Bhāratīya saṅkhyālakṣaṇāni – Characteristics of Hindu Numbers

Stumbling upon number gold

While rummaging through the Internet Archive’s many treasures of scanned literature, I had struck gold! (1) “The New Sanscrit Primer”, uploaded by Shri Ajit Gargeshwari (2), contains a very basic introduction to Sanskrit and can be finished by lay beginners in a matter of days.

The sankhyālakṣaṇāni section (2 p. 51) of the book teaches words for positive exponents of ten in Sanskrit. A handy śloka concisely makes remembering it easy:

This translates to “One, ten, (a) hundred, (a) thousand, ten thousand, (a) lakh, ten lakh, (a) crore, (are) in order. Ten crore, (a) hundred crore, (a) thousand crore, ten thousand crore, (a) lakh crore, ten lakh crore, (and) hence, (follow the above). (A) crore crore, ten crore crore, (a) hundred crore crore, (and) (a) thousand crore (are) (the) exponents of (the number) ten. (Thus,) was formulated by (the) elders for handling of (the) positions of (the) number(-ing) (system).” (Translation mine)

The book has a handy reference table right below the śloka in the same page (2 p. 51) for convenience (transliteration on the left mine):

Table 1: Exponents of ten (columns represent numeral in IAST, exponent as Sanskrit numeral, name in Devanagari script, and numerical representation respectively)

Words for numbers

Interestingly, the names for numbers beyond a crore don’t find any official recognition or usage in modern day India. This is because we do not have a need to use such large numbers in our daily lives. Also, some Indians have begun using the words million, billion, trillion, etc. to refer to large numbers by borrowing these terms from the West. No doubt, this is due to the excessive air-time of and subsequent high exposure to American and British based media material in English.

Scientists, mathematicians and engineers the world over use abbreviated and/or symbolic forms of the exponents of the number ten, among other base numbers, in their work. This makes it easy for them while referring to large numbers. It is, however, interesting to note that the ancients of Bhārata have had names for large numbers. This can logically mean that they had used those numbers for calculations in their lifetime.

But where do such large numbers find a place? Anyone can harbor a guess that such large numbers could have been useful for calculating the dimensions, tracing the trajectories, or counting the number of observable heavenly objects. But could they have used these large numbers elsewhere? Moreover, being an introductory text to beginners in Sanskrit, the book does not mention the references for the said words for numbers.

I will make a preliminary attempt at satisfying the curiosity of the above points based on ancient literature in this article.

Vālmīki Rāmāyaṇa

The Yuddha kāṇḍa of the Vālmīki Rāmāyaṇa describes in detail the war between Rāma and Rāvaṇa. In the twentieth sarga, Rāvaṇa sends Śuka, an asura (not to be confused with the son of Vedavyāsa), as an ambassador to Rāma’s side (before building the setu). (3) In the twenty fifth sarga, Śuka and another asura are sent by Rāvaṇa to spy on Rāma’s side after their crossing of the sea. (4) The spies are exposed by Vibhīṣaṇa to Rāma. But they are not hurt and are sent away with just a message to Rāvaṇa. Having been exposed to Rāma’s army twice, Śuka describes the enemy’s numbers to Rāvaṇa in the twenty eighth sarga (5). Thus, comes a detailed exposition on numeral names as tabulated below.

Exponent of ten Number Numeral name in Vālmīki Rāmāyaṇa (5)
7 100 x ṣatasahasra koṭi
12 ṣatasahasra x koṭi śaṅku
17 ṣatasahasra x śaṅku mahāśaṅku
22 ṣatasahasra x mahāśaṅku vṛnda
27 ṣatasahasra x vṛnda mahāvṛnda
32 ṣatasahasra x mahāvṛnda padma
37 ṣatasahasra x padma mahāpadma
42 ṣatasahasra x mahāpadma kharva
47 ṣatasahasra x kharva mahākharva
52 ṣatasahasra x mahākharva samudra
57 ṣatasahasra x samudra ogha
62 ṣatasahasra x ogha mahaugha

Table 2

Śuka warns Rāvaṇa that (as translated by Shri Murthy (5)) “This Sugreeva, the king of monkeys, having great strength and valour, always surrounded by a colossal army, is approaching you to make war, accompanied by the valiant Vibhishana and the ministers, as also a hundred thousand crores of Shankas (sic), a thousand Mahashankus, a hundred Vrindas, a thousand mahavrindas, a hundred padmas, a thousand Mahapadmas, a hundred kharves, samudras and Mahaughas of the same number, and a crore of Mahanghas (sic) whole army as such is identical of an ocean.”

