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Mathematics in Ancient and Medieval India c5.s7: Symbol for Zero and Decimal Place-value System

Mathematics in Ancient and Medieval India c5.s7

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Symbol for Zero and Decimal Place-value System

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² Bag, A. K. 'Symbol for Zero in Mathematical notation in India', *Boletin de la Academia Nacional de Ciencias*, Tome 48, Primer Congreso Argentino de Historia de la Ciencia, Primera Parte, p. 274-54, 1970.

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The concept of 'nothingness' or 'void' which is inherent and instinctive to man of all ages found its expression in the philosophical and literary works of all ancient civilised nations. The feeling of necessity to give a symbolical expression of this concept in mathematical calculation which would permit the representation of a number of only high order without any limit led ultimately to discovery of a symbol for what is now known as zero.

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The concept of 'nothingness' or 'void' in mathematical system was first introduced by the Babylonians ( 14th century B.C. ) who left an empty space in between two other numeral symbols along with their sexagesimal place-value system.³ But this gave rise to much confusion for giving any quantitative expression of the numerical value involved as there can be no fixed standard for an empty space to allow for its unambiguous repetitions. It therefore, helped very little in the symbolical expression of mathematical figures.

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³ Neugebauer, O. *The Exact Sciences in Antiquity*, p. 20, Copenhagen, 1951.

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In certain Greek astronomical Papyri Mss. ( 100 B.C. ) as well as Ptolemy's table of chords ( 150 A.D. ), symbols resembling that of modern zero were used to indicate the ending of a sentence or a marginal gap.¹ According to some scholars, this served as the origin of the introduction of the symbol for zero in mathematical notation. There is no justification of such a view as their application in mathematical calculation could not be traced in any of the ancient contemporary Greek mathematical works or thereafter.

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¹ Neugebauer, O. *Ibid.*, p. 10, 13-14.

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In India, the application of zero as a part of the numeral system, is characterised by its two types of uses as : A ) word-numerals, B ) symbolic-use.

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(A) *Word-numerals* : The word-numerals were based on a decimal scale of notation namely *eka*, *daśa*, *śata*, *sahasra*......upto 12 places from the time of *Saṃhitas* (1500 B.C.).² A large number like, *ṣaṣṭhīm sahasrā sapta śatāni navatim nava* ( 60, 799 ) was expressed by placing the word numerals of higher order to be followed by one of lower order in the scale. This successive placing of, *sahasra*, *śata*, *daśa*, *eka* for the expression of a large number in left to right order of gradually decreasing values obviously shows that the idea of decimal place-value was implied.¹ The scale is undoubtedly more convenient than the Babylonian sexagesimal scale. The *Śatapatha Brāhmaṇa*, *Taittirīya Brāhmaṇa*, *Chāndogya Upaniṣad*, *Vedāṅga Jyotiṣa* etc. had similar uses of numerals. The idea of decimal place-value developed when the decimal scale was associated with the value of the places in a right to left order (or left to right) in the scale with the introduction of symbol for zero. That means, the symbols or word numerals took values × 1, × 10, × 100.........when placed in 1st, 2nd, 3rd places......respectively. From the beginning of the Christian era, the word-numerals with their various synonyms began to be used to avoid repetition of the same word and to keep rythm of ślokas.

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² *Yajurveda Saṃhitā*, 17. 2; *Taittirīya Saṃhitā*, 4. 40. 11. 4 and 7. 2. 20. 1; *Maitrāyaṇī Saṃhitā*, 2. 8. 14; *Kāṭhaka Saṃhitā*, 17. 10; vide also *Hist. of Hindu Mathematics* by Datta and Singh, Part I, Motilal Banarasi Das, Lahore, p. 9, 1935.

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¹ Datta, B. ‘Vedic Mathematics’, *The Cultural Heritage of India*, old series, 3, p. 395-97, Ramkrishna Centenary Memorial, Belur Math, Calcutta.

