Avinash Sathaye Page 1 of 10 [email protected] Infinity in Classical Indian Mathematics By Avinash Sathaye Professor of Mathematics University of Kentucky 1. Introduction. Even a child learns to count 1, 2, 3 and so on. The idea which leads to imagining the end of this process, or a lack thereof, leads to the idea of infinity and takes a greater sophistication. In the same fashion, the idea to extend the count backwards and imagine a 0 or even a -1, -2, and so on needs a different imaginative process. Both these ideas are, of course old and by now, quite familiar to everybody. One has, thus gotten used to these numbers called the ``Intregers’’. Mathematicians in ancient India are credited with being the pioneers in both these inventions. The whole world routinely uses the decimal number system with the ten digits 0,1,2,3,4,5,6,7,8,9 serving to build arbitrarily large numbers with the power of their place value. This is the so-called Hindu Arabic system, developed in India and propagated through the Arabic Mathematical sources across Europe. Many of the standard techniques of algebraic calculations with integers written as decimal numbers are routinely taught in elementary schools and one no longer thinks about their power mainly due to their familiarity. These, however, were clearly developed in India; in view of the fact that without the place value system of number representation, they cannot exist! The idea of infinity is a different story. If you open Mathematical Books of today, you will find the idea of infinity mentioned in somewhat higher level courses. In Calculus related courses, you will find the idea of the ``real’’ infinity, the variables taking on larger and larger positive values. The resulting analysis of related variables leading to the notion of limits is at the heart of Calculus and thus, Modern Analysis. Similarly, the idea of a variable getting infinitesimally close to a finite number, like 0, is also at the heart of Calculus. You won’t, however, find either the infinite or the infinitesimal in an elementary book on algebra, let alone arithmetic! The only thing you may find in an algebra book is a very stern warning about not ever dividing by zero!Avinash Sathaye Page 2 of 10 [email protected] On the other hand, in the algebra books of old times in India, we find both the infinite and the infinitesimal treated routinely. They also include some rather intriguing exercises. These appear to be shear nonsense, if one approaches them armed with modern conventions without recognizing the novelty of approach used in these ancient books. Often, these exercises are discarded, calling them unfortunate blemishes on the otherwise brilliant achievements of the authors. My aim in this short essay is to analyze these problems in detail and propose that they may be based on a potentially powerful modern algebraic idea. I am not proposing that the old books had developed the full algebraic machinery. It certainly never materialized among the known mathematical texts or their subsequent commentaries. But that may be only due to insufficient follow-up activity and understanding. There is a very interesting example of this phenomenon in Indian Astronomy. Āryabhaṭa, the great mathematician and astronomer of the fifth had proposed a heliocentric model of the solar system as well as the notion that the earth was just a globe hanging in space. His ideas, however, were thoroughly rejected by the next mathematical genius Brahmagupta within a hundred years. The objections by Brahmagupta were the usual ``common sense’’ arguments for a flat earth, some of which persist even today. But due to the reputation of Brahmagupta, the Āryabhaṭa theories never became widespread in India until they were imported from the European sources centuries later. The ideas about infinity that I am proposing to discuss were certainly stated by Brahmagupta (sixth century) and the listed exercises below are to be found in the works of Bhāskarāchārya (II, of course) in the twelfth century. It is quite likely that Bhāskarāchārya’s ideas were not grasped by his commentators. At the same time, his work came to be the central text for algebra as well as astronomy during the rest of history of India. As a result, his original ideas were probably not pursued any further. There is yet another facet of the idea of the infinite, namely the counting infinity. In Modern Mathematics, this leads to the concepts of ordinal and cardinal numbers. In ancient Indian Mathematics, we find Jain texts discussing various such concepts of infinities. These texts are mainly religious or philosophical, but often carry a healthy amount of serious mathematics. They seem to introduce formal concepts of finite or enumerable, innumerable (very large but still finite) and infinite.Avinash Sathaye Page 3 of 10 [email protected] They even classify multidimensional concepts for infinity. It is possible that they might have come close to the ideas of modern cardinal (or at least ordinal) numbers. However, I have not yet succeeded in finding explicit pointers to advanced ideas similar to the algebraic ideas discussed here. So a similar evaluation of Jain theories of infinities will have to await further evidence. 2. Basic Definitions. To keep the discussion brief, I will give all the citations from Bhāskarāchārya’s Bījagaõita (his book on Algebra) with some cross reference from his Līlāvatī (his book on Arithmetic). There is a small difference in the numbering of the verses in different editions, but the reader should be able to locate them near the indicated citations. First, I collect the various defining properties of multiplication and division by zero.              . vadhādau viyat khasya khaṁ khena ghāte khahāro bhavet khena bhaktaśca rāśiḥ || bīj 2.18 A zero results when multiplied by zero, a ``khahara’’ (zero-divided) results when a number (rāshi) is divided by zero. In Līlāvatī, he gives more instruction about multiplying by zero.         yoge khaṁ kṣepasamaṁ vargādau khaṁ khabhājito rāśiḥ |       .