Antiquity of decimal system of numeration-1

 

              

                     THE ANTIQUITY OF THE DECIMAL SYSTEM OF NUMERATION 

    The decimal system of numeration, that is, the system of numeration, based on the number ten and its powers, has been acknowledged to be the most fundamental contribution of India to the development of mathematics.  Scholars generally believe that this system started with the concept of zero, which was first introduced in 598 A.D. by Brahmagupta in his Brahmasphutasiddhanta wherein he gives the rules for the arithmetical operations with zero.

 

The Vedic period

 

    But it must be mentioned that the decimal system is of very much greater antiquity, inasmuch as we find evidence for the use of this even in the Vedas, which according to traditional Indian scholars, emanated as the very breath of the Ultimate Reality, and hence for the advent of which no particular age can be prescribed.  It is said:

                    Anãdinidhãna Vedah Vãgucchrushtã  svayambhuvã jneyaysća”

i.e. the Vedas were eternal sounds emanating spontaneously from the breath of the Creator.  According to Colebrook:     

“According to the Hindus, numeration is of divine origin; the invention of nine figures (anka) with the device of places to make them suffice for all numbers, being ascribed to the beneficent Creator of the Universe”.

 

     Among the Vedas, the Rigveda has been acknowledged even by Western scholars as the most ancient.  We find a reference to the decimal system of numeration in the Ćamaka portion, wherein the devotee prays to Śiva to bless him with the various wherewithals for leading a happy and prosperous life.  He refers to these things in terms of different numbers, each number specifying a certain thing.  We find references first to odd numbers from 1 to 33, then to even numbers and then to multiples of 4 from 4 to 48, as follows:

     “eka ća me tisrasća me panća ća me sapta ća me nava ća ma ekãdaśa ća me trayodaśa ća me panćadaśa ća me saptadaśa ća  me navadaśa ća me ekavimśatisća me trayovimśatisća me pancavimśatisća me saptavimśatisća me navavimśatisća  me ekatrimśatisća me trayastrimśatisća me”.

 

    “ćatasrasća me ashtau ća me dwãdaśa ća me shodaśa ća me vimśatisća me ćaturvimśatisća me ashtavimśatisća me dwãtrimśatćća me shattrimśaćća me ćatvãrisća me ćatusćatvarisća me ashtaćatvarisća me”.

The devotee prays: “May the powers associated with these integers manifest in me in the form of the things that they signify”.

       The odd numbers from 1 to 33 signify the boons required from the Devas or angels, while the even numbers signify the things required for the wellbeing and prosperity of the 33 living beings in the world, and these are as follows:

                                                             -2-

 

                1 Nature or environment

                3 The three gunas, namely sattwa, rajas and tamas   

                5 The five elements, namely earth, water, fire, wind and sky

                7 The five sense organs and the mind and intellect or buddhi   

                9 The nine openings in the human body

11 The ten prãnas along with their conduit namely the Sushumnã     

13    Thirteen Devatas   

15  The various nerve fibres such as the Sangini

17    The seventeen important parts of the body

19    Nineteen herbs such as sangarasu 

21    Twenty-one sensitive spots in the human body   

23    The deities which control the major diseases

25    Twenty-five apsaras or beautiful women who inhabit the earth as they do the heavens

27    Twenty-seven gandharvas

29    Twenty-nine Vidyut Devatas

31    Thirty-one worlds

33    Thirty-three Devas

As regards the multiples of 4, from 4 to 48, they signify the following:

4        The four goals to be pursued in human life, namely dharma, artha,

       kãma and moksha

8        The four Vedas and the four Upavedas

12      The twelve parts of the śãstras

16    The sixteen siddhis to be attained

20    The basic elements, sensations like sound etc., both subtle and gross

24    The twenty-four letters of the Gayatrimantra

28    A metre in Sanskrit which consists of 28 letters per line

32    The Anushtup metre consisting of 32 letters per line

36    The Bruhati metre consisting of 36 letters per line

40    The Pankti metre containing 40 letters per line

44    The Trishtup metre containing 44 letters per line

48 The Jagati metre containing 48 letters per line.

    In the Rudrasookta of the Rigveda, we find:

             Gowrimimãya salilãni takshatyekapadi dwipadi, sã ćatushpadi

              Ashtapadi, navapadi, babhuvushi sahasrãkshaga paramã vyoman”.

