics.Partofitsappealistheoretical.Thenormaldistributioncanbeproventoarisewheneveralargenumberofmildlyrandomeffectsofsimilarsize,allactingindependently,areaddedtogether.Andmanythingsarelikethat.But not everything. That’s the second point I’d like to stress. The normal

distribution isnotnearlyasubiquitousas itonceseemed.Forabout100yearsnow,andespeciallyduringthepastfewdecades,statisticiansandscientistshavenoticed that plentyofphenomenadeviate from thispatternyet stillmanage tofollowapatternof theirown.Curiously, these typesofdistributionsarebarelymentioned in the elementary statistics textbooks, and when they are, they’reusuallytrottedoutaspathologicalspecimens.It’soutrageous.For,asI’ll trytoexplain,muchofmodernlifemakesalotmoresensewhenyouunderstandthesedistributions.They’rethenewnormal.Take thedistributionof city sizes in theUnitedStates. Insteadof clustering

around some intermediate value in a bell-shaped fashion, the vastmajority oftownsandcitiesaretinyandthereforehuddletogetherontheleftofthegraph.

Andthelargerthepopulationofacity,themoreuncommonacityofthatsizeis.Sowhen viewed in the aggregate, the distribution looksmore like anL-curvethanabellcurve.There’s nothing surprising about this. Everybody knows that big cities are

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

tobethesameiftheirpopulationsdifferbythesamefactor, ratherthanbythesameabsolutenumberofpeople(muchliketwopitchesanoctaveapartalwaysdifferbyaconstantfactorofdoublethefrequency).Andsupposewedolikewiseontheverticalaxis.

Thenthedatafallonacurvethat’salmostastraightline.Fromthepropertiesoflogarithms, we can then deduce that the original L-curvewas a power law, afunctionoftheform

wherexisacity’ssize,yishowmanycitieshavethatsize,Cisaconstant,andtheexponenta(thepowerinthepowerlaw)isthenegativeofthestraightline’sslope.Powerlawdistributionshavecounterintuitivepropertiesfromthestandpointof

conventional statistics.For example, unlikenormal distributions’, theirmodes,medians,andmeansdonotagreebecauseoftheskewed,asymmetricalshapesoftheirL-curves.PresidentBushmadeuseofthispropertywhenhestatedthathis2003 tax cuts had saved families an average of $1,586 each. Though that istechnically correct, hewas conveniently referring to themean rebate, a figurethat averaged in the whopping rebates of hundreds of thousands of dollars

receivedbytherichest0.1percentofthepopulation.Thetailonthefarrightofthe incomedistribution isknown to followapower law, and in situations likethis,themeanisamisleadingstatistictousebecauseit’sfarfromtypical.Mostfamilies, in fact, got less than $650 back. Themedianwas a lot less than themean.This example highlights themost crucial feature of powerlaw distributions.

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

averagedroppedby22percent inasingleday.Compared to theusual levelofvolatility in the stock market, this was a drop of more than twenty standarddeviations.Suchaneventisallbutimpossibleaccordingtotraditionalbell-curvestatistics; its probability is less than one in100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 (that’s10raisedtothe50thpower).Yetithappened. . .becausefluctuationsinstockprices don’t follow normal distributions. They’re better described by heavy-taileddistributions.Soareearthquakes,wildfires,andfloods,whichcomplicates the taskofrisk

management for insurance industries.Thesamemathematicalpatternholds forthenumbersofdeathscausedbywarsand terroristattacks,andeven formorebenignthingslikewordfrequenciesinnovelsandthenumberofsexualpartnerspeoplehave.Thoughtheadjectivesusedtodescribetheirprominenttailsweren’toriginally

meant to be flattering, such distributions have come towear themwith pride.Fat,heavy,andlong?Yeah,that’sright.Nowwho’snormal?

23.ChancesAre

HAVEYOUEVERhadthatanxietydreamwhereyousuddenlyrealizeyouhavetotake the final exam in some course you’ve never attended? For professors, itworkstheotherwayaround—youdreamyou’regivingalectureinacourseyouknownothingabout.That’swhatit’slikeformewheneverIteachprobabilitytheory.Itwasnever

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

probability that some event A happens, given (or conditional upon) theoccurrenceofsomeothereventB.It’saslipperyconcept,easilyconflatedwiththeprobabilityofBgivenA.They’renotthesame,butyouhavetoconcentratetoseewhy.Forexample,considerthefollowingwordproblem.Beforegoingonvacationforaweek,youaskyourspacyfriendtowateryour

ailingplant.Withoutwater, the plant has a 90 percent chance of dying. Evenwithproperwatering, ithasa20percentchanceofdying.And theprobabilitythatyourfriendwillforgettowaterit is30percent.(a)What’sthechancethatyour plantwill survive theweek? (b) If it’s deadwhenyou return,what’s thechance thatyour friend forgot towater it? (c) Ifyour friend forgot towater it,what’sthechanceit’llbedeadwhenyoureturn?Althoughtheysoundalike,(b)and(c)arenotthesame.Infact,theproblemtellsusthattheanswerto(c)is90percent.Buthowdoyoucombinealltheprobabilitiestogettheanswerto(b)?Or(a)?Naturally, the first few semesters I taught this topic, I stuck to the book,

inchingalong,playingitsafe.ButgraduallyIbegantonoticesomething.AfewofmystudentswouldavoidusingBayes’s theorem, the labyrinthine formula Iwas teaching them. Instead they would solve the problems by an equivalentmethodthatseemedeasier.Whattheseresourcefulstudentskeptdiscovering,yearafteryear,wasabetter

way to think about conditional probability. Their way comports with humanintuition instead of confounding it. The trick is to think in terms of naturalfrequencies—simple counts of events—rather than inmore abstract notions ofpercentages,odds,orprobabilities.As soonasyoumake thismental shift, thefoglifts.This is the central lesson ofCalculated Risks, a fascinating book by Gerd

Gigerenzer, a cognitive psychologist at the Max Planck Institute for HumanDevelopment in Berlin. In a series of studies about medical and legal issuesranging from AIDS counseling to the interpretation of DNA fingerprinting,Gigerenzer explores how people miscalculate risk and uncertainty. But ratherthanscoldorbemoanhumanfrailty,hetellsushowtodobetter—howtoavoidclouded thinking by recasting conditional-probability problems in terms ofnaturalfrequencies,muchasmystudentsdid.Inonestudy,GigerenzerandhiscolleaguesaskeddoctorsinGermanyandthe

United States to estimate the probability that a woman who has a positivemammogramactuallyhasbreastcancereven thoughshe’s ina low-riskgroup:fortytofiftyyearsold,withnosymptomsorfamilyhistoryofbreastcancer.Tomake the question specific, the doctors were told to assume the followingstatistics—couched in terms of percentages and probabilities—about theprevalence of breast cancer among women in this cohort and about themammogram’ssensitivityandrateoffalsepositives:Theprobabilitythatoneofthesewomenhasbreastcanceris0.8percent.Ifawomanhasbreastcancer,theprobabilityis90percentthatshewillhaveapositive mammogram. If a woman does not have breast cancer, theprobability is 7 percent that she will still have a positive mammogram.Imagineawomanwhohasapositivemammogram.Whatistheprobabilitythatsheactuallyhasbreastcancer?

Gigerenzer describes the reaction of the first doctor he tested, a department

chiefatauniversityteachinghospitalwithmorethanthirtyyearsofprofessionalexperience:[He]wasvisiblynervouswhiletryingtofigureoutwhathewouldtellthewoman.Aftermullingthenumbersover,hefinallyestimatedthewoman’sprobability of having breast cancer, given that she has a positivemammogram,tobe90percent.Nervously,headded,“Oh,whatnonsense.Ican’tdothis.Youshouldtestmydaughter;sheisstudyingmedicine.”Heknew that his estimate was wrong, but he did not know how to reasonbetter.Despitethefactthathehadspent10minuteswringinghismindforananswer,hecouldnotfigureouthowtodrawasoundinferencefromtheprobabilities.

Gigerenzer asked twenty-four otherGerman doctors the same question, and

theirestimateswhipsawedfrom1percentto90percent.Eightofthemthought

the chances were 10 percent or less; eight others said 90 percent; and theremaining eight guessed somewherebetween50 and80percent. Imaginehowupsettingitwouldbeasapatienttohearsuchdivergentopinions.As for the American doctors, ninety-five out of a hundred estimated the

woman’s probability of having breast cancer to be somewhere around 75percent.Therightansweris9percent.Howcanitbesolow?Gigerenzer’spointisthattheanalysisbecomesalmost

transparent if we translate the original information from percentages andprobabilitiesintonaturalfrequencies:Eightoutofevery1,000womenhavebreastcancer.Ofthese8womenwithbreast cancer, 7will have a positivemammogram.Of the remaining 992womenwho don’t have breast cancer, some 70 will still have a positivemammogram. Imagine a sample of women who have positivemammograms in screening. How many of these women actually havebreastcancer?

Sinceatotalof7+70=77womenhavepositivemammograms,andonly7ofthemtrulyhavebreastcancer,theprobabilityofawoman’shavingbreastcancergivenapositivemammogramis7outof77,whichis1in11,orabout9percent.Notice two simplifications in the calculation above. First, we rounded off

decimalstowholenumbers.Thathappenedinafewplaces,likewhenwesaid,“Of these 8 womenwith breast cancer, 7 will have a positivemammogram.”Reallyweshouldhavesaid90percentof8women,or7.2women,willhaveapositivemammogram.Sowesacrificedalittleprecisionforalotofclarity.Second, we assumed that everything happens exactly as frequently as its

probability suggests. For instance, since the probability of breast cancer is 0.8percent,exactly8womenoutof1,000inourhypotheticalsamplewereassumedto have it. In reality, this wouldn’t necessarily be true. Events don’t have tofollow their probabilities; a coin flipped 1,000 times doesn’t always come upheads500times.Butpretendingthatitdoesgivestherightanswerinproblemslikethis.Admittedly the logic is a little shaky—that’swhy the textbooks look down

their noses at this approach, compared to the more rigorous but hard-to-useBayes’s theorem—but the gains in clarity are justification enough. WhenGigerenzer tested another set of twenty-four doctors, this time using naturalfrequencies,nearlyallofthemgotthecorrectanswer,orclosetoit.Althoughreformulatingthedataintermsofnaturalfrequenciesisahugehelp,

conditional-probability problems can still be perplexing for other reasons. It’seasy to ask thewrong question or to calculate a probability that’s correct butmisleading.BoththeprosecutionandthedefensewereguiltyofthisintheO.J.Simpson

trialof1994–95.Eachofthemaskedthecourttoconsiderthewrongconditionalprobability.Theprosecutionspentthefirsttendaysofthetrialintroducingevidencethat

O.J. had a history of violence toward his ex-wife Nicole Brown. He hadallegedlybatteredher,thrownheragainstwalls,andgropedherinpublic,tellingonlookers, “This belongs tome.” Butwhat did any of this have to dowith amurder trial? The prosecution’s argument was that a pattern of spousal abusereflectedamotivetokill.Asoneoftheprosecutorsputit,“Aslapisapreludetohomicide.”Alan Dershowitz countered for the defense, arguing that even if the

allegations of domestic violence were true, they were irrelevant and shouldthereforebeinadmissible.Helaterwrote,“Weknewwecouldprove,ifwehadto, that an infinitesimal percentage—certainly fewer than 1 of 2,500—ofmenwhoslaporbeattheirdomesticpartnersgoontomurderthem.”In effect, both sideswere asking the court to consider theprobability that a

man murdered his ex-wife, given that he previously battered her. But as thestatisticianI.J.Goodpointedout,that’snottherightnumbertolookat.Therealquestionis:What’stheprobabilitythatamanmurderedhisex-wife,

given thathepreviouslybatteredherandshewasmurderedbysomeone?Thatconditionalprobabilityturnsouttobeveryfarfrom1in2,500.To see why, imagine a sample of 100,000 battered women. Granting

Dershowitz’snumberof1 in2,500,weexpectabout40of thesewomen tobemurdered by their abusers in a given year (since 100,000 divided by 2,500equals40).Wealsoexpect3moreofthesewomen,onaverage,tobekilledbysomeoneelse(thisestimateisbasedonstatisticsreportedbytheFBIforwomenmurdered in 1992; see the notes for further details). So out of the 43murdervictimsaltogether,40ofthemwerekilledbytheirbatterers.Inotherwords,thebattererwasthemurdererabout93percentofthetime.Don’t confuse this number with the probability that O.J. did it. That

probabilitywoulddependona lotofother evidence,pro andcon, suchas thedefense’sclaimthatthepoliceframedO.J.,andtheprosecution’sclaimthatthekillerandO.J.sharedthesamestyleofshoes,gloves,andDNA.Theprobabilitythatanyofthischangedyourmindabouttheverdict?Zero.

24.UntanglingtheWeb

INATIME longago, in thedarkdaysbeforeGoogle,searching theWebwasanexerciseinfrustration.Thesitessuggestedbytheoldersearchenginesweretoooftenirrelevant,whiletheonesyoureallywantedwereeitherburiedwaydowninthelistofresultsormissingaltogether.Algorithms based on link analysis solved the problem with an insight as

paradoxical as a Zen koan: AWeb search should return the best pages. Andwhat,grasshopper,makesapagegood?Apageisgoodifothergoodpageslinktoit.That sounds likecircular reasoning. It is . . .which iswhy it’s sodeep.By

grappling with this circle and turning it to advantage, link analysis yields ajujitsusolutiontosearchingtheWeb.The approach builds on ideas from linear algebra, the study of vectors and

matrices. Whether you want to detect patterns in large data sets or performgigantic computations involving millions of variables, linear algebra has thetoolsyouneed.AlongwithunderpinningGoogle’sPageRankalgorithm, it hashelped scientists classifyhuman faces, analyze thevotingpatterns ofSupremeCourtjustices,andwinthemillion-dollarNetflixPrize(awardedtothepersonorteam who could improve by more than 10 percent Netflix’s system forrecommendingmoviestoitscustomers).Foracasestudyoflinearalgebrainaction,let’slookathowPageRankworks.

And tobringout itsessencewithaminimumof fuss, let’s imaginea toyWebthathasjustthreepages,allconnectedlikethis:

ThearrowsindicatethatpageXcontainsalinktopageY,butYdoesnotreturnthefavor.Instead,YlinkstoZ.MeanwhileXandZlinktoeachotherinafrenzyofdigitalback-scratching.In this littleWeb,whichpage is themost important,andwhich is the least?

Youmightthinkthere’snotenoughinformationtosaybecausenothingisknownaboutthepages’content.Butthat’sold-schoolthinking.Worryingaboutcontentturnedouttobeanimpracticalwaytorankwebpages.Computersweren’tgoodatit,andhumanjudgescouldn’tkeepupwiththedelugeofthousandsofpagesaddedeachday.The approach taken byLarry Page and SergeyBrin, the grad studentswho

cofoundedGoogle,was to letwebpages rank themselves by votingwith theirfeet—or,rather,withtheirlinks.Intheexampleabove,pagesXandYbothlinktoZ,whichmakesZ the only pagewith two incoming links.So it’s themostpopularpageintheuniverse.Thatshouldcountforsomething.However,ifthoselinks come from pages of dubious quality, that should count against them.Popularitymeansnothingon itsown.Whatmatters ishaving links fromgoodpages.Whichbringsusbacktotheriddleofthecircle:Apageisgoodifgoodpages

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

thenoteson[>]foramorecompletestory.)Google’salgorithmassignsa fractionalscorebetween0and1 toeachpage.

ThatscoreiscalleditsPageRank;itmeasureshowimportantthatpageisrelative

totheothersbycomputingtheproportionoftimethatahypotheticalWebsurferwould spend there.Whenever there ismore than one outgoing link to choosefrom, the surfer selects one at random, with equal probability. Under thisinterpretation, pages are regarded as more important if they’re visited morefrequently(bythisidealizedsurfer,notbyactualWebtraffic).AndbecausethePageRanksaredefinedasproportions,theyhavetoaddupto

1 when summed over the whole network. This conservation law suggestsanother,perhapsmorepalpable,waytovisualizePageRank.Pictureitasafluid,a watery substance that flows through the network, draining away from badpagesandpoolingatgoodones.Thealgorithmseekstodeterminehowthisfluiddistributesitselfacrossthenetworkinthelongrun.Theansweremergesfromacleveriterativeprocess.Thealgorithmstartswith

aguess,thenupdatesallthePageRanksbyapportioningthefluidinequalsharesto the outgoing links, and it keeps doing that in a series of rounds untileverythingsettlesdownandallthepagesgettheirrightfulshares.Initiallythealgorithmtakesanegalitarianstance.Itgiveseverypageanequal

portion of PageRank. Since there are three pages in the example we’reconsidering,eachpagebeginswithascoreof1/3.

Next, these scores areupdated tobetter reflect eachpage’s true importance.

TheruleisthateachpagetakesitsPageRankfromthelastroundandparcelsitoutequallytoallthepagesitlinksto.Thus,afteroneround,theupdatedvalueofXwouldstillequal1/3,becausethat’showmuchPageRankitreceivesfromZ,theonlypagethatlinkstoit.ButY’sscoredropstoameasly1/6,sinceitgetsonlyhalfofX’sPageRank from theprevious round.Theotherhalfgoes toZ,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,onefindsthat thenumbersdon’tchangemuchfromone

roundtothenext.Bythen,Xhasa40.6percentshareofthetotalPageRank,Yhas19.8percent,andZhas39.6percent.Thosenumberslooksuspiciouslyclose

to 40 percent, 20 percent, and 40 percent, suggesting that the algorithm isconvergingtothosevalues.In fact, that’s correct. Those limiting values are what Google’s algorithm

woulddefineasthePageRanksforthenetwork.

TheimplicationisthatXandZareequallyimportantpages,eventhoughZhastwiceasmany linkscoming in.Thatmakes sense:X is just as important asZbecauseitgetsthefullendorsementofZbutreciprocateswithonlyhalfitsownendorsement.TheotherhalfitsendstoY.ThisalsoexplainswhyYfaresonlyhalfaswellasXandZ.Remarkably,thesescorescanbeobtaineddirectly,withoutgoingthroughthe

iteration.Just thinkabout theconditions thatdefinethesteadystate.Ifnothingchangesafteranupdateisperformed,wemusthavex′=x,y′=y,andz′=z.Soreplace the primed variables in the update equations with their unprimedcounterparts.Thenweget

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

oflinearalgebra.Thesteady-stateequationsabove,aswellastheearlierupdateequations with the primes in them, are typical examples of linear equations.They’re called linear because they’re related to lines. The variables x, y, z inthemappeartothefirstpoweronly,justastheydointhefamiliarequationforastraightline,y=mx+b,astapleofhigh-schoolalgebracourses.Linearequations,asopposedtothosecontainingnonlineartermslikex²oryz

orsinx,arecomparativelyeasy tosolve.Thechallengecomeswhen thereareenormousnumbersofvariablesinvolved,asthereareintherealWeb.Oneofthecentraltasksoflinearalgebra,therefore,isthedevelopmentoffasterandfasteralgorithms for solving such huge sets of equations. Even slight improvementshaveramificationsforeverythingfromairlineschedulingtoimagecompression.But thegreatest triumphof linearalgebra, fromthestandpointof real-world

impact,issurelyitssolutiontotheZenriddleofrankingwebpages.“Apageisgood insofar as good pages link to it.” Translated into symbols, that criterionbecomesthePageRankequations.Googlegotwhereitistodaybysolvingthesameequationsaswedidhere—

justwithafewbillionmorevariables...andprofitstomatch.

