Mathematics in Ancient and Medieval India c5.s6: Numerals
Mathematics in Ancient and Medieval India c5.s6
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Numerals
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The early symbolic numerals of the world were of various types. The characters varied to meet the linguistic conditions of the different cultural areas in different parts. Here main types of numerals used by the Ancient Babylonians, Egyptians, Chinese, Greeks, Romans, Indians and Arabians are tabulated below.
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TABLE OF NUMERALS
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TABLE
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( Contd. )
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| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 20 | 30 | 50 | 60 | 70 | ... | 100 | 200 | 300 | ... | 1000 | 2000 | 4000 | 5000 | 6000 | 10000 |
| (d) *Greek* Geometric (5th century B. C.) Attice system | | | | | | | | | | | | | | | | | | | | | | | | | | | |
| Alphabetic system ( 300 B. C. ) | | α | β | γ | δ | ε | | ζ | η | θ | ι | κ | λ | | | | | ρ | σ | τ | | α | β | | | | δ |
| (e) *Roman* | | I | II | | | V | | | | | X | | | L | | | | C | | | | M | | | | | |
| (f) *Indian* Kharoṣṭhī ( 250 B. C. ) | | I | II | III | × | I× | II× | | | XX | 7 | 3 | | 733 | | | | λ | λII | | | | | | | | |
| Brāhmī-Aśokān ( 250 B. C. ) | | - | = | | + | | 6 | | | | | | | C | | | | H | | | | | | | | | |
| Nānāghāt ( 150 B. C. ) | | = | | | | | | 4 | 7 | | α | α | O | | | | | 21 | 21 | | | 4 | | | | | |
| Nāsik ( 100 A. D. ) | | = | | | | h | 4 | 7 | y | ? | α | | ν | | | | J | τ | F | | 9 | 9 | | | | |
| Bakhshālī Ms. ( 400 A. D. ) | • | ~ | 3 | 3 | y | 4 | | 7 | 9 | → | place | value. | | | | | | | | | | | | | | |
| Sanskrit form (10th century A. D.) | 0 | 9 | 2 | 3 | 8 | y | ξ | υ | 2 | ε | → | place | value. | | | | | | | | | | | | | | |
| (g) *Modern Arabic* (10th century A.D. | • | 1 | ۲ | ۳ | ε | 0 | 4 | V | ∧ | 9 | → | place | value. | | | | | | | | | | | | | | |
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(a) *Babylonians* : The Babylonians wrote with stylus on clay tablets the result being wedge-shaped characters. The tablets were then baked in the sun. The numbers were usually expressed with the help of a sexagesimal scale. Practically 59 symbols from one to fifty-nine (made by two basic symbols for one and ten) were used. Whenever a symbol for sixty was required, it was replaced by 1 of the next higher order as follows : 1.10 = 1 × 60 + 10 = 70 2.12 = 2 × 60 + 12 = 132
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In this way a big member was expressed by 1. 41. 33. 59. 3. 45. This obviously involves an idea of place-value. The numerical system is not very happy, because, a comparatively greater amounts of repetition is unavoidable for obtaining a number below fiftynine. In many cases, the numerical value of a symbol had to be determined from the context.¹ For example, in certain tablet 71 is represented by symbols of 1.10.1 i.e. 1 × 60 + 10 + 1 = 71. The number 171½ was represented by symbols of 2.50. 1. ½ i. e. 2 × 60 + 50 + 1 + ½ = 171½. The symbol for 1.12 may be expressed as 1.12.0 = 4320 or as 1.12 = 72 or as 1; 12 = 1⅕ etc. written sexagesimally.
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¹ Smith, D. E. *Ibid.*, 2, p. 37-38; Neugebauer, 0. *The Exact Sciences in Antiquity*, p. 20, 26, Copenhagen, 1951.
