“Mazur” — // — : — page — # Enlightening Symbols “Mazur” — // — : — page — # Also by Joseph Mazur EuclidintheRainforest:DiscoveringUniversalTruth inLogicandMath(2006) Zeno’sParadox:UnravelingtheAncientMystery behindtheScienceofSpaceandTime(2007) What’sLuckGottoDowithIt?TheHistory,Mathematics, andPsychologyoftheGambler’sIllusion(2010) Edited Number:TheLanguageofScience(2007) “Mazur” — // — : — page — # Enlightening Symbols A Short History of Mathematical Notation and Its Hidden Powers Joseph Mazur PrincetonUniversityPress PrincetonandOxford “Mazur” — // — : — page — # Copyright©byJosephMazur Requestsforpermissiontoreproducematerialfromthisworkshouldbesentto Permissions,PrincetonUniversityPress PublishedbyPrincetonUniversityPress,WilliamStreet, Princeton,NewJersey IntheUnitedKingdom:PrincetonUniversityPress,OxfordStreet, Woodstock,OxfordshireOXTW press.princeton.edu AllRightsReserved LibraryofCongressCataloging-in-PublicationData Mazur,Joseph. Enlighteningsymbols:ashorthistoryofmathematicalnotationanditshidden powers/JosephMazur. pagescm Includesbibliographicalreferencesandindex. ISBN----(hardcover:alk.paper).Mathematical notation—History.I.Title. QA.M .(cid:156)—dc BritishLibraryCataloging-in-PublicationDataisavailable ThisbookhasbeencomposedinMinionandCandida Printedonacid-freepaper.“ PrintedintheUnitedStatesofAmerica “Mazur” — // — : — page v — # To my big brother, Barry, who taught me from 0 “Mazur” — // — : — page vi — # “Mazur” — // — : — page vii — # Contents Introduction ix Definitions xxi NoteontheIllustrations xxiii Part 1 Numerals 1 . CuriousBeginnings . CertainAncientNumberSystems . SilkandRoyalRoads . TheIndianGift . ArrivalinEurope . TheArabGift . LiberAbbaci . RefutingOrigins Part 2 Algebra 81 . SansSymbols . Diophantus’sArithmetica . TheGreatArt . SymbolInfancy . TheTimidSymbol . HierarchiesofDignity Contents vii “Mazur” — // — : — page viii — # . VowelsandConsonants . TheExplosion . ACatalogueofSymbols . TheSymbolMaster . TheLastoftheMagicians Part 3 The Power of Symbols 177 . RendezvousintheMind . TheGoodSymbol . InvisibleGorillas . MentalPictures . Conclusion AppendixALeibniz’sNotation AppendixBNewton’sFluxionofxn AppendixCExperiment AppendixDVisualizingComplexNumbers AppendixEQuaternions Acknowledgments Notes Index viii Contents “Mazur” — // — : — page ix — # Introduction Amathematician,amusician,andapsychologistwalkedintoabar... Severalyearsago,beforeIhadanythoughtsofwritingabookonthehistoryof symbols,IhadaconversationwithafewcolleaguesattheCavaTuracciolo,alittle winebar inthe villageof Bellagioon LakeComo. Thepsychologist declaredthat symbolshadbeenaroundlongbeforehumanshadaverballanguage,andthatthey areattherootsofthemostbasicandprimitivethoughts.Themusicianpointedout thatmodernmusicalnotationismostlyattributedtooneBenedictinemonkGuido d’Arezzo,wholivedattheturnofthefirstmillennium,butthatamoreprimitive formofsymbolnotationgoesalmostasfarbackasPhoenicianwriting.I,themath- ematician,astonishedmyfriendsbyrevealingthat,otherthannumerals,mathemat- icalsymbols—evenalgebraicequations—arerelativelyrecentcreations,andthatal- most all mathematical expressions were rhetorical before the end of the fifteenth century. “What?!”thepsychologistsnapped.“Whataboutmultiplication?Youmeanto tellusthattherewasnosymbolfor‘times’?” “Notbeforethesixteenth...maybeevenseventeenthcentury.” “Andequality?Whatabout‘equals’?themusicianasked. “Notbefore...oh...thesixteenthcentury.” “ButsurelyEuclidmusthavehadasymbolforaddition,”saidthepsychologist. “WhataboutthePythagoreantheorem,thatthingaboutaddingthesquaresofthe sidesofarighttriangle?” Introduction ix
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