PartSixFRONTIERS

25.TheLoneliestNumbers

ACCORDINGTOAmemorablesong fromthe1960s,one is the loneliestnumber,andtwocanbeasbadasone.Maybeso,but theprimenumbershaveitprettyroughtoo.PaoloGiordanoexplainswhy inhisbestsellingnovelTheSolitudeofPrime

Numbers. It’s the melancholy love story of two misfits, two primes, namedMattia and Alice, both scarred by childhood tragedies that left them virtuallyincapableofconnectingwithotherpeople,yetwhosenseineachotherakindreddamagedspirit.Giordanowrites,Primenumbersaredivisibleonlyby1andbythemselves.Theyholdtheirplace in the infiniteseriesofnaturalnumbers,squashed, likeallnumbers,betweentwoothers,butonestepfurtherthantherest.Theyaresuspicious,solitary numbers, which is why Mattia thought they were wonderful.Sometimeshethoughtthattheyhadendedupinthatsequencebymistake,that they’dbeen trapped, likepearls strungonanecklace.Other timeshesuspectedthat theytoowouldhavepreferredtobelikeall theothers, justordinarynumbers,butforsomereasontheycouldn’tdoit.[...]In his first year at university, Mattia had learned that, among prime

numbers, there are some that are evenmore special.Mathematicians callthem twin primes: pairs of prime numbers that are close to each other,almost neighbors, but between them there is always an even number thatpreventsthemfromtrulytouching.Numberslike11and13,like17and19,41 and43. If youhave thepatience to goon counting, youdiscover thatthese pairs gradually become rarer. You encounter increasingly isolatedprimes, lost in that silent,measured spacemadeonlyof ciphers, andyoudevelopadistressingpresentiment that thepairsencounteredupuntil thatpoint were accidental, that solitude is the true destiny. Then, just whenyou’re about to surrender, when you no longer have the desire to go oncounting, you come across another pair of twins, clutching each othertightly.Thereisacommonconvictionamongmathematiciansthathoweverfaryougo,therewillalwaysbeanothertwo,evenifnoonecansaywhereexactly,untiltheyarediscovered.Mattia thought thatheandAlicewere like that, twinprimes,aloneand

lost,closebutnotcloseenoughtoreallytoucheachother.

Here I’d like to explore some of the beautiful ideas in the passage above,particularly as they relate to the solitude of prime numbers and twin primes.These issuesarecentral tonumber theory, the subject thatconcerns itselfwiththestudyofwholenumbersandtheirpropertiesandthatisoftendescribedasthepurestpartofmathematics.Beforeweascendtowheretheairisthin,letmedispensewithaquestionthat

often occurs to practical-minded people: Is number theory good for anything?Yes. Almost in spite of itself, number theory provides the basis for theencryption algorithms used millions of times each day to secure credit cardtransactions over the Internet and to encode military-strength secretcommunications. Those algorithms rely on the difficulty of decomposing anenormousnumberintoitsprimefactors.Butthat’snotwhymathematiciansareobsessedwithprimenumbers.Thereal

reasonis that they’refundamental.They’re theatomsofarithmetic.Justas theGreek origin of theword “atom” suggests, the primes are “a-tomic,”meaning“uncuttable, indivisible.”And just as everything is composed of atoms, everynumberiscomposedofprimes.Forexample,60equals2×2×3×5.Wesaythat60isacompositenumberwithprimefactorsof2(countedtwice),3,and5.Andwhatabout1?Isitprime?No,itisn’t,andwhenyouunderstandwhyit

isn’t, you’ll begin to appreciate why 1 truly is the loneliest number—evenlonelierthantheprimes.Itdoesn’tdeservetobeleftout.Giventhat1isdivisibleonlyby1anditself,it

really should be considered prime, and for many years it was. But modernmathematicians have decided to exclude it, solely for convenience. If 1 wereallowedin,itwouldmessupatheoremthatwe’dliketobetrue.Inotherwords,we’veriggedthedefinitionofprimenumberstogiveusthetheoremwewant.The desired theorem says that any number can be factored into primes in a

unique way. But if 1 were considered prime, the uniqueness of primefactorizationwould fail. For example, 6would equal 2 × 3, but itwould alsoequal1×2×3and1×1×2×3and soon, and thesewouldallhave tobeacceptedasdifferentprimefactorizations.Silly,ofcourse,butthat’swhatwe’dbestuckwithif1wereallowedin.This sordid little tale is instructive; itpullsback thecurtainonhowmath is

done sometimes. The naive view is thatwemake our definitions, set them instone,thendeducewhatevertheoremshappentofollowfromthem.Notso.Thatwouldbemuchtoopassive.We’reinchargeandcanalterthedefinitionsasweplease—especiallyifaslighttweakleadstoatidiertheorem,asitdoeshere.Nowthat1hasbeenthrownunderthebus,let’slookateveryoneelse,thefull-

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

17,19,23,29.Rightoffthebat, thingsstartbadlywith2.It’safreak,amisfitamong misfits—the only prime with the embarrassment of being an evennumber. No wonder “it’s the loneliest number since the number one” (as thesongsays).Apartfrom2,therestoftheprimesareallodd...butstillquirky.Lookatthe

gaps between them. Sometimes they’re two spaces apart (like 5 and 7),sometimesfour(13and17),andsometimessix(23and29).To further underscorehowdisorderly theprimes are, compare them to their

straight-arrow cousins the odd numbers: 1, 3, 5, 7, 9, 11, 13, . . . The gapsbetweenoddnumbersarealwaysconsistent: twospaces,steadyasadrumbeat.So theyobeyasimple formula: thenthoddnumber is2n–1.Theprimes,bycontrast,marchtotheirowndrummer,toarhythmnooneelsecanperceive.Given the irregularities in thespacingof theprimes,numbers theoristshave

resortedtolookingatthemstatistically,asmembersofanensemble,ratherthandwelling on their idiosyncrasies. Specifically, let’s ask how they’re distributedamongtheordinarywholenumbers.Howmanyprimesarelessthanorequalto10?Or100?OranarbitrarynumberN?Thisconstructionisadirectparalleltothestatisticalconceptofacumulativedistribution.So imagine counting the prime numbers by walking among them, like an

anthropologist taking a census.Picture them standing there on thex-axis.Youstartatthenumber1andbeginwalkingtotheright,tallyingprimesasyougo.Yourrunningtotallookslikethis:

Thevaluesonthey-axisshowhowmanyprimesyou’vecountedbythetimeyoureachagivenlocation,x.Forallx’slessthan2,thegraphofyremainsflatat0,sincenoprimeshavebeencountedyet.Thefirstprimeappearsatx=2.Sothegraphjumpsupthere.(Gotone!)Thenitremainsflatuntilx=3,afterwhichitjumps up another step. The alternating jumps and plateaus form a strange,irregularstaircase.Mathematicianscallitthecountingfunctionfortheprimes.Contrastthisimagewithitscounterpartfortheoddnumbers.

Nowthestaircasebecomesperfectlyregular,followingatrendlinewhoseslopeis1/2.That’sbecausethegapbetweenneighboringoddnumbersisalways2.Isthereanyhopeoffindingsomethingsimilarfortheprimenumbersdespite

theirerraticcharacter?Miraculously,yes.Thekey is tofocuson the trend,notthedetailsofthestairsteps.Ifwezoomout,acurvebeginstoemergefromtheclutter.Here’sthegraphofthecountingfunctionforallprimesupto100.

Thestepsarelessdistractingnow.Thecurvelooksevensmootherifwecountalltheprimesouttoabillion:

First impressions to the contrary, this curve is not actually a straight line. Itdroops down slightly as it climbs. Its droopiness means that the primes arebecomingrarer.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.By finding functions that approximate the droopy curve, number theorists

have quantified how desolate the prime numbers truly are, as expressed by a

formulaforthetypicalspacingbetweenthem.IfNisalargenumber,theaveragegap between the primes near N is approximately equal to lnN, the naturallogarithm of N. (The natural logarithm behaves like the ordinary logarithmencounteredinhighschool,exceptit’sbasedonthenumbereinsteadof10.It’snaturalinthesensethatitpopsupeverywhereinadvancedmath,thankstobeingpartofe’sentourage.Formoreontheubiquityofe,seechapter19.)Although the lnN formula for the average spacing between primes doesn’t

worktoowellwhenNissmall,itimprovesinthesensethatitspercentageerrorgoes to zero asN approaches infinity.Toget a feel for thenumbers involved,supposeN=1,000.Itturnsoutthereare168primenumberslessthan1,000,sothe average gap between them in this part of the number line is 1,000/68, orabout5.9.Forcomparison, the formulapredictsanaveragegapof ln(1,000)≈6.9,which is toohighbyabout17percent.Butwhenwegomuchfartherout,saytoN=1,000,000,000,theactualandpredictedgapsbecome19.7and20.7,respectively,anoverestimateofonlyabout5percent.Thevalidity of the lnN formula asN tends to infinity is nowknown as the

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

howpoignant it is that twinprimesapparentlycontinue toexist in the farthestreaches of the number line, “in that silent, measured space made only ofciphers.” The odds are stacked against them. According to the prime numbertheorem, any particular prime nearN has no right to expect a potential matemuchcloserthanlnNaway,agulfmuchlargerthan2whenNislarge.Andyetsomecouplesdobeattheodds.Computershavefoundtwinprimesat

unbelievablyremotepartsofthenumberline.Thelargestknownpairconsistsoftwonumberswith100,355decimaldigitseach,snugglinginthedarkness.Thetwinprimeconjecturesayscoupleslikethiswillturnupforever.But as for finding another prime couple nearby for a friendly game of

doubles?Goodluck.

26.GroupThink

MYWIFEAND I have different sleeping styles—and ourmattress shows it. Shehoardsthepillows,thrashesaroundallnightlong,andbarelydentsthemattress,whileI lieonmyback,mummy-like,moldingacavernousdepression intomysideofthebed.Bedmanufacturers recommend flippingyourmattressperiodically, probably

withpeoplelikemeinmind.Butwhat’sthebestsystem?Howexactlyareyousupposedtoflipittogetthemostevenwearoutofit?BrianHayesexploresthisprobleminthetitleessayofhisbookGroupTheory

intheBedroom.Double-entendresaside,the“group”underdiscussionhereisacollection of mathematical actions—all the possible ways you could flip orrotatethemattresssothatitstillfitsneatlyonthebedframe.

Bylookingintomattressmathinsomedetail,Ihopetogiveyouafeelingfor

grouptheorymoregenerally.It’soneofthemostversatilepartsofmathematics.It underlies everything from the choreography of square dancing and thefundamental lawsofparticlephysics to themosaicsof theAlhambraand theirchaoticcounterparts,likethisimage:

As these examples suggest, group theory bridges the arts and sciences. Itaddresses something the two cultures share—an abiding fascination withsymmetry.Yetbecauseitencompassessuchawiderangeofphenomena,grouptheoryisnecessarilyabstract.Itdistillssymmetrytoitsessence.Normallywethinkofsymmetryasapropertyofashape.Butgrouptheorists

focusmoreonwhatyoucando toa shape—specifically,all thewaysyoucanchangeitwhilekeepingsomethingelseaboutit thesame.Moreprecisely,theylook for all the transformations that leave a shape unchanged, given certainconstraints.Thesetransformationsarecalledthesymmetriesoftheshape.Takentogether,theyformagroup,acollectionoftransformationswhoserelationshipsdefinetheshape’smostbasicarchitecture.In the case of a mattress, the transformations alter its orientation in space

(that’swhat changes)whilemaintaining its rigidity (that’s theconstraint).And

after the transformation is complete, the mattress has to fit snugly on therectangular bed frame (that’swhat stays the same).With these rules in place,let’s see what transformations qualify for membership in this exclusive littlegroup.Itturnsoutthereareonlyfourofthem.The first is the do-nothing transformation, a lazy but popular choice that

leavesthemattressuntouched.Itcertainlysatisfiesalltherules,butit’snotmuchhelpinprolongingthelifeofyourmattress.Still,it’sveryimportanttoincludeinthe group. It plays the same role for group theory that 0 does for addition ofnumbers, or that 1 does formultiplication.Mathematicians call it the identityelement,soI’lldenoteitbythesymbolI.Next come the three genuineways to flip amattress.To distinguish among

them,ithelpstolabelthecornersofthemattressbynumberingthemlikeso:

The first kind of flip is depicted near the beginning of this chapter. Thehandsomegentlemaninstripedpajamasistryingtoturnthemattressfromsidetosidebyrotatingit180degreesarounditslongaxis,inamoveI’llcallH,for“horizontalflip.”

Amorerecklesswaytoturnoverthemattressisaverticalflip,V.Thismaneuverswapsitsheadandfoot.Youstandthemattressuprightthelongway,sothatitalmost reaches the ceiling, and then topple it end over end. The net effect,besides the enormous thud, is that the mattress rotates 180 degrees about itslateralaxis,shownbelow.

Thefinalpossibilityistospinthemattresshalfaturnwhilekeepingitflatonthebed.

UnliketheHandVflips,thisrotation,R,keepsthetopsurfaceontop.Thatdifferenceshowsupwhenwelookatthemattress—nowimaginedtobe

translucent—fromaboveandinspectthenumbersatthecornersaftereachofthepossibletransformations.Thehorizontalflipturnsthenumeralsintotheirmirrorimages.Italsopermutesthemsothat1and2tradeplaces,asdo3and4.

The vertical flip permutes the numbers in a differentway and stands themontheirheads,inadditiontomirroringthem.

The rotation,however,doesn’tgenerateanymirror images. Itmerely turns thenumbersupsidedown,thistimeexchanging1for4and2for3.

Thesedetailsarenotthemainpoint.Whatmattersishowthetransformations

relate tooneanother.Theirpatternsof interactionencode thesymmetryof themattress.

To reveal those patterns with a minimum of effort, it helps to draw thefollowing diagram. (Images like this abound in a terrific book called VisualGroup Theory, by Nathan Carter. It’s one of the best introductions to grouptheory—ortoanybranchofhighermath—I’veeverread.)

Thefourpossiblestatesofthemattressareshownatthecornersofthediagram.Theupperleftstateisthestartingpoint.Thearrowsindicatethemovesthattakethemattressfromonestatetoanother.Forexample,thearrowpointingfromtheupperlefttothelowerrightdepicts

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

tofootandthendoingthatagainreturnsthemattresstoitsoriginalstate.Wecansummarize this property with the equation RR = I, where RR means “do Rtwice,”andIisthedo-nothingidentityelement.Thehorizontalandverticalfliptransformationsalsoundothemselves:HH=IandVV=I.Thediagramembodiesawealthofother information.For instance, it shows

that the death-defying vertical flip, V, is equivalent toHR, a horizontal flipfollowed by a rotation—amuch safer path to the same result. To check this,beginat the starting state in theupper left.Headdueeast alongH to thenextstate,andfromtheregodiagonallysouthwestalongR.Becauseyouarriveatthe

same state you’d reach if you followed V to begin with, the diagramdemonstratesthatHR=V.Notice,too,thattheorderofthoseactionsisirrelevant:HR=RH,sinceboth

roadsleadtoV.This indifference toorder is trueforanyotherpairofactions.Youshouldthinkofthisasageneralizationofthecommutativelawforadditionofordinarynumbers,xandy,accordingtowhichx+y=y+x.Butbeware:Themattressgroupisspecial.Manyothergroupsviolatethecommutativelaw.Thosefortunateenoughtoobeyitareparticularlycleanandsimple.Nowforthepayoff.Thediagramshowshowtogetthemostevenwearoutof

amattress.Anystrategythatsamplesallfourstatesperiodicallywillwork.Forexample,alternatingRandHisconvenient—andsinceitbypassesV,it’snottoostrenuous.Tohelpyourememberit,somemanufacturerssuggestthemnemonic“spininthespring,flipinthefall.”The mattress group also pops up in some unexpected places, from the

symmetryofwatermoleculestothelogicofapairofelectricalswitches.That’sone of the charms of group theory. It exposes the hidden unity of things thatwouldotherwiseseemunrelated...likeinthisanecdoteabouthowthephysicistRichardFeynmangotadraftdeferment.ThearmypsychiatristquestioninghimaskedFeynmantoputouthishandsso

hecouldexaminethem.Feynmanstuckthemout,onepalmup,theotherdown.“No, the other way,” said the psychiatrist. So Feynman reversed both hands,leavingonepalmdownandtheotherup.Feynman wasn’t merely playing mind games; he was indulging in a little

group-theoretichumor.Ifweconsiderallthepossiblewayshecouldhaveheldouthishands,alongwith thevarious transitionsamong them, thearrowsformthesamepatternasthemattressgroup!

But if all thismakesmattresses seemway too complicated,maybe the real

lessonhereisoneyoualreadyknew—ifsomething’sbotheringyou,justsleeponit.

27.TwistandShout

OUR LOCAL ELEMENTARY school invites parents to come and talk to theirchildren’sclasses.Thisgivesthekidsachancetohearaboutdifferentjobsandtolearnaboutthingstheymightnotbeexposedtootherwise.Whenmyturncame,Ishowedupatmydaughter’sfirst-gradeclasswithabag

full ofMöbius strips. The night before,mywife and I had cut long strips ofpaperandthengiveneachstripahalftwist,likethis,

beforetapingthetwoendstogethertomakeaMöbiusstrip:

Therearefunactivitiesthatasix-year-oldcandowiththeseshapesthatinvolvenothingmorethanscissors,crayons,tape,andalittlecuriosity.Asmywifeand Ipassedout theMöbiusstripsandart supplies, the teacher

asked theclasswhat subject they thoughtweweredoingnow.Oneboy raisedhishandandsaid,“Well,I’mnotsure,butIknowit’snotlinguistics.”Ofcourse,theteacherhadbeenexpectingananswerof“art,”ormaybe,more

precociously, “math.”Thebest answer,however,wouldhavebeen“topology.”

(In Ithaca it’s altogether possible that a first-grader could have come upwiththat.Butthetopologist’skidhappenedtobeinanotherclassthatyear.)Sowhat is topology? It’s a vibrant branch ofmodernmath, an offshoot of

geometry,butmuchmore loosey-goosey. In topology, twoshapesare regardedas the same if you can bend, twist, stretch, or otherwise deform one into theothercontinuously—thatis,withoutanyrippingorpuncturing.Unliketherigidobjects of geometry, the objects of topology behave as if theywere infinitelyelastic,asiftheyweremadeofanidealkindofrubberorSillyPutty.Topology shines a spotlight on a shape’s deepest properties—the properties

thatremainunchangedafteracontinuousdistortion.Forexample,arubberbandshaped like a square and another shaped like a circle are topologicallyindistinguishable. It doesn’t matter that a square has four corners and fourstraightsides.Thosepropertiesareirrelevant.Acontinuousdeformationcangetrid of them by rounding off the square’s corners and bending its sides intocirculararcs.

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

That twist endows the shape with its signature features. Most famously, aMöbiusstriphasonlyonesideandonlyoneedge.Inotherwords,itsfrontandback surfaces are actually the same, and so are its top and bottom edges. (Tocheckthis,justrunyourfingeralongthemiddleofthestriporitsedgeuntilyoureturntoyourstartingpoint.)Whathappenedhereisthatthehalftwisthookedthe former top and bottom edges of the paper into one big, long continuouscurve. Similarly, it fused the front to the back.Once the strip has been tapedshut,thesefeaturesarepermanent.YoucanstretchandtwistaMöbiusstripallyouwant,butnothingcanchangeitshalf-twistedness,itsone-sidedness,anditsone-edgedness.By having the first-graders examine the strange properties ofMöbius strips

thatfollowfromthesefeatures,Iwashopingtoshowthemhowmuchfunmathcouldbe—andalsohowamazing.

FirstIaskedthemeachtotakeacrayonandcarefullydrawalineallthewayaround the Möbius strip, right down the middle of the surface. With browsfurrowed, they began tracing something like the dashed line on the followingpage:

Afteronecircuit,manyof thestudentsstoppedandlookedpuzzled.Thentheybegan shouting to one another excitedly, because their lines had not closed asthey’dexpected.Thecrayonhadnotcomebacktothestartingpoint;itwasnowonthe“other”sideofthesurface.Thatwassurprisenumberone:youhavetogotwicearoundaMöbiusstriptogetbacktowhereyoustarted.Suddenlyoneboybeganmeltingdown.Whenhe realizedhiscrayonhadn’t

come back to its starting point, he thought he’d done something wrong. Nomatterhowmanytimeswereassuredhimthatthiswassupposedtohappen,thathewasdoingagreat job,andthatheshouldjustgoaroundthestriponemoretime,itdidn’thelp.Itwastoolate.Hewasonthefloor,wailing,inconsolable.

