Kuttaka: A case for learning Indian Mathematics

Note- Here's the link to Badri's course starting soon.

Ancient Indian Mathematicians had developed many aspects of mathematics over a long period, from at least 500 BCE onwards. India’s greatest invention was the place value number system and inclusion of zero in the place value. French mathematician and polymath Laplace said, the Indian place value system was “a profound and important idea which appears so simple to us now that we ignore its true merit, but its very simplicity, the great ease which it has lent to all computations, puts our arithmetic in the first rank of useful inventions, and we shall appreciate the grandeur of this achievement when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest minds produced by antiquity.”

The place value system allowed the Indians to deal with very large numbers and they made giant strides in the fields of Astronomy and Trigonometry. But while dealing with Astronomy, they came across a peculiar problem which today we call as “linear indeterminate equations in two variables”. The challenge was to find integer solutions to these problems.

These problems were not new. The Greeks had certainly known about this problem. The Chinese, in the 3rd century CE had posed similar problems, albeit with small numbers and had given solutions to them. These are known as Chinese Remainder problems. Typically, they are posed this way: “Find a number, which when divided by 5 leaves a remainder 3 and when divided by 7 leaves a remainder 1”. But the Chinese had not come up with a generalised way to solve these problems, particularly when it involved very large numbers.

Aryabhata of Kusumapura (also denoted as Aryabhata I) was an outstanding mathematician and astronomer who lived in the 5th & 6th century CE. Before his time, Indian mathematics was fairly well developed. But Aryabhata’s work called Aryabhateeyam is the first work which is assignable to a named author and a historical figure as opposed a mythical sage. It is also a work which we have received in full and has been embellished by many commentaries by other great mathematicians who have added significantly to the short sutra style of verses of Aryabhata.

If I tell you that much of the high school level Mathematics that we study today is right out of Aryabhata’s work, you will not believe me. Aryabhata introduces the decimal numbering system with place values, defines operations such as addition, subtraction, multiplication and division on whole numbers, then moves to fractions and defines all these operations, shows how problems with ratios are solved, deals with interest and principal problems, shows how squares and cubes, square roots and cube roots are calculated, deals with some 2D and 3D geometrical shapes and gives formulas to calculate perimeter, area and volume (and while doing the same makes a couple of big bloopers - errors which are rectified only in the 12th century CE by Bhaskara II), deals with problems on time and work, takes care of arithmetic progression, sum of successive natural numbers, the sum of their squares and cubes, in the process quietly slipping the solution to the quadratic equation - all this in about 5-6 pages of cramped, terse Sanskrit poetry.

That is work for many years of late primary and early secondary schooling. In fact in schools, we hardly touch cube root these days and probably square root calculation has also gone out of the window.

After dealing with all these, at the end of his Arithmetic section and before embarking on Astronomy and Trigonometry, he takes up the linear indeterminate equations in two variables.

There is no closed-form solution to this problem. In the space of just four lines, Aryabhata gives an elegant recursive algorithm to solve the problem. Any computer programmer would be proud of this algorithm today. It is so comprehensive and elegant, there is no way it can be bettered to this day. A few improvements have been made, by a namesake, Aryabhata II and later Bhaskara II to this algorithm but they are marginal.

Then Aryabhata takes up angles and sine function in the first quadrant and comes up with three different ways in which sine can be calculated for various angles - essentially coming up with a lookup table and a rational approximation to the irrational, in fact, the transcendental function.

The algorithm developed by Aryabhata for finding integral solutions to indeterminate equations in two variables is known to his successors as Kuttaka - a Sanskrit word that means pulverising, which is an apt word for his algorithm. Aryabhata takes up a larger Kuttaka problem and transforms it to a smaller Kuttaka problem in an ingenious way. He does this repeatedly which eventually leads to a simple equation for which an answer can be found easily. Then you have to climb up to arrive at the actual solution.

When the Arab mathematicians learnt Sanskrit and translated a lot of Indian work, they did not translate these algebraic manipulations because they couldn’t understand cryptic Sanskrit. It has taken us a lot of effort since the 18th century to translate and interpret correctly many of these algorithms and get them translated into English.

Nearly a millennium after Aryabhata presenting his algorithm, in the 17th and 18th century Europe many great mathematical minds started working on this problem. Great mathematicians such as Bachet and Euler rediscovered the exact same method.

In many cases, Indian methods have been replaced by an equivalent or superior European method. But there are quite a few places where the Indian methods continue to be better even to this day. It is for this reason that the students of mathematics should learn Indian mathematics. There is a feast of mathematical insight waiting for them.