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.