WithsometrepidationIaskedtheclasstotrythenextactivity.Whatdidtheythinkwouldhappenif theytooktheirscissorsandcutneatlydownthemidlineallthewayalongthelengthofthestrip?Itwillfallapart!Itwillmaketwopieces!theyguessed.Butaftertheytriedit

and something incredible happened (the strip remained in one piece but grewtwiceaslong),therewereevenmoresquealsofsurpriseanddelight.Itwaslikeamagictrick.After that it was hard to hold the students’ attention. They were too busy

trying theirownexperiments,makingnewkindsofMöbius stripswith twoorthree half twists in them and cutting them lengthwise into halves, thirds, orquarters,producingallsortsoftwistednecklaces,chains,andknots,allthewhileshoutingvariationsof“Hey, lookwhat I found!”But I still can’tgetover thatone littleboyI traumatized.Andapparentlymy lessonwasn’t thefirst tohavedrivenastudenttotears.ViHartwassofrustratedbyherboringmathcoursesinhighschoolthatshe

began doodling in class, sketching snakes and trees and infinite chains ofshrinkingelephants,insteadoflisteningtotheteacherdroningon.Vi,whocallsherselfa“full-timerecreationalmathemusician,”haspostedsomeofherdoodlesonYouTube.They’venowbeenwatchedhundredsofthousandsoftimes,andinthe case of the elephants, more than a million. She, and her videos, arebreathtakinglyoriginal.TwoofmyfavoriteshighlightthefreakypropertiesofMöbiusstripsthrough

an inventive use of music and stories. In the less baffling of the two, her“Möbius music box” plays a theme from a piece of music she composed,inspiredbytheHarryPotterbooks.

Themelodyisencodedasaseriesofholespunchedthroughatape,whichisthenfed throughastandardmusicbox.Her innovationwas to twist theendsof thetapeandjointhemtogethertoformaMöbiusstrip.Bycrankingthecrankonthemusic box, Vi feeds the tape through the device and themelody plays in thenormalfashion.Butaboutfiftyseconds into thevideo, the loopcompletesonecircuit, and because of the half twist in theMöbius strip, themusic box nowbeginsplayingwhatwasoriginallythebackofthepunchedtape,upsidedown.Hencethesamemelodybeginsagainbutnowwithallthenotesinverted.Highnotesbecomelownotes,andlownotesbecomehigh.They’restillplayedinthesameorder,butupsidedown,thankstothesomersaultsimposedbytheMöbiusstructure.Foranevenmorestrikingexampleofthetopsy-turvyimplicationsofMöbius

strips, in “Möbius Story: Wind and Mr. Ug,” Vi tells a bittersweet story ofunattainablelove.AfriendlylittletrianglenamedWind,drawnwithanerasablemarker,unknowinglylivesinaflatworldmadeoutofclearacetateandshapedlikeaMöbiusstrip.She’slonesomebuteverhopeful,eagertomeettheworld’sonly other inhabitant, a mysterious gent named Mr. Ug who lives one door

down.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

Möbius strips.Escher used them in his drawings of ants trapped in an eternalloop. Sculptors and stone carvers, like Max Bill and Keizo Ushio, haveincorporatedMöbiusmotifsintheirmassiveworks.Perhaps themostmonumentalofallMöbiusstructures is thatbeingplanned

for theNationalLibraryofKazakhstan. Itsdesign,by theDanisharchitecturalfirm BIG, calls for spiraling public paths that coil up and then down and inwhich “like a yurt the wall becomes the roof, which becomes floor, whichbecomesthewallagain.”

ThepropertiesofMöbiusstripsofferdesignadvantagestoengineersaswell.

For example, a continuous-loop recording tape in the shape of aMöbius stripdoublestheplayingtime.TheB.F.GoodrichCompanypatentedaMöbiusstripconveyor belt,which lasts twice as long as a conventional belt since itwearsevenly on “both” sides of its surface (you knowwhat Imean).OtherMöbiuspatents includenoveldesigns forcapacitors,abdominal surgical retractors,and

self-cleaningfiltersfordry-cleaningmachines.But perhaps the niftiest application of topology is one that doesn’t involve

Möbius strips at all. It’s a variationon the themeof twists and links, andyoumight find it helpful next time you have guests over for brunch on a Sundaymorning.It’stheworkofGeorgeHart,Vi’sdad.He’sageometerandasculptor,formerlyacomputerscienceprofessoratStonyBrookUniversityandthechiefofcontentatMoMath, theMuseumofMathematicsinNewYorkCity.Georgehasdevised away to slice abagel in half such that the twopieces are lockedtogetherlikethelinksofachain.

Theadvantage,besidesleavingyourguestsagog,isthatitcreatesmoresurfacearea—andhencemoreroomforthecreamcheese.

28.ThinkGlobally

THE MOST FAMILIAR ideas of geometry were inspired by an ancient vision—avisionof theworldasflat.FromthePythagoreantheoremtoparallel linesthatnever meet, these are eternal truths about an imaginary place, the two-dimensionallandscapeofplanegeometry.ConceivedinIndia,China,Egypt,andBabyloniamorethan2,500yearsago

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

us should try to learn a little about spherical geometry and its moderngeneralization, differential geometry. The basic ideas here are only about 200yearsold.PioneeredbyCarlFriedrichGaussandBernhardRiemann,differentialgeometry underpins such imposing intellectual edifices as Einstein’s generaltheory of relativity. At its heart, however, are beautiful concepts that can begraspedbyanyonewho’severriddenabicycle,lookedataglobe,orstretchedarubber band. And understanding them will help you make sense of a fewcuriositiesyoumayhavenoticedinyourtravels.For example, when I was little, my dad used to enjoy quizzing me about

geography.Which is farther north, he’d ask, Rome or NewYork City?MostpeoplewouldguessNewYorkCity,butsurprisinglythey’reatalmostthesamelatitude,withRomebeingjustabitfarthernorth.Ontheusualmapoftheworld(themisleadingMercatorprojection,whereGreenlandappearsgigantic),itlookslikeyoucouldgostraightfromNewYorktoRomebyheadingdueeast.Yetairlinepilotsnevertakethatroute.TheyalwaysflynortheastoutofNew

York, hugging the coast ofCanada. I used to think theywere staying close tolandforsafety’ssake,butthat’snotthereason.It’ssimplythemostdirectroutewhenyoutaketheEarth’scurvatureintoaccount.TheshortestwaytogetfromNewYorktoRomeistogopastNovaScotiaandNewfoundland,headoutovertheAtlantic,andfinallypasssouthofIrelandandflyacrossFrancetoarriveinsunnyItaly.

Thiskindofpathontheglobeiscalledanarcofagreatcircle.Likestraight

lines in ordinary space, great circles on a sphere contain the shortest pathsbetweenanytwopoints.They’recalledgreatbecausethey’rethelargestcirclesyou can have on a sphere.Conspicuous examples include the equator and thelongitudinalcirclesthatpassthroughtheNorthandSouthPoles.Another property that lines and great circles share is that they’re the

straightestpathsbetweentwopoints.Thatmightsoundstrange—allpathsonaglobe are curved, sowhat dowemeanby “straightest”?Well, somepaths aremorecurvedthanothers.Thegreatcirclesdon’tdoanyadditionalcurvingaboveandbeyondwhatthey’reforcedtodobyfollowingthesurfaceofthesphere.Here’s a way to visualize this. Imagine you’re riding a tiny bicycle on the

surfaceofaglobe, andyou’re trying to stayonacertainpath. If it’spartofagreat circle, you can keep the frontwheel pointed straight ahead at all times.That’sthesenseinwhichgreatcirclesarestraight.Incontrast,ifyoutrytoridealongalineoflatitudenearoneofthepoles,you’llhavetokeepthehandlebarsturned.

Ofcourse,assurfacesgo,theplaneandthesphereareabnormallysimple.Thesurfaceofahumanbody,oratincan,orabagelwouldbemoretypical—theyallhavefarlesssymmetry,aswellasvariouskindsofholesandpassagewaysthatmakethemmoreconfusingtonavigate.Inthismoregeneralsetting,findingtheshortest path between any two points becomes a lot trickier. So rather thandelving into technicalities, let’s stick to an intuitive approach. This is whererubberbandscomeinhandy.Specifically,imagineaslipperyelasticstringthatalwayscontractsasfarasit

can while remaining on an object’s surface. With its help, we can easilydetermine the shortest path betweenNewYork and Rome or, for that matter,betweenanytwopointsonanysurface.Tietheendsofthestringtothepointsofdepartureandarrivalandletthestringpullitselftightwhileitcontinuesclingingto the surface’s contours.When the string is as taut as theseconstraints allow,voilà!Ittracestheshortestpath.Onsurfaces justa littlemorecomplicated thanplanesorspheres,something

strangeandnewcanhappen:many locallyshortestpathscanexistbetweenthesametwopoints.Forexample,considerthesurfaceofasoupcan,withonepointlyingdirectlybelowtheother.

Thentheshortestpathbetweenthemisclearlyalinesegment,asshownabove,

and our elastic string would find that solution. So what’s new here? Thecylindrical shape of the can opens up new possibilities for all kinds ofcontortions.Supposewerequirethatthestringencirclesthecylinderoncebeforeconnectingtothesecondpoint.(ConstraintslikethisareimposedonDNAwhenitwraps around certain proteins in chromosomes.)Nowwhen the string pullsitselftaut,itformsahelix,likethecurvesonoldbarbershoppoles.

Thishelicalpathqualifiesasanothersolutiontotheshortest-pathproblem,in

the sense that it’s the shortest of the candidatepathsnearby. If younudge thestringalittle,itwouldnecessarilygetlongerandthencontractbacktothehelix.Youcouldsayit’sthelocallyshortestpath—theregionalchampionofallthosethatwraponcearoundthecylinder.(Bytheway,thisiswhythesubjectiscalleddifferentialgeometry:itstudiestheeffectsofsmalllocaldifferencesonvariouskindsofshapes,suchasthedifferenceinlengthbetweenthehelicalpathanditsneighbors.)Butthat’snotall.There’sanotherchampthatwindsaroundtwice,andanother

that goes around three times, and so on. There are infinitely many locallyshortest paths on a cylinder! Of course, none of these helices is the globallyshortestpath.Thestraight-linepathisshorterthanallofthem.Likewise,surfaceswithholesandhandlespermitmanylocallyshortestpaths,

distinguished by their pattern ofweaving around various parts of the surface.Thefollowingsnapshot fromavideoby themathematicianKonradPolthierofthe Free University of Berlin illustrates the non-uniqueness of these locallyshortestpaths,orgeodesics,onthesurfaceofanimaginaryplanetshapedlikeafigureeight,asurfaceknowninthetradeasatwo-holedtorus:

Thethreegeodesicsshownherevisitverydifferentpartsof theplanet, therebyexecuting different loop patterns. But what they all have in common is theirsuperiordirectnesscomparedtothepathsnearby.Andjustlikelinesonaplaneorgreatcirclesonasphere,thesegeodesicsarethestraightestpossiblecurvesonthe surface. They bend to conform to the surface but don’t bendwithin it.Tomakethisclear,Polthierhasproducedanotherilluminatingvideo.

Here,anunmannedmotorcycleridesalongageodesichighwayonatwo-holedtorus,followingthelayoftheland.Theremarkablethingisthatitshandlebarsare locked straight ahead; it doesn’t need to steer to stay on the road. Thisunderscores the earlier impression that geodesics, like great circles, are thenaturalgeneralizationofstraightlines.With all these flights of fancy, you may be wondering if geodesics have

anythingtodowithreality.Ofcoursetheydo.Einsteinshowedthatlightbeamsfollow geodesics as they sail through the universe. The famous bending ofstarlightaroundthesun,detectedintheeclipseobservationsof1919,confirmedthat light travels on geodesics through curved space-time, with the warpingbeingcausedbythesun’sgravity.

At amore down-to-earth level, themathematics of finding shortest paths iscritical to the routing of traffic on the Internet. In this situation, however, therelevant space is a gargantuanmaze of addresses and links, as opposed to thesmoothsurfacesconsideredabove,andthemathematicalissueshavetodowiththespeedofalgorithms—what’sthemostefficientwaytofindtheshortestpaththroughanetwork?Giventhemyriadofpotentialroutes,theproblemwouldbeoverwhelmingwereitnotfortheingenuityofthemathematiciansandcomputerscientistswhocrackedit.Sometimes when people say the shortest distance between two points is a

straight line, they mean it figuratively, as a way of ridiculing nuance andaffirmingcommonsense.Inotherwords,keepitsimple.Butbattlingobstaclescangiverisetogreatbeauty—somuchsothatinart,andinmath,it’softenmorefruitfultoimposeconstraintsonourselves.Thinkofhaiku,orsonnets,ortellingthestoryofyourlifeinsixwords.Thesameistrueofall themaththat’sbeencreatedtohelpyoufindtheshortestwayfromheretotherewhenyoucan’ttaketheeasywayout.Twopoints.Manypaths.Mathematicalbliss.

29.AnalyzeThis!

MATH SWAGGERS WITH an intimidating air of certainty. Like a Mafia capo, itcomesacrossasdecisive,unyielding,andstrong.It’llmakeyouanargumentyoucan’trefuse.Butinprivate,mathisoccasionallyinsecure.Ithasdoubts.Itquestionsitself

and isn’talwayssure it’s right.Especiallywhere infinity isconcerned. Infinitycan keep math up at night, worrying, fidgeting, feeling existential dread. Fortherehavebeen times in thehistoryofmathwhenunleashing infinitywroughtsuchmayhem,therewerefearsitmightblowupthewholeenterprise.Andthatwouldbebadforbusiness.In the HBO series The Sopranos, mob boss Tony Soprano consults a

psychiatrist,seekingtreatmentforanxietyattacks,tryingtounderstandwhyhismotherwantstohavehimkilled,thatsortofthing.Beneathatoughexteriorofcertaintyliesaveryconfusedandfrightenedperson.

Inmuchthesameway,calculusputitselfonthecouchjustwhenitseemedto

beatitsmostlethal.Afterdecadesoftriumph,ofmowingdownalltheproblemsthatstoodinitsway,itstartedtobecomeawareofsomethingrottenatitscore.The very things that had made it most successful—its brutal skill andfearlessnessinmanipulatinginfiniteprocesses—werenowthreateningtodestroyit. And the therapy that eventually helped it through this crisis came to beknown,coincidentally,asanalysis.Here’sanexampleofthekindofproblemthatworriedthemathematiciansof

the1700s.Considertheinfiniteseries

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

Disorientedbyaninfinitelylongexpressionlikethis,anoptimistmighthopethatsomeoftheoldrules—therulesforgedfromexperiencewithfinite sums—wouldstillapply.Forexample,weknowthat1+2=2+1;whenweaddtwoormore numbers in a finite sum, we can always switch their order withoutchangingtheresult:a+bequalsb+a(thecommutativelawofaddition).Andwhen there are more than two terms, we can always insert parentheses withabandon, grouping the terms however we like, without affecting the ultimateanswer.Forinstance,(1+2)+4=1+(2+4);adding1and2first,then4,givesthesameanswerasadding2and4first,then1.Thisiscalledtheassociativelawofaddition.Itworksevenifsomenumbersarebeingsubtracted,aslongasweremember that subtracting a number is the same as adding its negative. Forexample,considerathree-termversionoftheseriesabove,andask:Whatis1–1+1?Wecouldviewitaseither(1–1)+1or1+(–1+1),whereinthatsecondsetofparentheseswe’veaddednegative1insteadofsubtracting1.Eitherway,theanswercomesouttobe1.Butwhenwe try togeneralize theserules to infinite sums,a fewunpleasant

surpriseslieinstoreforus.Lookatthecontradictionthatoccursifwetrotouttheassociativelawandtrustinglyapplyitto1–1+1–1+1–1+∙∙∙.Ontheonehand, it appearswecanannihilate thepositive andnegative1sbypairingthemofflikeso:

On the other hand, we could just as well insert the parentheses like this andconcludethatthesumis1:

Neitherargumentseemsmoreconvincingthantheother,soperhapsthesumis

both0and1?Thatpropositionsoundsabsurdtoustoday,butatthetimesomemathematicianswerecomfortedbyitsreligiousovertones.Itremindedthemofthe theological assertion that God created the world from nothing. As themathematicianandpriestGuidoGrandiwrotein1703,“Byputtingparenthesesintotheexpression1–1+1–1+∙∙∙indifferentways,Ican,ifIwant,obtain0or1.Butthentheideaofcreationexnihiloisperfectlyplausible.”Nevertheless, it appears that Grandi favored a third value for the sum,

differentfromeither0or1.Canyouguesswhathethoughtitshouldbe?Thinkofwhatyou’dsayifyouwerekiddingbuttryingtosoundscholastic.Right—Grandi believed the true sum was . And far superior

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

Nowleavethefirst1ontheright-handsidealoneandlookatalltheotherterms.They harbor their own copy of S, positioned to the right of that first 1 andsubtractedfromit:

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

a new breed of analysts put all of calculus and its infinite processes (limits,derivatives,integrals,infiniteseries)onafirmfoundation,onceandforall.Theyrebuiltthesubjectfromthegroundup,fashioningalogicalstructureassoundasEuclid’sgeometry.Twooftheirkeynotionsarepartialsumsandconvergence.Apartialsumisa

running total.You simply add up a finite number of terms and then stop. Forexample,ifwesumthefirstthreetermsoftheseries1–1+1–1+∙∙∙,weget1–1+1=1.Let’scallthisS3.HeretheletterSstandsfor“sum”andthesubscript

3indicatesthatweaddedonlythefirstthreeterms.Similarly,thefirstfewpartialsumsforthisseriesare

Thusweseethatthepartialsumsbobblebackandforthbetween0and1,withnotendencytosettledownto0or1,to ,ortoanythingelse.Forthisreason,mathematicians todaywould say that the series 1 – 1+1 – 1+ ∙ ∙ ∙ does notconverge.Inotherwords, itspartialsumsdon’tapproachany limitingvalueasmoreandmoretermsareincludedinthesum.Thereforethesumoftheinfiniteseriesismeaningless.So supposewe keep to the straight and narrow—no dallyingwith the dark

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

byfacingdownthesenewdemons,theanalystsofthe1800sdiscovereddeepersecretsat theheartofcalculusandthenexposedthemtothelight.Thelessonslearnedhaveprovedinvaluable,notjustwithinmathbutformath’sapplicationstoeverythingfrommusictomedicalimaging.Considerthisseries,knowninthetradeasthealternatingharmonicseries:

Instead of one step forward, one step back, the steps now get progressivelysmaller.It’sonestepforward,butonlyhalfastepback,thenathirdofthestepforward,afourthofastepback,andsoon.Noticethepattern:thefractionswithodddenominatorshaveplussignsinfrontofthem,whiletheevenfractionshavenegativesigns.Thepartialsumsinthiscaseare

And ifyougo farenough,you’ll find that theyhome inonanumberclose to0.69. In fact, the series can be proven to converge. Its limiting value is thenaturallogarithmof2,denotedln2andapproximatelyequalto0.693147.So what’s nightmarish here? On the face of it, nothing. The alternating

harmonicseriesseemslikeanice,well-behaved,convergentseries,thesortyourparentswouldapproveof.And that’s what makes it so dangerous. It’s a chameleon, a con man, a

slippery sicko that will be anything you want. If you add up its terms in adifferentorder,youcanmakeitsumtoanything.Literally.Itcanberearrangedtoconvergetoanyrealnumber:297.126,or–42π,or0,orwhateveryourheartdesires.It’s as if the serieshadutter contempt for the commutative lawof addition.

