Antiquity of decimal system of numeration-2

 
                ANTIQUITY OF DECIMAL SYSTEM OF NUMERATION-2
 

As per the above commentary:

 

                                       Samudrah  = 108

                                        Madhyam  = 109.

 

And the total strength is said to be 1012  + 60003 + 36000 + 36000 x 102.

 

In the commentary by Rãghava, we find:

             Rakshasãm śatakotisahasrãni śatakotisankhyãkarakshãmseetyarthah,

             Rakshasãmayutãdeeni ća ayutãdisankhyãkarakshãmseetyartham Lankãyam santeeti śeshah”.

   

 As per the above commentary also, the strength is estimated to be the same as above, namely:

                        1012 + 60003 + 36 x 103 + 36 x 105 

          In the Kishkindhãkãnda (35.21), a physical parading of the monkey warriors takes place before Rama, and we find:

                 Kritasamsthã Saumitrena yathã purã

                  Adya tairvãnaraih sarvairãgantavaym mahãbalaih

                  Rikshakotisahasrani  golãngoolaśatani ća

                  Adya tvãmupay­ãsyanti jahi kopamarindama

                  Kotyoanekãhstu Kãkutstha kapeenãm deepatejasam”.

 

Commenting on this, Tilaka says:

                Rakshah senapekshayã svasenãdhikyamãha—śatakotirmadhyam

                   taddasagunãtmika sahasrakotisatasankhyã rikshah golãngoolãnam

                   śatani ća śatakotih anye kapayo anekãh kotyãh asankhyã ityarthah

 

Rãghava says in his commentary:

                   Rikshakotisahasrãni kotisahasrasankhyãkarikshah ityarthah

                    golãngoolaśatani anantasankhyakagolãngoola dairghyasadruśa

                    dairghyavisishtalangoolavisishtavãnaraviseshasćetyarthah

                    kapeenãmeka kotyasca adya asmin dine twãm upayãsyanti prãpsyanti

                    atah kopam jahi”

Here again we find:

                       Madhyam = śatakoti = 109

 

 The number of rikshas or bears is said to be 1010,  while the number of monkeys with tails as long as those of a cow is said to be 1011, and the number of others is several crores and hence  is difficult to count.                   

      In the Kishkindhãkãnda (37.20-24) also we find a further reference to the decimal system in the following verses where the messengers whom Sugreeva had

                                                                

 

 

                                                           -12-

 

sent to collect the monkey warriors came back and apprised Sugreeva about the number of monkey warriors whom they had mobilized and who were reaching him:

          “Tataste Anjanasakãśã girestasmãnmahãjavãh

            Tisrakotyah plavangamãnãm niryuryatra Rãgavah

            Astam gaććhati yatrãrkastasmingirivare sthitãh

            Taruhemamahãbhãsastasmãtkotyo daśa ćyut­ãh

            Kailãśaśikharebhyasća simhakesaravarćasãm

            Tatah kotisahasrãni vãnarãnãm upãgaman

            Phalamoolena jeevanto Himavantamupãśritãh

            Teshãm kotisahasrãnãm sahasram samavartata

            Angãrakasamãnãnãm bheemãnãm bheemakarmanãm

            Vindhyãtvãnarakoteenãm sahasrãnyapatan drutam

            Ksheerodavelãnilayãstamãla vanavãsinah

            Narekelãśanãsćaiva teshãm sankhyã na vartate”

   The number coming from the various regions was as follows:               

          Anjanagiri:  3 x 108

              Setting mountain:  10 x 107 = 108

              Kailãsa peaks:      103 x 107 = 1010

              Fruitbearing regions of the Himalayas: 107 x103 x 103 = 1013

              Vindhya region:  107 x 103 = 1010

         Besides the above, there was a large number, which could not be counted, of monkeys living near the milky ocean, living in mango groves and coconut groves and eating coconuts.

