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

FRACTIONS 173 Reduction to lowest terms.—A non-mathematical work, Tattvārthādhigama-Sūtra-Bhāṣya by Umāsvāti (c.150A.D.) casua- lly mentions as follows in the context of a philosophic discourse: Or, as when the expert mathematician, for the purpose of simplifying operations, removes common factors from the numerator and denominator of a fraction, there is no change in the value of the fraction, so....¹ Reduction to common denominator. Whenever we have to add or subtract fractions, we follow this reduction operation to a common denominator. Brahmagupta gives the reduction along with the similar processes : By the multiplication of the numerator and denomi- nator of each of the (fractional) quantities by other denominators, the quantities are reduced to a common denominator. In addition, the numerators are united, In subtraction their difference is taken.² Fractions in combination :—Since there was no proper symbolism available to these early Indian mathematicians, they divided combination of fractions into four classes : Bhāga, prabhāga, bhāgapavāha and bhāga-bhāga. (i) Bhāga has been mentioned by Brahmagupta (BrSpSi. XII, 8) thus : ( a/b ± c/d ± e/f ± ........ ) usually written as ┌───┬───┬───┐ ┌───┬────┬────┐ │ a │ c │ e │ or │ a │ .c │ .e │ ├───┼───┼───┤ ├───┼────┼────┤ │ b │ d │ f │ │ b │ d │ f │ └───┴───┴───┘ └───┴────┴────┘ where the dots denote subtraction. (ii) Prabhāga : The form ( a/b of c/d of e/f ......... ) This is written as ┌───┬───┬───┐ │ a │ c │ e │ ├───┼───┼───┤ │ b │ d │ f │ └───┴───┴───┘ (iii) Bhāgānubandha : The form ( a + b/c ) is written as ┌───┐ │ a │ ├───┤ │ b │ ├───┤ │ c │ └───┘

1. II, 52.
2. विपरीतच्छेदगुणा : राश्योश्छेदांशाः समच्छेदाः । संकलितेंऽशा योज्या व्यवकलितेंऽशान्तरं कार्यम् ॥ —BrSpSi. XII. 2.