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

176 BRAHMAGUPTA AND ARITHMETIC In the bhāgānubandha class, the whole number (rūpa- gaṇa) is multiplied by the denominator (of the frac- tion) should be increased by the numerator (of the fraction) or the upper denominator having been multiplied by the lower denominator, the initial numerator (i.e. the upper numerator) should be multi- plied by the sum of the lower numerator and denomi- nator.¹ (Pātīgaṇita, 39 cf. BrSpSi. XII. 9 (ii); GSS. (iii) 113 This means that (i) a + b/c = (ac + b)/c (ii) a/b + c/d of a/b (which was written by Indians in the style ┌───┐ │ a │ │ b │ │ c │ │ d │ └───┘ is equal to a(d + c)/bd Addition and Subtraction of Fractions In the Brāhmasphuṭa-siddhānta, Brahmagupta gives the rule for the addition and subtraction of fractions : If the denominators (cheda) of fractions are different then reduce these fractions to a common denomina- tor. Now for the additions, unite the numerators and take their difference in case of subtraction.² Brahmagupta and Mahāvīra give the method under Bhāga- jāti. Multiplication Brahmagupta says : The product of the numerators divided by the pro-

1. भागानुबन्धजातौ रूपगणश्छेद सङ्गुणः सांशः । अवरहरन्धोर्ध्वं हरेऽथोंशाद्युतहरघ्न आद्यंशः ॥ —Patīganita 39.
2. विपरीतच्छेदगुणाः राश्योश्छेदांशकाः समच्छेदाः । संकलितेंऽशा योज्या व्यवकलितेऽशान्तरं कार्यम् ॥ —BrSpSi. XII. 2