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History of Hindu Mathematics: A Source Book v1.c7.s10: 10. Word Numerals

History of Hindu Mathematics: A Source Book v1.c7.s10

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10. Word Numerals

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**Explanation of the System.** A system of expressing numbers by means of words arranged as in the place-value notation was developed and perfected in India in the early centuries of the Christian era. In this system the numerals are expressed by names of things, beings or concepts, which, naturally or in accordance with the teaching of the *Śâstras*, connote numbers. Thus the number one may be denoted by anything that is markedly unique, *e.g.*, the moon, the earth, etc.; the number two may be denoted by any pair, *e.g.*, the eyes, the hands, the twins, etc.; and similarly others. The zero is denoted by words meaning void, sky, complete, etc.

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The system is used in works on astronomy, mathematics and metrics, as well as in the dates of inscriptions and in manuscripts. The ancient Hindu mathematicians and astronomers wrote their works in verse. Consequently they strongly felt the need for a convenient method of expressing the large numbers that occur so often in the astronomical works and in the statement of problems in mathematics. The word numerals were invented to fulfil this need and soon became very popular. They are used even upto the present day, whenever big numbers have to be expressed in Sanskrit verse.

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The words denoting the numbers from one to nine and zero, with the use of the principle of place-value, give us a very convenient method of expressing numbers by word chronograms. To take a concrete case, the number 1,230 may be expressed in many ways:

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1. *kha-guṇa-kara-âdi*,
2. *kha-loka-karṇa-candra*,
3. *âkâśa-kâla-netra-dharâ*, etc.

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It will be observed that the same number can be expressed in hundreds of ways by word chronograms. This property makes the word numerals specially suitable for inclusion in metre. To secure still greater variety, the numbers beyond ten are also sometimes denoted by words.

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**List of Words.** The following is a list of words commonly used in this system to denote numbers:

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0 is expressed by *śûnya*, *kha*, *gagana*, *ambara*, *âkâśa*, *abhra*, *viyat*, *vyoma*, *antarikṣa*, *nabha*, *jaladharapatha*, *pûrṇa*, *randhra*, *viṣṇupada*, *ananta*, etc.
1 is expressed by *âdi*, *śaśi*, *indu*, *vidhu*, *candra*, *kalâdhara*, *himagu*, *śîtâṁśu*, *kṣapâkara*, *himâṁśu*, *sîtaraśmi*, *prâleyâṁśu*, *soma*, *śaśâṅka*, *mṛgâṅka*, *himakara*, *sudhâṁśu*, *rajanîkara*, *śaśadhara*, *śveta*, *abja*, *bhû*,

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*bhûmi*, *kṣiti*, *dharâ*, *urvarâ*, *go*, *vasundharâ*, *pṛthvî*, *kṣmâ*, *dharaṇî*, *vasudhâ*, *ilâ*, *ku*, *mahî*, *rûpa*, *pitâmaha*, *nâyaka*, *tanu*, etc.

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2 is expressed by *yama*, *yamala*, *aśvin*, *nâsatya*, *dasra*, *locana*, *netra*, *akṣi*, *dṛṣṭi*, *cakṣu*, *ambaka*, *nayana*, *îkṣaṇa*, *pakṣa*, *bâhu*, *kara*, *karṇa*, *kuca*, *oṣṭha*, *gulpha*, *jânu*, *jaṅghâ*, *dvaya*, *dvanda*, *yugala*, *yugma*, *ayana*, *kuṭumba*, *ravicandrau*, *naya*,¹ etc.

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¹ Method of comprehending things from particular standpoints—*dravyârthika* and *paryâyârthika*.

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3 is expressed by *râma*, *guṇa*, *triguṇa*, *loka*, *trijagat*, *bhuvana*, *kâla*, *trikâla*, *trigata*, *trinetra*, *haranetra*, *sahodarâḥ*, *agni*, *anala*, *vahni*, *pâvaka*, *vaiśvânara*, *dahana*, *tapana*, *hutâśana*, *jvalana*, *śikhin*, *kṛśânu*, *hotṛ*, *pura*, *ratna*² (Jaina), etc.

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² Used by Mahâvîra only; others take it for five.

