Mathematics in Ancient and Medieval India c5.s5.1

c5.s5.1

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