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A few synonyms of 0, 1, 2, 3, etc. may be given as follows :

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0 *śunya, kha, ākāśa, ambara, vyoma, nabha, pūrṇa* etc. 1 *kṣiti, dharā, pṛthvī, bhū, bhūmi, indu, candra, abja, vidhu, śaśāṅka* etc. 2 *yama, aśvin, dasra, akṣi, netra, nayana, bāhu, kara* etc. 3 *rāma, guṇa, agni, bhuvana, loka* etc.

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The application of nine word numerals on the decimal place-value scale was found in Sanskrit works and incriptions from 2nd century A.D. onwards. Several examples are given below :—

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(1) *Pauliśasiddhānta* (200 A.D.) *kha-kha-rūpa-aṣṭa*¹ 0 0 1 8 =8100.

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¹ *Pañcasiddhāntika* of Varāha, ed. by Thibaut, ch. 6, Verse 9.

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(2) *Sūryasiddānta* (400 A.D.)
(i) *nava-vasu-sapta-aṣṭa-kha-nava-aśvi*² 9 8 7 8 0 9 2 =2908789.
(ii) *svara-eka-pakṣa-ambara-svara-ṛtu*³ 7 1 2 0 7 6 =670217.

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² Vide *Pañcasiddhāntikā* of Varāhamihira (ed. by G. Thibaut) Ch. 9, v. 3.

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³ *Ibid*, Ch. 9, v. 2.

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(3) *Pañcasiddhāntika* (505 A.D.)
(i) *śunya-dvi-pañca-yama*⁴ 0 2 5 2 =2520
(ii) *muṇi-yama-yama-dvi* 7 2 2 2 =2227

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⁴ Ch. 1, v. 17. (Thibaut's edition).

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(4) *Inscription at Cambodia*⁵ (604 A.D.) *rasa-dasra-śarai śakendra varṣe* 6 2 5 =526 śaka year.

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⁵ Coedès, G. ‘A propos de l'origine des chiffres arabes’, *Bulletin of the School of Oriental Studies*, 6, p. 352, 1930-32.

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(5) *Inscription at Champa*⁶ (731 A.D.) *rāma-artha-ṣaṭkai śaka* 3 5 6 =653 śaka year

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⁶ Huber, E. ‘Études Indo-Chinoises’, *Bulletin de l'école Française, d'extrent orient*, 11, p. 266-67, 1911.

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(6) *Inscription at Java*⁷ (732 A.D.) *śruti-indriya-rasa śaka* 4 5 6 =654 śaka Such uses of word-numerals (i. e. same or synonymous words taking different values when placed at different places) are to be found in the *Brāhmasphuṭasiddhānta* and *Khaṇḍakhādyaka* of Brahmagupta (628 A D.) and in almost all later mathematical and astronomical works. The use of the word-numeral system is also found to occur in many of the inscriptions of Cambodia, Java, Champa, Malay etc. at about 7th century A. D.

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⁷ Coedès, G. *Ibid.*, p. 326.

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(B) *Symbolic-use* : A good number of inscription using numeral symbols are also known. Some of these inscriptions used a point-symbol (*śunya-bindu*) as well as circular symbol for zero. The earliest evidence of the symbol for zero is found in the *Bakhshālī Ms.*¹ (400 A.D.). A chronological list of the appearance of the symbols of numerals from one to nine along with or without that of zero fitted in a decimal place-value scale, is given below :

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¹ Vide Hoernlé, A.F.R. ‘The Bakhshālī Ms.’, *Indian Antiquary*, *17*, p. 33-48, 1888; Kaye, G. R. The *Bakhshālī Ms.*, Parts I & II, New Imperial Series No. 43, Calcutta, 1927; Datta, B. ‘The *Bakhshālī* Mathematics’, *Bulletin of the Calcutta Mathematical Society*, *21*, p. 1-60, 1929. The Bakhshālī Ms. was composed in the second century A. D. (Hoernlé, and Datta) and the present Ms. was probably a commentary.