        In the Kishkindhãkãnda (38.27-33) of Valmiki Ramayana, Sugreeva apprises Rama of the efforts made by him to mobilize his army and gives an idea of the forces raised by him:

           Vanaramukhyãsca śataśah śatrusoodana

             Praptãscãdya balinah prithivyãh sarvavãnarãn

             Rikshãsća vãnarãh soorãh golãngoolãsća Rãghava

             Kãntaravanadurganamabignã ghoradarśanah

             Devagandharvaputrãsća vãnarãh kãmaroopinah

             Swaih swaih parivrutãh sainyairvartante pathi Raghava”

             Śataih śatashasraisća kotibhisća plavamgamãh

             Ayutaisćãvrutah śankubhisća parantapa

             Arbudairarbudaśatairmadhyaisćantaisća vãnarãh

             Samudraisća parãrdhaisća harayo hariyoothapãh

             Ãgamishyanti te  rãjan mahendrasamavikramãh”            

 

   In the Tilaka commentary on the above, we find an explanation of the different terms used above as follows:

         Daśasahasram ayutam:   10,000    Ayutam    10 x 103 = 104

         Śatasahasram laksham:   100000   Laksha     102 x 103 = 105

                                                         -13-

                                                

          Lakshaśatam kotih:      10000000   Koti          102 x 105 = 107  

         Kotilaksham śankuh:                       Śanku       107 x 105 = 1012  

         Śankusahasram arbudam:                Arbudam 1012 x 103 = 1015

              Arbudadaśagunam madhyam:         Madhyam 1015x 10 =  1016

             Madhyãtdaśagunam antyam:           Antyam     1016  x 10 = 1017

   The commentator then goes on to say:

         Tato vimśatigunah samudrah, tatastrimśatigunah parãrddhah

           Parãrdhastenãsankhyatvamuktam”.

This means:

           Tatovimśatigunah samudrah:    Samudram:  1016 x 20

                Tatastrimśatigunah parãrddhah:    Parãrddha      1016 x 30

The understanding was that the term “Parãrddha” represented a fairly large quantity, which could not be counted.

     Commenting on the above, Raghava says:

          Hariyoothapãh harayah śatadibhirvãnarairãvrutã vartante

            tatra śatasahasre prasiddhe

            kotirlaksha śatamayutam daśasahasram

            śankuh kotilaksham arbudam śankusahasram

            madhyam arbudaddaśagunam, antyam madhyaddaśagunam

            samudrah antyaddaśagunah, parãrddha samudradtrimsadgunah iti”.

  The above confirms the calculations and the terminology for the different powers of ten shown earlier.        

      Govindaraja in his commentary says:

           “Sankhyãlakshanamuktam jyotisśãstre,

            eka daśa­satamasmãtsahasramayutam tatah param laksham,

            prayutam kotimathãrbudavrinde kharvam nikharvam ća.

            tasmãnmahãsarojam śankum śaritam pati twantam

            madhyam parãraddhamãhuryatottaradasagunam tathã jneyam”.

According to the above, however, there is a difference inasmuch as four other quantities such as vrindam, kharvam, nikharvam and mahasarojam are also introduced, each being ten times the previous one, as follows:

                              One          = 1    (eka)

                              Ten           = 101 (daśa)

                              Hundred   = 102  (śatam)

                              Thousand =  103 (sahasram)

                              Ten thousand=104 (ayutam)

                               Lakh             = 105 (laksham)

                               Prayutam      = 106

                                Koti             = 107

                               Arbudam     =  108

                                               Vrindam      =  109

                                Kharvam    =  1010

                                                Nikharvam  = 1011

                                                Mahasarojam= 1012

                                                                               -14-

                                                  Śanku         =    1013                                                          

                                                  Samudram =    1014

                                                 Antyam         =    1015

                                                  Madhyam   =    1016

                                 Parãrdham  =    1017

It will thus be seen that the two commentators have differed in the naming of the powers of ten. 

      In the commentary by Rãghava, we find almost the same enumeration as above, but with the following addition:

         Samudrah antyãtdaśagunah parãrddhah samudrãttrimsadgunah”.

