[close] [close] [close] BharatiKrishnaTirtha'sVedicmathematics FromWikipedia,thefreeencyclopedia Jumpto:navigation,search FortheactualmathematicsoftheVedicperiod,seethearticlesonSulbaSu trasandIndianmathematics. Vedicmathematicsisasystemofmathematicsconsistingofalistof16basicsu tras,oraphorisms.TheywerepresentedbyaHinduscholarandmathematician,Bh aratiKrishnaTirthajiMaharaja,duringtheearlypartofthe20thcentury.[1] TirthajiclaimedthathefoundthesutrasafteryearsofstudyingtheVedas,as etofsacredancientHindutexts.[2]However,Vedasdonotcontainanyofthe"V edicmathematics"sutras.[3][4] ThecalculationstrategiesprovidedbyVedicmathematicsaresaidtobecreative anduseful,andcanbeappliedinanumberofwaystocalculationmethodsinar ithmeticandalgebra,mostnotablywithintheeducationsystem.Someofitsmeth odssharesimilaritieswiththeTrachtenbergsystem. Contents [hide] *1TirthajiandthepromotionofVedicmathematics *2Originofthesystem *3Thesutras(formulaeoraphorisms) o3.1Subsutrasorcorollaries o3.2BackgroundinformationontheVedas *4Methodsdescribedinthesutras o4.1Allfromnineandthelastfromten +4.1.1Firstcorollary,whensquaringnumbers o4.2Byonemorethantheonebefore +4.2.1Method1:example:usingmultiplicationtocalculate1/1 9 +4.2.2Method2:example:usingdivisiontocalculate1/19 o4.3Multiplyingby11 o4.4Verticallyandcrosswise o4.5Transposeandapply o4.6Whenthesamuccayaisthesame,thatsamuccayaiszero o4.7Ifoneisinratio,theotheroneiszero *5Applications *6Seealso *7Notes *8References *9Externallinks [edit]TirthajiandthepromotionofVedicmathematics VedicmathematicswaspresentedbymaharajaTirthaji,whoisdescribedashaving therarecombinationoftheprobinginsightandrevealingintuitionofaYogiwi ththeanalyticalacumenandsynthetictalentofamathematician.[5]BorninIndi ain1884,Tirthajiwasconsideredanexceptionalscholar;byagetwentyhehad studiedatanumberofcollegesanduniversitiesthroughoutthecountry,beenaw ardedthetitleofSaraswatibytheMadrasSanskritAssociationforhisremarkab leproficiencyinSanskrit,andhadcompletedsevenmastersdegrees,includingS anskrit,Philosophy,English,Mathematics,HistoryandScience,withtheAmerica nCollegeofSciences.[6] Around1911,TirthajiresolvedtostudyseveralsectionsoftheAtharvavedatha thesaidhadbeendismissedbyOrientalists,Indologistsandantiquarianschola rsasnonsensical.[7]HewaspartofashrinkinggroupofIndianscholarswhobe lievedthattheVedasrepresentedaninexhaustiblemineofprofoundwisdombothsp iritualandsecular(Pratyagatmananda,1965).Tirthajiclaimedthatthereweres ectionsoftheAtharvavedalabeledganitasutrasormathematicalformulaethatmyst eriouslymadenoobviousreferencetomathematics.[8]Tirthajiexplainsthathe wasdeterminedtounderstandtheganitasutrareferences,andbeganstudyingancie ntlexiconsandlexicographyinmoredetail.[9]Withthisresolve,Tirthajiwent toSringeri,Karnataka,wherehebeganyearsofsolitarystudyandmeditation.[ 10] Eightyearslater,Tirthajiemergedclaimingtohavedeciphered16fundamentalm athematicalsutrasintheVedas,whichtodayhavebecomethefoundationofVedic mathematics.[11]AccordingtoTirthaji,thesutrascovereverybranchofmathem atics,fromarithmetictosphericalconics,andthatthereisnomathematicsbeyo ndtheirjurisdiction.