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(b) *Egyptians and the Semitic Groups* : The Egyptians used stone, papyrus, wood and pieces of pottery for writing. They in their Hieroglyphic system of writing possessed separate symbols for one, ten, hundred, thousand, ten thousand, hundred thousand and million. The Hieratic (religious) and Demotic (popular) system of writings had signs of symbols for one, five to ten, twenty, thirty,......to hundred. The Hieroglyphic and Hieratic numbers were commonly written from right to left and also from left to right and sometimes as in early inscriptions from top downwards. The numbers were expressed by repetition and addition of these symbols as necessary. As for example, the number 23 was represented by IIIUU i.e. 1+1+1+10+10=23; this was also written as UUIII. They therefore had little idea of place value.
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The Phoenician use symbols for one, ten, twenty and hundred; the Palmyrene for one, ten, twenty and hundred. There is no evidence that they had any idea of place value.
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(c) *Chinese* : Of the several systems used by the Chinese, those of Shang oracle bone form ( 14th to 11th century B.C. ), Bronze and Coin form ( 10th to 3rd century B.C. ) and the Counting rod forms ( 2nd century B.C. to 4th century A.D. ) deserve special notice in this connection.¹ Their symbolic numerals were written usually from top downwards with their values arranged in the descending order.
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¹ Needham, J. *The Science and Civilisation in China*, 3, p. 6-7, Tables 22 and 23, Cambridge, 1957.
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In the Shang oracle bone form, separate symbols for one, five, six, seven, eight, nine, ten, twenty, hundred, thousand etc. were used. For two to four, the symbol for one was repeated and other numbers were obtained by similar repetitions. For instance, 162 was expressed by the symbols of 100, 60 and 2 placed in the left to right order.
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In the Bronze and Coin form (10th century B. C. to 3rd century B. C. ) and other forms found on coins of Chou period (6th to 3rd century B.C.) more or less similar symbols as those of Shang oracle bone were employed. Neither improvement of method nor use of place-value was known to them.
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The Chinese in their Counting rod forms between 2nd century B.C. to 4th century A.D. used separate symbols for one, six, ten, sixty. Symbols for one was repeated to express numbers upto five and the values for seven to nine were represented by successive addition of symbols for one to that of six. For the numbers of the next higher order than 100, different symbols were used.
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(d) *Greeks* : The Greeks initially were deeply interested in numbers which were connected with geometric forms.¹ They had two other systems of notation—the Attice system and the Alphabetic system. The former, made of upright strokes and initials of several number names, was no better than the systems prevalent among the Egyptians and other contemporary civilisations. This system was replaced at about third century B.C. by the alphabetic system of numerals which were commonly known among the Hebrews, the Syrians, the Arabs, the Persians and other Semitic people. The alphabetic system had twenty-four letters in their common Ionic alphabet, to which they added three Phoenician alphabets to make it twenty-seven. These are divided into three groups, first nine alphabets denoting nine units, second nine alphabets denoting nine multiples of ten and the third nine alphabets denoting nine multiples of hundred. They wrote from left to right. In writing a compound number, the element of the higher denominations was put first. For examble, 43,678 was represented by the symbols of four myriad ( one myriad=ten thousands ), three thousand, six hundred, seventy, and eight maintaining an additive principle.
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¹ Smith, D. E. *History of Mathematics*, 2, p. 24-25, Dover Publication.
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(e) *Romans* : The Roman numbers were expressed in terms of symbols for one, five, ten, fifty, hundred, five hundreds, and thousand. The normal mode of writing was to put the greater elements in the beginning. For example, the number 1548 was written as MCCCCCXXXXVIII i.e. 1000+500+40+8. In certain cases, it was reversed to denote subtraction. But on a longer view, i.e. in a compound number of still greater value, it will be found that the normal mode has been preserved on the whole. The system had the advantage of using only few symbols for expressing any number in a very simple way. This is the reason why it received a considerable popularity and is still adopted occasionally in numbering the chapter and year of books.
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(f) *Indian* : It has already been discussed how the word numerals on a decimal scale were used by the Vedic Indians to express numbers. Regarding notational use, the earliest example of the use of symbols for the expression of numbers in India is provided by Kharoṣṭhī ( 250 B.C. ) and Brāhmī ( 250 B. C. ) inscriptions.¹
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¹ Bhau Daji, ‘The Ancient Sanscrit Inscriptions in the cave inscriptions’, *Journal of the Bombay Branch of the Royal Asiatic Society*, 8, p. 231, 1868; Smith, D. E. *History of Mathematics*, 2, p. 66, Dover Edition; Das, Sukumar Ranjan, ‘The Origin and Development of Numerals’, *Indian Historical Quarterly*, 3, p. 97-120, 356-57, Calcutta, 1927.
