### Antiquity of decimal system of numeration-2

ANTIQUITY OF DECIMAL SYSTEM OF NUMERATION-2

As per the above commentary:

Samudrah  = 108

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

Rikshakotisahasrani  golãngoolaśatani ća

Kotyoanekãhstu Kãkutstha kapeenãm deepatejasam”.

Commenting on this, Tilaka says:

śatani ća śatakotih anye kapayo anekãh kotyãh asankhyã ityarthah

Rãghava says in his commentary:

Rikshakotisahasrãni kotisahasrasankhyãkarikshah ityarthah

dairghyavisishtalangoolavisishtavãnaraviseshasćetyarthah

kapeenãmeka kotyasca adya asmin dine twãm upayãsyanti prãpsyanti

atah kopam jahi”

Here again we find:

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

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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

Devagandharvaputrãsća vãnarãh kãmaroopinah

Swaih swaih parivrutãh sainyairvartante pathi Raghava”

Śataih śatashasraisća kotibhisća plavamgamãh

Ayutaisćãvrutah śankubhisća parantapa

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

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Lakshaśatam kotih:      10000000   Koti          102 x 105 = 107

Kotilaksham śankuh:                       Śanku       107 x 105 = 1012

Śankusahasram arbudam:                Arbudam 1012 x 103 = 1015

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:

tatra śatasahasre prasiddhe

kotirlaksha śatamayutam daśasahasram

śankuh kotilaksham arbudam śankusahasram

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

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

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Śanku         =    1013

Samudram =    1014

Antyam         =    1015

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:

.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

Tatah kãnćanaśaiãbhastãrãyã veeryavãn pita

Tathãparena koteenãm sahasrena samanvitah

Pitã Rumãyãh samprãptah Sugreevaśvaśuro vibhuh

Bahuśahasairvãnarãnãm samanvitah

Golãngoolamahãrãjo Gavãksho Bheemavikramah

## Rukshãnãm bheemavegãnãm Dhumrah śatrunibarhanah

Vrutah kotisahsrãbhyãm dwãbhyãm samabhivartata

Mahãćalanibhairghoraih Panaso nãma yuthapah

Ãjagãma mahãveeryastisrubhih kotibhirvrutah

Neelãnjanaćayãkãro Neelo nãmãtha yuthapah

Tatah kãnćanaśailãbho Gavayo nãma yuthapah

Ãjagãma mahãveeryah koltibhirpanćabhirvrutah

Vrutah kotisahasrena Sugreevam  samupasthitah

## Maindasća Dwivdasćobhãśviputrau mahãbalau

Gajas­ća balavãn veerah koltibhitistrubhirvrutah

Ãjagãma mahãtejã Sugreevasya sameepatah

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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

Tatastãrãdyutistãro harirbheemavikramah

Ekãdasãnam koteenãmeeśwarastaisća samvrutah

Tato Rambhastvn uprãptastarunãdityasannibhah

Ayuten­ãvrutas­ćaiva sahasrena śatena ća

Tato yuthapatirveero Durmukho nãma vãnarah

# Kailãśasikharãkãrairvãnarairbheemavikramaih

Nalasćãpi mahãveeryah samvruto drumavãsibhih

Kotiśatena samprãpto sahasrena śatena ća

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

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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

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:

Ćaturangena sainyena yodhãstatrãpyanuttamãh

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Rathinasćasvavahãsća kulapatrãh supujitãh

Yatudhãna durãdharshãs sãgrakotisćã rakshasãm

Te mayã sakramah bhãgãh parikhãsćavapooritãh

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”

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

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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”

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

Their number was:

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

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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

Śatam kharvasahasrãnãm (mahakharvamiti smritam

Mahakharvasahasrãnãm..śatam) samudramityabheeyate

Śatam samudrasahasramogha ityabhidheeyate

(Śatamoghasahasrãnãm mahaugha iti visrutah)

Yevam kotisahasrena sankhãnam ca śatena ća

Mahãśankhasahasrena tathã vrindaśatena ća

Samudrena śatenaiva mahaughena tathaiva ća

Vibhishanena sacivai rakshasaih privaritah

Sugreevo vãnarendrastwam yuddhãrthamabhivartate

Mahãbalavruto nityam mahãbalaparãkramah

Sãranasya vaćah srutwã Rãvanam rakshãdhipam

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:

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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:

madhyam parãrdhamiti daśagunottara sanjnãh sankhyãyãh sthãnanãm

vyavaharãrtham kritah poorvaih”

We find here that:

abjam        = 109

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

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Ababa = 1070

(Koti)Ababa = 1077

Atata = 1084

Sougandika = 1091

Uppala = 1098

Kumuda =  10105

Pundarika = 10112

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

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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

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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