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(1) *Bakhshālī* Ms. (400 A.D.)

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| *Symbolic expression* | | *Number* |
| (i) [Figure:] | (17 verso) | 330 |
| (ii) [Figure:] | (22 recto) | 157500 |
| (iii) [Figure:] | (22 recto) | 947 |
| (iv) [Figure:] | (56 recto) | 846720 |

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(2) *Gurjara Grant inscription*¹ ( 595 A.D. )

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¹ Epigraphica Indica, 2, p. 20, 1894.

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[Figure:] 346

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(3) *Khmere inscription at Sambor*² ( 683 A.D. )

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² Coedès, G., *Ibid.*, 6, p. 326, Plate IV.

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[Figure:] ( śaka ) 605 ( śaka )

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(4) *Malay inscription at Palembang*³ ( 684 A.D. )

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³ Westenenk, L. C. *Djāwā* periodical, *1*, no. 1. p. 10, vide plate I, vide also *Acta orientalia*, 2, p. 12-21, 1924.

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[Figure:] ( śaka ) 606 ( śaka ) [Figure:] ( śaka ) 60 ( śaka )

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(5) *Malay inscription at Kotakapur*⁴ ( 716 A.D. )

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⁴ Kern, H. Inscriptie van Kota Kapur, *Bidragen tot de Taal-. Land-en Volkenkunde van Nederlandische...*, 67, p. 393-400, 1913; Kern, H. *Verspreide Geschriften*, 7, p. 208-209, 1917; Coedès, G. *Ibid.*, plate IV.

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[Figure:] ( śaka ) 608 ( śaka )

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(6) *Dinaya Sanskrit inscription at Java*⁵ ( 760 A.D. )

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⁵ Bosch, F. D. K. ‘De Sanskrit inscriptie op den steen van Dinaja (682 śaka)’, *Tijdschrift*, 57, p. 411, 1916; vide also 64, p. 227-29, 1924.

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[Figure:] ( śaka ) 2 & 8 & 6 / nayana-vasu-rase & śake =682 śaka

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(7) *Inscription at Po-nagar Champa*⁶ (813 A. D.)

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⁶ Finot, L. ‘Les inscriptions de Jaya Paramesvaravarman I, roi du Champa’, *Bulletin de l'école Française d'extrent orient*, 15, p. 47, 1915.

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(i) [Figure:] (śaka) 735 (śaka)
(ii) [Figure:] (śaka) 735 (śaka)

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(8) *Inscription at Bakul*¹ (839 A. D.)

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¹ Finot, L. *Ibid.*, p. 47.

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[Figure:] (śaka) 751 (śaka)

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We find here the use of word numerals with right to left order and symbolic numerals with a left to right order. The use of word numerals with left to right order has also been found in the *Mahāsiddhanta* of Āryabhaṭa II. These orders were so used for no obvious reason except possibly for emphasizing the idea that the decimal place-value concept remains unchanged in whatever order (right or left) the numerals are arranged when the number is always expressed by reading from higher to lower order in the scale. The symbolic use of zero is found to occur in the *Bakhshālī* Ms. (400 A. D.) of India as well as in number of inscriptions found at Sambor, Malay, Java, used in a decimal place-value system with their dates expressed in Śaka era of Indian origin. Some of these symbols including that of zero found in these inscriptions bear more or less close resemblance with those of the corresponding numbers in the *Bakhshālī* Ms. . The *Dinaya Sanskrit inscription* (see p. 72) found at Java (760 A. D.) however contained both word-numerals and the numeral symbols. This seems to suggest strongly a common origin of all these symbols obviously in India. Moreover, the evidence of the association of decimal scale e. g. *eka*, *daśa*, *śata* etc. with places is found in the works of Āryabhaṭa I (b. 476 A. D.) onwards. He writes that the value of the places are ten times of the preceding (*sthanāt-sthānaṃ daśaguṇaṃ syāt*, *Āryabhaṭīya*, Gaṇita, verse 2) in a numerical notation thereby expressing the idea of decimal place-value. That the decimal place-value concept was current before seventh century A.D., has also been attested by Vyāsa (before 700 A. D.) in his philosophical work, *Yogadarśanabhāṣya* (Ch 3, Sū. 13) thus : *yathaikā rekhā śatasthāne śataṃ daśasthāne daśa ekaṃ caikasthāne* i. e. "a (numerical) sign denotes hundred in the *śata* place, ten in the *daśa* place, and one in the *eka* place." The very similar language is used by Śaṅkarācārya (c. 800 A. D.) in his *bhāṣya* on *Vedānta Sūtra* (II. 2. 17). These illustrations establish without doubt that by the time of compositions of these two works, the decimal place-value concept must have been well established and was generally known. This system has appeared amongst the Arabs in the tenth century A. D. From there it went to Europe and it has now been adopted by all the civilised peoples of the world.

