ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

172 BRAHMAGUPTA AND ARITHMETIC
number. The remainder being zero, the cube-root
is exact.
Fractions
The concept of fractions in India can be traced to very
early times. In the Ṛgveda,¹ we find such terms as one-half
(ardha) and three-fourths (tri-pāda). In a passage of the
Maitrāyaṇī Saṁhitā² are mentioned the fractions one-sixteenth
(kalā), one-twelfth (kuṣṭha), one-eighth (śapha) and one-fourth
(pāda). In the Śulba Sūtras³ we have not only a mention of
fractions, but they have been used in the statement and solution
of problems of geometric nature. Here in the Śulba, unit frac-
tions are denoted by the use of cardinal number with the term
bhāga or aṁśa; thus pañca-daśa-bhāga (literally “fifteen parts”) is
equivalent to one-fifteenth, sapta-bhāga (literally, “seven parts”)
is equivalent to one-seventh, and so on... The use of ordinal
numbers with the term bhāga or aṁśa is also quite common :
thus pañcama bhāga stands for one-fifth. The composite fractions
like tri-aṣṭama stands for three-eighths and dvi-saptama for two-
sevenths. In the Bakhshālī Manuscript, the term tryaṣṭa occurs
for 3/8 and 3⅜ is called trayastrayasta (three-three-eighths).
The Sanskrit term for fraction is bhinna (literally meaning
‘broken’). Obviously the European terms as fractio, fraction,
roupt, rotto or rocto are translations of the same term; they are
derived from the Latin fractus (frangere) or ruptus meaning
‘broken’. The Indian term bhinna has a few more connotations;
it stands for such numbers of the form :
(a/b ± c/d), (a/b of c/d), (a/b ± c/d of a/b) or (a ± b/c).
These forms were termed jāti’ i.e., ‘classes’, and the Indian
treatises contain special rules for their reduction to proper frac-
tions. Śrīdhara and Mahāvīra each enumerate six jātis, while our
author, Brahmagupta, gives only five (Bhāskara II gives only
four). The need for division of fractions in ‘classes’ arose out
of the lack of proper symbolism to indicate mathematical opera-
tions. (Datta and Singh Arithmetic, p. 188). The only operational
symbol in use was a dot, standing for the negative sign.

1. Ṛv. X, 90,4
2. Mait S, III, 7,7.
3. B. Datta, Śulba, pp. 212ff.