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Srinivasa Ramanujan
A Remarkable Mathematical Genius
 
 
 
 Dr Subodh Mahanti

Ramanujan’s brief life and death are symbolic of conditions in India. Of our millions how few get any education at all; how many live on the verge of starvation.

Jawaharlal Nehru in his Discovery of India


Sheer intuitive brilliance coupled to long, hard hours on his slate made up for most of his educational lapse. This ‘poor and solitary Hindu pitting his brains against the accumulated wisdom of Europe’ as Hardy called him, had rediscovered a century of mathematics and made new discoveries that would captivate mathematicians for next century.

Robert Kanigel in The Man who Knew Infinity : A Life of the Genius Ramanujan



I still say to myself when I am depressed and find myself forced to listen to pompous and tiresome people, ‘Well, I have done one thing you could never have done, and that is to have collaborated with both Littlewood and Ramanujan on something like equal terms’.

Godfrey Harold Hardy

“Ramanujan’s life”, as Robert Kanigel, the author of a marvellous biography of Ramanujan, wrote, “can be made to serve as parable for almost any lesson you want to draw from it.” Ramanujan’s example stirred the imagination of many–particularly that of mathematicians. Thus, Subrahmanyan Chandrasekhar (1910-95), the Indian born astrophysicist, who got Nobel Prize in 1983, said : “I think it is fair to say that almost all the mathematicians who reached distinction during the three or four decades following Ramanujan were directly or indirectly inspired by his example.” Even those who do not know about Ramanujan’s work are bound to be fascinated by his life. As Kanigel wrote: “Few can say much about his work, and yet something in the story of his struggle for the chance to pursue his work on his own terms compels the imagination, leaving Ramanujan a symbol for genius, for the obstacles it faces, for the burden it bears, for the pleasure it takes in its own existence.”


Srinivasa Ramanujan

 

Ramanujan’s life is full of strange contrasts. He had no formal training in mathematics but yet “he was a natural mathematical genius, in the class of Gauss and Euler.” Probably Ramanujan’s life has no parallel in the history of human thought. Godfrey Harold Hardy, (1877-1947), who made it possible for Ramanujan to go to Cambridge and give formal shape to his works, said in one of his lectures given at Harvard Universty (which later came out as a book entitled Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work): “I have to form myself, as I have never really formed before, and try to help you to form, some of the reasoned estimate of the most romantic figure in the recent history of mathematics, a man whose career seems full of paradoxes and contradictions, who defies all cannons by which we are accustomed to judge one another and about whom all of us will probably agree in one judgement only, that he was in some sense a very great mathematician.”

Srinivasa Ramanujan Iyengar (best known as Srinivasa Ramanujan) was born on December 22, 1887, in Erode about 400 km from Chennai, formerly known as Madras where his mother’s parents lived. After one year he was brought to his father’s town, Kumbakonam. His parents were K. Srinivasa Iyengar and Komalatammal. He passed his primary examination in 1897, scoring first in the district and then he joined the Town High School. In 1904 he entered Kumbakonam’s Government College as F.A. student. He was awarded a scholarship. However, after school, Ramanujan’s total concentration was focussed on mathematics. The result was that his formal education did not continue for long. He first failed in Kumbakonam’s Government College. He tried once again in Madras from Pachaiyappa’s College but he failed again.

While at school he came across a book entitled A Synopsis of Elementary Results in Pure and Applied Mathematics by George Shoobridge Carr. The title of the book does not reflect its contents. It was a compilation of about 5000 equations in algebra, calculus, trigonometry and analytical geometry with abridged demonstrations of the propositions. Carr had compressed a huge mass of mathematics that was known in the late nineteenth century within two volumes. Ramanujan had the first one. It was certainly not a classic. But it had its positive features. According to Kanigel, “one strength of Carr’s book was a movement, a flow to the formulas seemingly laid down one after another in artless profusion that gave the book a sly seductive logic of its own.” Thisbook had a great influence on Ramanujan’s career. However, the book itself was not very great. Thus Hardy wrote about the book: “He (Carr) is now completely forgotten, even in his college, except in so far as Ramanujan kept his name alive”. He further continued, “The book is not in any sense a great one, but Ramanujan made it famous and there is no doubt it influenced him (Ramanujan) profoundly”. We do not know how exactly Carr’s book influenced Ramanujan but it certainly gave him a direction. `It had ignited a burst of fiercely single-minded intellectual activity’. Carr did not provide elaborate demonstration or step by step proofs. He simply gave some hints to proceed in the right way. Ramanujan took it upon himself to solve all the problems in Carr’s Synopsis. And as E. H. Neville, an English mathematician, wrote : “In proving one formula, as he worked through Carr’s synopsis, he discovered many others, and he began the practice of compiling a notebook.” Between 1903 and 1914 he had three notebooks.