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The Kharoṣṭhī was particularly script of North-Western India (including Afganisthan and the Northern Punjab). It disappeared in the 3rd century A.D. In the Kharoṣṭhī numerals, there were separate symbols for one, four, ten, twenty and hundred. They were written from right to left with lower elements before the higher. For example, the number 122 was expressed by symbols of 2, 20, 100, and 274 by 4, 10, 20, 20, 20, 200. But in cases of the hundred and its multiples, the symbols for the smaller elements were placed after the symbol for hundred to denote multiplication. All these were common with the principles underlying other systems of Semitic numeral notation which we have discussed before.
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The Brāhmī numeral system consisted of separate signs for one and four to nine, two and three were denoted by repetitions of the symbols for one. It had also separate symbols for ten and its multiples as well as four hundreds upto seventy thousands. The plan was indeed nearly the same, as those of the Demotic, Hieratic and the Chinese. The symbols were written from left to right beginning with symbols of higher values. For example, 274 was written with the help of the symbols for 200, 70 and 4. The Nānāghāt and the Nāsik cave inscriptions amongst few other inscriptions namely Kuṣāṇa (150 A.D.), Kṣatrapa (200 A.D.) and Gupta (c. 300-450 A.D.) are noteworthy. Both these types (vide Table) bear a close relation and resemble each other. According to many foreign scholars, both Nānāghāt (150 B.C.) and the Nāsik (100 A.D.) types of numerals form the basis of our third or the modern numeral system.
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This third numeral system, (given in the *Bakhshālī MS.*—vide Table) which contains only ten symbols from one to nine and the symbol for zero could express any number however large on decimal place-value system. In this system the values of the notational places, *eka*, *daśa*, *śata* etc. increase either from right to left or from left to right. The numbers are always read from higher to lower order. This system is identical with the system which as we have seen appeared later amongst the Arabs during the tenth century of the Christian era. From there it went to Europe. And this system has now been adopted by all the civilized peoples of the world.
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(g) *Arabic*: Bīrūnī (973 1048 A. D.) has referred to two types of notation of numbers namely, the alphabetic (*abjd*) system as *Hurūf al-jummal* or *Hiṣāb al-jummal* and the modern numerals as *al-arqam al-hind* (*Indian numerals*). Numerous applications of alphabetic numerals are also available in his *Chronology of the Ancient Nations*.¹ Bīrūnī in his *Kitāb al-tafhim li-awā'il sinā'at at-tanjīm* (The Book of Instructions in the Art of Astrology) has given both these notation of numbers. The alphabetic system (*Hurūf al-jummal*), as used by Bīrūnī was apportioned in orderly sequence to 26 alphabets though initially there was no particular order in their use. It was based on the values of the alphabets without use of decimal place order. This notation seems to have been used exclusively by Arabian astronomers. The other system (*al-arqam al-hind*) given by Bīrūnī and others is shown in the table (vide p. 60). In the tenth and a few following centuries it is found that several mathematical works were composed by the Arab scholars having titles containing the words, *al-hind*, *al-hindi*, *al-takht*, *al-ghobar*.¹ These contained methods and systems very closely resembling those of the Indians.
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¹ Sachau, E. C. *The Chronology of the Ancient Nation*, London, 1879; the work is a English translation ofthe Arabic original.
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¹ Datta, B. 'Testimony of Foreign Scholars on the Origin of the Decimal Place-value Numerals', *Bulletin of the Calcutta Mathematical Society*, 24, p. 193, 1932; Saidan, A. S. 'The Developement of Hindu Arabic Arithmetic', *Islamic Culture*, 39, p. 209-21, 1965.
Linked texts (1)
- Bakhshālī Manuscript — Sanskrit