Merely by adding its terms in a different order, you can change the answer—somethingthatcouldneverhappenforafinitesum.Soeventhoughtheoriginalseries converges, it’s still capable of weirdness unimaginable in ordinaryarithmetic.Rather than prove this astonishing fact (a result known as the Riemann

rearrangementtheorem),let’slookataparticularlysimplerearrangementwhosesum is easy to calculate. Suppose we add two of the negative terms in thealternatingharmonicseriesforeveryoneofitspositiveterms,asfollows:

Next,simplifyeachofthebracketedexpressionsbysubtractingthesecondtermfromthefirstwhileleavingthethirdtermuntouched.Thentheseriesreducesto

After factoring out from all the fractions above and collecting terms, thisbecomes

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

we’ve seen throughout this book, even the most abstruse and far-fetchedconcepts of math often find application to practical things. The link in thepresent case is that in many parts of science and technology, from signalprocessingandacousticstofinanceandmedicine,it’susefultorepresentvariouskindsofcurves, sounds, signals,or imagesas sumsof simplercurves, sounds,signals,orimages.Whenthebasicbuildingblocksaresinewaves,thetechniqueis known as Fourier analysis, and the corresponding sums are called Fourierseries.Butwhentheseriesinquestionbearssomeofthesamepathologiesasthealternatingharmonic seriesand its equallyderanged relatives, theconvergencebehavioroftheFourierseriescanbeveryweirdindeed.Here, for example, is a Fourier series directly inspired by the alternating

harmonicseries:

Togetasenseforwhatthislookslike,let’sgraphthesumofitsfirsttenterms.

Thispartialsum(shownasasolidline)isclearlytryingtoapproximateamuchsimplercurve,awaveshapedliketheteethofasaw(shownbythedashedline).Notice,however,thatsomethinggoeswrongneartheedgesoftheteeth.Thesinewaves overshoot themark there and produce a strange finger that isn’t in thesawtoothwaveitself.Toseethismoreclearly,here’sazoomnearoneofthoseedges,atx=π:

Supposewetrytogetridofthefingerbyincludingmoretermsinthesum.Noluck.Thefingerjustbecomesthinnerandmovesclosertotheedge,butitsheightstaysaboutthesame.

Theblamecanbe laidat thedoorstepof thealternatingharmonicseries. Its

pathologies discussed earlier now contaminate the associated Fourier series.They’reresponsibleforthatannoyingfingerthatjustwon’tgoaway.This effect, commonly called the Gibbs phenomenon, is more than a

mathematical curiosity. Known since the mid-1800s, it now turns up in ourdigitalphotographsandonMRIscans.TheunwantedoscillationscausedbytheGibbs phenomenon can produce blurring, shimmering, and other artifacts atsharp edges in the image. In a medical context, these can be mistaken fordamagedtissue,ortheycanobscurelesionsthatareactuallypresent.Fortunately,analystsacenturyagopinpointedwhatcausesGibbsartifacts(see

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

30.TheHilbertHotel

INFEBRUARY2010Ireceivedane-mailfromawomannamedKimForbes.Hersix-year-oldson,Ben,hadaskedheramathquestionshecouldn’tanswer,andshewashopingIcouldhelp:Today is the 100th day of school. He was very excited and told meeverythingheknowsaboutthenumber100,includingthat100wasanevennumber.Hethentoldmethat101wasanoddnumberand1millionwasanevennumber,etc.Hethenpausedandasked:“Isinfinityevenorodd?”

IexplainedtoKimthatinfinityisneitherevennorodd.It’snotanumberin

the usual sense, and it doesn’t obey the rules of arithmetic. All sorts ofcontradictionswouldfollowifitdid.Forinstance,Iwrote,“ifinfinitywereodd,2timesinfinitywouldbeeven.Butbothareinfinity!Sothewholeideaofoddandevendoesnotmakesenseforinfinity.”Kimreplied:Thankyou.Benwas satisfiedwith that answerandkindof likes the ideathatinfinityisbigenoughtobebothoddandeven.

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

Cantor’sgroundbreakingworkonset theory.Cantorwasparticularlyinterestedin infinite sets of numbers andpoints, like the set {1, 2, 3, 4, . . .} of naturalnumbersandthesetofpointsonaline.Hedefinedarigorouswaytocomparedifferentinfinitesetsanddiscovered,shockingly,thatsomeinfinitiesarebiggerthanothers.At thetime,Cantor’s theoryprovokednot justresistance,butoutrage.Henri

Poincaré,oneoftheleadingmathematiciansoftheday,calledita“disease.”Butanothergiantoftheera,DavidHilbert,sawitasalastingcontributionandlaterproclaimed,“NooneshallexpelusfromtheParadisethatCantorhascreated.”My goal here is to give you a glimpse of this paradise. But rather than

working directly with sets of numbers or points, let me follow an approachintroducedbyHilberthimself.Hevividlyconveyedthestrangenessandwonder

ofCantor’s theoryby tellingaparableaboutagrandhotel,nowknownas theHilbertHotel.It’salwaysbookedsolid,yetthere’salwaysavacancy.For the Hilbert Hotel doesn’t have merely hundreds of rooms—it has an

infinitenumberof them.Wheneveranewguestarrives, themanagershifts theoccupantofroom1toroom2,room2toroom3,andsoon.Thatfreesuproom1 for the newcomer and accommodates everyone else as well (thoughinconveniencingthembythemove).Nowsuppose infinitelymanynewguests arrive, sweatyand short-tempered.

Noproblem.Theunflappablemanagermovestheoccupantofroom1toroom2,room2toroom4,room3toroom6,andsoon.Thisdoublingtrickopensupalltheodd-numberedrooms—infinitelymanyofthem—forthenewguests.Later thatnight, anendless convoyofbuses rumblesup to reception.There

areinfinitelymanybuses,andworsestill,eachoneisloadedwithaninfinityofcrabby people demanding that the hotel live up to itsmotto, “There’s alwaysroomattheHilbertHotel.”Themanagerhasfacedthischallengebeforeandtakesitinstride.Firsthedoesthedoublingtrick.Thatreassignsthecurrentgueststotheeven-

numbered rooms and clears out all the odd-numbered ones—a good start,becausehenowhasaninfinitenumberofroomsavailable.But is that enough? Are there really enough odd-numbered rooms to

accommodatetheteeminghordeofnewguests?Itseemsunlikely,sincetherearesomethinglikeinfinitysquaredpeopleclamoringfortheserooms.(Whyinfinitysquared?Becausetherewasaninfinitenumberofpeopleoneachofaninfinitenumber of buses, and that amounts to infinity times infinity, whatever thatmeans.)Thisiswherethelogicofinfinitygetsveryweird.Tounderstandhowthemanagerisgoingtosolvehislatestproblem,ithelpsto

visualizeallthepeoplehehastoserve.

Of course, we can’t show literally all of them here, since the diagramwouldneed to be infinite in both directions. But a finite version of the picture isadequate.Thepoint is thatanyspecific buspassenger (youraunt Inez, say,onvacationfromLouisville)issuretoappearonthediagramsomewhere,aslongasweincludeenoughrowsandcolumns.Inthatsense,everybodyoneverybusisaccountedfor.Younamethepassenger,andheorsheiscertaintobedepictedatsomefinitenumberofstepseastandsouthofthediagram’scorner.The manager’s challenge is to find a way to work through this picture

systematically. He needs to devise a scheme for assigning rooms so thateverybodygetsoneeventually,afteronlya finitenumberofotherpeoplehavebeenserved.Sadly, the previous manager hadn’t understood this, and mayhem ensued.

Whenasimilarconvoyshoweduponhiswatch,hebecamesoflusteredtryingtoprocessallthepeopleonbus1thathenevergotaroundtoanyotherbus,leavingallthoseneglectedpassengersscreamingandfurious.Illustratedonthediagrambelow, thismyopicstrategywouldcorrespond toapath thatmarchedeastward

alongrow1forever.

Thenewmanager,however,haseverythingundercontrol.Insteadoftending

to just one bus, he zigs and zags through the diagram, fanning out from thecorner,asshownbelow.

He startswith passenger 1 on bus 1 and gives her the first empty room. Thesecondandthirdemptyroomsgotopassenger2onbus1andpassenger1onbus2, both of whom are depicted on the second diagonal from the corner of thediagram. After serving them, the manager proceeds to the third diagonal andhandsoutasetofroomkeystopassenger1onbus3,passenger2onbus2,andpassenger3onbus1.Ihopethemanager’sprocedure—progressingfromonediagonaltoanother—

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

Cantorusedit toprovethat thereareexactlyasmanypositivefractions(ratiosp/qofpositivewholenumberspandq)astherearenaturalnumbers(1,2,3,4,...).That’samuchstrongerstatementthansayingbothsetsareinfinite.Itsaysthey are infinite to precisely the same extent, in the sense that a one-to-onecorrespondencecanbeestablishedbetweenthem.

You could think of this correspondence as a buddy system in which eachnatural number is paired with some positive fraction, and vice versa. Theexistenceofsuchabuddysystemseemsutterlyatoddswithcommonsense—it’sthesortofsophistrythatmadePoincarérecoil.Foritimplieswecouldmakeanexhaustivelistofallpositivefractions,eventhoughthere’snosmallestone!And yet there is such a list. We’ve already found it. The fraction p/q

correspondstopassengerponbusq,andtheargumentaboveshowsthateachofthesefractionscanbepairedoffwithacertainnaturalnumber1,2,3,...,givenbythepassenger’sroomnumberattheHilbertHotel.The coup de grâce is Cantor’s proof that some infinite sets are bigger than

this. Specifically, the set of real numbers between 0 and 1 is uncountable—itcan’t be put in one-to-one correspondence with the natural numbers. For thehospitality industry, this means that if all these real numbers show up at thereception desk and bang on the bell, there won’t be enough rooms for all ofthem,evenattheHilbertHotel.Theproof is by contradiction.Suppose each real number couldbegiven its

ownroom.Thentherosterofoccupants,identifiedbytheirdecimalexpansionsandlistedbyroomnumber,wouldlooksomethinglikethis:

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

contradiction. For instance, to construct one that appears nowhere on the listshownabove,godownthediagonalandbuildanewnumberfromtheunderlineddigits:

Thedecimalsogeneratedis.6975...Butwe’renotdoneyet.Thenextstepistotakethisdecimalandchangeallits

digits, replacing each of them with any other digit between 1 and 8. Forexample,wecouldchangethe6toa3,the9toa2,the7toa5,andsoon.Thisnewdecimal.325...isthekiller.It’scertainlynotinroom1,sinceithas

a different first digit from the number there. It’s also not in room 2, since itssecond digit disagrees. In general, it differs from the nth number in the nthdecimalplace.Soitdoesn’tappearanywhereonthelist!The conclusion is that the Hilbert Hotel can’t accommodate all the real

numbers.Therearesimplytoomanyofthem,aninfinitybeyondinfinity.And with that humbling thought, we come to the end of this book, which

beganwithasceneinanotherimaginaryhotel.ASesameStreetcharacternamedHumphrey, working the lunch shift at the Furry Arms, took an order from aroomful of hungry penguins—“Fish, fish, fish, fish, fish, fish”—and soonlearnedaboutthepowerofnumbers.It’sbeenalongjourneyfromfishtoinfinity.Thanksforjoiningme.

Acknowledgments

Many friends andcolleagueshelped improve thisbookbygenerouslyofferingtheir sage advice—mathematical, stylistic, historical, andotherwise.Thanks toDougArnold,SheldonAxler,LarryBraden,DanCallahan,BobConnelly,TomGilovich, George Hart, Vi Hart, Diane Hopkins, Herbert Hui, Cindy Klauss,MichaelLewis,MichaelMauboussin,BarryMazur,EriNoguchi,CharliePeskin,Steve Pinker, Ravi Ramakrishna, David Rand, Richard Rand, Peter Renz,DouglasRogers,JohnSmillie,GrantWiggins,StephenYeung,andCarlZimmer.Othercolleaguescreatedimagesforthisbookorallowedmetoincludetheir

visual work. Thanks to Rick Allmendinger, Paul Bourke, Mike Field, BrianMadsen,NikDayman (Teamfresh),MarkNewman,KonradPolthier,ChristianRudderatOkCupid,SimonTatham,andJaneWang.I am immenselygrateful toDavidShipley for invitingme towrite theNew

YorkTimesseriesthatledtothisbook,andespeciallyforhisvisionofhowtheseries should be structured. Simplicity, simplicity, simplicity, urgedThoreau—andbothheandShipleywereright.GeorgeKalogerakis,myeditorattheTimes,wielded his pen lightly, moving commas, but only when necessary, whileprotecting me frommore serious infelicities. His confidence was enormouslyreassuring.KatieO’Brien on the production teammade sure themath alwayslookedrightandputupwiththerequisitetypographicalfussingwithgraceandgoodhumor.IfeelsofortunatetohaveKatinkaMatsoninmycornerasmyliteraryagent.

She championed this book from the beginning with an enthusiasm that wasinspirational.PaulGinsparg,JonKleinberg,TimNovikoff,andAndyRuinareaddraftsof

nearly every chapter, their only compensation being the pleasure of catchingclunkersandusingtheirbrilliantmindsforgoodinsteadofevil.Normallyit’sadragtobearoundsuchknow-it-alls,butthefactis,theydoknowitall.Andthisbookisthebetterforit.I’mtrulygratefulfortheireffortandencouragement.Thanks to Margy Nelson, the illustrator, for her playfulness and scientific

sensibility.Sheoftenfelttomelikeapartnerinthisproject,withherknackforfindingoriginalwaystoconveytheessenceofamathematicalconcept.Anywriterwould be blessed to haveAmandaCook as an editor.How can

anyone be so gentle andwise and decisive, all at the same time? Thank you,Amanda, for believing in this book and for helpingme shape every bit of it.EamonDolan,anotheroftheworld’sgreateditors,guidedthisproject(andme)

toward the finish line with a sure hand and infectious excitement. EditorialassistantsAshleyGilliamandBenHymanweremeticulousandfuntoworkwithand tookgood careof thebook at every stageof its development.CopyeditorTracyRoe taughtmeaboutappositives, apostrophes, andwordsaswords.Butmoreimportant(not“importantly”!),shesharpenedthewritingandthinkinginthesepages.ThanksaswelltopublicistMichelleBonanno,marketingmanagerAyeshaMirza, production editorRebeccaSpringer, productionmanagerDavidFutato,andtheentireteamatHoughtonMifflinHarcourt.Finally, let me add my most heartfelt thanks to my family. Leah and Jo,

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

Notes

1.FromFishtoInfinity

[>] Sesame Street: The video Sesame Street: 123 Count withMe (1997) isavailableforpurchaseonlineineitherVHSorDVDformat.[>] numbers.. .havelivesof theirown:Forapassionatepresentationof theideathatnumbershavelivesoftheirownandthenotionthatmathematicscanbeviewedasa formofart, seeP.Lockhart,AMathematician’sLament (BellevueLiteraryPress,2009).[>]“theunreasonableeffectivenessofmathematics”:Theessaythatintroducedthis now-famous phrase is E. Wigner, “The unreasonable effectiveness ofmathematics in the natural sciences,” Communications in Pure and AppliedMathematics,Vol.13,No.1(February1960),[>].Anonlineversionisavailableathttp://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html.For further reflectionson these ideasandon therelatedquestionofwhether

math was invented or discovered, see M. Livio, Is God a Mathematician?(Simon and Schuster, 2009), and R. W. Hamming, “The unreasonableeffectivenessofmathematics,”AmericanMathematicalMonthly,Vol.87,No.2(February 1980), available online at http://www-lmmb.ncifcrf.gov/~toms/Hamming.unreasonable.html.

2.RockGroups

[>]Theplayfulsideofarithmetic:AsIhopetomakeclear, thischapterowesmuchtotwobooks—oneapolemic,theotheranovel,bothofthembrilliant:P.Lockhart, A Mathematician’s Lament (Bellevue Literary Press, 2009), whichinspiredtherockmetaphorandsomeoftheexamplesusedhere;andY.Ogawa,TheHousekeeperandtheProfessor(Picador,2009).[>]achild’scuriosity:Foryoungreaderswholikeexploringnumbersandthepatterns they make, see H. M. Enzensberger, The Number Devil (HoltPaperbacks,2000).[>] hallmarkofanelegantproof:DelightfulbutmoreadvancedexamplesofvisualizationinmathematicsarepresentedinR.B.Nelsen,ProofswithoutWords(MathematicalAssociationofAmerica,1997).

3.TheEnemyofMyEnemy

[>]“Yeah,yeah”:FormoreofSidneyMorgenbesser’switticismsandacademicone-liners,seethesamplingatLanguageLog(August5,2004),“IfP,sowhynotQ?” online athttp://itre.cis.upenn.edu/%7Emyl/languagelog/archives/001314.html.[>] relationship triangles: Balance theory was first proposed by the socialpsychologist FritzHeider and has since been developed and applied by socialnetwork theorists, political scientists, anthropologists, mathematicians, andphysicists.Fortheoriginalformulation,seeF.Heider,“Attitudesandcognitiveorganization,” Journal of Psychology, Vol. 21 (1946), [>], and F. Heider,ThePsychology of Interpersonal Relations (John Wiley and Sons, 1958). For areviewofbalance theoryfromasocialnetworkperspective,seeS.Wassermanand K. Faust, Social Network Analysis (Cambridge University Press, 1994),chapter6.[>]polarizedstatesaretheonlystatesasstableasnirvana:Thetheoremthatabalancedstateinafullyconnectednetworkmustbeeitherasinglenirvanaofallfriendsor twomutuallyantagonisticfactionswasfirstproveninD.Cartwrightand F. Harary, “Structural balance: A generalization of Heider’s theory,”PsychologicalReview,Vol.63(1956),[>].Averyreadableversionofthatproof,andagentle introduction to themathematicsofbalance theory,hasbeengivenby two of my colleagues at Cornell: D. Easley and J. Kleinberg, Networks,Crowds,andMarkets(CambridgeUniversityPress,2010).In much of the early work on balance theory, a triangle of three mutual

enemies(andhencethreenegativesides)wasconsideredunbalanced.Iassumedthis implicitly when quoting the results about nirvana and the two-bloc statebeingtheonlyconfigurationsofafullyconnectednetworkinwhichalltrianglesarebalanced.However, some researchershavechallenged this assumptionandhave explored the implications of treating a triangle of three negatives asbalanced.Formoreonthisandothergeneralizationsofbalancetheory,seethebooksbyWassermanandFaustandbyEasleyandKleinbergcitedabove.[>]WorldWarI:TheexampleandgraphicaldepictionoftheshiftingalliancesbeforeWorldWarIarefromT.Antal,P.L.Krapivsky,andS.Redner,“Socialbalanceonnetworks:Thedynamicsoffriendshipandenmity,”PhysicaD,Vol.224 (2006), [>], available online at http://arxiv.org/abs/physics/0605183. Thispaper, written by three statistical physicists, is notable for recasting balance

theory in a dynamic framework, thus extending it beyond the earlier staticapproaches. For the historical details of the European alliances, see W. L.Langer,European Alliances and Alignments, 1871–1890, 2nd edition (Knopf,1956), andB.E. Schmitt,TripleAlliance andTripleEntente (HenryHolt andCompany,1934).

