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History of Hindu Mathematics: A Source Book v1.c7.s13: 13. The Place-Value Notation in Hindu Literature

History of Hindu Mathematics: A Source Book v1.c7.s13

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13. The Place-Value Notation in Hindu Literature

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**Jaina Canonical Works.** The earliest literary evidence of the use of the word "notational place" is furnished by the *Anuyogadvâra-sûtra*,³ a work written before the Christian era. In this work the total number of human beings in the world is given by “a number which when expressed in terms of the denominations, *koṭi-koṭi*, etc., occupies twenty-nine places (*sthâna*).” Reference to the “places of numeration” is found also in a contemporary work, the *Vyavahâra-sûtra*.¹

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³ The passage has been already quoted in detail (*vide supra* p. 12).

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¹ Ch. i.; *cf.* B. Datta, *Scientia*, July, 1931, p. 8.

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**Puranas.** The Purâṇas which are semi-religious and semi-historical works, also contain references to the notational places. These works were written for the purpose of spreading education on religious and historical matters amongst the common people. Reference to the place-value notation in these works shows the desire of their authors to give prominence to the system. The *Agni-Purâṇa*² says:

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² The *Agni-Purâṇa* contains also the use of the word numerals with place-value (*vide supra* p. 58).

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“In case of multiples from the units place, the value of each place (*sthâna*) is ten times the value of the preceding place.”

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The *Viṣṇu-Purâṇa*³ has similarly:

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³ vi. 3.

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“O *dvija*, from one place to the next in succession, the places are multiples of ten. The eighteenth one of these (places) is called *parârdha*.”

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The *Vâyu-Purâṇa*⁴ observes:

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⁴ ci. 102f.

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“These are the eighteen places (*sthâna*) of calculation; the sages say that in this way the number of places can be hundreds.”

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The above three works are the oldest among the Purâṇas and of these the *Agni* and the *Vâyu* Purâṇas in their present form are certainly as old as the fourth century A.D. The *Agni-Purâṇa* is referred by some scholars to the first or second century A.D.

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**Works on Philosophy.** The following simile has been used in *Vyâsa-Bhâṣya*¹ on the *Yoga-sûtra* of Patañjali:

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¹ iii. 13.

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"Thus the same stroke is termed one in the units place, ten in the tens place, and hundred in the hundreds place......."²

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² The translation is as given by J. H. Woods, *The Yoga System of Patañjali*, p. 216. In a foot-note, it is remarked: "Contrary to Mr. G. R. Kaye's opinion, the following passages show that the place-value system of decimals was known as early as the sixth century A.D." The above passage is also noted by Sir P. C. Ray in his *History of Hindu Chemistry*, Vol. II, p. 117.

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The same simile occurs in the *Śârîraka-Bhâṣya* of Śaṅkarâcârya:

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"Just as, although the stroke is the same, yet by a change of place it acquires the values, one, ten, hundred, thousand, etc...."³

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³ III. iii. 17; *cf.* B. Datta, *American Math. Monthly*, XXXIII 1926, pp. 220-1.

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The first of the above works cannot be placed later than the sixth century whilst the second one not later than the eighth. The quotations prove conclusively that in the sixth century, the place-value notation was so well known that it could be used as an illustration for a philosophical argument.

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**Literary Works.** A passage from the *Vâsavadattâ* of Subandhu comparing the stars with zero dots has already been mentioned. Several other instances of the use of zero are found in later literature, but they need not be mentioned here.⁴

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⁴ *E.g.*, the use of the *śûnya-bindu* in *Naiṣadha-carita* of Śrîharṣa (*c.* 12th century). *Cf.* B. Datta, *Ibid*, pp. 449-454.

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