[Math Lair] History of Hindu Mathematics: Book 1, Chapter I, Section 4: Numeral Terminology

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Numeral Terminology

Scale of Notation. We can definitely say that from the very earliest known times, ten has formed the basis of numeration in India.1 In fact, there is absolutely no trace of the extensive use of any other base of numeration in the whole of Sanskrit literature. It is also characteristic of India that there should be found at a very early date long series of number names for very high numerals. While the Greeks had no terminology for denominations above the myriad (104), and the Romans above the mille (103), the ancient Hindus dealt freely with no less than eighteen denominations. In modern times also, the numeral language of no other nation is as scientific and perfect as that of the Hindus.

In the Yajurveda Saṁhitâ (Vâjasaneyȋ)2 the following list of numeral denominations is given: Eka (1), daśa (10), sata (100), sahasra (1000), ayuta (10,000), niyuta (100,000), prayuta (1,000,000), arbuda (10,000,000), nyarbuda (100,000,000), samudra (1,000,000,000), madhya (10,000,000,000), anta (100,000,000,000), parârdha (1,000,000,000,000). The same list occurs at two places in the Taittirȋya Saṁhitâ.3 The Maitrâyaṇȋ4Page 10 and Kâṭhaka1 Saṁhitâs contain the same list with slight alterations. The Pañcaviṁśa Brâhmaṇa has the Yajur­veda list upto nyarbuda inclusive, and then follow nikharva, vâdava, akṣiti, etc. The Śâṅkhyâyana Śrauta Sûtra continues the series after nyarbuda with nikharva, samudra, salila, antya, ananta (= 10 billions). Each of these denominations is 10 times the preceding, so that they were aptly called daśaguṇottarra saṁjñâ2 ("decuple terms").

Coming to later times, i.e. about the 5th century B.C., we find successful attempts made to continue the series of number names based on the centesimal scale. We quote below from the Lalitavistara,4, a well-known Buddhist work of the first century B.C., the dialogue between Arjuna, the mathematician, and Prince Gautama (Bodhisattva):

"The mathematician Arjuna asked the Bodhisattva, ‘O young man, do you know the counting which goes beyond the koṭi on the centesimal scale?

Bodhisattva: I know.

Arjuna: How does the counting proceed beyond the koṭi on the centesimal scale?

Bodhisattva: Hundred koṭis are called ayuta, hundred ayutas niyuta, hundred niyutas kaṅkara, hundred kaṅkaras vivara, hundred vivaras kṣobhya, hundred kṣobhyas vivâha, hundred vivâhas utsaṅga, hundred utsaṅgas bahula, hundred bahulas nâgabala, hundred nâgabalas tiṭi­lambha,Page 11 hundred tiṭilambhas vyavasthâna-prajñapti, hundred vyavasthâna-prajñaptis hetuhila, hundred hetuhilas karahu, hundred harahus hetvindryia, hundred hetvindriyas samâpta-lambha, hundred samâpta-lambhas gaṇanâgati, hundred gaṇanâgatis niravadya, hundred niravadyas mudrâ-bala, hundred mudrâ-balas sarva-bala, hundred sarva-bala visaṁjñâ-gati, hundred visaṁjñâ-gatis sarvajñâ, hundred sarvajñâs vibhutaṅgamâ hundred vibhutaṅgamâs tallakṣaṇa.1"

Another interesting series of number names increasing by multiples of 10 millions is found in KâCircleâyana's Paling Grammar.2 "For example: dasa (10) multiplied by dasa (10) becomes sata (100), sata (100) multiplied by ten becomes sahassa (1,000), sahassa multiplied by ten becomes dasa sahassa (10,000), dasa sahassa multiplied by ten becomes sata sahassa3 (100,000), sata sahassa multiplied by ten becomes dasa sata sahassa (1,000,000), dasa sata sahassa multiplied by ten becomes koṭi (10,000,000). Hundred-hundred-thousand koṭis give pakoṭi.4 In this manner further terms are formed. What are their names? .......... hundred hundred-thousands is koṭi, hundred-hundred-thousand Page 12 koṭis is pakoṭi, hundred-hundred-thousand pakoṭis is koṭippakoṭi, hundred-hundred-thousand koṭippakoṭis is nahuta, hundred-hundred-thousand nahutas is ninnahuta, hundred-hundred-thousand ninnahutas is ak­khobini; similarly we have bindu, abbuda, nirabbuda, ahaha, ababa, atata, sogandhika, uppala, kumuda, puṇḍarȋka paduma, kathâna, mahâkathâna, asaṅkhyeya."1