4.Commuting

[>]revisitmultiplicationfromscratch:KeithDevlinhaswrittenaprovocativeseriesofessaysaboutthenatureofmultiplication:whatitis,whatitisnot,andwhycertainwaysofthinkingaboutitaremorevaluableandreliablethanothers.Hearguesinfavorofthinkingofmultiplicationasscaling,notrepeatedaddition,andshowsthatthetwoconceptsareverydifferentinreal-worldsettingswhereunits are involved. See his January 2011 blog post “What exactly ismultiplication?” at http://www.maa.org/devlin/devlin_01_11.html, as well asthree earlier posts from 2008: “It ain’t no repeated addition”(http://www.maa.org/devlin/devlin_06_08.html);“It’sstillnotrepeatedaddition”(http://www.maa.org/devlin/devlin_0708_08.html); and “Multiplication andthose pesky British spellings” (http://www.maa.org/devlin/devlin_09_08.html).Theseessaysgeneratedalotofdiscussionintheblogosphere,especiallyamongschoolteachers. If you’re short on time, I’d recommend reading the one from2011first.[>]shoppingforanewpairofjeans:Forthejeansexample,theorderinwhichthetaxanddiscountareappliedmaynotmatter toyou—inbothscenariosyouenduppaying$43.20—butitmakesabigdifferencetothegovernmentandthestore!Intheclerk’sscenario(whereyoupaytaxbasedontheoriginalprice),youwouldpay$4 in tax; inyour scenario, only$3.20.Sohowcan the final pricecome out the same? It’s because in the clerk’s scenario the store gets to keep$39.20,whereasinyoursitwouldkeep$40.I’mnotsurewhatthelawrequires,and it may vary from place to place, but the rational thing would be for thegovernment tochargesales taxbasedontheactualpayment thestorereceives.Only your scenario satisfies this criterion. For further discussion, seehttp://www.facebook.com/TeachersofMathematics/posts/166897663338316.[>]financialdecisions:ForheatedonlineargumentsabouttherelativemeritsofaRoth 401(k) versus a traditional one, andwhether the commutative law hasanything to dowith these issues, see the Finance Buff, “Commutative law ofmultiplication” (http://thefinancebuff.com/commutative-law-of-multiplication.html), and the SimpleDollar, “The newRoth 401(k) versus thetraditional 401(k): Which is the better route?”(http://www.thesimpledollar.com/2007/06/20/the-new-roth-401k-versus-the-traditional-401k-which-is-the-better-route/).[>]attendingMITandkillinghimselfdidn’tcommute:ThisstoryaboutMurray

Gell-Mann is recounted inG. Johnson,StrangeBeauty (Knopf, 1999), [>]. InGell-Mann’s own words, he was offered admission to the “dreaded”Massachusetts Institute of Technology at the same time as he was“contemplating suicide, as befits someone rejected from the Ivy League. Itoccurredtomehowever(anditisaninterestingexampleofnon-commutationofoperators) that I could tryM.I.T. first and kill myself later, while the reverseorder of eventswas impossible.” This excerpt appears inH. Fritzsch,MurrayGell-Mann:SelectedPapers(WorldScientific,2009),[>].[>]developmentofquantummechanics:ForanaccountofhowHeisenbergandDirac discovered the role of non-commuting variables in quantummechanics,seeG.Farmelo,TheStrangestMan(BasicBooks,2009),[>].

5.DivisionandItsDiscontents

[>]MyLeftFoot:AclipofthescenewhereyoungChristystrugglesvaliantlyto answer the question “What’s twenty-five percent of a quarter?” is availableonline at http://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.Thetranscriptof his conversation with customer service is available athttp://verizonmath.blogspot.com/2006/12/transcription-jt.html. The audiorecordingisathttp://imgs.xkcd.com/verizon_billing.mp3.[>] you’re forced toconclude that1mustequal .9999 . . . :For readerswhomaystillfindithardtoacceptthat1=.9999...,theargumentthateventuallyconvincedmewas this: theymust be equal, because there’s no room for anyotherdecimaltofitbetweenthem.(Whereasif twodecimalsareunequal, theiraverageisbetweenthem,asareinfinitelymanyotherdecimals.)[>] almost all decimals are irrational: The amazing properties of irrationalnumbers are discussed at a highermathematical level on theMathWorld page“IrrationalNumber,”http://mathworld.wolfram.com/IrrationalNumber.html.Thesense in which the digits of irrational numbers are random is clarified athttp://mathworld.wolfram.com/NormalNumber.html.

6.Location,Location,Location

[>]EzraCornell’sstatue:FormoreaboutCornell,includinghisroleinWesternUnionandtheearlydaysofthetelegraph,seeP.Dorf,TheBuilder:ABiographyof Ezra Cornell (Macmillan, 1952); W. P. Marshall, EzraCornell (KessingerPublishing, 2006); and http://rmc.library.cornell.edu/ezra/index.html, an onlineexhibitioninhonorofCornell’s200thbirthday.[>]systemsforwritingnumbers:Ancientnumbersystemsandtheoriginsofthedecimal place-value system are discussed in V. J. Katz, A History ofMathematics,2ndedition(AddisonWesleyLongman,1998),andinC.B.BoyerandU.C.Merzbach,AHistoryofMathematics,3rdedition(Wiley,2011).Forachattieraccount,seeC.Seife,Zero(Viking,2000),chapter1.[>]Romannumerals:MarkChu-Carrollclarifiessomeofthepeculiarfeaturesof Roman numerals and arithmetic in this blog post:http://scienceblogs.com/goodmath/2006/08/roman_numerals_and_arithmetic.php[>]Babylonians:AfascinatingexhibitionofBabylonianmathisdescribedbyN.Wade,“Anexhibitionthatgetstothe(square)rootofSumerianmath,”NewYork Times (November 22, 2010), online athttp://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.Thenyoucanuseall five fingers on your other hand to keep track of howmany sets of twelveyou’vecounted.Thebase60systemusedbytheSumeriansmayhaveoriginatedinthisway.Formoreonthishypothesisandotherspeculationsabouttheoriginsofthebase60system,seeG.Ifrah,TheUniversalHistoryofNumbers (Wiley,2000),chapter9.

7.TheJoyofx

[>] Jo realized something about her big sister, Leah: For sticklers, Leah isactually twenty-one months older than Jo. Hence Jo’s formula is only anapproximation.Obviously![>]“WhenIwasatLosAlamos”:FeynmantellsthestoryofBethe’strickfor

squaring numbers close to 50 in R. P. Feynman, “Surely You’re Joking, Mr.Feynman!”(W.W.NortonandCompany,1985),[>].[>]moneyinvestedinthestockmarket:Theidentityabouttheeffectofequalup-and-downpercentageswingsinthestockmarketcanbeprovensymbolically,bymultiplying1+xby1–x,orgeometrically,bydrawingadiagramsimilartothatusedtoexplainBethe’strick.Ifyou’reinthemood,trybothapproachesasanexercise.[>] sociallyacceptableagedifference inaromance:The“halfyourageplusseven”ruleabouttheacceptableagegapinaromanticrelationshipiscalledthestandardcreepinessruleinthisxkcdcomic:http://xkcd.com/314/.

8.FindingYourRoots

[>] their struggle to find the roots: The quest for solutions to increasinglycomplicatedequations,fromquadratictoquintic,isrecountedinvividdetailinM.Livio,TheEquationThatCouldn’tBeSolved(SimonandSchuster,2005).[>]doublingacube’svolume:Formoreabouttheclassicproblemofdoublingthe cube, see http://www-history.mcs.st-and.ac.uk/HistTopics/Doubling_the_cube.html.[>] square roots of negative numbers: To learn more about imaginary andcomplexnumbers,theirapplications,andtheircheckeredhistory,seeP.J.Nahin,AnImaginaryTale(PrincetonUniversityPress,1998),andB.Mazur,ImaginingNumbers(Farrar,StrausandGiroux,2003).[>] dynamics ofNewton’smethod: For a superb journalistic account of JohnHubbard’swork,seeJ.Gleick,Chaos(Viking,1987),[>].Hubbard’sowntakeonNewton’smethodappears in section2.8of J.HubbardandB.B.Hubbard,Vector Calculus, Linear Algebra, and Differential Forms, 4th edition (MatrixEditions,2009).For readerswhowant todelve into themathematicsofNewton’smethod, a

moresophisticatedbutstillreadableintroductionisgiveninH.-O.PeitgenandP.H.Richter,TheBeautyofFractals(Springer,1986),chapter6,andalsoseethearticle by A. Douady (Hubbard’s collaborator) entitled “Julia sets and theMandelbrotset,”startingon[>]ofthesamebook.[>]Theborderlandslookedlikepsychedelichallucinations::HubbardwasnotthefirstmathematiciantoaskquestionsaboutNewton’smethodinthecomplexplane; Arthur Cayley had wondered about the same things in 1879. He toolookedatbothquadraticandcubicpolynomialsand realized that the first casewaseasyandthesecondwashard.Althoughhecouldn’thaveknownaboutthefractals that would be discovered a century later, he clearly understood thatsomethingnastycouldhappenwhenthereweremorethantworoots.Inhisone-pagearticle“Desiderataandsuggestions:No.3—theNewton-Fourierimaginaryproblem,”American JournalofMathematics,Vol.2,No.1 (March1879), [>],availableonlineathttp://www.jstor.org/pss/2369201,Cayley’sfinalsentenceisamarvel of understatement: “The solution is easy and elegant in the case of aquadricequation,butthenextsucceedingcaseofthecubicequationappearstopresentconsiderabledifficulty.”[>] The structure was a fractal: The snapshots shown in this chapter were

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

That point then becomes the next z. This process is repeated until z comessufficientlyclosetoarootor,equivalently,untilz3–1comessufficientlyclosetozero,where“sufficientlyclose”isaverysmalldistance,arbitrarilychosenbythe person who programmed the computer. All initial points that lead to aparticular root are then assigned the samecolor.Thus red labels all thepointsthatconvergetooneroot,greenlabelsanother,andbluelabelsthethird.ThesnapshotsoftheresultingNewtonfractalwerekindlyprovidedbySimon

Tatham.Formoreonhiswork,seehiswebpage“FractalsderivedfromNewton-Raphsoniteration”(http://www.chiark.greenend.org.uk/~sgtatham/newton/).Video animations of the Newton fractal have been created by Teamfresh.

Stunninglydeepzoomsintootherfractals,includingthefamousMandelbrotset,areavailableattheTeamfreshwebsite,http://www.hd-fractals.com.[>]templebuildersinIndia:ForanintroductiontotheancientIndianmethodsfor finding square roots, see D.W. Henderson and D. Taimina,ExperiencingGeometry,3rdexpandedandrevisededition(PearsonPrenticeHall,2005).

9.MyTubRunnethOver

[>] my first word problem: A large collection of classic word problems isavailableathttp://MathNEXUS.wwu.edu/Archive/oldie/list.asp.[>] filling a bathtub: Amore difficult bathtub problem appears in the 1941movie How Green Was My Valley. For a clip, seehttp://www.math.harvard.edu/~knill/mathmovies/index.html. And while you’rethere, you should also checkout this clip from thebaseball comedyLittleBigLeague: 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:fiveplus three, that’s eight.”After a fewmore blunders, one player finally gets itright: hours.

10.WorkingYourQuads

[>] top ten most beautiful or important equations: For books about greatequations,seeM.Guillen,FiveEquationsThatChanged theWorld (Hyperion,1995);G.Farmelo, ItMustBeBeautiful (Granta,2002);andR.P.Crease,TheGreat Equations (W.W. Norton and Company, 2009). There are several listspostedonlinetoo.I’dsuggeststartingwithK.Chang,“Whatmakesanequationbeautiful?,” New York Times (October 24, 2004),http://www.nytimes.com/2004/10/24/weekinreview/24chan.html.For oneof thefew lists that include the quadratic equation, 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’s approach to the quadratic equation isexplainedinV.J.Katz,AHistoryofMathematics,2ndedition(AddisonWesleyLongman,1998),[>].

11.PowerTools

[>]Moonlighting:Thebanteraboutlogarithmsisfromtheepisode“InGodWeStronglySuspect.” It originally airedonFebruary11,1986,during the show’ssecond season. A video clip is available athttp://opinionator.blogs.nytimes.com/2010/03/28/power-tools/.[>] functions:Forsimplicity, I’vereferred toexpressions likex²as functions,thoughtobemorepreciseIshouldspeakof“thefunctionthatmapsxintox².”Ihopethissortofabbreviationwon’tcauseconfusion,sincewe’veallseenitoncalculatorbuttons.[>]breathtakingparabolas:ForapromotionalvideoofthewaterfeatureattheDetroit airport, created by WET Design, see http://www.youtube.com/watch?v=VSUKNxVXE4E.Several homevideos of it are also available atYouTube.One of the most vivid is “Detroit Airport Water Feature” by PassTravelFool(http://www.youtube.com/watch?v=or8i_EvIRdE).WillHoffmanandDerekPaulBoylehave filmedan intriguingvideoof the

parabolas (along with their exponential cousins, curves called catenaries, sonamed for the shapeofhangingchains) that are all aroundus in the everydayworld. See “WNYC/NPR’s Radio Lab presents Parabolas (etc.)” online athttp://www.youtube.com/watch?v=rdSgqHuI-mw. Full disclosure: thefilmmakers say this video was inspired by a story I told on an episode ofRadiolab (“Yellow fluff and other curious encounters,” available athttp://www.radiolab.org/2009/jan/12/).[>]foldapieceofpaperinhalfmorethansevenoreighttimes:ForthestoryofBritneyGallivan’sadventuresinpaperfolding,seeB.Gallivan,“Howtofoldapaper in half twelve times: An ‘impossible challenge’ solved and explained,”Pomona, CA: Historical Society of Pomona Valley, 2002, online athttp://pomonahistorical.org/12times.htm. For a journalist’s account aimed atchildren, see I. Peterson, “Champion paper-folder,”Muse (July/August 2004),[>],availableonlineathttp://musemath.blogspot.com/2007/06/champion-paper-folder.html.TheMythBustershaveattempted to replicateBritney’sexperimenton their television show(http://kwc.org/mythbusters/2007/01/episode_72_underwater_car_and.html).[>]Weperceivepitchlogarithmically:Forreferencesandfurtherdiscussionofmusicalscalesandour(approximately)logarithmicperceptionofpitch,seeJ.H.McDermott and A. J. Oxenham, “Music perception, pitch, and the auditory

system,” Current Opinion in Neurobiology, 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?Distinctintuitionsofthenumberscale inWestern andAmazonian indigene cultures,”Science,Vol. 320 (2008),pp. 1217–1220. Popular accounts of this study are available at ScienceDaily(http://www.sciencedaily.com/releases/2008/05/080529141344.htm) and in anepisodeofRadiolabcalled“Numbers”(http://www.radiolab.org/2009/nov/30/).

12.SquareDancing

[>] Pythagorean theorem: The ancient Babylonians, Indians, and Chineseappear to have been aware of the content of the Pythagorean theorem severalcenturies before Pythagoras and the Greeks. For more about the history andsignificanceofthetheorem,aswellasasurveyofthemanyingeniouswaystoprove it, see E.Maor,The Pythagorean Theorem (PrincetonUniversity Press,2007).[>] hypotenuse: On p. xiii of his book, Maor explains that the word“hypotenuse”means“stretchedbeneath”andpointsoutthatthismakessenseifthe right triangle is viewed with its hypotenuse at the bottom, as depicted inEuclid’s Elements. He also notes that this interpretation fits well with theChinesewordforhypotenuse:“hsien,astringstretchedbetweentwopoints(asinalute).”[>]buildingthesquaresoutofmanylittlecrackers:Childrenandtheirparentswill enjoy the edible illustrations of the Pythagorean theorem suggested byGeorgeHartinhispostfortheMuseumofMathematics“Pythagoreancrackers,”http://momath.org/home/pythagorean-crackers/.[>] another proof: If you enjoy seeing different proofs, a nicely annotatedcollectionofdozensofthem—withcreatorsrangingfromEuclidtoLeonardodaVincitoPresidentJamesGarfield—isavailableatAlexanderBogomolny’sblogCuttheKnot.Seehttp://www.cut-the-knot.org/pythagoras/index.shtml.[>] missing steps:With any luck, the first proof in the chapter should havegivenyouanAha!sensation.Buttomaketheargumentcompletelyairtight,wealso need to prove that the pictures aren’t deceiving us—in otherwords, theytruly have the properties they appear to have. A more rigorous proof wouldestablish,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).AnotherexcellentoptionisafreedownloadablePDFdocumentbyRichardFitzpatrickgivinghisownmoderntranslation of Euclid’s Elements, available athttp://farside.ph.utexas.edu/euclid.html.[>]ThomasJefferson:FormoreaboutThomasJefferson’sreverenceforEuclidand Newton and his use of their axiomatic approach in the Declaration ofIndependence, see I. B. Cohen, Science and the Founding Fathers (W. W.Norton and Company, 1995), [>], as well as J. Fauvel, “Jefferson andmathematics,” http://www.math.virginia.edu/Jefferson/jefferson.htm, especiallythe page about the Declaration of Independence:http://www.math.virginia.edu/Jefferson/jeff_r(4).htm.[>]equilateraltriangle:ForEuclid’sversionoftheequilateraltriangleproof,inGreek,seehttp://en.wikipedia.org/wiki/File:Euclid-proof.jpg.[>]logicalandcreative:Ihaveglossedoveranumberofsubtletiesinthetwoproofspresented in this chapter.For example, in the equilateral triangleproof,weimplicitlyassumed(asEucliddid)thatthetwocirclesintersectsomewhere—inparticular,atthepointwelabeledC.ButtheexistenceofthatintersectionisnotguaranteedbyanyofEuclid’saxioms;oneneedsanadditionalaxiomaboutthecontinuityofthecircles.BertrandRussell,amongothers,notedthislacuna:B. Russell, “The Teaching of Euclid,”Mathematical Gazette, Vol. 2, No. 33(1902), [>], available online at http://www-history.mcs.st-and.ac.uk/history/Extras/Russell_Euclid.html.Anothersubtletyinvolvestheimplicituseoftheparallelpostulateintheproof

thattheanglesofatrianglesumto180degrees.Thatpostulateiswhatgaveuspermissiontoconstructthelineparalleltothetriangle’sbase.Inothertypesofgeometry (known as non-Euclidean geometries), there might exist no lineparalleltothebase,orinfinitelymanysuchlines.In thosekindsofgeometries,whichareeverybit as logicallyconsistent asEuclid’s, theanglesof a triangledon’talwaysaddupto180degrees.Thus, thePythagoreanproofgivenhere ismorethanjuststunninglyelegant;itrevealssomethingdeepaboutthenatureofspace itself. For more commentary on these issues, see the blog post by A.Bogomolny, “Angles in triangle add to 180°,” http://www.cut-the-

knot.org/triangle/pythpar/AnglesInTriangle.shtml,andthearticlebyT.Beardon,“When the angles of a triangle don’t add up to 180 degrees,”http://nrich.maths.org/1434.

14.TheConicConspiracy

[>]parabolasandellipses:Forbackgroundonconicsectionsandreferencestothe vast literature on them, seehttp://mathworld.wolfram.com/ConicSection.html andhttp://en.wikipedia.org/wiki/Conic_section.Forreaderswithsomemathematicaltraining, a large amount of interesting and unusual information has beencollected by James B. Calvert on his website; 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).[>] parabolic mirror:You’ll gain a lot of intuition by watching the onlineanimationscreatedbyLouTalmananddiscussedonhispage“Thegeometryofthe conic sections,”http://rowdy.mscd.edu/~talmanl/HTML/GeometryOfConicSections.html. Inparticular, watch http://clem.mscd.edu/~talmanl/HTML/ParabolicReflector.htmlandfixonasinglephotonas itapproachesand thenbouncesoff theparabolicreflector.Thenlookatallthephotonsmovingtogether.You’llneverwanttotanyour facewithasun reflectoragain.Theanalogousanimation foranellipse isshownathttp://rowdy.mscd.edu/~talmanl/HTML/EllipticReflector.html.

