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In my pr ev i ous ar ti cl e, w e saw how Indi an schol ar s l ai d out thefoundati on of moder n mathemati cs. How ev er , i n my pr ev i ous ar ti cl e,w e saw about onl y the basi c number sy stem. In thi s ar ti cl e, w e w i l l goone step for w ar d to see the concept of l ar ge number s i n anci ent Indi anl i ter atur e i ncl udi ng Vedi c tex ts. In al l ear l y ci v i l i zati ons, the f i r st ex pr essi on of mathemati calunder standi ng appear s i n the for m of counti ng sy stems. N umber s i near l y ci v i l i zati on w er e ty pi cal l y done usi ng sy mbol s or ser i es of l i nessepar ated by space. How ev er , l ots of di f f i cul t i es w er e faced by thew ester n concept of number sy stem. Let us anal y ze thei r di f f i cul t i es Difficulties in Number system of western world In the W ester n w or l d, l ar ge number s w er e not i n use unti l qui ter ecentl y w i th the adv ent of moder n sci ence i n the ni neteenth centur y .In ancient times, Greeks number system were based on myriad (i.e10,000) and their largest was a myriad myriad, i.e 100,000,000

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Gr eek N umber sy stem Later , Ar chi medes (287 -21 2 BC) dev i sed a sy stem of l ar ge number s up to

, by usi ng pow er s of a my r i ad my r i ad. How ev er , Ar chi medes pr oposedhi s number i ng sy stem based on pow er of my r i ad and hence, w oul d hav efaced notati onal di f f i cul t i es i n dev i si ng number s sy stem based onpow er of my r i ad. So, he stopped at thi s number because of not bei ngabl e to der i v e any new or di nal number s l ar ger than 'my r i ad my r i adth'. Shor tl y , af ter Ar chi medes dev i sed a sy stem based on pow er of 1 0,Appol l oni ous of per ga dev i sed a mor e pr acti cal number sy stem w hi chw er e not based on pow er of 1 0 but on nami ng pow er of a my r i ad, forex ampl e,

Would be a myriad squared. Why there was a Need of a better number sy stemU si ng my r i ad, w ester n schol ar s w er e not abl e to der i v e al l possi bl el ar ge number s. Hence, w ester n schol ar s w er e faci ng di f f i cul t i es i nr epr esenti ng or der i v i ng many mathemati cal concepts. Hence, ther ew as a r i si ng need of a better , Mor e conci se, mor e consi stent and mor epr eci se number sy stem. Thi s i s w her e the need of adopti ng the Indi annumber sy stem came i nto pl ace. Actual l y , w hen the w ester ner s w er ethi nk i ng i n ones or tw o, Indi ans had i nv ented the w hol e pl ace v al uesy stem w i th number s ex tendi ng up to pow er of 621 . In fact, by the 7thcentury BCE, Indian scholars reintroduced the notion of infinity asthe quantity whose denominator is zero. Ther e w as no concept of i nf i ni ty i n w ester n w or l d and onl y by the end

of 1 3th centur y , Roman schol ar s i ntr oduced the concept ofr epr esenti ng l ar ge number s as mi l l i ons i .e. 1 ,000,000 w as ex pr essed asdecies centena m ilia , (ten hundr ed thousand); w hi ch w as ev i dentl ycal l ed as mi l l i on.


Let us now see how the Indians have dominatedthe number system Number Sy stem in Ramay ana It may be sur pr i si ng to k now that Indi an sages used l ar ge number s upto pow er of 1 062 and that too mi l l i ons of y ear s ago w i th the f i r st to use i t w er e Sage Val mi k i (Authorof Ramay ana) Fol l ow i ng v er se f r om Ramay ana sai d to be w r i tten at l east 1 2 mi l l i ony ear s agoin Tretây uga, pr esents a number sy stem up to pow er of 1 062, bi g enough to ber epr esent i nf i ni ty . Verse: Úatam úatasahsrânam, kotim âhurmanisinah Œatam kotisahasrânam úankurityabhidhiyate

Above verse can be precisely translated as Œatam œatasahsram = O ne Koti i e. Hundr ed hundr ed thousand = 1 00, 00,000 = 1 cr or e = 1 07 Œatam Kotisahsram = O ne Œanku i e. Hundr ed thousand cr or e = 1 00, 000, 0000,000 = Œanku =1 01 2 1 Koti = 1 07 = 1 cr or e 1 Œanku = 1012 = 1 lakh crore 1 Mahaœanku = 1017 1 Vr ndam = 1 022 1 Mahav r ndam = 1 027 1 Padmam = 1 032 1 Mahapadmam = 1 037


1 Khar v am = 1 042 1 Mahak har v am = 1 047 1 Samudr am = 1 052 1 Ougham = 1 057 1 Mahaugham = 1 062 Thi s number i s actual l y the count of the monk ey sol di er s w ho bui l t thehi stor i c Ram Sethu (Al so k now n as Adam Br i dge).