Here also we find reference to the numbers 1, 2, 4, 8, 9 and then 1000.

    In the Rigveda (11.4.8) we find reference to thousand as follows:

            Sahasradhã panćadaśãnyukthã yãvat

             Dyãvã Prithvi tãvat

             Sahasradhã mahimãnah sahasram yãvat

             Brahma nishthitam tãvati vãk”

The fifteen Ukthãs (the five karma indriyas, the five jnãna indriyas and the five bhutas) referred to here shine in thousands of forms in thousands of bodies.                                                                     

                                                                 

 

                                                         -3-

 

Brahma is coexpansive with the earth and the sky.  Its capabilities are also thousandfold.

     After reference to numbers from 1 to 9, in the Rigveda (9.31 to 34), we find references to the numbers from ten onwards up to seventeen:

             “Mitro navãksharena trivrutastomamudajayat tamujjesham

             Varuno daśãksharena virãjamudajayat tamujjesham

             Indra ekãdasãksharena trishtubhamudajayat tãmujjesham

             Viśwe Devãh dwãdaśãksharena jagateemudajayat tãmujjesham

             Vasavastrayodaśãksharena trayodaśastomamudajayat tamujjesham

             Rudrãsćaturdaśãksharena ćaturdaśastomamudajayat tamujjesham

             Ãdityãh panćadaśãksharena panćadaśastomamudajayat tamujjesham

             Aditih shodadaśãksharena shodaśtomamudajayat tamujjesham

             Prajãpatih saptadaśãksharena saptadaśastomamudajayat tamujjesham”.

 

  Later, we find in the Rigveda (14.23) references to odd numbers 15, 17, 18, 19, 32, 23, 24, 25, 31, 33, and then to the numbers 34, 36, and 48.

           Ãśustrivrudbhr­ãntah panćadaśo vyomã

            Saptadaśo dharuna ekavimśah praturttirashtãdaśastapo

            Navadaśo abheevartantah samviśo vaćo dwãtrimśah

            Sambharanastrayovimśo yonisćaturvimśah garbãh

            Panćavimśa ojãstrinavah kraturekatrimśah pratishthã

            Trayastrimśl vrudhrasya vishtapam ćatustrimśo nãkah

            Shatrimśl trivarto ashtãćatvãrimśo dhartram ćatushtomah.

Again, in Rigveda we find references to 3, 15, 17, 21, 24, 25, 27, 4, 40 and 33 in the following:

   Agnerbhãgo asi deekshãyã ãdhipatyam brahma sprutam trivrutstomah

    Indrasya bhãgo asi vishnorãdhipatyam kshatram sprutam panćadaśtomah

               Nrućakshasãm bhãgo asi dhãturãdhipatyam janitram sprutam saptadaśastomah

               Mitrasya bhãgo asi varunasyãdhipatyam divo vrushtivãta sprutam

                 Ekavimãsastomah

  Vasunãm bhãgo asi rudrãnãmãdhipatyam catushpãd sprutam ćaturvimśastomah

             Ãdityãnãm bhãgo asi marutãmãdhipatyam garbhãh sprutãh panćavimśastomah

             Adityai bhãgo asi pooshna ãdhipatyamojasprutam trinavastomah

  Devasya saviturbhãgo asi brihaspaterãdhipatyam sameećerdiśa

      sprutãsćatushtomah

  Yavãnãm bhãgo asyayavãnãmãdhipatyam  prajã sprutãsćatvãrimśastomah

  Rubhoonãm bhãgo asi Viśweshãm devãnãmãdhipatyam bhootam sprutam

      trayastrimśastomah.

