Srinivasa Ramanujan - His life and his genius

Srinivasa Ramanujan - His life and his genius

Prof. V. Krishnamurthy

(Expository address delivered on Sep.16, 1987 at Visvesvarayya Auditorium as part of the celebrations of Ramanujan Centenary by the IISC, Bangalore)

Srinivasa Ramanujan 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 namely, Town High School, Kumbakonam. At the age of 12, he borrowed Loney's Trigonometry, Part II, from a student of the B.A. Class, who was his neighbour. That student was amazed to find that this young boy, about 7 years his junior, had not only finished mastering the book at one reading but he had taught himself to do every problem in it. The book, though called Trigonometry had some of the advanced mathematics of that time in it. This book was Ramanujan's first contact with these advanced topics. Another book, called Carr's Synopsis, also passed through the hands of Ramanujan in those years and thus created for itself an imperishable record in history. Ramanujan was captivated by its contents. It brought forth all his powers, not because it was a great book, but because it was a compilation of about 6000 mathematical statements for which there were only sketchy proofs. The challenge was irresistible for Ramanujan. He started working out the proofs of results there in his own way out of his own thinking. Thus he became a mathematician before anybody could think of training him. Not only could he supply proofs to innumerable results there but he proceeded further to improve them and create his own theorems and results. He began writing theorem after theorem on the pages of quarto notebooks which are today collectively called Ramanujan's notebooks. This writing continued for almost ten years. Magic squares, Continued fractions, hypergeometric series, properties of prime numbers and composite numbers, partitions of numbers, elliptic integrals and many other such regions of mathematics were explored by him. Amounting to about 600 pages of writing, they contain about 3000 and odd theorems. He had to do all this by discovering much of it de novo, because his immediate neighbourhood contained no person or book knowledgeable in these areas.

He passed the matriculation examination in first class and this earned him a scholarship in the FA class at Govt. College, Kumbakonam - today called Kanitha Medhai Ramanujan College. His subjects were then: English, Mathematics, Physiology, Roman and Greek History and Sanskrit. But Mathematics absorbed all his time, interest and energy that he neglected the other subjects and thus failed in the annual exam. He lost his scholarship. He left Kumbakonam and got himself lost somewhere in Andhra. He came back to Govt. College Kumbakonam after a year but could not get the necessary attendance certificate. Later he completed the second year FA in Pachaiyappa's college, Madras and sat for the examination in December 1907. But again he failed in subjects other than mathematics. But his own personal researches continued on a slate and the writings in the notebooks continued.

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, Sir Francis Spring, also took interest in him. The clerk in the Madras Port Trust Office became the talk in the academic circles of Madras. Several attempts were made to get for him, who was not even an FA, a regular scholarship from the University of Madras. Mr. Ramachandra Rao, Prof. Griffith of the Madras Engg. College, Dr. Gilbert Walker, a senior wrangler and then Head of the India Meteorological Dept., Prof. B. Hanumantha Rao, Chairman of the Board of Studies of the University of Madras, Justice P.R. Sundaram Iyer, all had a role to play in the succession of events that finally brought Ramanujan to the University of Madras as a Research Scholar (the first such appointment in the history of the Univeristy) at the age of 26 on a stipend of Rs.75 per month.

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 Cambridge, a world-famous mathematician. His first letter to Prof. Hardy made history. It contained as an attachment, 120 theorems (without proofs), all originally discovered by him. Prof. Hardy and his colleague Prof. Littlewood spent two to three hours together on the letter. Several of the results completely floored the two experts. Even assuming some of them were wrong, they could see the serious mathematical mind behind it all. They decided the author was not a crach but a genius. History was made in that decision. They decided to support Ramanujan's mathematical work. Their efforts to bring him to England finally materialised in March 1914. Ramanujan was 27. From now on the mathematical career of Ramanujan is full of thrilling discoveries, exciting contacts with famous mathematicians, and epoch-making publications.