Monk ey sol di er s Bui l di ng Ram sethu W hi l e i nhabi tants of other conti nents w er e usi ng stones and f i nger s tocount, Vedi c sages counted i n tr i l l i ons & tr i l l i ons to measur e thecosmi c concepts of thi s uni v er se. Number system in Vedas: Many Vedi c tex ts poi nts to the deci mal number sy stem. Y ajurvedadescribes the number system with place value up to 18 places, the

highest called as parardha.

For Ex ampl e, af ter pr epar i ng br i ck s for a Vedi c r i tual , Ri shi (Sage) Medhâtithi pray s to the Lord of fire, A gni


Ver se: Imâ me A gna istakâ dhenava Santvekâ ãa desa ãa satam ãa Sahasram ćāyutam ãa niyutam ãa Prayutam ćārbudam ãa nyarbudam ãa Samudrasãa madhyam ćāntasãa Parârdhasãaita me agna ishtakâ Dhenavasantvamutrâmushmimlloke . Translation: Oh Agni ! Let these br i ck s be mi l k gi v i ng cow s to me Pl ease gi v e me one and ten and hundr ed and thousand Ten thousand and l ak h and ten l ak h and One cr or e and ten cr or e and hundr ed cr or e, A thousand cr or e and one l ak h cr or e i n thi s w or l d and other w or l ds too. ek a - 1 - one - 1 0º dasa - 1 0 - ten - 1 01 satam - 1 00 - hundr ed - 1 02 sahasr am - 1 000 - thousand - 1 03 ay utam - 1 0000 - ten thousand - 1 04 ni y utam - 1 00000 - one l ak h - 1 05 pr ay utam -1 000000 - ten l ak h - 1 06 - mi l l i on ar budam -1 0000000 - one cr or e - 1 07 - ten mi l l i on ny ar budam -1 00000000 - ten cr or e - 1 08 - hundr ed mi l l i on samudr am -1 000000000 - hundr ed cr or e- 1 09 - bi l l i on madhy am -1 000000000 - thousand cr or e- 1 01 0 - ten bi l l i on antam -1 00000000000 - ten thousand cr or e-1 01 1 - hundr ed bi l l i on par ar dham -1 000000000000- one l ak h cr or e - 1 01 2 - tr i l l i on Even the concept of Fibonacci number can be found in Vedic versetranslated as


The sun f l ow er smi l es at y ou w i th 34, 55 f l or ets. (34, 55 ar e the number si n the sequence of Fi bonacci number .)

Sacr ed tex t of v edas Concept of Infinity in vedasConcept of i nf i ni ty w as used r epeatedl y i n Vedi c er a. Latest bei ng thev i shw a r oop dar shan of l or d Kr i shna w her e l or d Kr i shna i s show n asananta, meani ng “i nf i ni ty ” or “hav i ng no end”. Some of the other w or dsused i n v edi c tex ts ar e purnam , as am k hyata and aditi. For Ex:WordAs am k hyata i s used i n Yajur Veda and Br i hadar any ak a U pani shad tor epr esent the number of my ster i es of Indr a as ananta . Fol l ow i ng v er sefr om y ajur v eda descr i bes the mathemati cal concept of i nf i ni ty .How ev er , thi s v er se (Shl ok a) i s mor e metaphy si cal than matahemi cal Verse: pûr namadah p ûr nami dam pûr nât pûr namudacy ate


p ûr nâsy a pûr namaday a p ûr namev âv asi shy ate Tr ansl ati on: Fr om i nf i ni ty i s bor n i nf i ni ty .W hen i nf i ni ty i s tak en out of i nf i ni ty , l ef t ov er i s onl y i nf i ni ty . Another Vedi c v er se w i th concept of i nf i ni ty i s Om purnam adah purnam idam Purnat purnam udacyate Purnas ya purnam adaya Purnam evâvaúisòyate Translation OM- the Compl ete W hol e;pur nam- per fectl y compl ete; adah-that; pur nam-per fectl y compl ete; i dam- thi s phenomenal w or l d; pur nat-the al l -per fect; pur nam-compl ete uni t;udacy ate- i s pr oduced; pur nasy a- Compl ete W hol e; pur nam-compl etel y ; aday a-hav i ng been tak en aw ay ; pur nam-the compl ete; ev a- ev en; av asi sy ate-i s r emai ni ng. Athar v eda has some v er ses si gni fy i ng the concept of one, ar i thmeti cpr ogr essi on, and ar i thmeti c ser i es. Bel ow ar e the number s al ong w i ththe ter m used i n Vedas and other anci ent Indi an tex ts Numerals greater then 100