 

     While specifying the presiding deities of various things, we find references in the Rigveda again to the odd numbers from 1 to 33:

     Ekayãstuvata prajã adheeyanta prajãpatiradhipatirãseet

     Tisrubhistuvata brahmãsrujyata brahmanaspatiradhipatirãseet

 

                                                         -4-

 

     Panćabhistuvata bhootãnyasrujyanta bhootãnãm patiradhipatirãseet

     Saptabhistuvata saptarushayo asrujanta bhootãnam patiradhipatirãseet

     Navabhistuata pitaro asrujyanta jyantãdiradhipatirãseet

     Ekadaãsbhistuvata rutavo asrujyantãruttavãh adhipataya ãsan

     Trayodaśabhihstuvata mãsã asrujyanta samvatsaro adhipatirãseet

     Panćadaśabhistuvata kshtramasrujayantendro adhipatirãseet

     Saptadaśabhistuvata grãmyã paśavo asrujyanta brihaspatiradhipatirãseet

     Navadaśabhistuvata śudrãryavasrujyettãmahorãtre adhipatiee ãstãm

     Ekavimãsatyãstuvataikaśaphãh paśavo asrujyanta varuno adhipatirãseet

     Trayovimśatyãstuvata kshudrãh paśavo asrujyanta pooshãdhipatirãseet

     Panćavimsatyãstuvata ãranyãh paśavo asrujyanta vãyuradhipatirãseet

     Saptavimśatyãstuvatadyãvãprithivi nyautãm vasavo rudrã adityã

        amushyãmasta evãdhipatya ãsan

     Navavimśatyśtuvata  vanaspatayo asrujyanta somo adhipatirãseet

     Ekatrimśatã stuvata prajã asrujyanta yavãsćãyavãdhipatiya ãsan

     Trayastrimśatã stuvata bhootãnyaśãjyan prajapatih

         parameshtyadhipatirãseet.

 

            In the Rigveda (15.3), we find references to 16 and 44:   

                  “Shodaśãstoma ojo dravinam ća

                  Ćatusćatvãrimsastoma varća dravinam”.

            In the Rudram portion of Rigveda (16.53 et seq), we find references to 1000:

                 “Sahasrãni sahasraśo bahvostava hetayah

                  Tãsãmeeśãno bhagavah parãćeenã mukhãkrudhi

                  Asankhyãtã sahasrãni yo rudrãdhibhoomyãm

                  Teshãm sahasrayojane ava dhanvãni tanmasi”.

 

            We find references again to 1000 in the Rigveda (17.74. 15.55 and 18.62):

                   Yãmasya kanvo aduhatprapeenam sahasradhãrã payasã maheergãm

 

                   Yena vahasi sahasram, yenãgne sarvavedasam

 

                   Tenemam yajnam no naya svardeveshu gantave”.

             Reference to 100 is found in Rigveda (19.37)

                  “Punantu mã pitarah somyãsah punantu  mã pitãmahãh

                    Punantu mã prapitãmhãh pavitrena śatãyushãh

                    Punantu mã pitãmhãh punantu prapitãmahãh

                    Pavitrena śatãyushã viśwamãyurvyaśnavananai”.

 

    In the Purushasookta, an ancient Vedic hymn to the Lord Almighty the Creator,

we find reference to the third power of ten, namely thousand, many times:

             Sahasrseershã purushah sahasrãkshah sahasrapãt”

 

                                                                -5-

 

The Creator is described as having thousand heads, thousand eyes and thousand feet.

    When Vedic benediction is given, the following is recited:

            Satamãnam bhavati satãyuh purusha ãyush devendriyo pratitishthati”

Here we find a reference to hundred years as the normal age of a human being and prosperity is prayed for in multiples of hundred.

     When a married woman is blessed, the following Vedic benediction is given:

            Dasãsyãm putrãnãm patim ekãdaśam kridhi”.

That is, she is blessed by saying: “May you be the mother of ten children and consider your husband as your eleventh child”.  Here we find reference                            to ten and eleven.

 

     In Krishnayajurveda Taittriyasamhita (2.4.13), while describing the rules for the Tredãtaveeyashtih, the following reference to 1000 is found:

          Yadvã idam kim ća tadasmai tat prãyaććhatrućah sãmãni yajuoogumshi sahasram vã asmai tat prãyaććhat tasmãt sahasradakshinãm”.    

 

     In the Vajasaneyi Samhita of Yajurveda, (xvii.2), where the devotee prays for a large number of cows to be given to him, we find reference to the decimal system of numeration with various names given to the successive powers of ten, as follows:

 

       Imã me agna ishtakãh dhenavah santvekã ća daśa ća daśa ća śatam ća śatam ća sahasram ća sahasram ćãyutam ćãyutam ća niyutam ća niyutam ća prayutam ća prayutam ćãrbudam ća nyarbudam ća samudras­ća madhyam ćantasća parãrdhasćaivã me agna ishtakãh dhenavah”.