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4 is expressed by *veda*, *śruti*, *samudra*, *sâgara*, *abdhi*, *ambhodha*, *ambhodhi*, *jaladhi*, *udadhi*, *jalanidhi*, *salilâkara*, *viṣanidhi*, *vâridhi*, *payodhi*, *payonidhi*, *ambudhi*, *kendra*, *varṇa*, *âśrama*, *yuga*, *turya*, *kṛta*, *aya*, *âya*, *diś*, *bandhu*, *koṣṭha*, *gati*, *kaṣâya*, etc.

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5 is expressed by *bâṇa*, *śara*, *śastra*, *sâyaka*, *iṣu*, *bhûta*, *parva*, *prâṇa*, *pavana*,³ *pâṇḍava*, *artha*, *viṣaya*, *mahâbhûta*, *tatva*, *bhâva*, *indriya*, *ratna*, *karaṇîya*,⁴ *vrata*, etc.

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³ See *SiŚe*, i. 27; *SiŚi*, *gaṇitâdhyâya*, x. 2. Used also for 7 (See the quotations by Bhaṭṭotpala in his commentary on *Bṛhatsaṁhitâ*, ch. ii). In Al-Bîrûnî's list it is erroneously put for 9.

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⁴ That which ought to be done; according to the Jainas—*ahiṁsâ*, *sunṛta*, *asteya*, *brahmacarya*, and *aparigraha*.

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6 is expressed by *rasa*, *aṅga*, *kâya*, *ṛtu*, *mâsârdha*, *darśana*, *râga*, *ari*, *śâstra*, *tarka*, *kâraka*, *lekhya*, *dravya*,⁵ *khara*, *kumâravadana*, *ṣaṇmukha*, etc.

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⁵ Used by Mahâvîra.

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7 is expressed by *naga*, *aga*, *bhûbhṛt*, *parvata*, *śaila*, *acala*, *adri*, *giri*, *ṛṣi*, *muni*, *yati*, *atri*, *vâra*, *svara*, *dhâtu*, *aśva*, *turaga*, *vâji*, *haya*, *chandaḥ*, *dhî*, *kalatra*, *tatva*,¹ *dvîpa*, *pannaga*,² *bhaya*,³ *mâtṛkâ*, *vyasana*, etc. 8 is expressed by *vasu*, *ahi*, *nâga*, *gaja*, *danti*, *dvirada*, *diggaja*, *hastin*, *ibha*, *mâtaṅga*, *kuñjara*, *dvîpa*, *puṣkarin*, *sindhura*, *sarpa*, *takṣa*, *siddhi*, *bhûti*, *anuṣṭubha*, *maṅgala*, *anîka*, *karman*,⁴ *durita*, *tanu*,⁵ *dik*,⁶ *mada*,⁷ etc. 9 is expressed by *aṅka*, *nanda*, *nidhi*, *graha*, *randhra*, *chidra*, *dvâra*, *go*,⁸ *upendra*, *keśava*, *târkṣyadhvaj*, *durgâ*, *padârtha*,⁹ *labdha*, *labdhi*, etc. 10 is expressed by *diś*, *dik*, *diśâ*, *âśâ*, *aṅgulî*, *paṅkti*, *kakubh*, *râvaṇaśira*, *avatâra*, *karman*, etc. 11 is expressed by *rudra*, *îśvara*, *mṛḍa*, *hara*, *îśa*, *bhava*, *bhaarga*, *śulin*, *mahâdeva*, *akṣauhiṇî*, etc. 12 is expressed by *ravi*, *sûrya*, *ina*, *arka*, *mârtaṇḍa*, *dyumaṇi*, *bhânu*, *âditya*, *divâkara*, *mâsa*, *râśi*, *vyaya*, etc. 13 is expressed by *viśvedevâḥ*, *viśva*, *kâma*, *atijagatî*, *aghaṣa*, etc. 14 is expressed by *manu*, *vidyâ*, *indra*, *śakra*, *loka*,¹⁰ etc. 15 is expressed by *tithi*, *ghasra*, *dina*, *ahan*, *pakṣa*,¹¹ etc. 16 is expressed by *nṛpa*, *bhûpa*, *bhûpati*, *aṣṭi*, *kalâ*, etc. 17 is expressed by *atyaṣṭi*, etc.

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¹ Used by Mahâvîra because the Jainas recognise seven *tatvas*; used for five by others.

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² Used by Mahâvîra.

