Bhāskarācārya's treatment of the concept of infinity (Q2818658)

The author has dedicated his article to Bhāskarācārya, one of the greatest Indian mathematicians, on the occasion of his 900th birthday in 2014. His aim is to analyze Bhāskara's exercises on the concept of infinity.NEWLINENEWLINEBhāskara elucidates the operations of zero both in his arithmetic text {\textit{Līlāvatī}} and in his algebra text {\textit{Bījagaṇita}}. The author gives the relevant definitions for the terms {\textit{khahara}} (that which has zero as its divisor) and {\textit{khaguṇa}} (that which has zero as its multiplier) which require specific algebraic manipulations. For convenience, some formal definitions using modern terminology are also given. Modern notation is used for easy reference. The author then takes upon three examples which require our special attention -- one from the {\textit{Līlāvatī}} and two from the {\textit{Bījagaṇita}}. The first example is justified by the author by taking the equation as tending to zero, and it therefore can be treated as a limit. Here, Bhāskara has added that this is useful for astronomical calculations. The other two examples are from the {\textit{Bījagaṇita}}. Here, Bhāskara does not talk about their utility. The author notes that the first one can be solved only if the winning rule (which he has explained earlier with regard to infinity) is used and the rule of idempotence thereafter. The last example is worked out without the winning rule but with the rule of idempotence.NEWLINENEWLINEIt is remarkable that the great 20th century mathematician Ramanujan seems to have thought on similar lines as is evident from the book [Ramanujan. The man and the mathematician. Bombay: Asia Publishing House (1967), pp. 82--83] by \textit{S. R. Ranganathan} and also reported by Parthasarathi Mukhopadhyay. His thoughts on zero and infinity are very similar. The author has summarized Ramanujan's conversation with P. C. Mahalanobis: ``Ramanujan spoke of zero as representative of the Absolute ({\textit{nirguṇa brahman}}), something which has no attributes and no description. Infinity on the other hand, was totality of all possibilities capable of being manifest in reality. Further, the product of zero and infinity would supply the whole set of infinite numbers. In other words he was probably thinking of {\textit{khahara}}\dots''.NEWLINENEWLINEIn algebra books of ancient India, infinity and infinitesmal are treated routinely. Bhāskara is no exception. Only, his rules and examples are rather intriguing. They cannot be dismissed as ``unfortunate blemishes'' nor can they be termed as forerunners to modern calculus. Though he might have intended further elaboration, Bhāskara did not return to the subject again. The author has rightly observed that Bhāskara's ideas on zero and infinity seem to point to new algebraic concepts, and new mathematical systems based on these ideas give scope for further research on the subject.