                     

                               1 = 10o (eka)

              10 = 101  (daśa)

            100 = 102   (śata)

          1000 = 103  (sahasra)

        10000 = 104   (ayuta)

    1,00,000 = 105  (niyuta)

  10,00,000 = 106  (prayuta)

               10,000,000 = 107   (arbuda)

d             100,000,000 = 108   (nyarbuda)

            1,000,000,000  =  109  (samudra)

          10,000,000,000  =  1010 (madhya)

        100,000,000,000  =  1011 (anta)

                 1,000,000,000,000 =  1012  (parãrdha)

 

       Here, we find that the term Parãrdha means 1012.  Brahma, the Creator is said to have a life span of two parãrdhas, as we find in the following:

                        Parãrdhadwayajeevinah ãdyabrahmanah …”

 

 

                                                           -6-

 

      In Krishna Yajurveda Taittriyasamhitã  (5.7.3), we find reference to 100 as follows:

         “Yannityukhyam samindha ishtaka evaitã upadhattam

           Vanaspatyãh suvargasya lokasya samashtyai

           Śatãyudhãya śataveeryãya śatoktaye abhimatishãhe

           Śatam yo na śarado ajeetamindro neshadati duritãni viśwã.

     In Taittriyasamhita (4.4.11.3) in Rutaveyshtakopadhãnam, the devotee prays for a large number of cows here and now and in this world, their numbers running in powers of ten and ending with parãrdha:

       Ãdityãnãm te adhipatayastebhyo namaste no mrudayantu te yam dwishmo yasća no dweshti tam vo jambhe dadhãmi sahasrasya pramã asi sahasrasya pratimã asi sahasrasya vimã asi sahasrayonmã asi sãhasro asi sahasrasya twe mã me agna ishtakã dhenavah santwekã ća śatam ća sahasram ćãyutam ća niyutam ća prayutam ćãrbudam ća nyarbudam ća samudrsća madhyam ćãntasća parãrdhasćemã agna ishtakã dhenavah santu shashtih sahasramayutamksheeyamãnã rutasthãh sthateevrudho ghrutasćuto madhusćuta oorjaswateeh swadhãvineestã me agna ishtakã dhenavah santu virãjo nãma kãmadughã amushãsmilloke”.

     In the Taittriyasamhita (7.2.3 to 20), the powers of ten with their distinct names are referred to again as follows:

   “ayugmmekãdhikãnãm sanghah, dwayãdi yugmasankhyãsanghah, tryãditrayottarasankhyãsanghah, ćaturãdi ćaturuttarasankhyãsanghah, panćãdi panćottarasankhyãsanghah, daśãdi daśottarasankhyãsanghah.

       

    First, we find reference to the mantras to be recited from the 6th to the 11th  nights, and on the 12th day during daytime, and in that connection, the following is what we find.  We find first mention of ãhutis being offered for the deities signified by the numbers from 1 to 20, and then 29, 39, 49, 59, 69, 79, 89, 99, 100 and 200. Then we have reference to the odd numbers from 1 to 19, and the  numbers ending in 9 as we proceed from 20 to 100 (as before), and finally 100.  Then, we have reference to even numbers from 2 to 20 and then 98 and 100.  Then, we have reference to odd numbers from 3 to 19 and the numbers ending in 9 as we proceed from 20 to 100 as before, and 100. Then, we have multiples of 4 from 4 to 20, and then 96 and 100; then we have multiples of 5 up to 20 and then 95 and 100.  Then we have multiples of 10 from 10 to 100.  Then, we have 50 and multiples of 100 from 100 to 1000.  Finally, we have:

      śatamãrabhya daśaguno vruddhiyuktãsankhyã sanghah”.

        Dasãya swãhã:        10

        Śatãya swãhã:         102

        Sahasrãya swãha:    103

        Ayutãya swãhã:        104

        N iyutãya swãhã:      105

                                                                      -7-                                                     

                 

                   Prayutãya swãhã:     106

        Arbudãya swãhã:      107

        Nyarbudãya swahã:  108

        Samudrãya swãhã:    109

        Madhyãya swãhã:     1010

        Antãya swãhã:          1011

        Parãrddhãya swãhã:   1012

Here, we find that parãrdha term has been used to denote 1012 .