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³ Used by Mahâvîra.

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⁴ Used by Mahârvîra for 8 and by others for 10.

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⁵ Used by Mahâvîra.

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⁶ This word has been used for 8 as well as for 10. The use of *diś* or *dik* for 4 also occurs.

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⁷ Used by Mahâvîra only.

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⁸ This has been used for 1 also.

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⁹ Used by Mahâvîra only.

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¹⁰ Also used for 3.

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¹¹ Also used for 2.

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18 is expressed by *dhṛti*, etc. 19 is expressed by *atidhṛti*, etc. 20 is expressed by *nakha*, *kṛti*, etc. 21 is expressed by *utkṛti*, *prakṛti*, *svarga*, etc. 22 is expressed by *kṛti*, *jâti* (?), etc. 23 is expressed by *vikṛti*. 24 is expressed by *gâyatrî*, *jina*, *arhat*, *siddha*, etc. 25 is expressed by *tatva*,¹ etc. 27 is expressed by *nakṣatra*, *uḍu*, *bha*, etc. 32 is expressed by *danta*, *rada*, etc. 33 is expressed by *deva*, *amara*, *tridaśa*, *sura*, etc. 48 is expressed by *jagatî*, etc. 49 is expressed by *tâna*, etc.

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¹ Generally used for 5; also for 7 by Mahâvîra.

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**Word Numerals without Place-value.** In the *Veda* we do not find the use of names of things to denote numbers, but we do find instances of numbers denoting things. For instance, in the *Ṛgveda* the number 'twelve' has been used to denote a year² and in the *Atharvaveda* the number 'seven' has been used to denote a group of seven things (the seven seas, etc.).³ There are instances, however, of fractions having been denoted by word symbols, *e.g.*, *kalâ* = 1/16, *kuṣṭha* = 1/12, *śapha* = ¼.

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² "*Deva hitim jugupurdvâdaśasya ṛtum narona praminantyete*......" (vii. 103, 1).

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³ "*Om ye trisapta pariyante*...." (i. 1, 1).

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The earliest instances of a word being used to denote a whole number are found about 2,000 B.C., in the *Śatapatha Brâhmaṇa*⁴ and *Taittirîya Brâhmaṇa*.⁵ The *Chândogya Upaniṣad* also contains several instances. In the *Vedâṅga Jyotiṣa*¹ (1,200 B.C.) words for numerals have been used at several places. The *Śrauta-sûtras* of *Kâtyâyana*² and *Lâtyâyana*³ have the words *gâyatrî* for 24 and *jagatî* for 48.

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⁴ The word *kṛta* has been used for 4. "*catuṣtomena kṛtena ayânâm*...." (xiii. 3. 2. 1).

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⁵ "*Ye vai catvâraḥ stomâḥ kṛtaṃ tat*...." (i. 5. 11. 1).

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¹ *rûpa* = 1, *aya* = 4, *guṇa* = *yuga* = 12, *bhasamûha* = 27. See (*YJ.* 23, *ÂJ.* 31), (*YJ.* 13, *ÂJ.* 4), (*ÂJ.* 19), (*YJ.* 25) and (*YJ.* 20) respectively.

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² Weber's edition of *Kâtyâyana Śrauta Sûtra*, p. 1015.

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³ ix. 4. 31.

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At this early stage, however, the word symbols were nothing more than curiosities; their use to denote numbers was rare. Moreover, we find evidences of a certain indefiniteness in the numerical significance attached to certain words. For instance, in the same work, the *Aitareya Brâhmaṇa*, the word *virâṭ* has been used to denote 10 at one place and 30 at another. The principle of place-value being unknown, the word symbols could not be used to denote large numbers, which were usually denoted in terms of the numerical denominations or by breaking the number into parts.⁴ The use of the word symbols without place-value is found in the *Piṅgala Chandaḥ-sûtra* composed before 200 B.C. The principle of place-value seems to have been applied to the word numerals between 200 B.C. and 300 A.D.

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⁴ *E.g.*,

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**Word Numerals with Place-value.** The earliest instance of the use of the word numerals with place-value in its current form is found in the *Agni-Purâṇa*,⁵

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⁵ *Agni-Purâṇa*, Baṅgabâsî ed., Calcutta (1314 B.S.), chs. 122-23, 131, 140, 141, 328-335. According to Pargiter, probably the greatest Puranic scholar of modern times, "the *purâṇas* cannot be later than the earliest centuries of the Christian era." (*JRAS*, 1912, pp. 254-55). The *Agni-Purâṇa* is admitted by all scholars to be the earliest of the *Purâṇas*.

