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History of Hindu Mathematics: A Source Book v1.c7.s4: 4. Numeral Terminology

History of Hindu Mathematics: A Source Book v1.c7.s4

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4. Numeral Terminology

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**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.

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

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In the *Yajurveda Saṃhitā* (*Vājasaneyī*)² the following list of numeral denominations is given: *Eka* (1), *daśa* (10), *śata* (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), *parārdha* (1,000,000,000,000). The same list occurs at two places in the *Taittirīya Saṃhitā*.³ The *Maitrāyaṇī*⁴ and *Kâṭhaka*¹ *Saṁhitâs* contain the same list with slight alterations. The *Pañcaviṁśa Brâhmaṇa* has the *Yajurveda* list upto *nyarbuda* inclusive, and then follow *nikharva*, *vâḍava*, *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 *daśaguṇottara saṁjñâ*² ("decuple terms").

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² *Yajurveda Saṃhitā*, xvii. 2.

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³ iv. 40. 11. 4; and vii. 2. 20. 1.

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

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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 *vâḍava* intervenes.

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² Cf. Bhâskara II, *L*, p. 2.

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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.³ 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):

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

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⁴ *Lalitavistara*, ed. by Rajendra Lal Mitra, Calcutta, 1877, p. 168.

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

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Bodhisattva: I know.

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

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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* *tiṭi-* *lambha*, hundred *titilambhas* *vyavasthâna-prajñapti*, hundred *vyavasthâna-prajñaptis* *hetuhila*, hundred *hetuhilas* *karahu*, hundred *karahus* *hetvindriya*, hundred *hetvindriyas* *samâpta-lambha*, hundred *samâpta-lambhas* *gaṇanâgati*, hundred *gaṇanâgatis* *niravadya*, hundred *niravadyas* *mudrâ-bala*, hundred *mudrâ-balas* *sarva-bala*, hundred *sarva-balas* *visamjñâ-gati*, hundred *visamjñâ-gatis* *sarvajñâ*, hundred *sarvajñâs* *vibhutamgamâ*, hundred *vibhutamgamâs* *tallakṣaṇa*.¹”

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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.

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Another interesting series of number names increasing by multiples of 10 millions is found in Kâccâyana's Pali 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* *sahassa*³ (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 the further terms are formed. What are their names? .......... hundred hundred-thousands is *koṭi*, hundred-hundredthousand *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 *akkhobhiṇi*; similarly we have *bindu*, *abbuda*, *nirabbuda*, *ahaha*, *ababa*, *atata*, *sogandhika*, *uppala*, *kumuda*, *puṇḍarîka* *paduma*, *kathâna*, *mahâkathâna*, *asaṅkhyeya*.”

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² “Grammaire Pâlie de Kâccâyana,” *Journ. Asiatique*, Sixieme Serie, XVII, 1871, p. 411. The explanations to sūtras 51 and 52 are quoted here.

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³ Also called *lakkha* (*lakṣa*).

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

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In the *Anuyogadvâra-sûtra*² (*c.* 100 B.C.), a Jaina canonical work written before the 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 *Śîrṣaprahelikâ*. 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.

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² *Sûtra* 142.

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

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**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 Âryabhaṭa 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), *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.

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

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¹ *Ā*, ii. 2.

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In most of the mathematical works, the denominations are called “names of places,” and eighteen of these are generally enumerated. Śrīdhara (750) gives the following names:² *eka*, *daśa*, *śata*, *sahasra*, *ayuta*, *lakṣa*, *prayuta*, *koṭi*, *arbuda*, *abja*, *kharva*, *nikharva*, *mahāsaroja*, *śaṅku*, *saritā-pati*, *antya*, *madhya*, *parārdha*, and adds that the decuple names proceed even beyond this. Mahāvīra (850) gives twenty-four notational places:³ *eka*, *daśa*, *śata*, *sahasra*, *daśa-sahasra*, *lakṣa*, *daśa-lakṣa*, *koṭi*, *daśa-koṭi*, *śata-koṭi*, *arbuda*, *nyarbuda*, *kharva*, *mahākharva*, *padma*, *mahā-padma*, *kṣoṇi*, *mahā-kṣoṇi*, *śaṅkha*, *mahā-śaṅkha*, *kṣiti*, *mahā-kṣiti*, *kṣobha*, *mahākṣobha*.

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² *Triś*, R. 2-3; the term used is *daśaguṇāḥ saṁjñāḥ*, i.e., “decuple names.”

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³ *GSS*, i. 63-68; “The first place is what is known as *eka*; the second is *daśa*” etc.

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Bhāskara II's (1150) list agrees with that of Śrīdhara except for *mahāsaroja* and *saritāpati* which are replaced by their synonyms *mahāpadma* and *jaladhi* respectively. He remarks that the names of places have been assigned for practical use by ancient writers.⁴

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⁴ *L*, p. 2.

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Nārāyaṇa (1356) gives a similar list in which *abja*, *mahāsaroja* and *saritāpati* are replaced by their synonyms *saroja*, *mahābja* and *pārāvāra* respectively.

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**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 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.

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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 formation 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.

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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.

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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* was deleted and we have *ûna-vimśati*, *ûna-trimśat*, etc.—forms which are used upto the present day. The alternative expressions *nava-daśa* (nine-ten), *nava-vimśati* (nine-twenty), etc., were also sometimes used.¹

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² *Taittirîya Saṁhitâ*, vii. 2. 11.

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¹ 19 = nava-daśa (*Vâjasaneyî Saṁhitâ*, xiv. 23; *Taittirîya Saṁhitâ*, xiv. 23. 30). 29 = nava-viṁśati (*Vâjasaneyî Saṁhitâ*, xiv. 31). 99 = nava-navati (*Ṛgveda*, i. 84. 13).

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Practically the whole of Sanskrit literature is in verse, so that for the sake of metrical convenience, various devices were resorted to in the formation of number expressions, the most common being the use of the additive² method. The following are a few examples of common occurrence taken from mathematical works :

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² 3339 = trîṇi śatâni trîsahasrâṇi triṁśa ca nava ca, i.e., “three hundreds and three thousands and thirty and nine.” (*Ṛgveda*, iii. 9. 9; also x. 52. 6.)

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*Subtractive:*
(1) the number 139 is expressed as 40+100-1;³
(2) 297 is expressed as 300-3.⁴

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³ *GSS*, i. 4: *cattvâriṁśascaikona śatâdhika* (“forty increased by one-less-hundred”).

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⁴ *L*, p. 4, Ex. 1: *Trihînasya śata-trayasya* (“three less three hundred”).

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*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).⁷

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⁵ *Â*, ii. 3: *dvi-navaka*.

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⁶ *Triś*, Ex. 43: *tri-navaka* (“three nines”), *dvi-ṣaṭ* (“two sixes”).

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⁷ *GSS*, i. 28: *tryaśîtimiśrâṇi catuśśatâni catussahasraghna nagânvitâni* (“eighty-three combined with four hundred and four thousand multiplied by seven”).

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The expression of the number 12345654321 in the form "beginning with 1 upto 6 and then diminishing in order" is rather interesting.¹

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¹ *GSS*, i. 27: *ekâdiṣaḍantâni krameṇa hînâni*.

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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.

Cites (15)