.While the other terms are the same as above, when it comes to Parãrddha, however, we find:

           Parãrddha = Samudra x 30 =  1016 x 30

                                      

      In the Kishkindhãkãnda (39.17-39) we find that Valmiki narrates the names of the various monkey warriors who had presented themselves before Sugreeva in obedience to his command, and describes the strength of each one of them.

      Here also, we find a reference to the decimal system of numeration with powers of ten:

   “Nimeshãntaramãtrena tatastairhariyutapaih

    Kotiśataparivãraih kãmaropibhirãvrutah

    Kotisahasrairdaśabhih śreemãn parivrutã sadã

    Veerah Śatabalirnãma vãnarah pratyadruśyata

    Tatah kãnćanaśaiãbhastãrãyã veeryavãn pita

    Anekairdaśasahasraih kotibhih pratyadruśyata

    Tathãparena koteenãm sahasrena samanvitah

    Pitã Rumãyãh samprãptah Sugreevaśvaśuro vibhuh

    Bahuśahasairvãnarãnãm samanvitah

    Pitã Hanumatah śreemãn kesari pratyadruśyata

    Golãngoolamahãrãjo Gavãksho Bheemavikramah

    Vrutah kotisahasrena vãnarãnãm pratyadruśyata

    Rukshãnãm bheemavegãnãm Dhumrah śatrunibarhanah

    Vrutah kotisahsrãbhyãm dwãbhyãm samabhivartata

    Mahãćalanibhairghoraih Panaso nãma yuthapah

    Ãjagãma mahãveeryastisrubhih kotibhirvrutah

    Neelãnjanaćayãkãro Neelo nãmãtha yuthapah

    Adruśyata mahãkãyah kotibhirdaśabhirvrutah

    Tatah kãnćanaśailãbho Gavayo nãma yuthapah

    Ãjagãma mahãveeryah koltibhirpanćabhirvrutah

    Dareemukhasća balavãn yutapo abhyãyau tadã

    Vrutah kotisahasrena Sugreevam  samupasthitah

    Maindasća Dwivdasćobhãśviputrau mahãbalau

    Kotikotisahasrena vãnarãnãmadruśyata

    Gajas­ća balavãn veerah koltibhitistrubhirvrutah

    Ãjagãma mahãtejã Sugreevasya sameepatah

   

                                                          -15-

 

    Ruksharãjo mahãtejã Jãmbavãnnãma nãmatah

    Kotibhirdaśabhih prãptah Sugreevasya vaśe sthitah

    Rumanvãnnãma vikrãnto vãnaro vãnreśwaram

    Ãyayau balavãnstoornam kotiśatasam­ãvrutah

    Tatah kotisahasrãnãm sahsrena śatena ća

    Prushtato anugatah prãpto haribhirgandhamãdanah

    Tatah padmasahasrena vrutah śankuśatena ća

    Yuvarãjo Angadah prãpthah pitrutulyaparãkramah

    Tatastãrãdyutistãro harirbheemavikramah

    Panćabhirharikotibhirdooratah pratyadruśyata

    Indrajãnuh kapiveero yuthapah pratyadruśyata

    Ekãdasãnam koteenãmeeśwarastaisća samvrutah

    Tato Rambhastvn uprãptastarunãdityasannibhah

    Ayuten­ãvrutas­ćaiva sahasrena śatena ća

    Tato yuthapatirveero Durmukho nãma vãnarah

    Pratyadruśyata kotibhyãm  dwãbhyãm parivruto balee

    Kailãśasikharãkãrairvãnarairbheemavikramaih

    Vrutah kotisahasrena Hanumãn pratyadruśyata

    Nalasćãpi mahãveeryah samvruto drumavãsibhih

    Kotiśatena samprãpto sahasrena śatena ća

    Tato Dadhimukhah śreemãn kotibhirdãsabhirvrutah

    Samprãpto abhimatastasya Sugrevasya mahãtmanah

    Śarabhah Kumudo Vahnirvãnaro Ramha eva ća

    Yete ćãnye ća bahavo vãnarãh kãmaroopinah

    Ãvrutya prithveem sarvãn parvatãnsća vanãni ća

    Yuthapãh samanuprãptãsteshãm sankhyã na vidyate

 