15.SineQuaNon

[>]sunrisesandsunsets:ThechartsshowninthetextareforJupiter,Florida,using data from 2011. For convenience, the times of sunrise and sunset havebeenexpressedrelativetoEasternStandardTime(theUTC-05:00timezone)allyear long toavoid theartificialdiscontinuitiescausedbydaylight-saving time.You can create similar charts of sunrise and sunset for your own location atwebsites such as http://ptaff.ca/soleil/?lang=en_CA orhttp://www.gaisma.com/en/.Students seem to find these charts surprising (for instance, some of them

expectthecurvestobetriangularinappearanceinsteadofroundedandsmooth),whichcanmakeforinstructiveclassroomactivitiesatthehigh-schoolormiddle-school level.For apedagogical case study, seeA.Friedlander andT.Resnick,“Sunrise, sunset,”MontanaMathematics Enthusiast, Vol. 3,No. 2 (2006),[>],available athttp://www.math.umt.edu/tmme/vol3no2/TMMEvol3no2_Israel_pp249_255.pdf.Deriving formulas for the times of sunrise and sunset is complicated, both

mathematically and in terms of the physics involved. See, for example, T. L.Watts’s webpage “Variation in the time of sunrise” athttp://www.physics.rutgers.edu/~twatts/sunrise/sunrise.html. Watts’s analysisclarifieswhy the timesofsunriseandsunsetdonotvaryassimplesinewavesthroughout theyear.Theyalso includea secondharmonic (a sinewavewithaperiodofsixmonths),mainlyduetoasubtleeffectoftheEarth’stiltthatcausesasemiannualvariationinlocalnoon,thetimeofdaywhenthesunishighestinthesky.Happily,thistermisthesameintheformulasforbothsunriseandsunsettimes.Sowhenyousubtractonefromtheothertocomputethelengthoftheday(thenumberofhoursbetweensunriseandsunset),thesecondharmoniccancelsout.What’sleftisverynearlyaperfectsinewave.MoreinformationaboutallthiscanbefoundontheWebbysearchingfor“the

EquationofTime.”(Seriously—that’swhatit’scalled!)AgoodstartingpointisK.Taylor’swebpage“Theequationoftime:Whysundialtimediffersfromclocktime depending on time of year,”http://myweb.tiscali.co.uk/moonkmft/Articles/EquationOfTime.html, or theWikipediapagehttp://en.wikipedia.org/wiki/Equation_of_time.[>] trigonometry:ThesubjectislovinglysurveyedinE.Maor,TrigonometricDelights(PrincetonUniversityPress,1998).

[>]patternformation:Forabroadoverviewofpatternsinnature,seeP.Ball,The Self-Made Tapestry (Oxford University Press, 1999). The mathematicalmethods in this field are presented at a graduate level in R. Hoyle, PatternFormation (CambridgeUniversity Press, 2006). Formathematical analyses ofzebra stripes, butterfly-wingpatterns, andother biological examples of patternformation, see J. D. Murray, Mathematical Biology: II. Spatial Models andBiomedicalApplications,3rdedition(Springer,2003).[>]cosmicmicrowavebackground:Theconnectionsbetweenbiologicalpatternformation and cosmology are one of the many delights to be found in JannaLevin’sbookHowtheUniverseGotItsSpots(PrincetonUniversityPress,2002).It’s structuredas a seriesofunsent letters tohermother and rangesgracefullyover the history and ideas of mathematics and physics, interwoven with anintimatediaryofayoungscientistassheembarksonhercareer.[>]inflationarycosmology:Forabriefintroductiontocosmologyandinflation,seetwoarticlesbyStephenBattersby:“Introduction:Cosmology,”NewScientist(September 4, 2006), online at http://www.newscientist.com/article/dn9988-introduction-cosmology.html, and “Best ever map of the early universerevealed,” New Scientist (March 17, 2006), online athttp://www.newscientist.com/article/dn8862-best-ever-map-of-the-early-universe-revealed.html. The case for inflation remains controversial, however.Its strengths and weaknesses are explained in P. J. Steinhardt, “The inflationdebate: Is the theory at the heart of modern cosmology deeply flawed?”ScientificAmerican(April2011),[>].

16.TakeIttotheLimit

[>]Zeno:ThehistoryandintellectuallegacyofZeno’sparadoxesarediscussedinJ.Mazur,Zeno’sParadox(Plume,2008).[>]circlesandpi:Foradelightfullyopinionatedandwittyhistoryofpi,seeP.Beckmann,AHistoryofPi(St.Martin’sPress,1976).[>]Archimedesusedasimilarstrategy:ThePBStelevisionseriesNova ranawonderful episode about Archimedes, infinity, and limits, called “InfiniteSecrets.” It originally aired on September 30, 2003. The program website(http://www.pbs.org/wgbh/nova/archimedes/) offers many online resources,includingtheprogramtranscriptandinteractivedemonstrations.[>]methodofexhaustion:ForreaderswishingtoseethemathematicaldetailsofArchimedes’smethod of exhaustion,Neal Carothers has used trigonometry(equivalenttothePythagoreangymnasticsthatArchimedesreliedon)toderivetheperimetersof the inscribedandcircumscribedpolygonsbetweenwhich thecircle is trapped; see http://personal.bgsu.edu/~carother/pi/Pi3a.html. PeterAlfeld’s webpage “Archimedes and the computation of pi” features aninteractiveJavaappletthatletsyouchangethenumberofsidesinthepolygons;see http://www.math.utah.edu/~alfeld/Archimedes/Archimedes.html. TheindividualstepsinArchimedes’soriginalargumentareofhistoricalinterestbutyou might find them disappointingly obscure; seehttp://itech.fgcu.edu/faculty/clindsey/mhf4404/archimedes/archimedes.html.[>] pi remains as elusive as ever: Anyone curious about the heroiccomputationsofpitoimmensenumbersofdigitsshouldenjoyRichardPreston’sprofile of the Chudnovsky brothers. Entitled “The mountains of pi,” 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

[>] Every year about a million American students take calculus: D. M.Bressoud,“Thecrisisofcalculus,”MathematicalAssociationofAmerica(April2007), available athttp://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’s law: Several articles, videos, and websites present the details ofSnell’s law and its derivation from Fermat’s principle (which states that lighttakesthepathofleasttime).Forexample,seeM.Golomb,“Elementaryproofsfor the equivalence of Fermat’s principle and Snell’s law,” AmericanMathematical Monthly, Vol. 71, No. 5 (May 1964), pp. 541–543, andhttp://en.wikibooks.org/wiki/Optics/Fermat%27s_Principle. Others providehistoricalaccounts;seehttp://en.wikipedia.org/wiki/Snell%27s_law.Fermat’s principlewas an early forerunner to themore general principle of

least action. For entertaining and deeply enlightening discussions of thisprinciple, including its basis in quantummechanics, 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—exceptforthoseveryclose to the classical path where the action is minimized (or, more precisely,made stationary). There the quantum interference becomes constructive,rendering those paths exceedingly more likely to be observed. This, inFeynman’saccount,iswhynatureobeysminimumprinciples.Thekeyisthatwelive in the macroscopic world of everyday experience where the actions arecolossal compared to Planck’s constant. In that classical limit, quantumdestructive interference becomes extremely strong and obliterates nearlyeverythingelsethatcouldotherwisehappen.

18.ItSlices,ItDices

[>]oncology:Formoreaboutthewaysthatintegralcalculushasbeenusedtohelpcancerresearchers,seeD.Mackenzie,“Mathematicalmodelingofcancer,”SIAMNews,Vol.37(January/February2004),andH.P.Greenspan,“Modelsforthe growth of a solid tumor by diffusion,” Studies in Applied Mathematics(December1972),pp.317–340.[>] solid common to two identical cylinders: The region common to twoidentical circular cylinders whose axes intersect at right angles is knownvariously as a Steinmetz solid or a bicylinder. For background, seehttp://mathworld.wolfram.com/SteinmetzSolid.html andhttp://en.wikipedia.org/wiki/Steinmetz_solid.TheWikipediapage also includesa very helpful computer animation that shows the Steinmetz solid emerging,ghostlike, from the intersecting cylinders. Its volume can be calculatedstraightforwardlybutopaquelybymoderntechniques.An ancient andmuch simpler solutionwas known to bothArchimedes and

Tsu Ch’ung-Chih. It uses nothing more than the method of slicing and acomparisonbetweentheareasofasquareandacircle.Foramarvelouslyclearexposition, seeMartinGardner’s column “Mathematical games: Somepuzzlesbasedoncheckerboards,”ScientificAmerican,Vol.207(November1962), [>].And for Archimedes and Tsu Ch’ung-Chih, see: Archimedes, The Method,English translation by T. L. Heath (1912), reprinted by Dover (1953); and T.Kiang,“AnoldChinesewayoffindingthevolumeofasphere,”MathematicalGazette,Vol.56(May1972),[>].Moreton Moore points out that the bicylinder also has applications in

architecture:“TheRomansandNormans,inusingthebarrelvaulttospantheirbuildings,were familiarwith thegeometryof intersectingcylinderswhere twosuch vaults crossed one another to form a cross vault.” For this, as well asapplications to crystallography, see M. Moore, “Symmetrical intersections ofrightcircularcylinders,”MathematicalGazette,Vol.58(October1974),[>].[>]Computergraphics:InteractivedemonstrationsofthebicylinderandotherproblemsinintegralcalculusareavailableonlineattheWolframDemonstrationsProject (http://demonstrations.wolfram.com/). To play them, you’ll need todownload the free Mathematica Player(http://www.wolfram.com/products/player/), which will then allow you toexplorehundredsofotherinteractivedemonstrationsinallpartsofmathematics.

The bicylinder demo is athttp://demonstrations.wolfram.com/IntersectingCylinders/.Mamikon Mnatsakanian at Caltech has produced a series of animations thatillustrate the Archimedean spirit and the power of slicing. My favorite ishttp://www.its.caltech.edu/~mamikon/Sphere.html, which depicts a beautifulrelationship among the volumes of a sphere and a certain double-cone andcylinderwhoseheightandradiusmatchthoseofthesphere.Healsoshowsthesamethingmorephysicallybydraininganimaginaryvolumeofliquidfromthecylinder and pouring it into the other two shapes; seehttp://www.its.caltech.edu/~mamikon/SphereWater.html.Similarlyelegantmechanicalarguments in theserviceofmatharegiven inM.Levi,TheMathematicalMechanic(PrincetonUniversityPress,2009).[>] Archimedes managed to do this: For Archimedes’s application of hismechanicalmethodtotheproblemoffindingthevolumeofthebicylinder,seeT.L.Heath,ed.,Proposition15,TheMethodofArchimedes,RecentlyDiscoveredbyHeiberg(CosimoClassics,2007),[>].On [>] of the same volume, Archimedes confesses that he views his

mechanical method as a means for discovering theorems rather than provingthem:“certainthingsfirstbecamecleartomebyamechanicalmethod,althoughtheyhadtobedemonstratedbygeometryafterwardsbecausetheirinvestigationbythesaidmethoddidnotfurnishanactualdemonstration.But it isofcourseeasier,whenwehavepreviouslyacquired,by themethod, someknowledgeofthe questions, to supply the proof than it is to find it without any previousknowledge.”ForapopularaccountofArchimedes’swork,seeR.NetzandW.Noel,The

ArchimedesCodex(DaCapoPress,2009).

19.Allaboute

[>]whatise,exactly:Foranintroductiontoallthingseandexponential,seeE.Maor, e: The Story of a Number (Princeton University Press, 1994). ReaderswithabackgroundincalculuswillenjoytheChauvenetPrize–winningarticlebyB.J.McCartin,“e:Themasterofall,”MathematicalIntelligencer,Vol.28,No.2 (2006), [>]. A PDF version is available athttp://mathdl.maa.org/images/upload_library/22/Chauvenet/mccartin.pdf.[>]about13.5percentoftheseatsgotowaste:Theexpectedpackingfractionforcouplessittinginatheateratrandomhasbeenstudiedinotherguisesinthescientific literature. It first arose in organic chemistry—see P. J. Flory,“Intramolecular reaction between neighboring substituents of vinyl polymers,”

JournaloftheAmericanChemicalSociety,Vol.61(1939),pp.1518–1521.Themagicnumber1/e² appearson the top rightcolumnofp.1519.Amore recenttreatmentrelatesthisquestiontotherandomparkingproblem,aclassicpuzzleinprobability theory and statistical physics; seeW. H. Olson, “AMarkov chainmodelforthekineticsofreactantisolation,”JournalofAppliedProbability,Vol.15, No. 4 (1978), pp. 835–841. Computer scientists have tackled similarquestions in their studies of “randomgreedymatching” algorithms for pairingneighboringnodesofnetworks;seeM.DyerandA.Frieze,“Randomizedgreedymatching,”RandomStructuresandAlgorithms,Vol.2(1991),[>].[>]howmanypeopleyoushoulddate:Thequestionofwhentostopdatingandchoose a mate has also been studied in various forms, leading to suchdesignations as the fiancée problem, the marriage problem, the fussy-suitorproblem, and the sultan’s-dowry problem. But the most common term for itnowadays is the secretary problem (the imagined scenario being that you’retryingtohirethebestsecretaryfromagivenpoolofcandidates;youinterviewthecandidatesoneatatimeandhavetodecideonthespotwhethertohiretheperson or say goodbye forever). For an introduction to the mathematics andhistory of this wonderful puzzle, seehttp://mathworld.wolfram.com/SultansDowryProblem.html andhttp://en.wikipedia.org/wiki/Secretary_problem. For more details, see T. S.Ferguson,“Whosolvedthesecretaryproblem?”StatisticalScience,Vol.4,No.3(1989), [>]. A clear exposition of how to solve the problem is given athttp://www.math.uah.edu/stat/urn/Secretary.html. For an introduction to thelargersubjectofoptimalstoppingtheory,seeT.P.Hill,“Knowingwhentostop:How togamble ifyoumust—themathematicsofoptimal stopping,”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: Chaos in the three-body problem is discussed in I. Peterson,Newton’sClock(W.H.Freeman,1993).[>] “madehisheadache”:For thequote abouthow the three-bodyproblemmadeNewton’sheadache,seeD.Brewster,MemoirsoftheLife,Writings,andDiscoveriesofSirIsaacNewton(ThomasConstableandCompany,1855),Vol.2,[>].

21.StepIntotheLight

[>] vector calculus: A great introduction to vector calculus and Maxwell’sequations, and perhaps the best textbook I’ve ever read, is E. M. Purcell,Electricity and Magnetism, 2nd edition (Cambridge University Press, 2011).AnotherclassicisH.M.Schey,Div,Grad,Curl,andAllThat,4thedition(W.W.NortonandCompany,2005).[>]Maxwelldiscoveredwhatlightis:Thesewordsarebeingwrittenduringthe150th anniversaryofMaxwell’s 1861paper “Onphysical linesof force.”See,specifically, “Part III. The theory of molecular vortices applied to staticalelectricity,” Philosophical Magazine (April and May, 1861), [>], available athttp://en.wikisource.org/wiki/On_Physical_Lines_of_Force and scanned fromtheoriginalathttp://www.vacuum-physics.com/Maxwell/maxwell_oplf.pdf.The original paper is well worth a look. A high point occurs just below

equation 137, where Maxwell—a sober man not prone to theatrics—couldn’tresistitalicizingthemostrevolutionaryimplicationofhiswork:“Thevelocityoftransverse undulations in our hypothetical medium, calculated from theelectromagneticexperimentsofM.M.KohlrauschandWeber,agreessoexactlywiththevelocityoflightcalculatedfromtheopticalexperimentsofM.Fizeau,that we can scarcely avoid the inference that light consists in the transverseundulations of the same medium which is the cause of electric and magneticphenomena.”[>] airflow around a dragonfly as it hovered: For Jane Wang’s work ondragonfly flight, see Z. J. Wang, “Two dimensional mechanism for insecthovering,” Physical Review Letters, Vol. 85, No. 10 (September 2000), pp.2216–2219,andZ.J.Wang,“Dragonflyflight,”PhysicsToday,Vol.61,No.10(October 2008), [>]. Her papers are also downloadable fromhttp://dragonfly.tam.cornell.edu/insect_flight.html.Avideoofdragonflyflightisat the bottom of http://ptonline.aip.org/journals/doc/PHTOAD-ft/vol_61/iss_10/74_1.shtml.[>]HowIwishIcouldhavewitnessedthemoment:Einsteinalsoseemstohavewished he were a fly on the wall in Maxwell’s study. As he wrote in 1940,“Imagine[Maxwell’s]feelingswhenthedifferentialequationshehadformulatedprovedtohimthatelectromagneticfieldsspreadintheformofpolarizedwavesandatthespeedoflight!Tofewmenintheworldhassuchanexperiencebeenvouchsafed.” See p. 489 in A. Einstein, “Considerations concerning the

fundamentsof theoreticalphysics,”Science,Vol.91 (May24,1940),pp.487–492 (available online at http://www.scribd.com/doc/30217690/Albert-Einstein-Considerations-Concerning-the-Fundaments-of-Theoretical-Physics).[>] the legacy of his conjuringwith symbols:Maxwell’s equations are oftenportrayedasatriumphofpurereason,butSimonSchaffer,ahistorianofscienceat Cambridge, has argued they were driven as much by the technologicalchallenge of the day: the problem of transmitting signals along underseatelegraph cables. See S. Schaffer, “The laird of physics,” Nature, Vol. 471(2011),[>].

22.TheNewNormal

[>]theworldisnowteemingwithdata:Forthenewworldofdatamining,seeS.Baker,TheNumerati(HoughtonMifflinHarcourt,2008),andI.Ayres,SuperCrunchers(Bantam,2007).[>] Sports statisticians crunch the numbers: M. Lewis,Moneyball (W. W.NortonandCompany,2003).[>] “Learn some statistics”:N.G.Mankiw, “A course load for the gameoflife,”NewYorkTimes(September4,2010).[>]“Takestatistics”:D.Brooks,“Harvard-bound?Chinup,”NewYorkTimes(March2,2006).[>] central lessons of statistics: For informative introductions to statisticsleavened by fine storytelling, see D. Salsburg, The Lady Tasting Tea (W. H.Freeman,2001),andL.Mlodinow,TheDrunkard’sWalk(Pantheon,2008).[>]Galtonboard:Ifyou’veneverseenaGaltonboardinaction,checkoutthedemonstrationsavailableonYouTube.Oneofthemostdramaticofthesevideosmakes use of sand rather than balls; see http://www.youtube.com/watch?v=xDIyAOBa_yU.[>] heightsof adultmenandwomen:You can findyour placeon theheightdistribution by using the online analyzer athttp://www.shortsupport.org/Research/analyzer.html. Based on data from 1994,itshowswhatfractionoftheU.S.populationisshorterortallerthananygivenheight. For more recent data, see M. A. McDowell et al., “Anthropometricreference data for children and adults: United States, 2003–2006,” NationalHealth Statistics Reports, No. 10 (October 22, 2008), available online athttp://www.cdc.gov/nchs/data/nhsr/nhsr010.pdf.[>]OkCupid:OkCupidisthelargestfreedatingsiteintheUnitedStates,with7millionactivemembersasofsummer2011.Theirstatisticiansperformoriginalanalysisontheanonymizedandaggregateddatatheycollectfromtheirmembersand then post their results and insights on their blog OkTrends(http://blog.okcupid.com/index.php/about/). The height distributions arepresented in C. Rudder, “The big lies people tell in online dating,”http://blog.okcupid.com/index.php/the-biggest-lies-in-online-dating/. Thanks toChristianRudderforgenerouslyallowingmetoadapttheplotsfromhispost.[>]Powerlawdistributions:MarkNewmangivesasuperbintroductiontothistopic inM.E. J.Newman, “Power laws, Pareto distributions andZipf’s law,”

ContemporaryPhysics,Vol.46,No.5(2005),pp.323–351(availableonlineathttp://www-personal.umich.edu/~mejn/courses/2006/cmplxsys899/powerlaws.pdf). Thisarticle includes plots of word frequencies in Moby-Dick, the magnitudes ofearthquakes inCalifornia from1910 to 1992, the networth of the 400 richestpeople in the United States in 2003, and many of the other heavy-taileddistributionsmentionedinthischapter.Anearlierbutstillexcellenttreatmentofpower laws is M. Schroder, Fractals, Chaos, Power Laws (W. H. Freeman,1991).[>] 2003 tax cuts: I’ve borrowed this example from C. Seife, Proofiness(Viking, 2010). The transcript of President Bush’s speech is available athttp://georgewbush-whitehouse.archives.gov/news/releases/2004/02/print/20040219-4.html. Thefigures used in the text are based on the analysis by FactCheck.org (anonpartisanprojectoftheAnnenbergPublicPolicyCenteroftheUniversityofPennsylvania), available online athttp://www.factcheck.org/here_we_go_again_bush_exaggerates_tax.html, andthis analysis published by the nonpartisan Tax Policy Center:W. G. Gale, P.Orszag,andI.Shapiro,“Distributionaleffectsofthe2001and2003taxcutsandtheir financing,” http://www.taxpolicycenter.org/publications/url.cfm?ID=411018.[>] fluctuations in stock prices: B. Mandelbrot and R. L. Hudson, The(Mis)Behavior ofMarkets (BasicBooks, 2004);N.N. Taleb,The Black Swan(RandomHouse,2007).[>]Fat,heavy,andlong:Thesethreewordsaren’talwaysusedsynonymously.Whenstatisticiansspeakofalongtail,theymeansomethingdifferentthanwhenbusiness and technology people discuss it. For example, in Chris Anderson’sWired article “The long tail” from October 2004(http://www.wired.com/wired/archive/12.10/tail.html) and in his book of thesamename,he’sreferringtothehugenumbersoffilms,books,songs,andotherworksthatareobscuretomostofthepopulationbutthatnonethelesshavenicheappealandsosurviveonline.Inotherwords,forhim,thelongtailisthemillionsof little guys; for statisticians, the long tail is the very few big guys: thesuperwealthy,orthelargeearthquakes.The difference is that Anderson switches the axes on his plots, which is

somewhatlikelookingthroughtheotherendofthetelescope.Hisconventionisoppositethatusedbystatisticiansintheirplotsofcumulativedistributions,butithas a long tradition going back toVilfredoPareto, an engineer and economistwhostudiedtheincomedistributionsofEuropeancountriesinthelate1800s.In

anutshell,Anderson andParetoplot frequencyas a functionof rank,whereasZipf and the statisticians plot rank as a function of frequency. The sameinformationisshowneitherway,butwiththeaxesflipped.This leads to much confusion in the scientific literature. See

