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Srinivasa Ramanujan
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A Remarkable Mathematical Genius Dr Subodh Mahanti
Ramanujans 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. Ramanujans example stirred the imagination of manyparticularly 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 Ramanujans 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 Ramanujans 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 Ramanujans 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 mothers parents lived. After one year he was brought to his fathers 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 Kumbakonams Government College as F.A. student. He was awarded a scholarship. However, after school, Ramanujans total concentration was focussed on mathematics. The result was that his formal education did not continue for long. He first failed in Kumbakonams Government College. He tried once again in Madras from Pachaiyappas 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 Carrs 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 Ramanujans
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 Carrs 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 Carrs
Synopsis. And as E. H. Neville, an English mathematician, wrote : In
proving one formula, as he worked through Carrs synopsis, he discovered
many others, and he began the practice of compiling a notebook. Between
1903 and 1914 he had three notebooks.
 Komalatammal: Ramanujan's mother 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.
 Godfrey Harold Hardy
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 worlds 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.
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 Ramanujans 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.
 Bertrand Russell 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.
 A postage stamp released by Govt.of India in 1962 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
Ramanujans 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.
 Evariste Galois 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 Ramanujans 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.
Subrahmanyan Chandrasekhar Much of Ramanujans 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.
 The statue of Ramanujan made by the American Architect Paul Grandlund 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.
 Leonhard Euler 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 Ramanujans work. Opinions may differ as to the importance of Ramanujans 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 denyprofound and invincible originality. The Norwegian mathematician Atle Selberg, one of the great number theorists of this century wrote : Ramanujans 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 Ramanujans 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 we may cite the cases of Niels Henrik Abel (1809-29)
and Evarista Galois (1811-32). Abel solved one of the great mathematical
problems of his day - finding a general solution for a class equations
called quintiles. Abel solved the problem by proving that such a solution
was impossible. Galois pioneered the branch of modern mathematics known
as group theory. What is important is that we should recognise the greatness
of such people and take inspiration from their work.
 Niels Henrik Abel Even after more than 80 years of the death of Ramanujan the situation is not very different as far the rigidity of the education system. Today also a Ramanujan is not likely to get a chance to pursue his career. This situation remains very much similar as described by JBS Haldane (1982-1964), a British born geneticist and philosopher who spent last part of his life in India. Haldane said : Today in India Ramanujan could not get even a lectureship in a rural college because he had no degree. Much less could he get a post through the Union Public Service Commission. This fact is a disgrace to India. I am aware that he was offered a chair in India after becoming a Fellow of the Royal Society. But it is scandalous that Indias great men should have to wait for foreign recognition. If Ramanujans work had been recognised in India as early it was in England, he might never have emigrated and might be alive today. We can cast the blame for Ramanujans non-recognition on the British Raj. We cannot do so when similar cases occur today... Nehrus statement given at the beginning is very much valid even today.
And for these very reasons the story of Ramanujan should be told and
retold to our younger people particularly to those who aspire to do something
extraordinary but feel dejected under the prevailing circumstances.
And in this connection it is worthwhile to remember what Chandrasekhar
had to say: I can recall the gladness I felt at the assurance that one
brought up under circumstances similar to my own could have achieved what
I could not grasp. The fact that Ramanujans early years were spent in
a scientifically sterile atmosphere, that his life in India was not without
hardships that under circumstances that appeared to most Indians as nothing
short of miraculous, he had gone to Cambridge, supported by eminent mathematicians,
and had returned to India with very assurance that he would be considered,
in time as one of the most original mathematicians of the century these
facts were enough, more than enough, for aspiring young Indian students
to break their bands of intellectual confinement and perhaps soar the
way what Ramanujan had.
 Janaki : Ramanujan's wife As someone has written Ramanujan did mathematics for its own sake, for thrill that he got in seeing and discovering unusual relationships between various mathematical objects. Today Ramanujans work has some applications in particle physics or in the calculation of pi up to a very large number of decimal places. His work on Riemans Zeta Function has been applied to the pyrometry, the investigations of the temperature of furnaces. His work on the Partition Numbers resulted in two applications new fuels and fabrics like nylons. But then highlighting the importance of the application side Ramanujans work is really not very important. Ramanujan died of tuberculosis in Kumbakonam on April 26, 1920. He was only 32 years old. It was always maths ... Four days before he died he was scribbling, said Janaki, his wife. The untimely death of Ramanujan was most unfortunate particularly so when we take into account the circumstances under which he died. As Times Magazine rightly wrote: There is something peculiarly sad in the spectacle of genius dying young, dying with the first sweets of recognition and success tasted, but before the full recognition of powers that lie within. The only Ramanujan Museum in the country, founded by Shri P. K. Srinivasan, a mathematics teacher, operates from March 1993 in the Avvai Academy, Royapuram, Madras. The achievement of Ramanujan was so great that those who can really grasp the work of Ramanujan may doubt that so prodigious a feat had ever been accomplished in the history of thought. Further Reading1. Ramanujan: Twelve Lectures on the Subjects Suggested by His Life and Work by G. H. Hardy, Chelsea Publishing Co, New York, 1940. 2) The Man Who Knew Infinity : A Life of the Genius Ramanujan by R. Kanigel, Abacus Books, London, 1992. 3) Ramanujans Notebooks (Part I&II) by B.C. Berndt Springer, New York, 1985-1989. 4) Ramanujan:The Man and the Mathematician by S.R. Ranganathan, Asia Publishing House, Bombay, 1967. 5) Srinivasa Ramanujan : A Mathematical Genius by K. Srinivasa Rao; East West Books (Madras) Pvt. Ltd. 1998. 6) Srinivasa Ramanujan, Suresh Ram, National Book Trust India, 1989. 7) Ganit Jagater Bismay Ramanujan by Satyabachi Sar, Gyan Bichitra Prakashani, Agartala, 2000. A well-written book in Bengali.
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