Valmiki says that within a few minutes, leaders of monkey warriors accompanied by hundreds of crores of monkey warriors were to be seen there, and the list is given as follows along with their strength:

          Śatabali: 107 x 103 x 101  = 1011

          Tãrãs father: Many times 107 x 103 x 101 = Many times 1011

          Rumã’s father: 107 x 103 = 1010

               Hanuman’s father (Kesari): Several thousands

          Gavãksha: 107 x 103 = 1010

           Dhumra: 2 x 107 x 103 = 2 x 1010

          Panasa: 30 x 107 = 3 x 108

               Neela: 107 x 101 = 108

               Gavaya: 5 x 107

          Dareemukha: 107 x 103 = 1010

               Mainda and Dwivida: 107 x 107 x 103 = 1017

          Gaja: 10 x 107 = 3 x 108

              Jãmbavãn: 107 x 101 = 108

                                                                             -16-

 

 

               Rumanvãn: 107 x 102  = 109

          Gandhamãdana: 107 x 103 x 103 x 102  = 1015

          Angada: 1014 x 103 + 1013 x 102 = 1017 + 1015 = 101 x 1015

          Tãrah: 5 x 107

          Indrajãnu: 11 x 107

          Rambah: 104 x 103 x 102 = 109

          Durmukha: 2 x 107

               Hanumãn: 107 x 103 = 1010

               Nala: 107 x 102 x 103 x 102  = 1014

          Dadhimukha: 107 x 101 = 108

          Śarabha, Kumuda, Vahni & Ramha: Uncountable strength

 

       Again, in Kishkindhãkãnda, (65.3-10), we find that the monkey warriors were mentioning for how many yojanas they could cross the ocean, and here again we find a reference to multiples of ten:

                   Gaja :                    10 yojanas

                   Gavaksha:             20 yojanas

                   Gavaya:                 30 yojanas

                   Śarabha:                40 yojanas    

                   Gandhamãdhana:  50 yojanas

                   Mainda:                 60 yojanas

                   Dwivida:               70 yojanas

                   Sushena:                80 yojanas

                   Jãmbavãn:             90 yojanas

                   Angada:               100 yojanas (but he does not know whether he 

                                                                       would be able to come back)              

      In the Yuddhakãnda (27.4-5) of Valmiki Ramayana, we find Śuka giving an idea to Ravana about the military strength of Sugreeva and his army of monkey warriors:

                       Yeshãm kotisahasrãni nava panća ća sapta ća

                         Tathã śankhasahasrãni tathã vrindaśatãni ća

                         Yete Sugreevasaćivãh Kishkindhanilayãh sadã”.

The strength is mentioned in terms of powers of ten:

                          107 (9 + 5 + 7) + (śanku or 1012) x 103  + (vrinda or 109) x 102

 

      In the Yuddhãkãnda (3.24-29), we find Hanuman describing to Rama the strength of Ravana’s army as he was able to see it in Lanka, and while doing so, he also explains the system of numeration based on powers of ten, along with the special names given to the different powers of ten by our ancients:

         Ayutam rakshasãmatra purvadwãram samãśritam

          soolahasta duradharshah sarve khadgãgrayodhinah

          Niyutam rakshasãmatra dakshinadwaramãśritam

          Ćaturangena sainyena yodhãstatrãpyanuttamãh

                                                          -17-

 

          Prayutam rakshasãmatra pasćimadwãramãśritam

          ćarmakhadgadharãh sarve tathã sarvãstrakovidãh

          Nyarbudam rakshasmatrã uttaradwãramãśritam

          Rathinasćasvavahãsća kulapatrãh supujitãh

          Śataśoatha sahasrãni madhyamam skandhamaśritah

          Yatudhãna durãdharshãs sãgrakotisćã rakshasãm

          Te mayã sakramah bhãgãh parikhãsćavapooritãh

          Dagdhã ća nagari Lankã prakãrãsćavãsaditãh”

Commenting on the above, Govindaraja says:

 Vibhãgam ća balau dhãrayeti Ramapraśnasyottaramãha—ayutamiti. Atra          sarve purvadwarasthãh ayutam daśasahasram

           agre senãgre yudhyanta ityagrayoginah khadgairãgrayodhinah sarve

           ayutankhayakãh”.