           

     In the Maitrãyanisamhita (11.8.4), the list is given as above, but after ayuta, we have prayuta, then again ayuta, then nyarbuda, samudra, madhya, anta and parãrddha.

     In the Kãtakasamhita (xvii.10) we have a similar list but with niyuta and prayuta having interchanged places.  In the same samhita, (39.6) after nyarbuda, a new term bãdava is introduced.  Otherwise, the same list is seen with slight alterations as indicated above.

    In the Pancavimśa Brahmana, the Yajurveda list is given up to nyarbuda, followed by nikharva, vadava, akshiti etc.

    In the Sãnkhyãyana Srautasutra, after nyuarbuda, we have nikharva, samudra, salila, antya and ananta.

 

                                             The Purãnic period            

 

    In the Agnipurãna we find reference to these powers of ten, in ch.122, 123,131, 140, 141, 328-335. According to some scholars, the date of the Agnipurana cannot be placed later than the sixth century.  But the references given above prove conclusively that in the 6th century, at any rate, the place value notation based on powers of ten was very much prevalent and was so well known that even philosophers could use it before laymen and scholars could resort to them ith ease to clinch their point.

    In the Vishnupurãna, (6.3) we find Parãśara explaining to Maitreya that a parãrdha is 1017 based on the system of numeration in increasing powers of ten according to place value in succession :

                         Śri Matitreya uvãća:

                               Parãrdhasankhyãm Bhagavan mamãćakshva yayã tu sah

                               Dwiguneekritayã jneyah prãkrutah pratisamćarah

                          Śri Parãśara uvãća:

                               Sthãnãtsthãnam daśagunamekasmãdganyate dwija

                               Tato ashtãdaśame bhage parãrdhambhidheeyate”.

 

            “In case of multiples from the units place, the value of each place (sthãna) is ten times the value of the preceding place. More than one place to the next in succession, the places are successive multiples of ten.    The number in the eighteenth place is called the parãrddha”.

 

                                                          -8-

 

This reckoning gives the following in powers of 10 starting from the zeroth power:

                         Ekam:        100 =1

              Dasa:         101

              Śatam:       102 

              Sahasram:  103 

              Ayutam:      104

              Niyutam:       105

                     Prayutam:   106

              Arbudam:    107

                     Nyarbudam: 108

                     Vrinda:         109

              Kharva:        1010    

                     Nikharva:     1011

                     Śankha:         1012

              Padma:         1013

              Samudram:   1014

              Madhyam:     1015

              Antam:          1016

              Parãrddham: 1017

 

 

   In the Vãyupurãna (63.92-104), we find reference to eighteen place values, based on successive powers of ten:

            Kasmãdesha parãrddhasćakasćaisha para ućyate

             Yetadveditumiććhãmastanno nigadasattama

             Srunudhwam me parãrddhasća parisankhyãm parasya ća

             Eka daśa satasćaiva sahasrasćaiva sankhyayã

             Vijneyeyam ãsahasrantu sahasr­ãni dasãyutam

             Ekam śatasahasrantu niyutam proćyate budhaih

             Tathã śatasahasrãnãmarbudam kotirućyate

             Arbudam daśakotyastu hyabjam kotiśatam viduh

             Sahasramapi koteenãm kharvamãhurmaneeshinah

             Daśakotisahasrãni nikharvamiti tam viduh

             Śatam kotisahasrãnãm Śankurityabhidheeyate

             Sahasrantu sahasrãnam koteenãm dasadhã punah

             Gunitãni samudram vai prãhuh sankhyãvido janãh

             Koteenãm sahasramayutamipyayam Madhya ućyate

              Kotisahasraniutã sa ćãnta iti sanjitah

              Kotikotisahasrãni parãrddha iti keertyate

              Parãrddhadwigunamćãpi  paramãhurmaneeshinah

              Śatamãhuh paridrudham sahasramparipadmakam

              Vijneyamayutam tasminniyutam prayutam tatah

              Arbudam nirbudamćaiva Khurbudamća tatah smrutam

                                                        -9-

 

 

               Kharvamćaivca nikharvamća śanku padmam tathaiva ća

               Samudram madhyamamćaiva parãrddhamaparam tatah

               Evamshtãdaśaitãni sth­ãnani gananãvidhau                                             

               Śatãneeti vijãneeyãt sanjitãni maharushibhih

               Kalpasankhyã pravruttasya parãrddham Brahmanah smrutam

               Tãvććhesho api kãlo asya tasyãnte parimrujyate

               Para Yesha Parãrddhamća sankhyãtah sankhyayã mayã

 

           “These are the eighteen places of calculation.  The sages say that in this

             way  the number of places can be hundreds”.