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*Daśâyutânâmayutaṁ sahasrâṇi ca viṁśatiḥ*
*Kotyaḥ ṣaṣṭiśca ṣaṭ caiva yo'smin rajan-mṛidhe hatâḥ*

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that is, 10 (10000) + 10000+20 (1000) + 60 (10,000,000) + 6 (10,000,000)—*Mahâbhârata*, *Strîparva*, xxvi. 9.

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a work which belongs to the earliest centuries of the Christian era. Bhaṭṭotpala in his commentary on the *Bṛhat-saṁhitâ* has given a quotation from the original *Puliśa-siddhânta*¹ (*c.* 400) in which the word system is used. The number expressed in this quotation is *kha* (0) *kha* (0) *aṣṭa* (8) *muni* (7) *râma* (3) *aśvi* (2) *netra* (2) *aṣṭa* (8) *śara* (5) *râtripâḥ* (1) = 1,582,237,800. There are in this work² several other quotations from the *Puliśa-siddhânta*, which contain word numerals. Later astronomical and mathematical manuals such as the *Sûrya-siddhânta* (*c.* 300), the *Pañca-siddhântikâ*³ (505), the *Mahâ-* and *Laghu-Bhâskarîya*⁴ (522), the *Brâhma-sphuṭa-siddhânta*⁵ (628), the *Triśatikâ*⁶ (*c.* 750), and the *Gaṇita-sâra-saṁgraha*⁷ (850), all make use of the word notation.⁸

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¹ *Bṛhat-saṁhitâ*, ed. by S. Dvivedi, Benares, p. 163.

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² *Ibid*, pages 27, 29, 49, 51, etc. We are, however, not sure whether those quotations are from the original work or from a later redaction of the same.

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³ i. 8; viii. 1, etc.

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⁴ See *MBh*, ch. 7 and *LBh*, ch. 1.

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⁵ i. 51-55, etc.

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⁶ R. 6, Ex. 6, etc.

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⁷ ii. 7, 9, etc.

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⁸ In the face of the evidence adduced here, G. R. Kaye's assertion, (*Indian Mathematics*, Calcutta, 1915, p. 31) that the word numerals were introduced into India in the ninth century from the east, shows his ignorance of Indian mathematical works, or is a deliberate misrepresentation.

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**Word Numerals in Inscriptions.** The earliest epigraphic instances of the use of the word numerals are met with in two Sanskrit inscriptions⁹ found in Cambodia which was a Hindu colony. They are dated 604

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⁹ R. C. Mazumdar, *Ancient Indian colonies in the far east*,—*Campa*, Vol. I, Lahore, 1927; see inscriptions Nos. 32, 39; also 40, 41, 43 and 44.

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A.D. and 625 A.D. Their next occurrence is found in a Sanskrit inscription of Java, belonging to the 8th century.¹

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¹ *IA*, XXI, p. 48.

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In India proper, although they were in use amongst the astronomers and mathematicians from the 3rd or 4th century A.D. onwards, it did not become the fashion to use them in inscriptions till a much later date. The earliest Hindu inscriptions using these numerals are dated 813 A.D.² and 842 A.D.³ In the following century they are used in the plates issued by the Eastern Chalukya Amma II, in 943 A.D.⁴ In later times the epigraphic instances become more frequent. The notation is also found in several manuscripts in which dates are given in verse.⁵

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² The Kaḍab plates, *IA*, XII, p. 11; declared by Fleet to be suspicious (Kanarese Dynasties, *Bombay Gazetteer*, I, ii, 399, note 7); cf. Bühler, *l.c.*, p. 86, note 4.

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³ The Dholpur Inscription, *Zeitschrift der Deutschen Morgenländischen Gesellschaft*, XL, p. 42.

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⁴ *IA*, VII, p. 18.

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⁵ Bühler, *l.c.*, p. 86, note 7.