 “Śataśa iti. Atheti kãrtsnye satdaśah sahasrãni anekasahasrãni  durãdharshãh yatudhãnah rãkshasãh madhyamam skandam nagaramadhyamasthãnam. Asritã sarvasankhyãmãha sãgreti. Rakshasãm  sãgrakotih purnakotih atha Lankãyãm”   

Tilaka also says in his commentary:

                  “Niyutam laksham”

                  “Prayutam daśalaksham”   

                  “Sãgra kotih sapadakotih”.

Raghava in his commentary says:

 Ye sarve sarvastrakovidãh teshãm rakhasãm prayutam daśalaksham pasćimadwãramãśritam”.

Here also we find references to power of ten as follows:

                           Ayutam:   104

                                        Niyutam:  105

                           Prayutam:106

The deployment of forces as described by Hanuman in and around Lanka was as follows:   

                                  

                                                                     108

                                                         ________________________       

                                                         |                                               |

                                                         |                          |                     | 
                                                         |                 Hundreds             |

                                                         |               of thousands,        | 104

                                                  106   |               more than one      |

                                                         |               crore  or 107          |

                                                         |                                              |

                                                         |_______________________|                                                                                                                                                                                                            

                                                                             

                                                                              105

 

                                                               -18-

 

Thus, we find that the decimal system of numeration has been prevalent since the time of the Vedas and has percolated down to the era of the Puranas and the epics in Sanskrit literature.

 

     In the Yuddhakãnda (26.12, 33, 37-38) of Valmiki Ramayana, we find a spy of Ravana giving an idea of the strength of Rama’s army as follows:

              Yuthapãnãm sahasrãnãm śatena parivãritah

                Rãjan śatasahasrãni ćatvãrimśaććaiva ća”

  

                Śatam śatasahasrãnãm trimśaćća haripungavãh”

 

              “Rãjan satatamadhyãste Śarabho nãma yuthapah

                Yetasya balinah sarve vihãrã nãma yuthapãh”

He says that the army consisted of many leaders who numbered hundreds of                                                        

thousands, that is, many times the number:

                                          102 x 103

 Then, he said that the number of the prominent ones among the monkey warriors was:

                                      30 x 102 x 102 x 103  = 3 x 108.

Then he said that the number of warriors under the command of the leader Śarabha was:

                                     40 x 102 x 103 = 4 x 106

In Yuddhakãnda (27.25 and 33) we find the spy describing further the strength of the different types of warriors as follows:

                    Vrutãh kotisahasrena hareenãm samupasthitah”

 

                    “Yete smitamukhãh ghorã golãngoola mahãbalãh

                      Śatam Śatasahasrãni drushtvã vai setubandhanam”

Then, he described the number of the monkey warriors assembled in the task of building the bridge over the ocean, and said they were surrounded by:

                                    107 x 103 = 1010 monkey warriors, while the number engaged in the task of construction of the bridge was:

                                    102 x 102 x 103 = 107.

 

 

         In the Yuddhakãnda, (28.4-5) we find the spy of Ravana mentioning the strength of the ministers and advisers of Sugreeva, who were residents of Kishkindhã, in terms of the sum of powers of 10, as follows:

      :         

                   Yeshãm kotisahasrãni nava panća ća sapta ća

                    Tathã śankhasahasrãni tathã vrindaśatãni ća

                    Yete Sugreevasaćivãh kishkindhãnilayãh sadã”.

Their number was:

                    21 x 107 x 103 + 1012 x 103 + 109  x 102.