The powers of ten on this reckoning and their names as given above are as follows:

                             Ekam = 1

                             Daśa  = 101

                                           Śatam = 102

                             Sahasram =  103

                                           Ayutam     =  104

                             Niyutam    =  105

                             Prayutam  =  106                            

                                        Koti           =   107

                                                            Arbudam   =   108

                                                            Abjam        =   109

                                                             Kharvam   =   1010

                                            Nikharvam=   1011

                              Śanku       =    1012

                              Samudram

                            (Saritãm patih) =   1013

                              Madhyam  =    1014

                              Antam        =    1015

                               Parãrddham  =  1017

                               Param           =  2 x 1017

     While summing up the narration, the successive powers of ten are described with slightly different names as: Eka, Daśa, Śatam, Sahasram, Ayutam, Niyutam, Prayutam, arbudam, nirbudam, khurbudam, kharvam, nikharvam, śanku, padmam, samudram, madhyam,  parãrddham. (Since in this there is no mention of  anta here, parãrddha according to this summing up comes out to be only 1016).

 

   In the Vyãsabhãshya on Patanjali’s Yogasutras, (iii.13) we find the statement that the successive place values are successive multiples by ten:

              “Thus, the same stroke is termed one in the units place, ten in the

               tens place, and hundred in the hundreds place”.

   In the Sãreerakabhãshya (III.iii.17), Ãdi Śankara also refers to the decimal     system based on powers of ten, when he says:

 

                                                                -10-

 

              “Thus, although the stroke is the same, yet by a change of place, it 

              acquires the values 1, then 10, then 100 and so on”.

The Rãmayana period

    In the Ramayana, one of the greatest epics of India, containing a record of contemporary history during the period of Rama’s reign, which is more a historical novel with religion and spirituality in the background, where normally one does not expect to find a discourse on systems of numeration, we find that there is a reference to the system of numeration based on powers of ten, which shows that the people who were living in the contemporary society and with that civilization were so advanced that the decimal system was something which people could handle with consummate ease, and which scholars and commentators could refer to, to clinch their arguments or points during debate.

    In the Valmiki Ramayana, whose period is definitely very much earlier than 598 A.D., we find a reference to the decimal system of numeration.  Indeed, we find that even in those earlier times people were aware of this system even in countries like Lanka, the kingdom of Ravana. 

    In the Kishkindhãkãnda (35.14-16), we find Vãli’s wife Tãrã pleading with Lakshmana not to be angry with Sugreeva for the delay that had taken place in the search for Sita, and she then gave a clue to the strength of Ravana’s army:

             Samãneshyati Sugreevah Seetayã saha Rãghavam

             Śaśankamiva rohinyã hatva tam rãkshãsadhamam

             Śatakotisahasrãni Lankãyãm kila rãkshasã

             Ayutãni ća shadtrimśatsahasrani śatani ća

             Ahatva tamsća durdharsãn rãkshasãn kãmaroopinah

             Asakyam Rãvanam hantum yena sã Mythili hritã

             Te na sakyã rane hantumasahãyena Lakshmana

             Ravanah kroorakarma ća Sugreevena viśeshatah

             Yevamãkhyatavãn Vãli sa hyabigno hareeśwarah

             Ãmastu na me vyaktah sravanãtbraveemi yat”

             

          Explaining how she, though confined to the household, came to know of the strength of Ravana’s army, Tãrã explained that she was merely recollecting what Vãli had told her about the army strength of Ravana, but Vãli had not told her how he had estimated the same.            

          The commentator Tilaka explains the system of numeration referred to above as follows:

         

            Yesha sankhyã vivakshitaiva.

              Daśakotih samudrah śatakotirmadhyam madhyanãm sahasram tathã shastisahasrãdhikatrilakshnãni tadupari, shadtrimsatsahasrãni tadupari tãvanti Śatãneetyarthah”.

      

                                                            

Comments