SarahBannerHistoryofMathematicsShanyuJi
MathematicalSymbolsandTheirEffectonFacingInfinityandIrrationality
Increasingcomplexityofasubjectorideapromptsadesirefromthehumanmindto
simplify.Itstrivestobreakdowntheideaintoseveralentities,soitmaykeeptrackofall
parametersanddetails.Withtheadvancementofmathematicscamemoreunknownsand
moresolutions.Ratherthanassigningeachmathematicalsituationitsownindependent
representation,individualsdesiredtorepresentmathematicalentitiesinamoregeneralform.
Anewmathematicallanguagebegantoform,oneinwhichsymbolsandabbreviationstookthe
placeofwords.Itsintroductionopenedthedoorstounderstandinggroundbreaking
mathematicalconcepts.Notably,theintroductionofsymbolstothemathematicalworld
promptedincreasedexposuretoirrationalandinfiniteentities,makingtheirexistencedifficult
toignoreformuchlonger.
DiophantusofAlexandria,labeledbysomeasthe“fatherofalgebra”,isnotedtobe
amongthefirsttousesymbolsinhiscalculations.Diophantineequationscontainedinteger
coefficients,andwerestrictlysolvedforintegersolutions.Heviewedirrationalsolutionsas
impracticalandillogical.iFigure1depictsasolvedproblemextractedfromDiophantus’work,
Arithmetica..ii
SarahBannerHistoryofMathematicsShanyuJi
Intheproblem,Diophantusassigns‘x’asanunknownnumericalquantity,notspecifyingthe
stateofitsrationality.Fromtheequationintheproblem,hederives32x2=1.Heacknowledges
theexistenceofanirrationalsolutionbutproceedstosolvefortherationalsolution.Preceding
Diophantus,usingwordstodescribemathematicalexercisespromoteda‘real-life’viewofthe
situationwhich,inturn,madeiteasiertojustifytheabsurdityofirrationalnumbers.However,
generalizingthesituationviatheuseofsymbols,asportrayedinDiophantus’problem,
counteractswhateverounceofjustificationstillheld.DespiteDiophantus’unwillingnesstodeal
withirrationalentities,hisuseofsymbolstorepresentgeneralpolynomicsituationswasafirst
steptowardfacingirrationalityheadon.
TheIndiansemployedtheirownversionofsymbolsandabbreviations.Brahmagupta,an
Indianmathematicianandastronomer,ismostnotableforhisdevelopmentofthesymbolfor
zero.iiiPreviously,zerowasconsideredamereplaceholderbytheBabyloniansandalackof
valuebytheGreeksandRomans.iiiHowever,Brahmaguptaacknowledgedzeroasalegitimate
numericalentityandpromptlyassigneditasymbolicrepresentation.ivThedevelopmentofa
symboltorepresentzerowascrucial,asitsassignmentasanumberwasaforeignandabstract
Figure1:ProblemextractedfromDiophantus''Arithmetica'inmodernnotation.
SarahBannerHistoryofMathematicsShanyuJiconcepttomathematicians.WhereasBrahmaguptadenotedzeroasameredotplacedbeneath
numbersv,theIndiansymbolultimatelymorphedintotheillustrationinFigure2iii.
Therecognitionofzeroasanumber,followedbyitsnecessarysymbolicrepresentation,
ultimatelypavedthewaytowardadiscoverymadebyBhaskaraIIapproximately500years
later.BhaskaraIIunderstoodthatonedividedbyone-halfyieldedtwo,onedividedbyone-third
yieldedthree,andsoonandsoforth.Thus,hecametotheconclusionthatonedividedbyzero,
wouldyieldinfinity.viThiswasakeydiscoveryforIndianmathematicsandsignaledfurther
developmentofanunderstandingofinfinityandirrationality.