Number Sanskrit


200 Dvisata

300 T risata

356 sat pancasat tr isata

400 Catursata

500 Pancasata

1000 Sahasra

2000 Dvisahasra

3000 T risahasra

4000 Catursahasra

10 ,000 dasasahasra, ayuta

20 ,000 Vimsatsahasra

30 ,000 T rimsatsahasra

100,000 satasahasra, laksha, lak

200 ,000 dvi-sata-sahasra

300,000 tr i-sata-sahasra

1 ,000 ,000 prayuta, niyuta

10 ,000 ,000 koti, krore

100 ,000 ,000 arbuda, vrnda, nyarbuda

Numerals from Billion and above

Number Sanskrit

1,000 ,000 ,000 abja, shatakoti, maharbuda, nikharva, nikarvaka, badva

10,000 ,000 ,000 Kharva


100,000 ,000 ,000 nikharva, akshita1 ,000 ,000 ,000 ,000 mahaapadma, antya, antyam, nikharva

10 ,000 ,000 ,000 ,000 sha.nku

100,000 ,000 ,000 ,000 Jaladhi

1000 ,000 ,000 ,000 ,000 A ntya

10 ,000 ,000 ,000 ,000 ,000 Madhya

100,000 ,000 ,000 ,000 ,000 Paraardha

Word-Numeral Decimal System Indian scholars expressed all large numbers using the decimalnumber system. The highest power of 10 named today is 1030 (Deca). But ancient Indian mathematicians had exact names forpowers up to 1053.

Word-Numeral

Ek am

Dashk am

1 Shatam

1 Shahashr am

1 0 Dash Shahashr am

Lak sha

Dash Lak sha

Koti hi

Ay utam


N i y utam

Kank ar am

pak oti

Vi v ar am

Par ar adahaa

k s hobhya

N i v ahata or v iv aha

U tsangaha or k otippak oti

Bahul am

N aagbaal aha

Ti tl ambam

nahuta

Vy av asthaanapr agnapti hi or titlam bha

Hetuhel l am or v yav as thanapajnapati

Kar ahuhu

Hetv i ndr eey am or ninnahuta

Sampaata Lambhaha or hetv indriya

Gananaagati hi or s am aptalam bha

N i r av ady am or gananagati


ak k hobini

Mudr aabal am or nirav adya

Sar aabal am

Vi shamagnagati hi

Sar v agnaha or bindu

Vi bhutangaama

Tal l ak shanaam

abbuda

nirabbuda

ahaha

ababa

atata

s oganghik a

uppala

k um uda

pundarik a

padum a

k athana

m ahak athana


as am k hyeya

dhv ajagranis ham ani

Numbers above infinity Bel ow ar e some of the number s r epr esenti ng Inf i ni ty

§ bodhisattva (बोिधस व or बोिधस ) -

1 037218383881977644441306597687849648128

§ lalitav is tarautra (लिलतातुलनातारासू ) -10200infinities

§ m ats ya (म य) -10600infinities

k urm a (कुरमा) -102000infinities

v araha (वरहा) -103600infinities

naras im ha (नरिस हा) -104800infinities

v am ana (वामन) -105800infinities

paras huram a (परशुराम) -106000infinities

ram a (राम) -106800infinities

k hris hnaraja (कृ णराज) -10infinities

k aik i (काईक or काइक ) -108000infinities

balaram a (बलराम) -109800infinities

das av atara (दशावतारा) -1010000infinities

bhagav atapurana (भागवतपुराण) -1018000infinities

av atam s ak as utra (अवता सकासु ा) -1030000infinities

m ahadev a (महादेव) -1050000infinities

prajapati ( जापित) -1060000infinities

jyotiba ( योितबा) -1080000infinities

Large number in other texts Many anci ent Indi an tex ts tal k s about Lor d Br ahma bei ng the cr eator ofthi s uni v er se and hav e of ten menti on Lor d Br ahma to be tr i l l i ons ofy ear s ol d. Bel ow i s the i mage ex tr acted f r om such tex ts?