While Ramanujan made up his mind to pursue mathematics forgetting everything else but then he had to work under extreme hardship. He could not even buy enough paper to record the proofs of his results. Once he said to one of his friends, “when food is problem, how can I find money for paper? I may require four reams of paper every month.” In fact Ramanujan was in a very precarious situation. He had lost his scholarship. He had failed in examination. What is more, he failed to prove a good tutor in the subject which he loved most.

At this juncture, Ramanujan was helped by R. Ramachandra Rao, then Collector of Nellore. Ramchandra Rao was educated at Madras Presidency College and had joined the Provincial Civil Service in 1890. He also served as Secretary of the Indian Mathematical Society and even contributed solution to problem posed in its Journal. The Indian Mathematical Society was founded by V. Ramaswami Iyer, a middle-level Government servant, in 1906. Its Journal put Ramanujan on the world’s mathematical map. Ramaswami Iyer met Ramanujan sometime late in 1910. Ramaswami Iyer gave Ramanujan notes of introduction to his mathematical friends in Chennai (then Madras). One of them was P.V. Seshu Iyer, who earlier taught Ramanujan at the Government College. For a short period (14 months) Ramanujan worked as clerk in the Madras Port Trust which he joined on March 1, 1912. This job he got with the help of S. Narayana Iyer.

Ramanujan’s name will always be linked to Godfrey Harold Hardy, a British mathematician. It is not because Ramanujan worked with Hardy at Cambridge but it was Hardy who made it possible for Ramanujan to go to Cambridge. Hardy, widely recognised as the leading mathematician of his time, championed pure mathematics and had no interest in applied aspects. He discovered one of the fundamental results in population genetics which explains the properties of dominant, and recessive genes in large mixed population, but he regarded the work as unimportant.

Encouraged by his well-wishers, Ramanujan, then 25 years old and had no formal education, wrote a letter to Hardy on January 16, 1913. The letter ran into eleven pages and it was filled with theorems in divergent series. Ramanujan did not send proofs for his theorems. He requested Hardy for his advice and to help getting his results published. Ramanujan wrote : “I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras on a salary of only £ 20 per annum. I have had no university education but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at mathematics. I have not trodden through the conventional regular course which is followed in a university course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as “startling“… I would request you to go through the enclosed papers. Being poor, if you are convinced that there is anything of value I would like to have my theorems published. I have not given the actual investigations nor the expressions that I get but I have indicated the lines on which I proceed. Being inexperienced I would very highly value any advice you give me “. The letter has become an important historical document. In fact, ‘this letter is one of the most important and exciting mathematical letters ever written’. At the first glance Hardy was not impressed with the contents of the letter. So Hardy left it aside and got himself engaged in his daily routine work. But then he could not forget about it. In the evening Hardy again started examining the theorems sent by Ramanujan. He also requested his colleague and a distinguished mathematician, John Edensor Littlewood (1885-1977) to come and examine the theorems. After examining closely they realized the importance of Ramanujan’s work. As C.P. Snow recounted, ‘before mid-night they knew and knew for certain’ that the writer of the manuscripts was a man of genius’. Everyone in Cambridge concerned with mathematics came to know about the letter. Many of them thought `at least another Jacobi in making had been found out’. Bertrand Arthur William Russell (1872-1970) wrote to Lady Ottoline Morell. “I found Hardy and Littlewood in a state of wild excitement because they believe, they have discovered a second Newton, a Hindu Clerk in Madras … He wrote to Hardy telling of some results he has got, which Hardy thinks quite wonderful.”

Fortunately for Ramanujan, Hardy realised that the letter was the work of a genius. In the next three months Ramanujan received another three letters from Hardy. However, in the beginning Hardy responded cautiously. He wrote on 8 February 1913. To quote from the letter. “I was exceedingly interested by your letter and by the theorems which you state. You will however understand that, before I can judge properly of the value of what you have done it is essential that I should see proofs of some of your assertions … I hope very much that you will send me as quickly as possible at any rate a few of your proofs, and follow this more at your leisure by more detailed account of your work on primer and divergent series. It seems to me quite likely that you have done a good deal of work worth publication; and if you can produce satisfactory demonstration I should be very glad to do what I can to secure it” .

In the meantime Hardy started taking steps for bringing Ramanujan to England. He contacted the Indian Office in London to this effect. Ramanujan was awarded the first research scholarship by the Madras University. This was possible by the recommendation of Gilbert Walker, then Head of the Indian Meteorological Department in Simla. Gilbert was not a pure mathematician but he was a former Fellow and mathematical lecturer at Trinity College, Cambridge. Walker, who was prevailed upon by Francis Spring to look through Ramanujan’s notebooks wrote to the Registrar of the Madras University : “The character of the work that I saw impressed me as comparable in originality with that of a Mathematical Fellow in a Cambridge College; it appears to lack, however, as might be expected in the circumstances, the completeness and precision necessary before the universal validity of the results could be accepted. I have not specialised in the branches of pure mathematics at which he worked, and could not therefore form a reliable estimate of his abilities, which might be of an order to bring him a European reputation. But it was perfectly clear to me that the University would be justified in enabling S. Ramanujan for a few years at least to spend the whole of his time on mathematics without any anxiety as to his livelihood.”