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

23.ChancesAre

[>]probabilitytheory:ForagoodtextbooktreatmentofconditionalprobabilityandBayes’stheorem,seeS.M.Ross,IntroductiontoProbabilityandStatisticsforEngineersandScientists,4thedition(AcademicPress,2009).ForahistoryofReverendBayesandthecontroversysurroundinghisapproachtoprobabilisticinference,seeS.B.McGrayne,TheTheoryThatWouldNotDie(YaleUniversityPress,2011).[>] ailing plant: The answer to part (a) of the ailing-plant problem is 59percent. The answer to part (b) is 27/41, or approximately 65.85 percent. Toderivetheseresults,imagine100ailingplantsandfigureouthowmanyofthem(onaverage)getwateredornot,andthenhowmanyofthosegoontodieornot,based on the information given. This question appears, though with slightlydifferentnumbersandwording,asproblem29onp.84ofRoss’stext.[>] mammogram:The studyof howdoctors interpretmammogram results isdescribed in G. Gigerenzer, Calculated Risks (Simon and Schuster, 2002),chapter4.[>] conditional-probability problems can still be perplexing: For manyentertaining anecdotes and insights about conditional probability and its real-world applications, as well as how it’s misperceived, see J. 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“When batterer becomes murderer,” Nature, Vol. 381 (1996), p. 481. Goodphrasedhisanalysis intermsofoddsratiosandBayes’stheoremratherthaninthe more intuitive natural frequency approach used here and in Gigerenzer’sbook. (Incidentally, Good had an interesting career. In addition to his manycontributions toprobability theoryandBayesian statistics,hehelpedbreak theNazi Enigma code duringWorldWar II and introduced the futuristic concept

nowknownasthetechnologicalsingularity.)Foranindependentanalysisthatreachesessentiallythesameconclusionand

thatwasalsopublishedin1995,seeJ.F.MerzandJ.P.Caulkins,“Propensitytoabuse—propensitytomurder?”Chance,Vol.8,No.2(1995),p.14.Theslightdifferencesbetween the two approaches are discussed in J.B.Garfield andL.Snell,“Teachingbits:Aresourceforteachersofstatistics,”JournalofStatisticsEducation, Vol. 3, No. 2 (1995), online athttp://www.amstat.org/publications/jse/v3n2/resource.html.[>] Dershowitz countered for thedefense:Here is howDershowitz seems tohave calculated that fewer than 1 in 2,500 batterers per year go on tomurdertheirpartners.On[>]ofhisbookReasonableDoubts,hecitesanestimatethatin1992,somewherebetween2.5and4millionwomen in theUnitedStateswerebattered by their husbands, boyfriends, and ex-boyfriends. In that same year,according to the FBI Uniform Crime Reports (http://www.fbi.gov/about-us/cjis/ucr/ucr), 913 women were murdered by their husbands, and 519 werekilled by their boyfriends or ex-boyfriends. Dividing the total of 1,432homicidesby2.5millionbeatingsyields1murderper1,746beatings,whereasusing the higher estimate of 4 million beatings per year yields 1 murder per2,793 beatings. Dershowitz apparently chose 2,500 as a round number inbetweentheseextremes.What’s unclear is what proportion of the murdered women had previously

beenbeatenbythesemen.ItseemsthatDershowitzwasassumingthatnearlyallthehomicidevictimshadearlierbeenbeaten,presumablytomakethepointthatevenwhentherateisoverestimatedinthisway,it’sstill“infinitesimal.”A few years after the verdict was handed down in the Simpson case,

Dershowitz and the mathematician John Allen Paulos engaged in a heatedexchangevialetterstotheeditoroftheNewYorkTimes.Theissuewaswhetherevidenceofahistoryofspousalabuseshouldberegardedasrelevanttoamurdertrial in lightofprobabilistic arguments similar to thosediscussedhere.SeeA.Dershowitz,“Thenumbersgame,”NewYorkTimes(May30,1999),archivedathttp://www.nytimes.com/1999/05/30/books/l-the-numbers-game-789356.html,and J. A. Paulos, “Once upon a number,”New York Times (June 27, 1999),http://www.nytimes.com/1999/06/27/books/l-once-upon-a-number-224537.html.[>] expect3moreof thesewomen,onaverage, tobekilledbysomeoneelse:AccordingtotheFBIUniformCrimeReports,4,936womenweremurderedin1992. Of thesemurder victims, 1,432 (about 29 percent) were killed by theirhusbands or boyfriends. The remaining 3,504 were killed by somebody else.Therefore,consideringthatthetotalpopulationofwomenintheUnitedStatesatthat timewas about 125million, the rate atwhichwomenweremurdered by

someone other than their partners was 3,504 divided by 125,000,000, or 1murderper35,673womenperyear.Let’s assume that this rate of murder by nonpartners was the same for allwomen, battered or not. Then in our hypothetical sample of 100,000 batteredwomen, we’d expect about 100,000 divided by 35,673, or 2.8, women to bekilled by someone other than a partner. Rounding 2.8 to 3, we obtain theestimategiveninthetext.

24.UntanglingtheWeb

[>]searchingtheWeb:ForanintroductiontoWebsearchandlinkanalysis,seeD. Easley and J. Kleinberg, Networks, Crowds, and Markets (CambridgeUniversity Press, 2010), chapter 14. Their elegant exposition has inspiredmytreatmenthere.ForapopularaccountofthehistoryofInternetsearch,includingstories about the key characters and companies, see J. Battelle, The Search(PortfolioHardcover,2005).Theearlydevelopmentoflinkanalysis,forreaderscomfortablewith linear algebra, is summarized in S.Robinson, “The ongoingsearchforefficientWebsearchalgorithms,”SIAMNews,Vol.37,No.9(2004).[>]grasshopper:Foranyoneconfusedbymyuseoftheword“grasshopper,”itis an affectionate nickname for a student who hasmuch to learn from a Zenmaster. In the television seriesKung Fu, onmany of the occasions when theblindmonkPoimpartswisdomtohisstudentCaine,hecallshimgrasshopper,harkingback to their first lesson, a scenedepicted in the1972pilot film (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:Therecognitionofthecircularityproblemforrankingwebpages and its solutionvia linear algebragrewout of two linesof researchpublished in 1998. One was by my Cornell colleague Jon Kleinberg, thenworkingasavisitingscientistatIBMAlmadenResearchCenter.Forhisseminalpaper on the “hubs and authorities” algorithm (an alternative form of linkanalysisthatappearedslightlyearlierthanGoogle’sPageRankalgorithm),seeJ.Kleinberg,“Authoritativesourcesinahyperlinkedenvironment,”ProceedingsoftheNinthAnnualACM-SIAMSymposiumonDiscreteAlgorithms(1998).TheotherlineofresearchwasbyGooglecofoundersLarryPageandSergey

Brin. Their PageRankmethodwas originallymotivated by thinking about theproportion of time a random surferwould spend at each page on theWeb—a

process with a different description but one that leads to the same way ofresolvingthecirculardefinition.ThefoundationalpaperonPageRankisS.BrinandL. Page, “The anatomy of a large-scale hypertextualWeb search engine,”Proceedings of the Seventh InternationalWorldWideWebConference (1998),[>].Assooftenhappensinscience,strikinglysimilarprecursorsoftheseideashad

already been discovered in other fields. For this prehistory of PageRank inbibliometrics, psychology, sociology, and econometrics, see M. 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’sbooksandonlinevideolectureswouldbeafineplaceto start: G. Strang, Introduction to Linear Algebra, 4th edition (Wellesley-CambridgePress,2009),andhttp://web.mit.edu/18.06/www/videos.html.[>] linear algebra has the tools you need: Some of the most impressiveapplications of linear algebra rely on the techniques of singular valuedecomposition andprincipal component analysis.SeeD. James,M.Lachance,andJ.Remski,“Singularvectors’subtlesecrets,”CollegeMathematicsJournal,Vol.42,No.2(March2011),[>].[>]PageRankalgorithm:AccordingtoGoogle,theterm“PageRank”referstoLarry Page, not webpage. Seehttp://web.archive.org/web/20090424093934/http://www.google.com/press/funfacts.html[>] classify human faces: The idea here is that any human face can beexpressedasacombinationofasmallnumberoffundamentalfaceingredients,or eigenfaces. This application of linear algebra to face recognition andclassification was pioneered by L. Sirovich andM. Kirby, “Low-dimensionalprocedure for the characterization of human faces,” Journal of the OpticalSocietyofAmericaA,Vol.4(1987),pp.519–524,andfurtherdevelopedbyM.Turk and A. Pentland, “Eigenfaces for recognition,” Journal of CognitiveNeuroscience, Vol. 3 (1991), [>], which is also available online athttp://cse.seu.edu.cn/people/xgeng/files/under/turk91eigenfaceForRecognition.pdfFor a comprehensive list of scholarly papers in this area, see the Face

RecognitionHomepage(http://www.face-rec.org/interesting-papers/).[>]votingpatternsofSupremeCourtjustices:L.Sirovich,“Apatternanalysisof the second Rehnquist U.S. Supreme Court,” Proceedings of the NationalAcademyofSciences,Vol.100,No.13(2003),pp.7432–7437.Forajournalisticaccount of this work, see N.Wade, “A mathematician crunches the SupremeCourt’snumbers,”NewYorkTimes (June24,2003).For adiscussionaimedat

legal scholars by amathematician and now law professor, see P.H.Edelman,“The dimension of the Supreme Court,”Constitutional Commentary, Vol. 20,No.3(2003),pp.557–570.[>]NetflixPrize:ForthestoryoftheNetflixPrize,withamusingdetailsaboutitsearlycontestantsandtheimportanceofthemovieNapoleonDynamite,seeC.Thompson, “If you liked this, you’re sure to love that—Winning the Netflixprize,”NewYorkTimesMagazine(November23,2008).TheprizewaswoninSeptember2009,threeyearsafterthecontestbegan;seeS.Lohr,“A$1millionresearch bargain forNetflix, andmaybe amodel for others,”NewYorkTimes(September 22, 2009). The application of the singular value decomposition totheNetflixPrizeisdiscussedinB.Cipra,“Blockbusteralgorithm,”SIAMNews,Vol.42,No.4(2009).[>]skippingsomedetails:Forsimplicity,Ihavepresentedonlythemostbasicversion of the PageRank algorithm. To handle networkswith certain commonstructuralfeatures,PageRankneeds tobemodified.Suppose,forexample, thatthenetworkhassomepagesthatpointtoothersbutthathavenonepointingbacktothem.Duringtheupdateprocess,thosepageswill losetheirPageRank,asifleakingorhemorrhagingit.Theygiveittoothersbutit’sneverreplenished.Sothey’llallendupwithPageRanksofzeroandwillthereforebeindistinguishableinthatrespect.At theother extreme, considernetworks inwhich somepages, orgroupsof

pages, hoardPageRank by being clubby, by never linking back out to anyoneelse.SuchpagestendtoactassinksforPageRank.Toovercometheseandothereffects,BrinandPagemodifiedtheiralgorithm

asfollows:Aftereachstepintheupdateprocess,allthecurrentPageRanksarescaleddownbyaconstantfactor,sothetotalislessthan1.Whateverisleftoveristhenevenlydistributedtoallthenodesinthenetwork,asifbeingraineddownon everyone. It’s the ultimate egalitarian act, spreading the PageRank to theneediestnodes.JoethePlumberwouldnotbehappy.For a deeper look at the mathematics of PageRank, with interactive

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

25.TheLoneliestNumbers

[>] one is the loneliest number:HarryNilssonwrote the song “One.”ThreeDogNight’scoverof itbecameahit, reachingnumber5on theBillboardHot100,andAimeeMannhasaterrificversionofitthatcanbeheardinthemovieMagnolia.[>] The Solitude of Prime Numbers: P. Giordano, The Solitude of PrimeNumbers(PamelaDormanBooks/VikingPenguin,2010).Thepassageexcerptedhereappearson[>].[>] number theory: For popular introductions to number theory, and themysteriesofprimenumbers inparticular, themostdifficult choice iswhere tobegin.Thereareat least threeexcellentbooks tochoosefrom.Allappearedataround the same time, and all centered on the Riemann hypothesis, widelyregarded as the greatest unsolved problem in mathematics. For some of themathematicaldetails,alongwiththeearlyhistoryoftheRiemannhypothesis,I’drecommend J. Derbyshire, Prime Obsession (Joseph Henry Press, 2003). Formoreemphasisonthelatestdevelopmentsbutstillataveryaccessiblelevel,seeD. Rockmore, Stalking the Riemann Hypothesis (Pantheon, 2005), andM. duSautoy,TheMusicofthePrimes(HarperCollins,2003).[>] encryptionalgorithms:For theuseofnumber theory incryptography,seeM.Gardner,Penrose Tiles to TrapdoorCiphers (MathematicalAssociation ofAmerica, 1997), chapters 13 and 14. The first of these chapters reprintsGardner’sfamouscolumnfromtheAugust1977issueofScientificAmericaninwhichhetoldthepublicaboutthevirtuallyunbreakableRSAcryptosystem.Thesecond chapter describes the “intense furor” it aroused within the NationalSecurityAgency.Formore recent developments, see chapter 10of duSautoy,TheMusicofthePrimes.[>] primenumber theorem:Alongwith thebooksbyDerbyshire,Rockmore,andduSautoymentionedabove, therearemanyonlinesourcesof informationabouttheprimenumbertheorem,suchasChrisK.Caldwell’spage“Howmanyprimesarethere?”(http://primes.utm.edu/howmany.shtml),theMathWorldpage“Prime number theorem”(http://mathworld.wolfram.com/PrimeNumberTheorem.html),andtheWikipediapage “Prime number theorem”(http://en.wikipedia.org/wiki/Prime_number_theorem).[>] CarlFriedrichGauss:ThestoryofhowGaussnoticedtheprimenumber

theorem at age fifteen is told on [>] of Derbyshire,PrimeObsession, and ingreater detail by L. J. Goldstein, “A history of the prime number theorem,”AmericanMathematicalMonthly,Vol.80,No.6(1973),pp.599–615.Gaussdidnotprovethetheorembutguesseditbyporingovertablesofprimenumbersthathe had computed—by hand—for his own amusement. The first proofs werepublishedin1896,aboutacenturylater,byJacquesHadamardandCharlesdelaValléePoussin,eachofwhomhadbeenworkingindependentlyontheproblem.[>]twinprimesapparentlycontinuetoexist:HowcantwinprimesexistatlargeN,inlightoftheprimenumbertheorem?Thetheoremsaysonlythattheaveragegap is lnN. But there are fluctuations about this average, and since there areinfinitelymanyprimes,someofthemareboundtogetluckyandbeattheodds.Inotherwords,eventhoughmostwon’t findaneighboringprimemuchcloserthanlnNaway,someofthemwill.For readerswhowant to see some of themath governing “very small gaps

betweenprimes”beautifullyexplainedinaconciseway,seeAndrewGranville’sarticle on analytic number theory in T.Gowers,The PrincetonCompanion toMathematics(PrincetonUniversityPress,2008),pp.332–348,especiallyp.343.Also,there’saniceonlinearticlebyTerryTaothatgivesalotofinsightinto

twin primes—specifically, how they’re distributed and why mathematiciansbelievethereareinfinitelymanyofthem—andthenwadesintodeeperwaterstoexplain the proof of his celebrated theorem (with BenGreen) that the primescontain arbitrarily long arithmetic progressions. See T. Tao, “Structure andrandomness in the prime numbers,”http://terrytao.wordpress.com/2008/01/07/ams-lecture-structure-and-randomness-in-the-prime-numbers/.For further details and background information about twin primes, see

http://en.wikipedia.org/wiki/Twin_prime andhttp://mathworld.wolfram.com/TwinPrimeConjecture.html.[>]anotherprimecouplenearby:I’mjustkiddingaroundhereandnottryingtomake a serious point about the spacing between consecutive pairs of twinprimes.Maybesomewhereoutthere,fardownthenumberline,twosetsoftwinshappentobeextremelyclosetogether.Foranintroductiontosuchquestions,seeI. Peterson, “Prime twins” (June 4, 2001),http://www.maa.org/mathland/mathtrek_6_4_01.html.In any case, themetaphorof awkward couples as twinprimeshasnot been

lost on Hollywood. For light entertainment, you might want to rent a moviecalledTheMirror Has Two Faces, a Barbra Streisand vehicle costarring JeffBridges.He’sahandsomebutsociallycluelessmathprofessor.She’saprofessorintheEnglishliteraturedepartment,aplucky,energetic,buthomelywoman(or

at least that’s howwe’re supposed to see her)who liveswith hermother andgorgeoussister.Eventually, the twoprofessorsmanagetoget togetherfor theirfirstdate.When theirdinnerconversationdrifts to the topicofdancing(whichembarrasseshim),hechangesthesubjectabruptlytotwinprimes.Shegetstheideaimmediatelyandasks,“Whatwouldhappenifyoucountedpastamillion?Wouldtherestillbepairslikethat?”Hepracticallyfallsoffhischairandsays,“Ican’tbelieveyouthoughtofthat!Thatisexactlywhatisyettobeproveninthetwinprimeconjecture.”Laterinthemovie,whentheystartfallinginlove,shegiveshimabirthdaypresentofcufflinkswithprimenumbersonthem.

