Srinivasa Ramanujan - His life and his genius
(Expository address delivered on Sep.16, 1987 at Visvesvarayya Auditorium as part of the celebrations of Ramanujan Centenary by the IISC, Bangalore)
was a mathematical genius not only of the twentieth century but for all time.
He was an enigma to his teachers even at school because of his prodigious
memory and unusual mathematical talent which began to show even before he was
ten years old. That was the age when he topped the whole district at the
Primary examination and this procured him a half-fee concession at school
He passed the
matriculation examination in first class and this earned him a scholarship in
the FA class at
In 1910 Ramanujan, now
23, heard of the Indian Mathematical
Society and its founder Prof. V. Ramaswami Iyer, a Deputy Collector. To
Ramaswamy Iyer goes the credit of being the first among those who recognised the mathematical
genius in Ramanujan and also acted on that recognition. Ramanujan was
introduced to Prof. Seshu Iyer, who introduced him to Dr. Ramachandra Rao,
Collector of Nellore. Ramachandra Rao
looked into his notebooks, and
also listened to him and decided that
Ramanujan was a remarkable man. Ramachandra Rao supported him for some time.
After a few months he accepted a clerk's job in the office of the Madras Port
Trust in March 1912. By this time the Chairman of the Madras Port Trust,
Ramanujan thus became
a professional mathematician and
remained as such for the rest of his life, which was not much, unfortunately, -
it was only seven years. Upon the suggestion of Prof. Seshu Iyer and others he
had then already started a correspondence with Prof. G.H. Hardy, a Fellow of
the Royal Society and Cayley lecturer in Mathematics at
The Hindu of Madras in its edition of May 13, 1914,
reported as follows: Mr. S. Ramanujan of
The common man who knows very little of mathematics may not be able to grasp the full impact of Ramanujan's genius and intuition for discovery and creativity except through statements of appreciation from others. Instead of quoting these appreciations and encomia we give below a few anecdotes and make them simple enough to be understood (at least partially) by the layman so that the latter can have a first-hand knowledge of the super-genius that Ramanujan was.
Dr. P. C. Mahalanobis, who later in Free India became
Nehru's right hand man of National Planning, was a student at
Here the reader must know what a continued fraction is. A simple infinite continued fraction, for example, is
1 + 1/(2 +1/(3 + 1/(4 +1/(5 + … ))) ….)
If you truncate it at each step, you get what are called the successive convergents of the continued fraction, thus:
1 ; 1 + 1/2 = 3/2 ; 1 + 1/(2 + 1/3) = 1 + 3/7 = 10/7 ;
1 + 1/(2 + 1/(3 + 1/4)) = 1 + 1 /(2 + 4/13) = 1 + 13/30 = 43/30 ;
and so on, endlessly.
Now let us try to understand Ramanujan's flash of an answer to the problem in Strand Magazine, though we do not have the exact problem which was the subject of conversation between Mahalonobis and Ramanujan, let us concoct a simplified version of it, for the lay reader to understand it. The problem is to find the door numbers of the two houses, call these numbers x and y, such that a mathematical relation, say
x2 - 10 y2 = +1 or -1
is satisfied. Mahalanobis tries it and comes out with the answer x = 3 and y = 1 in no time. Now Ramanujan, as soon as he hears the statement of the problem, dictates the following continued fraction:
3 + 1/(6 + 1/(6 + 1/(6 + 1/(6 + … ))) … )
Here 3/1 is the first convergent. The numbers 3 and 1 are the first answers to x and y which Mahalanobis has himself discovered by trial and error. Now Ramanujan's continued fraction answer not only gives this first elementary answer but gives an infinity of answers on the assumption that the street may have an infinite number of houses in it! Thus the second convergent, 3 + 1/6 = 19/6 gives the pair, 19 and 6 for x and y. This can be verified to be a true answer.
192 - 10 x 62 = + 1.