 

                                                            -19-

 

       In the Yuddhakãnda (28.33-42), we find Suka explaining to Ravana the strength of Rama’s army as gathered from the information passed on by spies, and here also we find an explanation about the different powers of ten and the special names assigned to them:

           Śatam satasahasrãnãm kotimãhurmaneeshinah

            Śatam kotisahasrãnãm sankha ityabhidheeyate

            Śatam sankhasahasrãnãm mahãsankha iti smritam

            Mahãsankhasahasrãnãm śatam vrindhamiti smritam

            Śatam vrindasahasrãnãm mahãvrindamiti smritam

            Mahãvrindasahasrãnãm śatam padmamiti smritam

            Mahapadmasahasrãnãm śatam kharvamihoćyate

            Śatam kharvasahasrãnãm (mahakharvamiti smritam

            Mahakharvasahasrãnãm..śatam) samudramityabheeyate

            Śatam samudrasahasramogha ityabhidheeyate

            (Śatamoghasahasrãnãm mahaugha iti visrutah)

            Yevam kotisahasrena sankhãnam ca śatena ća

            Mahãśankhasahasrena tathã vrindaśatena ća

            Mahãvrindasahasrena tathã padmaśatena ća

            Mahapadmasahasrena tathã kharvaśatena ća

            Samudrena śatenaiva mahaughena tathaiva ća

            Yevam kotimahaughena samudrasadrusena ća

            Vibhishanena sacivai rakshasaih privaritah

            Sugreevo vãnarendrastwam yuddhãrthamabhivartate

            Mahãbalavruto nityam mahãbalaparãkramah

            Sãranasya vaćah srutwã Rãvanam rakshãdhipam

            Balamadiśya tatsarvam suko vakyamabraveet”.

The powers of ten which are mentioned here according to one version are as follows:

              (1)  Koti =                            =102 x (102 x 103) = 107

                    Sankha =                       = 102 x (107 x 103) = 1012 

                     Mahasankha =              =102  x (1012 x 103) = 1017

                                Vrindam =                    =10 2 x (1017 x 103) = 1022

                                Mahavrindam               =102 x (1022 x 103) = 1027

                               Padmam                        = 102 x (1027 x 103) = 1032

                               Mahapadmam               = 102 x (1032 x 103) = 1037

                               Kharvam                       = 102  x (1037  x 103) = 1042

                               Mahakharvam               = 102 x (1042 x 103) = 1047

                               Samudram                     = 102 x (1047 x 103) = 1052

                               Ogha                              = 102 x (1052x 103) = 1057

           

             (2) According to another version, the names of powers of ten are as follows:

 

 

                                                              -20-

 

        Up to Kharvam, the calculation is the same as above, but thereafter there is some variation:

                    Kharvam                       = 102 x (1037 x 103) = 1042

                              Samudram                    =  102 x (1042 x 103) = 1047

                               Ogha                            =  102  x (1047 x 103) = 1052

                               Mahaugha                   =  102 x (1052 x 103)  = 1057

        (1) According to the system indicated in first version, the strength of Rama’s army is estimated to be:

    1010 + 1014 + 1020 + 1024 + 1030 + 1034 + 1040 + 1044 + 1054 + 1049 + 1057  + 1064 + 4 + 1 +1.

The number 4 above refers to the Ministers of Vibhishana and the two ones refer to Vibhishana and Sugreeva.

        (2) According to the system indicated in second version, the strength of Rama’s army comes out to be as follows:                                                        

 

    1010 + 1014 + 1020 + 1024 + 1030 + 1034 + 1040 + 1044 + 1049 + 1064  + 4 + 1 + 1 

              

                                                        Later periods                                                      

 

     Coming to later years, Bhãskarãćãrya in the introduction to his Lilavati enumerates the various powers of ten which had been brought into use long ago by our ancients for purposes of daily affairs as follows:

            “Ekadaśaśatasahasrãyutalakshaprayutakotayah kramaśah arbudamabjam

               kharvanikharvamahãpadmaśankhavastasmãt jaladhi saćantyam

               madhyam parãrdhamiti daśagunottara sanjnãh sankhyãyãh sthãnanãm

               vyavaharãrtham kritah poorvaih”

We find here that:

               abjam        = 109

               jaladhi (samudram): 1014

However, we find no reference to kumudam flower here.  But one finds some reference to kumudam in some of the Jain works.  For instance, in the Swopanjabhãshya by Umãswati on the Tattvarthãdhikasutra (iv.15), there is reference to the word “kumudam”. 