<http://en.wikipedia.org/wiki/File:HinduMeasurements.svg>Hi ndu uni ts of t i me i n a l ogar i thmi c scal e W hen the w or l d w as thi nk i ng of one and tw o, Indi an sages w er e deal i ngi n tr i l l i ons w hi ch i s v er y ev i dent i n thei r r el i gi ous thought. Forex ampl e, i n Vedas w hi ch i s at l east ten thousand y ear s ol d (as percar bon dati ng), w e f i nd Sansk r i t w or ds and v er ses cor r espondi ng to

pow er s of 1 0 up to a tr i l l i on and ev en 1 062. (Sansk r i t w or ds cr or es andl ak hs, r efer r i ng to 1 0,000,000 and 1 ,00,000 r especti v el y , ar e i ncommon use ev en today ). Large numbers in Surya Prajnapti Another mathemati cal tex t Sur y a Pr ajnapti (w r i i ten ar ound 400 BC)separ ates al l number s i n thr ee sets: enumer abl e, i nnumer abl e, andi nf i ni te. Each of these w as fur ther subdi v i ded i nto thr ee or der s as: � Inf i ni te: near l y i nf i ni te, tr ul y i nf i ni te, i nf i ni tel y i nf i ni te � Innumer abl e: near l y i nnumer abl e, tr ul y i nnumer abl e andi nnumer abl y i nnumer abl e Enumer abl e: l ow est, i nter medi ate and hi ghest Concept of Transfinite number The concept of i nf i ni ty i s used i n mathemati cal theor y of l i mi ts. It i sdef i ned a number gr eater than any f i ni te number . How ev er , i n the l ate1 9th centur y , mathemati ci ans star ted study i ng tr ansf i ni te number .Tr ansf i ni te number i s a number w hi ch i s not onl y gr eater than anyf i ni te number but al so gr eater than i nf i ni ty . Inci dental l y , i t w as theIndi an mathemati ci an ‘Jai na’ w ho f i r st consi der ed the concept oftr ansf i ni te number s i n 400 BCE. Large number in Taittiriya-Samhita Tai tt i r i y a-Samhi ta w r i tten befor e 1 000 B.C uses ter mi nol ogy fornumber s up to or der 1 0 ** 1 9 as 1 , 1 0, 1 0**1 , 1 0**2, 1 0**3, 1 0**4, 1 0**5, 1 0**6, 1 0**7 , 1 0**8, 1 0**9,1 0**1 0, 1 0**1 1 , 1 0**1 2, 1 0**1 3, 1 0**1 4, 1 0**1 5, 1 0**1 6, 1 0**1 7 , 1 0**1 8, 1 0**1 9. How ev er the ter mi nol ogy used i n the medi ev al age v ar i ed sl i ghtl y


http://en.wikipedia.org/wiki/File%3AHinduMeasurements.svg

than those used i n Tai tt i r i y a-Samhi ta. Al so, as di scussed abov e, theVal mi k i Ramay ana has speci f i ed the ter mi nol ogy for number s up tothe or der of 1 0 **60. How ev er , Poi nt to be noted her e i s that ter m‘samudr a’ her e denotes 1 0**9 w her eas i n the Tai tt i r i y a-Samhi ta,‘Samudr a’ r epr esents 1 0**50 and many other ter ms for l ar ge number sused by v al mi k i di f fer s f r om those used i n Tattr i y a-Samhi ta. Al so,ex cept thi s contex t, al l other numer i cal ter ms used i n v al mi k iRamay ana ar e same as those used i n Tai tt i r i y a-Samhi ta e.g. N i y uta andN y ar buda, Ar buda, Madhy a, Anty a, Samudr a and Par ar dha. Pr obabl y ,because of l i mi ted use of sequenti al l ar ge number s, some of thenumer i cal ter ms di sappear ed i n the l ater l i ter atur e and those w hi chw er e found do not r epr esent the same as used i n anci ent l i ter atur e. Forex : Samar anganasutr adhar a (1 1 th centur y A.D.) uses padma for 1 0**1 3, ,Sank u for 1 0 ** 1 2, Khar v a for 1 0**1 0 and Vr nda for 1 0**9. In the Tai tt i r i y a U pani shad, ther e i s a secti on w hi ch si gni f i es thequasi -mathemati cal r el ati onshi p betw een the bl i ss of a Br ahman(Enl i ghtened i ndi v i dual ) and the bl i ss of a y oung man w ho hav e al lmater i al i st i c pl easur e of thi s w or l d. It say s that a y oung handsome manw ho i s f i t , str ong, heal thy , and i s the k i ng of the w or l d, i s one uni t ofhuman bl i ss. Fur ther , many human attr i butes ar e used to pr ov i de aser i es of mul ti pl i cati on to measur e human bl i ss i n 1 001 0 uni ts equal tospi r i tual enl i ghtenment or sal v ati on (Mok sha).