TheintroductionandspreadoftheHindu-ArabicNumeralSystemplayedakeyrolein
developingmethodstocopewithirrationalentities.PrecedingtheintroductionoftheHindu-
ArabicNumeralSystem,manyemployedtheRomanNumeralsystem.However,its
cumbersomeandspecificnaturecouldbelikenedtothatofwords,notashorthandnotation.vii
TheHindu-ArabicNumeralSystemwasmuchsimpler.Itcontainedonlytenfigures,including
zero,andcouldrepresentanynumberusinganycombinationoffigures.viiiDespitethesimple
andgeneralnatureoftheHindu-ArabicNumeralSystem,manycontinuedtousetheRoman
NumeralSystem.However,certainindividuals,suchasSimonStevin,madethetransition.As
outlinedinhisworkLaDisme,SimonStevinextendedtheHindu-ArabicNumeralSystemto
introducetheconceptofdecimalfractions.viiHisintroduction,inturn,pavedthewayforany
Figure2:Indiansymbolforzero.
SarahBannerHistoryofMathematicsShanyuJinumber,regardlessofitswholeandpartialcomponents,tobewrittenasacombinationof
figures.Perhaps,certainnumbersthatwerethoughttocontinueforeverwerethusproven
otherwise.Itisalsoconceivablethatirrationalnumberscouldnowbeapproximatedmore
accurately.Irrationalnumbers,whichpreviouslycouldonlyberepresentedasaratiooftwo
integers,couldnowberepresentedbyafamiliarcombinationoffigures.Understandingofthe
irrationalnumberwouldinherentlyincrease,forindividualscouldphysicallyseeevery
componentthatdefinedtheirrationalnumber.Stevin’sintroductionofdecimalfractions
removedsomeofthefearassociatedwithirrationalnumbers,asitwasnowapparentthatthey
werejustaninfinitecombinationoffamiliarwholeintegers.
Adirectrelationshipexistsbetweentheintroductionandmodificationofmathematical
symbolsandarequirementtofaceinfiniteandirrationalentities.Previously,each
mathematicalsituationreserveditsown,uniquerepresentation,allowingthedeveloperto
modifytherepresentationhowhepleased.Mostnotably,afearofirrationalentitieswould
drivehimtoaltertherepresentationtoservetwocriteria:1)Suitthatparticularsituation2)
Disregardirrationalsolutions.However,symbolsandabbreviationsgeneralizedmathematicsso
thatarepresentationnolongerappliedtoasinglesituationbutmanysituations.Thus,
disregardingirrationalsolutionsbecamemoredifficult.Thecopingmethodsmaynothave
developeduntilhundredsofyearsfollowingtheintroductionofthenewsymbolic
representation.However,thetransitiontosymbolstogeneralizemathematicalsituations
undoubtedlyplayedaroleinfacingthefearofinfinityandirrationality.
SarahBannerHistoryofMathematicsShanyuJi
i"Diophantus-HellenisticMathematics-TheStoryofMathematics."Diophantus-Hellenistic
Mathematics-TheStoryofMathematics.Web.03Oct.2016.
ii"DiophantusofAlexandria;aStudyintheHistoryofGreekAlgebra."DiophantusofAlexandria;
aStudyintheHistoryofGreekAlgebra.Web.02Oct.2016.
iii"IndianMathematics-TheStoryofMathematics."IndianMathematics-TheStoryof
Mathematics.Web.01Oct.2016.
iv“IndianBrahmagupta–TheStoryofMathematics.”IndianBrahmagupta–TheStoryof
Mathematics.Web.01Oct.2016.
vBy879AD,ZeroWasWrittenAlmostasWeNowKnowIt,anOval-butinThisCaseSmaller
thantheOtherNumbers.AndThankstotheConquestofSpainbytheMoors,ZeroFinally
ReachedEurope;bytheMiddleoftheTwelfthCentury,TranslationsofAl-Khowarizmi'sWork
HadWeavedTheirWaytoEngland."TheHistoryofZero."TheHistoryOfZero.Web.02Oct.
2016.
vi"BhāskaraII."-NewWorldEncyclopedia.Web.30Sept.2016.
viiJi,Shanyu.“Hindu-ArabicNumeralSystem.”HistoryofMathematics,Math
4388,30September2016,UniversityofHouston,Houston,TX.Lecture.
viii"Hindu-ArabicNumerationSystem."Basic-mathematics.com.Web.01Oct.2016.
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