The Hindu of Madras in its edition of May 13, 1914, reported as follows: Mr. S. Ramanujan of Madras, whose work in Higher Mathematics has excited the wonder of Cambridge, is now in residence at Trinity. He will read mainly with the two Fellows of the College - Mr. Hardy and Mr. Littlewood. They are going through masses of work he has already done, and are making some surprising discoveries in it!

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 Cambridge when Ramanujan joined it. As a senior student Mahalanobis used to visit Ramanujan. One day Ramanujan invited him for lunch in his room and Mahalanobis arrived when Ramanujan was still busy at the cooking. Mahalanobis pulled a chair, sat near where Ramanujan was cooking, and began reading the Strand Magazine. That was First World War Time (1914). The magazine contained a problem in its puzzle section and it attracted Mahalanobis. It related to two officers who were billeted in Paris in two houses in the same street. The door numbers, the problem went on, were related by a mathematical expression and the problem was to find the door numbers. Mahalanobis could easily solve the problem in a few minutes by trial and error and he wanted to share his excitement with Ramanujan. So Mahalanobis says, while Ramanujan was still stirring the pan, 'Ramanujan, here is a problem for you'. Ramanujan says, 'Tell me the problem'. Mahalanobis describes the problem and no sooner than he has finished describing it, Ramanujan, while still stirring the pan, says, 'Take down the solution'. And Ramanujan dictates a Continued Fraction.

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 Cambridge produced from Ramanujan, 37 research papers, a few of them jointly with Hardy. Here is an anecdote pertaining to the longest of those research papers. It is a 64-page paper entitled 'Highly Composite Numbers' and it was published in the Proceedings of the London Mathematical Society in 1915.

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 Cambridge. W.N. Bailey and S. Pollard were two such friends who later became well-known mathematicians. Bailey reports of this incident: 'He (Ramanujan) started at the beginning and quickly turned over the pages as he explained the ideas and the arguments very briefly. Pollard wrestled with the argument and was rewarded by a severe headache. I gave up the struggle earlier and was not troubled by a headache'!

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 University of Madras rose to the occasion and made a permanent provision for Ramanujan by granting him an unconditional allowance of 250 pounds a year for five years from April 1, 1919 the date of expiry of the overseas scholarship that he was then drwing. The University was also to be moved by Prof. Littlehailes, the new Director of Public Instruction for the creation of a University Professorship of Mathematics and Ramanujan to be offered that professorship, but alas, fate decided otherwise. Ramanujan spent his fifth year abroad in nursing homes and sanatoria. He returned to India in April 1919 and continued to suffer his incurable illness. All the time his mind was totally absorbed in Mathematics and he continued his productive work. Thus arose the so-called Lost notebook of Ramanujan. His three notebooks have now been edited and published with commentaries in five volumes by Bruce Berndt in a marathon work extending for 12 years in the eighties and nineties. The 'Lost Notebook' has been published by the Narosa publishing house in 1988.

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 n^{5 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, Davenport and Salis brought it down to 5 5/6. Rankin in 1939 brought it down to 5 4/5. That is where it stood till 1974. The tantalizing 5 1/2 which Ramanujan had conjectured was still a dream. But then, in 1974 a French mathematician, Deligne, while working in some high-flown areas of Modern Algebraic Geometry, a subject which was non-existent before 1950, using some powerful techniques from that area, was able to bring the elusive number to 5 1/2 and thus Ramanujan's conjecture stood vindicated and proved formally. Deligne actually obtained Ramanujan's conjecture as a special case of conceptually advanced results in algebraic geometry. For this work Deligne was awarded the Fields Medal (the analogue of a Nobel prize) in 1978. In this work of Deligne, in addition to the mathematicians named above, other great mathematicians like Petersson, Selberg, Andre Weil, Serre and Eichler had paved the way for Deligne's result. Can mankind ever guess how Ramanujan ever hit upon the number 5 1/2, for his Tau-function, which finally needed in its proof, the significant work done by such a galaxy of mathematicians, not to speak of the entire ammunition from the 20^th century armoury of modern algebraic geometry, a subject which was not even born in Ramanujan's time?

Here is another which is more astounding. The reduction process of reducing all matter to some primitives started in the 17^th 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 21^st 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?

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