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# From zero to size zero

About 900 years ago, Bhaskar II or Bhaskaracharya posed the following problem: In a forest, a number of apes equal to the square of one-eighth of the total apes in the pack are playing noisy games. The remaining twelve apes, who are of a more serious disposition, are on a nearby hill and irritated by the shrieks coming from the forest. What is the total number of apes in the pack? Quaint, but if you really get down to it, fairly complicated for a layman. Of course, any 10th-grader of today will solve it, but nine centuries ago Bhaskaracharya was triumphantly discovering the 'quadratic equation', and the fact that there are two possible answers to the ape problem: 16 and 48.

The story of Indian mathematics is much misunderstood, having been subjected to two opposing pressures. The first is the coloniser's pressure — most of the past was ridiculed and dismissed. Most Western texts talk of Greek or Arab influence on Indian mathematics, portraying it as a largely one-way process.

The second pressure comes as a reaction to the first. Ultra-nationalists claim that everything was already known to the great Indian sages and others have simply plagiarised it.

The truth lies somewhere in between. There is, however, no doubt that it was in India that the system of expressing every possible number using a set of 10 symbols (0 to 9) with each numeral having a place value and an absolute value was developed. This not only simplified mathematics but paved the way for breathtaking numbers to be imagined and handled.

Most modern scholars accept that the beginning of mathematics in the Indian subcontinent can be traced to the Indus Valley civilisation, circa 2, 500 BC. Although details are not known as the script is still undeciphered, the bricks made in precise 3:2:1 ratio, the ruler found in Lothal with markings of the 'Indus inch' (equal to 1. 32 present day inches) and other such evidence clearly points to a level of mathematics quite similar to that developed in almost all ancient civilisations from Egypt and Mesopotamia to the Chinese.

The Indus civilisation imploded for unclear reasons. With the rise of the Vedic civilisation came a revival of mathematics with the practical purpose of constructing sacrificial altars and other paraphernalia. The only surviving source for this chunk of history comes from the Sulbasutras, appendices to the Vedas, that describe geometrical measurements. They were written by a clutch of mathematicians — Baudhayana (800 BC), Manava (750 BC), Apastamba (600 BC), and Katyayana (200 BC) — about whom practically nothing is known. Around the middle of the third century BC, Brahmi numerals from 1 to 9 began to appear. This would have great significance later.

With the decline of the Vedic age, there began what was till very recently thought of as the Dark Age for mathematics in India. This interregnum of nearly a millennium, however, saw a silent but no less significant advance of mathematics, initiated and led by Jain mathematicians. "The main topics of Jaina mathematics in around 150 BC were: the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations. More surprisingly, the Jains developed a theory of the infinite containing different levels of infinity, a primitive understanding of indices, and some notion of logarithms to base 2", say O'Connor and Robertson of the St Andrew Scotts University, Scotland.

Like the earlier Vedic mathematicians, but much more confidently, the Jains loved large numbers. Their cosmology talked of 2 588 years — that's an unimaginably large number with 174 digits. They calculated that the number of all human beings that had ever existed was 2 96, another humongous number. They thought of infinity as being of five types: "infinite in one direction, infinite in two directions, infinite in area, infinite everywhere and perpetually infinite".

The classical age of Indian mathematics is usually said to begin from 500 AD because of one man — Aryabhata. Heading a school of astronomy and mathematics in Kusumpura (probably near Patna) he made a remarkable break from the past, replacing the superstitious theories around eclipses with modern ones, introducing trigonometry and solutions to indeterminate equations. Several mathematicians flourished in Kusumpura, continuing Aryabhata's work. A similar school developed in Ujjain where, in 700 AD, came another Titan, Brahmagupta, who introduced negative numbers and developed the use of zero. In these schools, and in certain families, mathematics was used for astronomical calculations. This led to much devotion to the subject as superior techniques guaranteed better grasp over the stars.

Another school of the classical age — the Kerala-based mathematician-astronomers — flourished from the 14th century onwards, after Madhava of Cochin. He is credited with having nearly worked out a system of calculus, as also the Taylor series rediscovered in England 300 years later. The classical age is said to have lasted till 1200 AD.

Passing through the Mughal period mathematicians like Kamalakara and Sawai Jai Singh's chief mathematician Jagannath Samrat, the modern period has seen the emergence of Srinivasa Ramanujan, considered to be a genius.

Despite this rich tradition of mathematics, and despite the worldwide resurgence of interest in the subject driven by its applications in information technology, India appears to be fast losing interest in mathematical research. A survey of mathematical papers published by Subbaiah Arunachalam showed that India's contribution to total mathematical research papers published in the world was 2. 3 per cent in 1984 and 2. 1 per cent in 2000. Experts say that the main reason behind this is lack of motivated and qualified teachers at the higher education level.

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