26.GroupThink

[>]mattressmath:ThemattressgroupistechnicallyknownastheKleinfour-group.It’soneofthesimplestinagiganticzooofpossibilities.Mathematicianshavebeenanalyzinggroupsandclassifyingtheirstructuresforabout200years.Foranengagingaccountofgrouptheoryandthemorerecentquesttoclassifyallthefinitesimplegroups,seeM.duSautoy,Symmetry(Harper,2008).[>] group theory: Two recent books inspired this chapter: N. Carter,VisualGroup Theory (Mathematical Association of America, 2009); and B. Hayes,Group Theory in the Bedroom (Hill and Wang, 2008). Carter introduces thebasicsofgrouptheorygentlyandpictorially.HealsotouchesonitsconnectionstoRubik’scube,contradancingandsquaredancing,crystals,chemistry,art,andarchitecture.AnearlierversionofHayes’smattress-flippingarticleappearedinAmericanScientist,Vol.93,No.5(September/October2005),p.395,availableonline at http://www.americanscientist.org/issues/pub/group-theory-in-the-bedroom.Readersinterestedinseeingadefinitionofwhata“group”isshouldconsult

anyoftheauthoritativeonlinereferencesorstandardtextbooksonthesubject.Agood place to start is the MathWorld pagehttp://mathworld.wolfram.com/topics/GroupTheory.html or theWikipedia pagehttp://en.wikipedia.org/wiki/Group_(mathematics). The treatment I’ve givenhereemphasizessymmetrygroupsratherthangroupsinthemostgeneralsense.[>]chaoticcounterparts:MichaelFieldandMartinGolubitskyhavestudiedtheinterplaybetweengroup theory andnonlinear dynamics. In the courseof theirinvestigations, they’ve generated stunning computer graphics of symmetricchaos, many of which can be found on Mike Field’s webpage(http://www.math.uh.edu/%7Emike/ag/art.html). For the art, science, andmathematicsofthistopic,seeM.FieldandM.Golubitsky,SymmetryinChaos,2ndedition(SocietyforIndustrialandAppliedMathematics,2009).[>] the diagram demonstrates thatHR =V: Aword about some potentiallyconfusingnotationusedthroughoutthischapter:InequationslikeHR=V,theHwaswrittenon the left to indicate that it’s the transformationbeingperformedfirst. Carter uses this notation for functional composition in his book, but thereadershouldbeawarethatmanymathematiciansusetheoppositeconvention,placingtheHontheright.[>] Feynmangotadraftdeferment:For theanecdoteaboutFeynmanandthe

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

27.TwistandShout

[>]Möbiusstrips:Art,limericks,patents,parlortricks,andseriousmath—youname it, and if it has anything todowithMöbius strips, it’s probably inCliffPickover’s jovial book The Möbius Strip (Basic Books, 2006). An earliergenerationfirstlearnedaboutsomeofthesewondersinM.Gardner,“TheworldoftheMöbiusstrip:Endless,edgeless,andone-sided,”ScientificAmerican,Vol.219,No.6(December1968).[>]funactivitiesthatasix-year-oldcando:Forstep-by-stepinstructions,withphotographs, of some of the activities described in this chapter, see “How toexplore a Möbius strip” at http://www.wiki-how.com/Explore-a-Möbius-Strip.JulianFlerongivesmanyotherideas—Möbiusgarlands,hearts,paperclipstars—in “Recycling Möbius,”http://artofmathematics.wsc.ma.edu/sculpture/workinprogress/Mobius1206.pdf.Forfurtherfunwithpapermodels,seeS.Barr’sclassicbookExperiments in

Topology(Crowell,1964).[>]topology:ThebasicsoftopologyareexpertlyexplainedinR.CourantandH.Robbins(revisedbyI.Stewart),WhatIsMathematics?2ndedition(OxfordUniversity Press, 1996), chapter 5. For a playful survey, seeM.Gardner,TheColossal Book of Mathematics (W. W. Norton and Company, 2001). Hediscusses Klein bottles, knots, linked doughnuts, and other delights ofrecreational topology in part 5, chapters 18–20. A very good contemporarytreatment is D. S. Richeson,Euler’sGem (Princeton University Press, 2008).Richesonpresentsahistoryandcelebrationoftopologyandanintroductiontoitsmain concepts, usingEuler’s polyhedron formula as a centerpiece.At amuchhigher level, but still within comfortable reach of peoplewith a collegemathbackground,seethechaptersonalgebraictopologyanddifferentialtopologyinT. Gowers, The Princeton Companion to Mathematics (Princeton UniversityPress,2008),pp.383–408.[>] the intrinsic loopiness of a circle and a square: Given that a circle andsquarearetopologicallyequivalentcurves,youmightbewonderingwhatkindsof curves would be topologically different. The simplest example is a linesegment. To prove this, observe that if you travel in one direction around acircle,asquare,oranyotherkindofloop,you’llalwaysreturntoyourstartingpoint,but that’snot true for travelingona linesegment.Since thisproperty isleftunchangedbyalltransformationsthatpreserveanobject’stopology(namely,

continuous deformations whose inverse is also continuous), and since thisproperty differs between loops and segments,we can conclude that loops andsegmentsaretopologicallydifferent.[>] ViHart: Vi’s videos discussed in this chapter, “Möbiusmusic box” and“Möbius story: Wind and Mr. Ug,” can be found on YouTube and also athttp://vihart.com/musicbox/ and http://vihart.com/blog/mobius-story/. For moreof her ingenious and fun-loving excursions intomathematical food, doodling,balloons, beadwork, and music boxes, see her website athttp://vihart.com/everything/. She was profiled in K. Chang, “Bending andstretching classroom lessons tomakemath inspire,”New York Times (January17, 2011), available online athttp://www.nytimes.com/2011/01/18/science/18prof.html.[>] artists have likewise drawn inspiration: To see images of the MöbiusartworkbyMauritsEscher,MaxBill,andKeizoUshio,searchtheWebusingtheartist’snameand“Möbius”assearchterms.IvarsPetersonhaswrittenabouttheuse of Möbius strips in literature, art, architecture, and sculpture, withphotographs and explanations, at his Mathematical Tourist blog:http://mathtourist.blogspot.com/search/label/Moebius%20Strips.[>]NationalLibraryofKazakhstan:Thelibraryiscurrentlyunderconstruction.Foritsdesignconceptandintriguingimagesofwhatitwilllooklike,gotothewebsiteforthearchitecturalfirmBIG(BjarkeIngelsGroup),http://www.big.dk/.Click on the icon for ANL, Astana National Library; it appears in the 2009column (the fourth column from the right) when the projects are arranged intheir default chronological order. The site contains forty-one slides of thelibrary’s internalandexternal structure,museumcirculation, thermalexposure,andsoon,allofwhichareunusualbecauseofthebuilding’sMöbiuslayout.Fora profile of Bjarke Ingels and his practice, see G. Williams, “Open sourcearchitect: Meet the maestro of ‘hedonistic sustainability,’”http://www.wired.co.uk/magazine/archive/2011/07/features/open-source-architect.[>]Möbiuspatents:SomeofthesearediscussedinPickover,TheMöbiusStrip.You can find hundreds of others by searching for “Möbius strip” in GooglePatents.[>]bagel:Ifyouwanttotrycuttingabagelthisway,GeorgeHartexplainshistechnique at his website http://www.georgehart.com/bagel/bagel.html. Or youcan see a computer animation by Bill Giles here:http://www.youtube.com/watch?v=hYXnZ8-ux80. If you prefer to watch ithappeninginrealtime,checkoutavideobyUltraNurdcalled“MöbiusBagel”(http://www.youtube.com/watch?v=Zu5z1BCC70s). But strictly speaking, this

shouldnotbecalledaMöbiusbagel—apointofconfusionamongmanypeoplewho have written about or copied George’s work. The surface on which thecreamcheeseisspreadisnotequivalenttoaMöbiusstripbecauseithastwohalftwists in it, not one, and the resulting surface is two-sided, not one-sided.Furthermore, a true Möbius bagel would remain in one piece, not two, afterbeing cut in half. For a demonstration of how to cut a bagel in this genuineMöbiusfashion,seehttp://www.youtube.com/watch?v=l6Vuh16r8o8.

28.ThinkGlobally

[>] avisionof theworldas flat:By referring toplanegeometryas flat-earthgeometry, Imightseemtobedisparaging thesubject,but that’snotmy intent.Thetacticoflocallyapproximatingacurvedshapebyaflatonehasoftenturnedouttobeausefulsimplificationinmanypartsofmathematicsandphysics,fromcalculus to relativity theory. Plane geometry is the first instance of this greatidea.NordoImeantosuggestthatalltheancientsthoughttheworldwasflat.For

anengagingaccountofEratosthenes’smeasurementof thedistancearound theglobe, see N. Nicastro,Circumference (St.Martin’s Press, 2008). For a morecontemporary approach that you might like to try on your own, RobertVanderbeiatPrincetonUniversityrecentlygaveapresentationtohisdaughter’shigh-schoolgeometryclassinwhichheusedaphotographofasunset toshowthat the Earth is not flat and to estimate its diameter.His slides are posted athttp://orfe.princeton.edu/~rvdb/tex/sunset/34-39.OPN.1108twoup.pdf.[>] differential geometry: A superb introduction to modern geometry wascoauthoredbyDavidHilbert,oneofthegreatestmathematiciansofthetwentiethcentury. This classic, originally published in 1952, has been reissued as D.Hilbert and S. Cohn-Vossen, Geometry and the Imagination (AmericanMathematical Society, 1999). Several good textbooks and online courses indifferential geometry are listed on the Wikipedia pagehttp://en.wikipedia.org/wiki/Differential_geometry.[>]mostdirectroute:Foraninteractiveonlinedemonstrationthatletsyouplotthe shortest route between any two points on the surface of the Earth, seehttp://demonstrations.wolfram.com/GreatCirclesOnMercatorsChart/. (You’llneed to download the freeMathematica Player, whichwill then allow you toexplore hundreds of other interactive demonstrations in all parts ofmathematics.)[>] Konrad Polthier: Excerpts from a number of Polthier’s fascinatingeducational videos about mathematical topics can be found online athttp://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).The imagesshown

inthetextarefromtheDVDTouchingSoapFilms(Springer,1995),byAndreasArnez,KonradPolthier,MartinSteffens,andChristianTeitzel.[>] shortest path through a network: The classic algorithm for shortest-pathproblemsonnetworkswascreatedbyEdsgerDijkstra.Foranintroduction,seehttp://en.wikipedia.org/wiki/Dijkstra’s_algorithm. Steven Skiena has posted aninstructive animation of Dijkstra’s algorithm athttp://www.cs.sunysb.edu/~skiena/combinatorica/animations/dijkstra.html.Nature can solve certain shortest-path problems by decentralized processes

akin to analog computation. For chemical waves that solve mazes, see O.Steinbock,A.Toth,andK.Showalter,“Navigatingcomplexlabyrinths:Optimalpaths from chemical waves,” Science, Vol. 267 (1995), p. 868. Not to beoutdone,slimemoldscansolvethemtoo:T.Nakagaki,H.Yamada,andA.Toth,“Maze-solvingbyanamoeboidorganism,”Nature,Vol.407(2000),p.470.ThisslimycreaturecanevenmakenetworksasefficientastheTokyorailsystem:A.Teroetal.,“Rules forbiologically inspiredadaptivenetworkdesign,”Science,Vol.327(2010),p.439.[>] tellingthestoryofyourlifeinsixwords:Delightfulexamplesofsixwordmemoirs are given at http://www.smithmag.net/sixwords/ andhttp://en.wikipedia.org/wiki/Six-Word_Memoirs.

29.AnalyzeThis!

[>]analysis:Analysisgrewoutoftheneedtoshoreupthelogicalfoundationsof calculus. William Dunham traces this story through the works of elevenmasters, from Newton to Lebesgue, in W. Dunham, The Calculus Gallery(Princeton University Press, 2005). The book contains explicit mathematicspresentedaccessiblyforreadershavingacollege-levelbackground.Atextbookin a similar spirit is D. Bressoud,A Radical Approach to Real Analysis, 2ndedition (Mathematical Association of America, 2006). For a morecomprehensivehistoricalaccount,seeC.B.Boyer,TheHistoryoftheCalculusandItsConceptualDevelopment(Dover,1959).[>]vacillatingforever:Grandi’sseries1–1+1–1+1–1+∙∙∙isdiscussedinameticulouslysourcedWikipediaarticleaboutitshistory,withlinkstofurtherthreads about itsmathematical status and its role inmath education.All thesemay be reached from the main page “Grandi’s series”:http://en.wikipedia.org/wiki/Grandi’s_series.[>] Riemann rearrangement theorem: For a clear exposition of theRiemannrearrangementtheorem,seeDunham,TheCalculusGallery,pp.112–115.[>] Strange, yes. Sick, yes: The alternating harmonic series is conditionally

convergent,meaning it’sconvergentbutnotabsolutelyconvergent (the sumoftheabsolutevaluesofitstermsdonotconverge).Foraserieslikethat,youcanreorder thesumtogetany realnumber.That’s theshocking implicationof theRiemannrearrangementtheorem.Itshowsthataconvergentsumcanviolateourintuitiveexpectationsifitdoesnotconvergeabsolutely.In the much-better-behaved case of an absolutely convergent series, all

rearrangements of the series converge to the same value. That’s wonderfullyconvenient. Itmeans that an absolutely convergent series behaves like a finitesum.Inparticular,itobeysthecommutativelawofaddition.Youcanrearrangethetermsanywayyouwantwithoutchangingtheanswer.Formoreonabsoluteconvergence,seehttp://mathworld.wolfram.com/AbsoluteConvergence.htmlandhttp://en.wikipedia.org/wiki/Absolute_convergence.[>] Fourier analysis: Tom Körner’s extraordinary book Fourier Analysis(CambridgeUniversity Press, 1989) is a self-described “shopwindow” of theideas, techniques, applications, and history of Fourier analysis. The level ofmathematicalrigorishigh,yetthebookiswitty,elegant,andpleasantlyquirky.ForanintroductiontoFourier’sworkanditsconnectiontomusic,seeM.Kline,MathematicsinWesternCulture(OxfordUniversityPress,1974),chapter19.[>] Gibbs phenomenon: The Gibbs phenomenon and its tortuous history isreviewedbyE.HewittandR.E.Hewitt,“TheGibbs-Wilbrahamphenomenon:AnepisodeinFourieranalysis,”ArchivefortheHistoryofExactSciences,Vol.21(1979),[>].[>]digitalphotographsandonMRIscans:TheGibbsphenomenoncanaffectMPEG and JPEG compression of digital video:http://www.doc.ic.ac.uk/~nd/surprise_96/journal/vol4/sab/report.html. When itappearsinMRIscans, theGibbsphenomenonisknownastruncationorGibbsringing;seehttp://www.mr-tip.com/serv1.php?type=art&sub=Gibbs%20Artifact.Formethods tohandle theseartifacts, 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) displaying sharp edges or other mild types of jumpdiscontinuities, the partial sums of the sine waves were proven to convergepointwise but not uniformly to the original function. Pointwise convergencemeansthatatanyparticularpointx,thepartialsumsgetarbitrarilyclosetotheoriginal function as more terms are added. So in that sense, the series doesconverge, as one would hope. The catch is that some points are much morefinicky than others. The Gibbs phenomenon occurs near the worst of those

points—theedgesintheoriginalfunction.Forexample,considerthesawtoothwavediscussedinthischapter.Asxgets

closer to the edge of a sawtooth, it takesmore andmore terms in theFourierseriestoreachagivenlevelofapproximation.That’swhatwemeanbysayingtheconvergenceisnotuniform.Itoccursatdifferentratesfordifferentx.In this case, the non-uniformity of the convergence is attributable to the

pathologies of the alternating harmonic series,whose terms appear as Fouriercoefficientsforthesawtoothwave.Asdiscussedabove,thealternatingharmonicseries converges, but only because of the massive cancellation caused by itsalternatingmixofpositiveandnegativeterms.Ifallitstermsweremadepositiveby taking their absolute values, the series would diverge—the sum wouldapproachinfinity.That’swhythealternatingharmonicseriesissaidtoconvergeconditionally,notabsolutely.Thisprecarious formofconvergence then infectstheassociatedFourierseriesandrendersitnon-uniformlyconvergent,leadingtotheGibbsphenomenonanditsmockingupraisedfingerneartheedge.Incontrast,inthemuchbettercasewheretheseriesofFouriercoefficientsis

absolutelyconvergent, theassociatedFourierseriesconvergesuniformlyto theoriginalfunction.ThentheGibbsphenomenondoesn’toccur.Formoredetails,see http://mathworld.wolfram.com/GibbsPhenomenon.html andhttp://en.wikipedia.org/wiki/Gibbs_phenomenon.The bottom line is that the analysts taught us to be wary of conditionally

convergent series. Convergence is good, but not good enough. For an infiniteseriestobehavelikeafinitesuminallrespects,itneedsmuchtighterconstraintsthan conditional convergence can provide. Insisting on absolute convergenceyields thebehaviorwe’dexpect intuitively,bothfor theseries itselfandfor itsassociatedFourierseries.

30.TheHilbertHotel

[>] Georg Cantor’s: For more about Cantor, including the mathematical,philosophical, and theological controversies surrounding his work, see J. W.Dauben,GeorgCantor(PrincetonUniversityPress,1990).[>] set theory: Ifyouhaven’t read ityet, I recommend thesurprisebestsellerLogicomix, a brilliantly creative graphic novel about set theory, logic, infinity,madness, and the quest for mathematical truth: A. Doxiadis and C. H.Papadimitriou, Logicomix (Bloomsbury, 2009). It stars Bertrand Russell, butCantor,Hilbert,Poincaré,andmanyothersmakememorableappearances.[>] DavidHilbert: The classic biography ofDavidHilbert is amoving andnontechnical account of his life, his work, and his times: C. Reid, Hilbert(Springer,1996).Hilbert’scontributionstomathematicsaretoonumeroustolisthere,butperhapshisgreatestishiscollectionoftwenty-threeproblems—allofwhichwereunsolvedwhenheproposedthem—thathethoughtwouldshapethecourse of mathematics in the twentieth century. For the ongoing story andsignificanceoftheseHilbertproblemsandthepeoplewhosolvedsomeofthem,seeB.H.Yandell,TheHonorsClass(AKPeters,2002).Severaloftheproblemsstillremainopen.[>]HilbertHotel:Hilbert’sparableoftheinfinitehotelismentionedinGeorgeGamow’severgreenmasterpieceOneTwoThree...Infinity(Dover,1988),[>].Gamowalsodoesagoodjobofexplainingcountableanduncountablesetsandrelatedideasaboutinfinity.Thecomedicanddramaticpossibilitiesof theHilbertHotelhaveoftenbeen

explored by writers of mathematical fiction. For example, see S. Lem, “TheextraordinaryhotelorthethousandandfirstjourneyofIontheQuiet,”reprintedin Imaginary Numbers, edited by W. Frucht (Wiley, 1999), and I. Stewart,ProfessorStewart’sCabinetofMathematicalCuriosities(BasicBooks,2009).Achildren’s book on the same theme is I. Ekeland, The Cat in Numberland(CricketBooks,2006).[>]anyotherdigitbetween1and8:Atinyfinesseoccurredintheargumentfortheuncountabilityof the realnumberswhenI required that thediagonaldigitswere to be replaced by digits between 1 and 8. This wasn’t essential. But Iwantedtoavoidusing0and9tosidestepanyfussinesscausedbythefactthatsomerealnumbershavetwodecimalrepresentations.Forexample,.200000...equals .199999 . . . Thus, if we hadn’t excluded the use of 0s and 9s as

replacement digits, it’s conceivable the diagonal argument could haveinadvertentlyproducedanumberalreadyonthelist(andthatwouldhaveruinedtheproof).Bymyforbiddingtheuseof0and9,wedidn’thavetoworryaboutthisannoyance.[>] infinity beyond infinity: For amoremathematical but still very readablediscussionofinfinity(andmanyoftheotherideasdiscussedinthisbook),seeJ.C.Stillwell,YearningfortheImpossible(AKPeters,2006).Readerswishingtogo deeper into infinitymight enjoyTerryTao’s blog post about self-defeatingobjects, http://terrytao.wordpress.com/2009/11/05/the-no-self-defeating-object-argument/. In a very accessible way, he presents and elucidates a lot offundamental arguments about infinity that arise in set theory, philosophy,physics, computer science, game theory, and logic. For a survey of thefoundationalissuesraisedbythesesortsofideas,seealsoJ.C.Stillwell,RoadstoInfinity(AKPeters,2010).

Credits

[>]: “Sesame Workshop”®, “Sesame Street”® and associated characters,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,[>]–[>],[>],[>]–[>]analysis

calculusand,[>]–[>],[>]–[>]complex,[>]link,[>]numerical,[>]–[>]

Anderson,Chris,[>]Archimedes(Greekmathematician),[>]–[>],[>],[>],[>],[>],[>],[>],[>],[>]arithmetic, [>], [>]. See also addition; 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,[>]computers

algorithmsand,[>]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,[>]–[>]powerlawdistributions,[>]–[>]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

StevenStrogatz is theSchurmanProfessor ofAppliedMathematics atCornellUniversity. A renowned teacher and one of the world’s most highly citedmathematicians, he has been a frequent guest on National Public Radio’sRadiolab. He is the author of Sync and The Calculus of Friendship, and therecipient of a lifetime achievement award for math communication. He alsowroteapopularNewYorkTimesonlinecolumn,“TheElementsofMath,”whichformedthebasisforthisbook.