Evidence from other Sanskrit texts

Exponent of ten Numeral name in Yajurveda saṃhita & Taittirīya saṃhita (6) Numeral name in Pañcaviṃṣa brāhmaṇa (6) Numeral name in Sāṅkhyāyana ṣrautasūtra (6) Numeral name in Triṣatikā of Śrīdharācārya (6) (7)
0 eka eka eka eka
1 daṣa daṣa daṣa daṣa
2 ṣata ṣata ṣata ṣata
3 sahasra sahasra sahasra sahasra
4 ayuta ayuta ayuta ayuta
5 niyuta niyuta niyuta lakṣa
6 prayuta prayuta prayuta prayuta
7 arbuda arbuda arbuda koṭi
8 nyarbuda nyarbuda nyarbuda arbuda
9 samudra nikharva nikharva abja
10 madhya vādava samudra kharva
11 anta akṣiti salila nikharva
12 parārdha antya mahāsaroja
13 ananta ṣaṅku
14 saritāṃpati(samudra)
15 antya
16 madhya
17 parārdha

Table 3

Śrīdharācārya mentions that the numeral names extend beyond parārdha! (6) (7)

Evidence from Buddhist texts

Datta and Singh (6) present data from Lalitavistara, a Buddhist text, where Arjuna, an astrologer councilor or “a very eminent master of arithmetic” (8), asks Bodhisattva about numbers beyond a koṭi (crore):

Exponent of ten Number Numeral name in Lalitavistara (6) (8)
7 1,00,00,000 koṭi
9 100 x koṭi ayuta
11 100 x ayuta niyuta
13 100 x niyuta kaṅkara
15 100 x kaṅkara vivara
17 100 x vivara kṣobhya
19 100 x kṣobhya vivāha
21 100 x vivāha utsaṅga
23 100 x utsaṅga bahula
25 100 x bahula nāgabala
27 100 x nāgabala tiṭilambha
29 100 x tiṭilambha vyavasthāna prajñapti
31 100 x vyavasthāna prajñapti hetuhila
33 100 x hetuhila karahu
35 100 x karahu hitvindriya
37 100 x hetvindriya samāpta lambha
39 100 x samāpta lambha gaṇanāgati
41 100 x gaṇanāgati niravadya
43 100 x niravadya mudrā bala
45 100 x mudrā bala sarva bala
47     100 x sarva bala visaṃjñāgati
49 100 x visaṃjñāgati sarvajña
51 100 x sarvajña vibhutaṅgama
53 100 x vibhutaṅgama tallakṣaṇa

Table 4

Amazingly, the Buddha speaks of a number which is the fifty third exponent of 10 (one followed by 53 zeroes!). Rajendralala Mitra, the translator of original source (8) of Datta and Singh (6), while referring to tallakṣaṇa, clearly mentions that “The names are mostly new to Hindu Sanskrit arithmetic.” This explains why this set of numeral names is relatively unheard of.

Datta and Singh (6) also mention a series of numeral names in Kāccāyana’s Pāli grammar:

Exponent of ten Number Numeral name in Kāccāyana’s Pāli grammar (6)
1 10 dasa
2 10 x dasa sata
3 10 x sata sahassa
4 10 x sahassa dasa sahassa
5 10 x dasa sahassa sata sahassa (or lakkha)
6 10 x sata sahassa (or lakkha) dasa sata sahassa
7 10 x dasa sata sahassa koṭi
14 koṭi x koṭi pakoṭi
21 koṭi x pakoṭi koṭippakoṭi
28 koṭi x koṭippakoṭi nahuta
35 koṭi x nahuta ninnahuta
42 koṭi x ninnahuta akkhobhini
49 koṭi x akkhobhini bindu
56 koṭi x bindu abbuda
63 koṭi x abbuda nirabbuda
70 koṭi x nirabbuda ahaha
77 koṭi x ahaha ababa
84 koṭi x ababa atata
91 koṭi x atata sagandhika
98 koṭi x sagandhika uppala
105 koṭi x uppala kumuda
112 koṭi x kumuda puṇḍarīka
119 koṭi x puṇḍarīka paduma
126 koṭi x paduma kathāna
133 koṭi x kathāna mahākathāna
140 koṭi x mahākathāna asaṅkhyeya

Table 5

This asaṅkhyeya exceeds tallakṣaṇa and reaches the one hundred and fortieth exponent of ten (one followed by 140 zeroes)!