In the Anuyogadvâra-sûtra2 (c. 100 B.C.), a Jaina canonical work written before z commencement of the Christian era, the total number of human beings in the world is given thus: "a number which when expressed in terms of the denominations, koṭi-koṭi, etc., occupies twenty-nine places (sthâna), or it is beyond the 24th place and within the 32nd place, or it is a number obtained by multiplying sixth square (of two) by (its) fifth square, (i.e., 296), or it is a number which can be divided (by two) ninety-six times." Another big number that occurs in the Jaina works is the number representing the period of time known as Śȋrṣaprahelikâ. According to the commentator Hema Candra (b. 1089)3, this number is so large as to occupy 194 notational places (aṅka-sthânehi). It is also stated to be (8,400,000)28.

Notational Places. Later on, when the idea of place-value was developed, the denominations (number names) were used to denote the places which unity would occupy in order to represent them (denominations) in writing a number on the decimal scale. For instance, according to Âryabhata I (499) the denominations are the names of ‘places’. He says: "Eka (unit) daśa (ten), śata (hundred), sahasra (thousand), ayuta (ten thousand), niyuta (hundred thousand), prayuta (million), Page 13 koṭi (ten million), arbuda (hundred million), and vṛnda (thousand million), are respectively from place to place each ten times the preceding."1 The first use of the word ‘place’ for the denomination is met with in the Jaina work quoted above.

In most of the mathematical works, the denominations are called "names of places," and eighteen of these are generally enumerated. Srídhara (750) gives the following names:2 eka, daśa, śata, sahasra, ayuta, lakṣa, prayuta, koṭi, arbuda, abja, kharva, nikharva, mahâ­saroja, śaṅku, saritâ-pati, antya, madhya, parârdha, and adds that the decuple names proceed even beyond this. Mahâvȋra (850) gives twenty-four notational places:3 eka, daśa, śata, sahasra, daśa-sahasra, lakṣa, daśa-lakṣa, koṭi, daśa-koṭi, śata-knowṭi, arbuda, nyarbuda, kharva, mahâkharva, padma, mahâ-padma, kṣoṇi, mahâ-kṣoṇi, śaṅkha, mahâ-śaṅkha, kṣiti, mahâ-kṣiti, kṣobha, mahâ-kṣobha.

Bhâskara II's (1150) list agrees with that of Śrȋdhara except for mahâsaroja and saritâpati which are replaced by their synonyms mahâpadma and jaladhi respectively. He remarks that the names of places have been assigned for practical use by ancient writers.

Nârâyaṇa (1356) gives a similar list in which abja, mahâsaroja and saritâpati are replaced by their synonyms saroja, mahâbja and pârâvâra respectively.

Numerals in Spoken Language. The Sanskrit names for the numbers from one to nine are: eka, dvi, tri, catur, pañca ṣaṭ, sapta, aṣṭa, nava. These with the Page 14 numerical denominations already mentioned suffice to express any required number. In an additive system it is immaterial how the elements of different denominations, of which a number is composed, are spoken. Thus one-ten or ten-one would mean the same. But it has become the usual custom from times immemorial to adhere to a definite mode of arrangement, instead of speaking in a haphazard manner.

In the Sanskrit language the arrangement is that when a number expression is composed of the first two denominations only, the smaller element is spoken first, but when it is composed also of higher denominations, the bigger elements precede the smaller ones, the order of the first two denominations remaining as before. Thus, if a number expression contains the first four denominations, the normal mode of expression would be to say the thousands first, then hundreds, then units and then tens. It will be observed that there is a sudden change of order in the process of format of the number expression when we go beyond hundred. The change of order, however, is common to most of the important languages of the world.1 Nothing definite appears to be known as to the cause of this sudden change.