[12] Afterdiscoveringthesutras,TirthajitraveledaroundIndiapresentingVedicma thematics,andevenlecturedintheUnitedStatesandEnglandin1958(Trivedi, 1965).Inadditiontolecturing,Tirthajialsowrotesixteenvolumes,oneforea chbasicsutra,explainingtheirapplications.[13]Beforetheywerepublished,t hemanuscriptswerelostirretrievably.Thebook"VedicMathematics"carriesas mallbiographicalsketchoftheLateswamijibyoneofhisdisciples.Inthisin troductionitisstatedthat"..theSwamijiusedtosaythathehadwrittensix teenvolumesonthesesutras,oneforeachsutraandthatthemanuscriptswered epositedatthehouseofoneofhisdisciples...".Theintroductionfurtherstat esthatSwamijiwasunperturbedbythelossofmanuscriptsandindicatedthathe wouldrewritethesixteenvolumes.[clarificationneeded][14]Beforefallingill anddyingin1960,Tirthajiwasabletorewriteonlythefirstofthesixteenv olumeshehadcomposed.[15]ThistextsimplytitledVedicMathematics,andpubli shedin1965hasbecomethebasisforallstudyinthearea.[16] [edit]Originofthesystem TherehasbeenmuchcontroversyamongIndianscholarsaboutTirthajisclaimsthat themathematicsisVedicandthatitencompassesallaspectsofmathematics(Ka nsara,2000). First,TirthajisdescriptionofthemathematicsasVedicismostcommonlycritici sedonthebasisthat,thusfar,noneofthesutrascanbefoundinanyextantV edicliterature(Williams,2000).WhenchallengedbyProfK.S.Shuklatopointo utthesutrasinquestionintheParishishtaoftheAtharvaveda,Shuklareported thattheswamijisaidthatthesixteensutraswerenotinthestandardeditions oftheParishishta,andthattheyoccurredinhisownParishishtaandnotanyo ther.[17][18]Consideringthelackofreferencestothesutras,coupledwiththe factthatthelanguagestyledoesnotseemVedic,someproposethatthesutras weresimplycomposedbyTirthajihimself CriticshavequestionedwhetherthissubjectdeservesthenameVedicorindeedm athematics.TheypointtothelackofevidenceofanysutrasfromtheVedicperi odconsistentwiththesystem,theinconsistencybetweenthetopicsaddressedby thesystem(suchasdecimalfractions)andtheknownmathematicsofearlyIndia ,thesubstantialextrapolationsfromafewwordsofasutratocomplexarithmet icstrategies,andtherestrictionofapplicationstoconvenient,specialcases. Theyfurthersaythatsucharithmeticasisspedupbyapplicationofthesutra scanbeperformedonacomputerorcalculatoranyway,makingtheirknowledgera therirrelevantinthemodernworld.[19] Theyarealsoworriedthatitdeflectsattentionfromgenuineachievementsofan cientandmodernIndianmathematicsandmathematicians,andthatitspromotionb yHindunationalistsmaydamagemathematicseducationinIndia.[20] Inresponsetocriticismsthatthesutrascannotbelocatedwithinthetexts,se veralpeoplehaveexplainedhowtextualreferencesshouldnotbethebasisfore valuatingtheVedicityofthemathematics(Agrawala,1992).SomeproposethatVe dicmathematicsisdifferentthanotherscientificworkbecauseitisnotpragma ticallyworkedout,butisbasedonadirectrevelation,oranintuitionalvisual isationoffundamentalmathematicaltruths(Agrawala,1992;Pratyagatmananda,196 5).Tirthajihasbeendescribedashavingthesamereverentialapproachtowardsth eVedasastheancientrishisthatformedthem.Thus,itseemsasthoughsomebe lievethatTirthajimaynothavefoundthesutraswithintheVedas,butthathe receivedthemspirituallyastherishisdid,whichshouldvalidatethemasVedic .ThecontroversyabouttheVedicityofthemathematicsisfurtherconfusedbyt hedoublemeaningofveda.Sincevedacanbetranslatedtomeanknowledge,itisa