The fundamentals of Vedic mathematics
NO ONE raises an eyebrow when children are required to memorise multiplication tables till 19. Then, why should anyone throw a fit if students are taught how to multiply 199 by 199 without resorting to multiplication tables, simply because the method used is Vedic mathematics?
The resurgence of interest in Vedic maths came about as a result of Jagadguru Swami Sri Bharathi Krishna Tirthaji Maharaj publishing a book on the subject in 1965. The erstwhile Bharatiya Janata Party government in Uttar Pradesh, Madhya Pradesh, Rajasthan and Himachal Pradesh then introduced Vedic maths into the school syllabus, but this move was perceived as an attempt to impose Hindutva, because Vedic philosophy was being projected as the repository of all human wisdom. The subsequent hue and cry over the teaching of Vedic maths is mainly because it has come to be identified with fundamentalism and obscurantism, both considered the polar opposite of science. The critics argue that belief in Vedic maths automatically necessitates belief in Hindu renaissance.
But is this argument valid? It has long been known that the richness of Indian mathematics extends beyond the discovery of zero. Krishna Tirtha is credited with the discovery of 16 mathematical formulae that were part of the parishishta (appendix) of the Atharva Veda, one of the four Vedas (See box). Tirtha's simple formulae make intricate mathematical calculations possible. Besides speeding up a number of mathematical procedures, Tirtha's formulae cover factorisation; highest common factors; simultaneous, quadratic, cubic and biquadratic equations; partial fractions, elementary geometry, and differential and integral calculus (See box). But Tirtha is not without his critics, even apart from those who consider Vedic maths is "unscientific".
A standard criticism is that the Vedic maths text is limited to only middle- and high-school formulations and the emphasis is on a series of problem-solving tricks. The critics also point out that the Atharva Veda appendix, containing Tirtha's 16 mathematical formulae, is not to be found in any of the existing texts. A third criticism is already the most pertinent: The book is badly written.
Criticism of Vedic maths found a powerful voice in S G Dani, mathematics professor at the Tata Institute of Fundamental Research in Bombay. He wrote recently in The Times of India: "The book (Tirtha's Vedic Mathematics) apparently gathered wider respectability around the mid-1980s following a statement in Parliament by the then human resources minister. Inclusion of such spurious material as Vedic not only corrupts the intellectual process of a proper study of history, but is also unhealthy for society in view of their being prone to abuse in various ways. It is absurd and outrageous to build up a false framework of history to whip up pride; our mathematical heritage offers plenty to be proud of, without resorting to such gimmickry."
Part of the controversy over Vedic maths stems from the widespread acceptance of the book as genuinely Vedic and because politicians have hopped on to its bandwagon. But positioned on the other side of the equation are mathematicians like Dani. "The baseless myths about the antiquity and super-capabilities of the book had better be cleared," warns Dani, "lest we spoil a whole generation of children by making them end up with a wrong approach to both history and mathematics."
Deceptive features
In assessing the strengths and weaknesses of Vedic mathematics, the works of Aryabhata I (AD 475) and subsequently Brahmagupta (622), Bhaskara II (1150) and perhaps even Sangama Grama Madhava and Narayana Pandita (14th century) should be taken into account. This means it would probably be more appropriate to refer to Indian rather than Vedic mathematics.
Indeed, much of Tirtha's work seems to be of the high-school variety, but appearances are deceptive. The Indian tradition of mathematics is essentially inductive and so rigorous, proof is rarely stated explicitly. This inductive-cum-intuitive approach (as opposed to a deductive one) also is evident in some of Srinivasa Ramanujan's work. When I S Bhanu Murthy attempted to prove some of Tirtha's formulae, he found a few profound theorems of the number theory were involved.
So, not all of Tirtha's work can be dismissed as elementary. Much of it is arithmetical in nature and enhances computational skills that have considerable pedagogical value.
Tirtha's work contains sufficient examples to discount complaints that it is merely a bag of computational tricks, but it is true that the so-called appendix to the Atharva Veda is not to be found in any extant text. Unfortunately, Tirtha never set down in complete form the 16 formulae that were supposed to lie beyond the computational methods used. The only references are to abbreviated forms. It is also true that the techniques in Tirtha's book relating to division and recurring decimals are not to be found in the work of early Indian mathematicians. However, Tirtha's techniques concerning squares, square roots, cubes and cube roots follow the familiar work of Aryabhata I, Sridhara (750) and Bhaskara II.
Two possibilities arise from all this: Either Tirtha discovered lost parts of the Atharva Veda or he must have evolved the formulae himself, which would make him a greater mathematician than he claimed to be. In either case, it is irrelevant to debate whether the formulae form part of the appendix of the Atharva Veda; what is important is whether the formulae are useful -- and on this point, there can be no dispute.
Rich tradition
Even beyond Tirtha's work, there is a case being made for a closer examination of ancient Indian mathematics. Because Aryabhata I's method of determining square and cube roots and Sridhara's and Bhaskara II's formula to determine cubes are all far quicker than the usual methods, why shouldn't they be used? Why shouldn't students be taught Bhaskara II's and Brahmagupta's methods of solving equations?
Many aspects of geometry were discussed by ancient Indian mathematicians and pi is a case in point. Several series were set down as estimations for pi, with Lilavati's approximations being 22/7 and 3927/1250.
Ancient Indian mathematics does not represent all that is wonderful in the subject. In fact, many aspects of modern mathematics were beyond the ken of the mathematicians of yore. But a rational and scientific approach is needed to cull from these early works those that is useful and relevant to today's needs. To dismiss Vedic maths as fundamentalist and obscurantist violates the scientific spirit and merely reflects a dogmatic attitude, which is as much an enemy of science as fundamentalism.
Vedic maths is a historical legacy that belongs to all of humankind, for it is possible to accept it without having to believe in the Hindu rashtras.