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, (18771947), 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 singleminded 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
middlelevel 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 wellwishers, 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 (18851977)
to come and examine the theorems. After examining closely they realized
the importance of Ramanujan’s work. As C.P. Snow recounted,
‘before midnight 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 (18721970) 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 wellwishers
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 wellknown 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 (170789);
geometric theory  which uses such geometrical methods of analysis
as Cartesian coordinates, 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 mocktheta
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