       As Colebrooke has pointed out in his introduction to Lilavati of Bhãskarãćãrya, in Kacććayana’s Pali Grammar, we find further development of the decimal system of numeration, with names given to further higher powers of ten, up to 10140 as follows:

                                     Pakoti = 1014

                                     Kotippakoti = 1021

                                     Nahuta = 1028

                                     Ninnahuta = 1035

                                     Akkhobini =  1042

                                     Bindu = 1049

                                                           -21-

 

                                     Adbuda = 1056

                                     Niradbuda = 1063                                                    

                                     Ababa = 1070

                                    (Koti)Ababa = 1077

                                     Atata = 1084

                                     Sougandika = 1091

                                     Uppala = 1098

                                     Kumuda =  10105

                                     Pundarika = 10112

                                 Paduma  = 10119

                                                                     Kathana = 10126

                                               Mahãkathana = 10133

                                               Asankhyeya = 10140   

Thus, we find that even during the ancient days, the scholars and mathematicians had become great adepts in the decimal system of numeration and had soared to such great heights that they could conceive of such high powers of ten as 10140 ,  a quantity which they described as uncountable.                                                          

 

                                                                  

Ancient Tamil literature

 

       Perhaps, Tolkappiyar has referred to “kumudam” as “ãmbal” in Tamil, following the Jain enumeration.

 

     It is interesting to find that reference to the decimal system has been prevalent right from the days of Tolkãppiyam, the most ancient Tamil literature, whose antiquity is again shrouded in mystery.  We find mention therein of Tamil equivalents for the names assigned to different powers of ten.                                                      

   

    For instance, words ending in ai, am, pal, preceded by the word “yezh” signify not only things but also numbers. Thus, the words “tamarai (lotus), “vellam (ocean), “ãmbal” (a flower) represent numbers besides the things that they signify.

    For example, in Duraikanći, we have:

                  pal vellam meekkora”,

and in Aingurunooru, we have:

                 “vella varambinoozhi”.

and in these we have  a reference to “vellam”.

    In Patitrupattu, we have:

              Aruviyambal ayira vellavoozhi

where we find reference to “ãmbal” and “vellam”.

According to commentators, 1000 vellams = ãmbal

       In Paripãdal, we find:

                Neitalum kuvalaiyum ãmbalum śankamum

                  Maiyil kamalamum vellamum nudaliya

                                                          -22- 

 

                Sei kuriyeettam kazhippiya vazhimurai”.

 

Here, we find reference to kamalam (lotus), vellam (ocean), and “ãmbal” (flower), but no clue is available about what numbers they signify.

      In Diwãkaram, numbers in increasing powers of ten have been referred to in the form of aphorisms by correlating Sanskrit terminology with Tamil equivalents.

                                                              

                 “Ekam onre daśam pattãgum

                  Nooru śatame sahasram ãyiram

                  Padinãyurappeyar ayutamãgum”.

The word “kamalam” (lotus) corresponds to the Sanskrit word “Padmam” meaning lotus, while “vellam” corresponds to the Sanskrit word “Samudram (ocean).  As we have seen already, these terms denote very big numbers.

  

      In the face of these numerous references to the system of numeration based on the powers of ten, with special names for the higher powers of ten, it is clear that the decimal system of numeration was known to the ancient Indians and was prevalent even in the days of antiquity symbolized by the Vedic period.  Besides, it has also found mention in the Tamil and Jain literature of yore.

 

      It is interesting to note that historians of the Mohenjadaro and Harappa civilizations also refer to the prevalence of the decimal system of numeration, based on the evidence in the excavations relating to that period.  The deciphering of the Indus valley seals may perhaps throw some further light on the unquestionable antiquity of the decimal system.

     

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