Ver ses of Tai tt i r i y a-Samhi ta Large numbers in Jainism Jai n schol ar s had a bi g fasci nati on of l ar ge number s. In fact , they w er ethe f i r st , sci enti f i c thi nk er to theor i ze that “al l i nf i ni t i es ar e not thesame or equal ”. Thi s i dea w as establ i shed i n the moder n w or l d onl y i nthe l ate ni neteenth centur y w hen Cantor i ni t i ated hi s theor y of sets. Besi de concept of i nf i ni t i es, Jai n schol ar s w er e al so aw ar e of i ndi cesr el ated theor y , though they di d not used any conv eni ent notati on as w euse i n mathemati cs today . Instead of usi ng notati on, Jai n used si mpl estatements to speci fy theor y of i ndi ces. Instead of cal l i ng squar e r ootsand squar es, they used the ter m “f i r st squar e r oot” or “second squar e”etc. For ex : consi der the fol l ow i ng statement f r om thei r tex ts: The f i r stsquar e r oot mul ti pl i ed by the second squar e r oot i s the cube of the


second squar e r oot. Accor di ng to w hat w e l ear ned about i ndi ces, thi sstatement can be denoted

a1/2 x a1/4 = (a1/4 )3

Large numbers in Buddhism Buddhi st l i ter atur e al so descr i bes f i ni te, i nf i ni te, deter mi nate andi ndeter mi nate number s. Buddhi st mathemati ci ans cl assi f i edmathemati cs as Gar na (Si mpl e Mathemati cs) or Sank hy an (Hi gherMathemati cs). N umber s w er e of thr ee ty pes: Asank hey a (uncountabl e),Sank hey a (countabl e), and Anant (i nf i ni te). Accor di ng to Lal i tav i star a Sutr a<http://en.w i k i pedi a.or g/w i k i /Lal i tav i star a_Sutr a> (a Buddhi st w or k ),ther e w as a contest i nv ol v i ng w r i t i ng, ar i thmeti c and ar cher y , w her ebuddha w as defeated by gr eat mathemati ci an Ar juna w ho show ed of fhi s mathemati cal sk i l l s by ci t i ng the w or ds r epr esenti ng number s i npow er of ten up to 1 'tal l ak shana', equal s to 1 053. It al so states that thesew or ds w er e just one of the ser i es of counti ng sy stem w hi chgeometr i cal l y ex pands up to 1 0421 , that i s, a 1 fol l ow ed by 421 zer os.

The Decimal System in Harappa N o one k now s the ex act per i od of i nv enti on of number sy stem and themoder n w or l d ask s for pr oof of Vedas bei ng mi l l i ons of y ear s ol d. So, l etus go on the basi s of ev i dences. As per the ev i dences, a deci mal sy stemex i sted dur i ng the pr e-Har appa per i od. Ar chaeol ogi cal ev i dences w i thw ei ghts cor r espondi ng to r ati os of 0.05, 0.1 , 0.2, 0.5, 1 , 2, 5, 1 0, 20, 50, 1 00,200, and 500 hav e been found i n Har appa and Mohenjo-Dar o. How ev er ,the most notabl e char acter i sti c of these w ei ghts and measur es ar e thei rr emar k abl e accur acy . For Ex : A br onze r od mar k ed w i th 0.367 i nches ex actl y measur es to .367i nches. Scal es w i th uni t of .367 i nches w er e used i n those day s forpr oper pl anni ng and constr ucti on of tow ns, r oads, dr ai ns, pal aces,tow er s as per the gui del i nes l ai d out by the ar chi tect/Ki ng. Thi s


http://en.wikipedia.org/wiki/Lalitavistara_Sutra

di scov er y of an accur ate w ei ght based sy stem poi nts to the ex i stence oftr ade and commer ce i n the Har appa soci ety .

Stones used as W ei ghts dur i ng Har appa per i od Af ter the Har appa per i od, many Indi an mathemati ci ans i ntr oducedadv anced concepts of pl ace v al ues sy stem, i ntegr al cal cul us,di f fer enti al equati ons etc w hi ch w e w i l l see i n our nex t ar ti cl es. I w i l l end thi s ar ti cl e w i th the fol l ow i ng quotati on f r om G.B. Hal sted: “The impo rtance o f the creatio n o f the zero mark can nev erb e exag g erated. This g iv ing to airy no thing , no t merely alo cal hab itatio n and a name, a picture, a s ymb o l, b ut helpfulpo w er, is the characteris tic o f the Hindu race fro m w hence its prang . It is lik e co ining the Nirv ana into dynamo s . Nos ing le mathematical creatio n has b een mo re po tent fo r theg eneral o n-g o o f intellig ence and po w er.”


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