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**Origin and Early History.** It should be noted that the arrangement of words, representing the numbers zero and one to nine, in a word chronogram is contrary to the arrangement that is followed when the same number is written with numerical signs. This fact has misled some scholars to think that the decimal notation and the word numerals were evolved at two different places. G. R. Kaye has gone so far as to suggest that the word numerals were imported into India from the east. This suggestion is incorrect for the simple reason that in no language other than Sanskrit do we find any early use of the word system. Moreover, in no country other than India do we find any trace of the use of a word system of numeration as far back as the fourth century A.D., at which period it was in common use amongst the astronomers and mathematicians of India.

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During the earlier stages of the development of this system, we find that instead of the word symbols, the number names were used, being arranged from left to right just as the numerical signs. An instance of this is found in the Bakhshâlî Manuscript¹ (*c.* 200), where the number

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¹ Folio 58, recto. The dots indicate some missing figures. The problem apparently required the expression of a big number in numerical denominations. We do not find a problem of this type in any of the later works. Cf. B. Datta, "The Bakhshālī Mathematics." *BCMS*, XXI, p. 21.

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is expressed as

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*Ṣaḍviṁśaśca* (26) *tripañcâśa* (53) *ekonatriṁśa* (29) *evacha*
*Dvâṣa*[*ṣṭi*] (62) *ṣaḍviṁśa* (26) *catuḥcatvâriṁśa* (44) *saptati* (70)
*Catuḥṣaṣṭi* (64) *na*[*navati*] (99) ... *ṁsanantaram*
*Triraśîti* (83) *ekaviṁśa* (21) *aṣṭa* (8) ... *pakaṁ*

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In the same manuscript, however, the contrary arrangement is used when the number 54 is expressed as *catuḥ* (4) *pañca* (5).² Jinabhadra Gaṇi (575) has used word symbols with the left to right arrangement to express numbers.³ It seems, therefore, that in the beginning opinion was divided as to which method of arrangement should be followed in the word system.

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² Folio 27, recto.

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³ *Bṛhat-kṣetra-samâsa*, i. 69ff.

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The extensive use of the word numerals by early mathematicians such as Puliśa, Varâhamihira, Lalla and others appears to have set the fashion to write the word numerals with a right to left arrangement, which was generally followed by later writers.

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No explanation as to why the right to left arrangement was preferred in the word system is to be found in any of the ancient works. The following explanation suggests itself to us, and we believe that it is not far from the truth: The different words forming a number chronogram were to be so selected that the resulting word expression would fit in with the metre used. To facilitate the selection the number was first written down in numerical figures. The selection of the proper words would then, naturally, begin with the figure in the units place, and proceed to the left just as in arithmetical operations. This is in accordance with the rule "*aṅkânâṃ vâmato gatiḥ*," *i.e.*, 'the numerals proceed to the left,' which seems to have been very popular with the Indian mathematicians. The right to left arrangement is thus due to the desire of the mathematicians to look upon the process of formation of the word chronogram as a sort of arithmetical operation.

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**Date of Invention.** The use of the word numerals in the *Agni-Purâṇa* which was composed in the 4th century A.D. or earlier, shows that the word system of numerals must have become quite common in India at that time, the *Purâṇas* being works meant for the common folk. That it was a well developed system in the fourth century is also shown by its extensive use in the *Sûrya-siddhânta* and the *Puliśa-siddhânta*. Its invention consequently must be placed at least two centuries earlier. This would give us the period, 100 A.D. to 200 A.D., as the time of its invention. This conclusion is supported by the epigraphic use of the word notation in 605 A.D., in Cambodia, which shows that by the end of the 6th century A.D., the knowledge of the system had spread over an area roughly of the size of Europe.

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It must be pointed out here that the decimal place-value notation and the word numerals were not invented at the same time. The decimal notation must have been in existence and in common use amongst the mathematicians long before the idea of applying the place-value principle to a system of word names could have been conceived. Thus we find that in the beginning (*c.* 200), the place-value principle, as is to be expected, was used with the number names. The word symbols were then substituted for the number names for the sake of metrical convenience. The right to left procedure was finally adopted because of the mathematicians’ desire to look upon the formation of the word numeral as a sort of mathematical operation.

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The above considerations place the invention of the decimal place-value notation at a period, at least two or three centuries before the invention of the word system. The word notation, therefore, points to the 1st century B.C. as the time of invention of the place-value notation. This conclusion agrees with that arrived on epigraphic evidence alone.

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