lsopossiblethatVedicmathematicssimplyreferstothefactthatthesutrasar esupposedtopresentallknowledgeofmathematics.Tirthajisdefinitionofveda doesnotclearlyclarifywhetherheusesittorepresentallknowledgeortheVedi ctexts;rather,itseemsthatheusesittorefertoboth. [edit]Thesutras(formulaeoraphorisms) Vedicmathematicsisbasedonsixteensutraswhichserveassomewhatcrypticins tructionsfordealingwithdifferentmathematicalproblems.Belowisalistoft hesutras,translatedfromSanskritintoEnglish: *"Byonemorethanthepreviousone"[21] *"Allfrom9andthelastfrom10"[22] *"Verticallyandcrosswise(multiplications)"[23] *"Transposeandapply"[24] *"Transposeandadjust(thecoefficient)"[25] *"IftheSamuccayaisthesame(onbothsidesoftheequation,then)thatS amuccayais(equalto)zero"[26][27] *BytheParavartyarule[28][29] *"Ifoneisinratio,theotheroneiszero."[30] *"Byadditionandbysubtraction."[31] *Bythecompletionornoncompletion(ofthesquare,thecube,thefourthp ower,etc.)[32][33] *Differentialcalculus[34][35] *Bythedeficiency *Specificandgeneral *Theremaindersbythelastdigit *"Theultimate(binomial)andtwicethepenultimate(binomial)(equalszero ),"[36] *"Onlythelastterms,"[37][38][39][40] *Byonelessthantheonebefore *Theproductofthesum *Allthemultipliers [edit]Subsutrasorcorollaries *"Proportionately"[41] *Theremainderremainsconstant *"Thefirstbythefirstandthelastbythelast"[42] *For7themultiplicandis143 *Byosculation *Lessenbythedeficiency *"Whatevertheextentofitsdeficiency,lessenitstillfurthertothatve ryextent;andalsosetupthesquare(ofthedeficiency)".[43] *"Byonemorethanthepreviousone"[44] *"Lasttotalingten"[45][46] *Thesumoftheproducts *"By(alternative)eliminationandretention(ofthehighestandlowestpow ers)"[47] *Bymereobservation,[48][49] *Theproductofthesumisthesumoftheproducts *Ontheflag Editor'slistof16Sutrasand13SubsutrasorcorollariesfromtheVedas[50] Tocheckthefactorizationofpolynomialexpressions:"Theproductofthesumof thecoefficients(SC)inthefactorsisequaltothesumofthecoefficientsin theproduct."TheSCoftheproduct(theexpression)=ProductoftheSC(inth efactors).Additionally,thissubsutrahelpstofillinthegapswhensomefac torsareknown.[51] [edit]BackgroundinformationontheVedas Thewordvedahastwobasicmeanings.Thefirst,aliteraltranslationoftheSans kritword,isknowledge(Veda).Thesecond,andmostcommonmeaningoftheword,r eferstothesacredancientliteratureofHinduism,theVedas,acollectionofh ymns,poetryandHinduceremonialformulae(Veda).Believedtobeoneoftheold esthumanwrittenrecords,theVedasdatebackover4000years(Gaskell,2000). Traditionally,theywerepasseddownorallyandadaptedfromgenerationtogener ationbysacredsagescalledrishis,beforeeventuallyemergingwritteninVedic ,anancientformofSanskrit. TheVedasaredividedintofourmainsections:theRigveda,Samaveda,Yajurve daandtheAtharvaveda,knowncollectivelyastheSamhitas(Veda).Thefirstth ree,theRigveda,Samaveda,andYajurvedaarebasicallyritualhandbooksthat wereusedbypriestsduringtheVedicperiod(1500500BCE)(Veda).Vedicmathema ticsisapparentlypartofthefourthVeda,Atharvaveda,whichisdistinctfrom theothersinseveralways.First,unlikethereligiousfocusoftheotherVeda s,theAtharvavedacontainshymns,spellsandmagicalincantationsforpersonal anddomesticuse(Veda).Also,theAtharvaveda,whichwaswrittenlaterthant