Evidence from Jaina texts

Datta and Singh (6) cite two Jaina texts which refer to large numbers:

296 turns out to be 7.9228163 x 1028, i.e. a number with 29 places.

What’s the source of the handy śloka ?

Remember the śloka that got me probing deeper into this in the first place? Let me remind you:

After combing through the introductory pages of several interesting treatises on ancient Indian mathematics, I finally stumbled upon the source of this śloka. It is Śrī Bhāskarācārya’s iconic treatise on mathematics, Līlāvati. (9)

Some observations

  1. The numeral names mentioned in the above sources are mostly similar. The differences are mainly beyond the lakṣa (lakh) or koṭi (crore). This may suggest that the usage of names of numerals below these two quanitities were almost standardized. Perhaps, the common citizens did not have much to do beyond these quantities as abundance as an idea or physical reality may only be possible in case of the intellectuals or the kings respectively.
  2. Parārdha denotes 1018 in the Līlāvati (9), 1012 in Yajurveda saṃhita & Taittirīya saṃhita (6), and 1017 in Triṣatikā of Śrīdharācārya (6)(7). Interestingly, beginner learners of Tarka will notice the numeral parārdha in the the introduction to saṅkhyā, one of the 24 guṇas (qualities), in Annambhaṭṭa’s Tarkasaṅgraha:

Shri V. N. Jha, in his English translation to the Tarkasaṅgraha (10), translates parārdha to 1016. It is not clear where from Shri Jha gets this number. Nevertheless, the number denoted by parārdha could have been any number between 1012 and 1018.

  1. General knowledge buffs show off to their “uninformed” peers that “googol” refers to 10100 and hence, the etymology of the popular INTERNET search engine. But there are several Indic numbers which they have not yet heard of. Mahaugha (1062), tallakṣaṇa (1053), and asaṅkhyeya (10140) are only scratches on the surface. There may be many more numbers of their kind hidden in ancient manuscripts.

Conclusions

Numbers have been in use ever since humans began counting. So are the names for numbers. The diversity in the names for numerals in India suggests the ingenuity of the mathematicians of the time as they denote the possible variety of uses they put these numbers to. An attempt to understand a small page in an introductory Sanskrit reader can teach a lot more when probed in the right direction.

Bibliography

  1. Karnataka Samskrita University. Internet Archive. [Online] [Cited: June 27, 2017.] https://archive.org/details/karnatakasamskritauniversity.
  2. Savant, Abaji Ramachandra. The New Sanskrit Primer. Belgaum : Savant, Abaji Ramachandra, 1899.
  3. Murthy, K. M. K. Book VI : Yuddha Kanda – Book Of War Chapter [Sarga] – 20. Valmiki Ramayana. [Online] December 2003. [Cited: June 29, 2017.] http://www.valmikiramayan.net/yuddha/sarga20/yuddha_20_frame.htm.
  4. Murthy, K.M.K. Book VI : Yuddha Kanda – Book Of War Chapter [Sarga] 25. Valmiki Ramayana. [Online] March 2004. [Cited: June 29, 2017.] http://www.valmikiramayan.net/yuddha/sarga25/yuddha_25_frame.htm.
  5. —. Book VI : Yuddha Kanda – Book Of War Chapter [Sarga] 28. Valmiki Ramayana. [Online] May 2004. [Cited: June 29, 2017.] http://www.valmikiramayan.net/yuddha/sarga28/yuddhaitrans28.htm.
  6. Datta, Bibhutibusan and Singh, Avadesh Narayan. History of Hindu Mathematics – A source book. Calcutta : Asia Publishing House, 1935. pp. 9-13. Vol. 1.
  7. śrīdharācārya and ācārya, sudyumn. Triṣatikā. naī dillī : rāṣṭrīya saṃskṛt saṃsthān, 2004. 81-86111-09-3.
  8. Mitra, Rajendralala. Lalita-vistara. Calcutta : J. W. Thomas Baptist Mission Farm, 1881.

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