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Mathematics in Ancient and Medieval India c5.s5: Fraction

Mathematics in Ancient and Medieval India c5.s5

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Fraction

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Few fractions are mentioned in the Vedic literature. *Ardha*, *pāda*, *sapha* and *kala* denote ½, ¼, ⅛, 1/16 respectively. But only the first two are common. The word *pāda* was derived from four parts of a stanza; a stanza in Sanskrit literature is divided into four parts from the Vedic times, each of which is called a *pāda*. The idea of a fraction of 1/16 was derived from the increasing visibility of the moon from the new moon to fullmoon covering the period of sixteen days and thus the moon is said to possess 16 *kalas* or parts. Mention of the number of fractions are also found in the *Ṛgveda*, *Taittirīya Saṃhita*, *Taittirīya Brāhmaṇa* etc.¹ In the *śulbas*², the fraction is denoted by the term *bhāga*, e.g., *pañcamabhaga*=⅕, *daśamabhaga*=1/10 etc. A number followed by a fraction was denoted in the *śulba* as follows : *ardhacaturtha* =3½, *ardhapañcaśat*=49½ etc. Four fundamental arithmetical operations with elementary fractions were known to the *Śulbakāras*. Some expressions other than unit fractions were also used in the *Śulbasūtras*¹ (c. 600 B.C. ), *Arthaśāstra*² of Kauṭilya (3rd century B. C. ), *Vedāṅgajyotiṣa*³ (200 B.C. ), *Sūryaprajñapti*⁴ (200 B.C. ), etc. In the *Śulba-sūtras*, fractions were expressed in language. The *Bakhshālī Ms.* (4th century A.D. )⁵ first expressed operations of fraction in terms of symbols both with and without the dividing lines. This is followed in the *Brāhmasphuṭasiddhānta* of Brahmagupta (628 A.D.) and *Gaṇitasārasaṁgraha* of Mahāvīra (c. 850 A.D.) and in later works. For addition and subtraction of fractions, Brahmagupta suggested the operation of reduction to a common denominator. The works of Mahāvīra

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¹ Datta, B. and Singh, A. *History of Hindu Mathematics*, Part I, Motilal Banarasi Das, Lahore, p. 185, 1935.

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² Datta, B. *The Science of Śulba*, University of Calcutta, p. 212-16, 1932.

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¹ *caturtha-saviśeṣa saptama*=1/7(¼√2. (*Āpastambaśulba*, 19. 4; 19. 7); *caturdaśa prakramān triṁśca prakrama saptabhāgān*=14 3/7 prakramas (*Kātyāyanaśulba*, 6. 2) etc.

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² Shamasastry, R. *Kauṭilya's Arthaśāstra* (Eng. tr. ), fifth ed., p. 113–21, Mysore, 1956.

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³ *kalā daśa saviṁśa nāḍikā*=10 1/20 (*Vedāṅga Jyotiṣa*, v. 38).

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⁴ *do joyanāhaṁ addhaduyātvaliśate sitayabhāge*=2(42⅛)/183 (*Sūryaprajñapti*, sūtra 18).

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⁵ 11 / 1 yu 5 / 1 means 11+5; 2 / 1 / 2 represents 2½; 1 / 1 / 2 as 1½;

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|1 & 1 & 1 & 1 / 1 & 1 & 1 & 1 / 2+ & 3 & 4+ & 5| bha|36 / 1| means 36/((1-½)(1+⅓)(1-¼)(1+⅕))

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and Śrīdhara contain special rule for their reductions to proper fractions. The fraction was denoted as *bhinna* by Mahāvīra (c. 850 A. D.) and Śrīdhara (c. 903 A. D.). The numbers of the form $(a/b±c/d), (a/b of c/d), (a/b±c/d of a/b)$ were termed *jāti* (i.e., classes).

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Amongst other nations¹, Babylonians (c. 2000 B.C.) on their cuneiform clay tablets included some special symbols for ⅔, 2/18, 4/18 and 5/7 etc. Egyptians, Greeks as well as the Romans dealt with unit fraction. Chang Chiu-Chien (6th century A.D.) remarked in the preface of his *Arithmetical classic* that the chinese found it difficult to express a fractional number and to perform any operation with them. Even as late as 10th century A.D. Rabbi Saadia ben Joshep al-Fayyaumi, a Hebrew writer living in Egypt made wide use of the unit fraction in his computations.

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¹ Smith, D. E. *History of Mathematics*, 2, Dover Publication, p. 210-215.

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