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The Chinese (8th century A. D.) left a gap or some vacant space similar to the Babylonians where a zero was required. This is found on some Thang Mss. of the Tunhuang cave-temples. One of the Ms. rolls, namely *Li-Chhêng Suan Ching*¹ contains the number 405 both in written form and in rod numerals like 𝄀𝄀𝄀𝄀 𝄀𝄀𝄀𝄀𝄀. A symbol for zero in the usual circular form appeared only in 1247 A. D. in a work *Su Shu Chiu Chang* of Chhin Chiu-Shao.² Thus it gives : | ≡ ╥ 000 as 147000. Needham, from the consideration of amalgamation of Indian, Chinese and of the local residence of Java, Champa, Malay etc. due to colonisation, concluded that the origin of the symbol for zero should be attributed to the mixed culture of the Indians and the Chinese. But such an argument is quite untenable in view of the fact that the use of the symbol for zero fitted in a decimal place-value scale appears in China only in Ms. of the 13th century A. D. Furthermore, there is no evidence that the use of word-numerals including that of zero in a decimal place-value scale was ever made by the Chinese or even by any other nations except by the Indian prior to their symbolic use. This has also been clearly recognised by the French scholar Coedès.¹

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¹ Needham, J. *Science and Civilisation in China*, 3, p. 9, Cambridge, 1959.

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² Smith, D. E. *Hist. of Mathematics*, 2, p. 42, Dover Edition.

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¹ Coedès, G. *Ibid.*, 6, p. 325.

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There is no doubt that the Arabs derived their knowledge of decimal place-value from India. Al-Bīrūnī (1030 A. D.), who stayed in India quite a number of years and is very much well known for his encyclopaedical knowledge, made the following observations in his *India* (Eng. tr. Sachau, *1*, p. 174), “the numerical signs which we use are derived from the finest of the Hindu signs.” The earliest use of zero in Arabia by a circular symbol was found in a 10th century A. D. manuscript.² In other Mss. during the same period, the forms such as ᴑ, ᵂ and ⎴ were used for zero. The dot symbol (•), similar to that of the Indians, was also used by Bīrūnī in his table. The modern Arabic symbol 5 is represented by a symbol that looks somewhat like the modern zero and the zero is denoted by a dot. The Indian name *śunya* was taken over by the Arabs as *aṣ-ṣifr* or *ṣifr*. This was subsequently changed to zephirum (1202 A. D. - Fibonacci), *tziphra* (1340 A. D. - Planudes) and *zenero*, *zepiro* (c. 16th century A. D. Italy).³

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² Smith, D. E. *History of Math.*, 2, p. 74, Table Column III, Dover Edition.

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³ Smith, D. E. *Ibid.*, p. 71.

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The zero in its circular symbol appeared in Europe in a 13th century A. D. manuscript, reproduced in *Della vita e delle opere di Leonardo Pisaro* (Rome, 1852). Next, it is found to have been used in European Manuscripts from 14th century A. D. onwards.¹

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¹ Smith, D. E. *Ibid.*, p. 71.

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