The third convergent, 3 + 1/(6+1/6) = 3 + 6/37 = 117 / 37 gives the answer x = 117 and y = 37 which also satisfies the relation specified by the problem. The fourth convergent can be calculated to be 721/228 and this pair x = 721 and y = 228 can also be verified to satisfy the relation. And so on it goes. Every convergent gives a pair of numbers, which constitute an answer. The greatness of Ramanujan was that as soon as he heard of the problem, without effort he decided that the answer could be given in the form of a continued fraction and immediately gave the continued fraction, which not only solved the problem but gave an infinity of solutions to it!
Four years at
A prime number is a number p which is exactly divisible by no number except 1 and p. So 1 and p are the only (two) divisors of p. 1,2,3,5,7,11,13,17,19,23, … are prime numbers. A composite number is one which has not only 1 and itself as a divisor, but also at least one more divisor. The number 4 is divisible by 1, 2 and 4. It has three divisors. So 4 is a composite number. 4, 6, 8,9,10,12,14,15,16,18,20, … are all composite numbers. A highly composite number is a composite number which has more divisors than all its predecessors. 4 is one such. Because, the number of divisors of 4 is 3 and no number less than 4 has so many divisors. The list of highly composite numbers goes as follows:
4, 6, 12, 24, 36, 60, 120, …
Ramanujan has analysed the properties of highly
composite numbers very deeply and produced this paper. When he was writing this
paper for publication, that was in 1914, he showed the manuscript to some of
the mathematics seniors in
In 1918 Ramanujan was elected Fellow of the Royal Society and
in the same year was also elected Fellow of Trinity College, both honours
coming as a first to any Indian. The
Ramanujan's innovations have not been surpassed either before or after him. His results continue to have unforeseen impact in fields of mathematics which were not even born in his time. Here are two unparalleled anecdotes in this context:
One of the creations of Ramanujan, which goes back to his notebooks is the tau-function. The function is so important it is nowadays called Ramanujan's tau-function. It is a function of the variable n, but the definition is rather complicated for a non-mathematician; so we will not attempt it here. But what is relevant to our story is the question of how big the function grows for larger and larger values of n. Ramanujan himself conjectured that
Tau(n) £ K n5 1/2 + e
But he offered no proof. So mathematician after
mathematician tried to prove it. Hardy could prove the result with 8 in the
place of 5 1/2. That was certainly a
weaker result than Ramanujan's, but at least it had a proof. So other people
went on trying to bridge the gap between 8 and 5 1/2. The problem was to
estimate the growth of the function more correctly and bring it down to 5 1/2 from 8. Ramanujan himself now
offered a proof to bring it down to 7. Hardy and Littlewood brought it down to
6 in 1918. Each such effort was a major effort and resulted in an important
technical paper. Kloosterman in 1927 brought it down to 5 7/8. In 1933,
Here is another which is more astounding. The reduction process of reducing all matter to some primitives started in the 17th century and is still going on. The list of elementary and sub-elementary particles seems to be a receding horizon. We have photons for conveying electro-magnetic force, gravitons whose exchange is what causes gravitation, quarks which combine in threes to make protons and neutrons and gluons which are messenger particles. Further research has led scientists to believe that the ultimate is what they call a string as tiny and probably tinier than a length of 1/(10 raised to the power 31) millimetres with no mass but with intrinsic tension and the capacity to oscillate as well as vibrate. The electron is one mode of vibration of the string, the quark is another mode of vibration and graviton, still another. Interaction between particles is a matter of strings breaking or joining together or an open string forming a loop. The latest in this list is a superstring. The deeper the physicists go into these the more they find many advanced tools of mathematics are needed, first for the formulation of these objects, and secondly for the understanding of their behaviour. It appears the strings live in a world of ten dimensions. The physicists are comfortable with four dimensions - three of space and one of time. But the extra dimensions that the strings need seem to be lost in a peculiar curling up which is where they need all the mathematics in the world. Much of this mathematics is awaiting the 21st century for its development. There are 26-dimensional models of string theory and we are told Ramanujan's work is exactly what is needed here!
It is no surprise therefore that as years pass by, his fame increases and the influence of his work spreads into new areas. Valmiki was eulogised by Brahma the Creator on his famous epic with the words: So long as mountains and rivers survive in this world, that long this Ramayana will survive. We can also say of Ramanujan: So long as there is mathematical science and scientific activity in the world, that long we will hear of Ramanujan and his genius. When comes such another?