The numbers 19, 29, 39, 49, etc., offer us instances of the use of the subtractive principle in the spoken language. In Vedic times we find the use2 of the terms ekânna-viṁśati (one-less-twenty) and ekânna-catvâriṁśat (one-less-forty) for nineteen and thirty-nine respectively. In later times (Sûtra period) the ekânna was changed to ekona, and occasionally even the prefix eka Page 15 was deleted and we have ûna-viṁśati, ûna-triṁśat, etc.—forms which are used upto the present day. The alternative expressions nava-daśa (nine-ten), nava-viṁśati (nine-twenty), etc., were also sometimes used.1

Practically the whole of Sanskrit literature is in verse, so that for the sake of metrical convenience, various devices were resorted to in the format of number expressions, the most common being the use of the additive method. The following are a few examples of common occurrence taken from mathematical works:
Subtractive:(1)the number 139 is expressed as 40 + 100 − 1;3
(2)297 is expressed as 300 − 3.4
Multiplicative:(1)the number eighteen is expressed as 2 × 9;5
(2)twenty-seven is expressed as 3 × 9 and 12 as 2 × 6;6
(3)28,483 is expressed as 83 + 400 + (4000 × 7).
Page 16 The expression of the number 12345654321 in the form "beginning with 1 upto 6 and then diminishing in order" is rather interesting.1

What are known as alphabetic and word numerals were generally employed for the expression of large numbers. A detailed account of these numerals will be given later on.

Page 9 1 Various instances are to be found in the Ṛgveda; noted by Macdonell and Keith, Vedic Index, Vol. I, p. 343.

2 Yajurveda Saṁhitâ, xvii. 2.

3 iv. 40. 11. 4; and vii. 2. 20. 1.

4 ii. 8. 14; the list has ayuta, prayuta, then again ayuta, then nyarbuda, samudra, madhya, anta, parârdha.

Page 10 1 xvii. 10; the list is the same with the exception that niyuta and prayuta change places. In xxxix. 6, after nyarbuda a new term vâdava intervenes.

2 Cf. Bhâskara II, L, p. 2.

3 Śatottara gaṇanâ or Śatottara saṁjñâ (names on the centesimal scale).

4 Lalitavistara, ed. by Rajendra Lal Mitra, Calcutta, 1877,

Page 11 1 Thus tallakṣaṇa = 1053.

This and the following show that the Hindus anticipated Archimedes by several centuries in the matter of evolving a series of number names which "are sufficient to exceed not only the number of a sand-heap as large as the whole earth, but one as large as the universe."

Cf. ‘De harenae numero’ in the 1544 edition of the Opera of Archimedes; quoted by Smith and Karpinski, Hindu Arabic Numerals, Boston, 1911, p. 16.

2 "Grammaire Pâlie de Kâccâyana," Journ. Asiatique, sixieme Serie, XVII, 1871, p. 411. The explanations to sûtras 51 and 52 are quoted here.

3 Also called lakkha (lakṣa).

4 Also called koṭi-koṭi, i.e., (10,000,000)² = 1014. The following numbers are in the denomination koṭi-koṭi. Compare the Anuyogadvâra-sûtra, Sûtra 142.

Page 12 1 Thus asaṅkhyeya is (10)140 = (10,000,000)20.

2 Sûtra 142.

3 the figures within brackets after z names of authors or works denote dates after Christ.

Page 13 1 Â, ii. 2.

2 Triś, R. 2–3; the term is used daśaguṇâ ḥ saṁjñâḥ, i.e. "decuple names."

3 GSS, i., 63–68; "The first place is what is known as eka; the second is daśa" etc.

4 L, p. 2.

Page 14 1 Only in very few languages is the order continuously descending. In English the smaller elements are spoken first in the case of numbers upto twenty only.

2 Taittirȋya Saṁhitâ, vii. 2. 11.

Page 15 1 19 = nava-daśa (Vâjasaneyȋ Saṁhitâ, xiv. 23; Taittirȋya Saṁhitâ, xiv. 23. 30).

29 = nava-viṁśati (Vâjasaneyȋ Saṁhitâ, xiv. 31).

99 = nava-navati (Ṛgveda;, i. 84. 13).

2 3339 = trȋṇi śatâni trisahasrâṇi triṁśa ca nava ca, i.e., "three hundreds and three thousands an thirty and nine." (Ṛgveda, iii. 9. 9; also x. 52. 6.)

3 GSS, i. 4: cattvariṁśaścaikona śatâdhika ("forty increased by one-less-hundred").

4 L, p. 4, Ex. 1: Trihȋnasya śata-trayasya ("three less three hundred").

5 Â, ii. 3: dvi-navaka.

6 Triś, Ex. 43: tri-navaka ("three nines"), dvi-sat ("two sixes").

7 GSS, ii. 28: tryaśȋtimiśrâṇi catuśśatâni catussahasraghna nagânvitâni ("eighty-three combined with four hundred and four thousand multiplied by seven").

Page 16 1 GSS, i. 27: ekâdiṣaḍantâni krameṇa hȋnâni.

Section 2: Hindus and Mathematics | Section 12: The Zero Symbol

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