Ramanujan was not very eager to travel abroad. In fact he was quite apprehensive. However, many of his well-wishers prevailed upon him and finally Ramanujan left Madras by S.S. Navesa on March 17, 1914. Ramanujan reached Cambridge on April 18, 1914. When Ramanujan reached England he was fully abreast of the recent developments in his field. This was described by J. R. Newman in 1968: “Ramanujan arrived in England abreast and often ahead of contemporary mathematical knowledge. Thus, in a lone mighty sweep, he had succeeded in recreating in his field, through his own unaided powers, a rich half century of European mathematics. One may doubt whether so prodigious a feat had ever been accomplished in the history of thought.”

Today it is simply futile to speculate about what would have happened if Ramanujan had not come in contact with Hardy. It could happen either way. But then Hardy should be given due credit for recognizing Ramanujan’s originality and helping him to carry out his work. Hardy himself was very clear about his role. “Ramanujan was”, Hardy wrote, “my discovery. I did not invent him — like other great men, he invented himself — but I was the first really competent person who had the chance to see some of his work, and I can still remember with satisfaction that I could recognize at once what I treasure I had found.”

It may be noted that before writing to Hardy, Ramanujan had written to two well-known Cambridge mathematicians viz., H.F. Baker and E.W. Hobson. But both of them had expressed their inability to help Ramanujan.

Ramanujan was awarded the B.A. degree in March 1916 for his work on ‘Highly composite Numbers’ which was published as a paper in the Journal of the London Mathematical Society. He was the second Indian to become a Fellow of the Royal Society in 1918 and he became one of the youngest Fellows in the entire history of the Royal Society. He was elected “for his investigation in Elliptic Functions and the Theory of Numbers.” On 13 October 1918 he was the first Indian to be elected a Fellow of Trinity College, Cambridge.

Much of Ramanujan’s mathematics comes under the heading of number theory — a purest realm of mathematics. The number theory is the abstract study of the structure of number systems and properties of positive integers. It includes various theorems about prime numbers (a prime number is an integer greater than one that has not integral factor). Number theory includes analytic number theory, originated by Leonhard Euler (1707-89); geometric theory - which uses such geometrical methods of analysis as Cartesian co-ordinates, vectors and matrices; and probabilistic number theory based on probability theory. What Ramanujan did will be fully understood by a very few. In this connection it is worthwhile to note what Hardy had to say of the work of pure mathematicians: “What we do may be small, but it has certain character of permanence and to have produced anything of the slightest permanent interest, whether it be a copy of verses or a geometrical theorem, is to have done something beyond the powers of the vast majority of men.” In spite of abstract nature of his work Ramanujan is widely known.

Ramanujan was a mathematical genius in his own right on the basis of his work alone. He worked hard like any other great mathematician. He had no special, unexplained power. As Hardy, wrote: “I have often been asked whether Ramanujan had any special secret; whether his methods differed in kind from those of other mathematicians; whether there was anything really abnormal in his mode of thought. I cannot answer these questions with any confidence or conviction; but I do not believe it. My belief that all mathematicians think, at bottom, in the same kind of way, and that Ramanujan was no exception.”

Of course, as Hardy observed Ramanujan “combined a power of generalization, a feeling for form and a capacity for rapid modification of his hypotheses, that were often really startling, and made him, in his peculiar field, without a rival in his day.

Here we do not attempt to describe what Ramanujan achieved. But let us note what Hardy had to say about the importance of Ramanujan’s work. “Opinions may differ as to the importance of Ramanujan’s work, the kind of standard by which it should be judged and the influence which it is likely to have on the mathematics of the future. It has not the simplicity and the inevitableness of the greatest work; it would be greater if it were less strange. One gift it shows which no one will deny—profound and invincible originality.”

The Norwegian mathematician Atle Selberg, one of the great number theorists of this century wrote : “Ramanujan’s recognition of the multiplicative properties of the coefficients of modular forms that we now refer to as cusp forms and his conjectures formulated in this connection and their later generalization, have come to play a more central role in the mathematics of today, serving as a kind of focus for the attention of quite a large group of the best mathematicians of our time. Other discoveries like the mock-theta functions are only in the very early stages of being understood and no one can yet assess their real importance. So the final verdict is certainly not in, and it may not be in for a long time, but the estimates of Ramanujan’s nature in mathematics certainly have been growing over the years. There is doubt no about that.”

Often people tend to speculate what Ramanujan would have achieved if he had not died or if his exceptional qualities were recognised at the very beginning. There are many instances of such untimely death of gifted persons, or rejection of gifted persons by the society or the rigid educational system. In mathematics