Srinivasa Aiyangar
Ramanujan 


Born: 22 Dec 1887 in Erode, Tamil
Nadu , India Died: 26 April 1920 in Kumbakonam, Tamil Nadu ,
India Srinivasa Ramanujan was one of India's
greatest mathematical geniuses. He made substantial contributions to
the analytical theory of numbers and worked on elliptic functions,
continued fractions, and infinite series.
Ramanujan was born in his grandmother's house in Erode, a small
village about 400 km southwest of Madras. When Ramanujan was a year
old his mother took him to the town of Kumbakonam, about 160 km
nearer Madras. His father worked in Kumbakonam as a clerk in a cloth
merchant's shop. In December 1889 he contracted smallpox.
When he was nearly five years old, Ramanujan entered the primary
school in Kumbakonam although he would attend several different
primary schools before entering the Town High School in Kumbakonam in
January 1898. At the Town High School, Ramanujan was to do well in
all his school subjects and showed himself an able all round scholar.
In 1900 he began to work on his own on mathematics summing geometric
and arithmetic series. Ramanujan was shown how to
solve cubic equations in 1902 and he went on to find his own method
to solve the quartic. The following year, not knowing that the
quintic could not be solved by radicals, he tried (and of course
failed) to solve the quintic. It was in the Town
High School that Ramanujan came across a mathematics book by G S Carr
called Synopsis of elementary results in pure mathematics. This book,
with its very concise style, allowed Ramanujan to teach himself
mathematics, but the style of the book was to have a rather
unfortunate effect on the way Ramanujan was later to write down
mathematics since it provided the only model that he had of written
mathematical arguments. The book contained theorems, formulae and
short proofs. It also contained an index to papers on pure
mathematics which had been published in the European Journals of
Learned Societies during the first half of the 19th century. The
book, published in 1856, was of course well out of date by the time
Ramanujan used it. By 1904 Ramanujan had begun to
undertake deep research. He investigated the series (1/n) and
calculated Euler's constant to 15 decimal places. He began to study
the Bernoulli numbers, although this was entirely his own independent
discovery. Ramanujan, on the strength of his good
school work, was given a scholarship to the Government College in
Kumbakonam which he entered in 1904. However the following year his
scholarship was not renewed because Ramanujan devoted more and more
of his time to mathematics and neglected his other subjects. Without
money he was soon in difficulties and, without telling his parents,
he ran away to the town of Vizagapatnam about 650 km north of Madras.
He continued his mathematical work, however, and at this time he
worked on hypergeometric series and investigated relations between
integrals and series. He was to discover later that he had been
studying elliptic functions. In 1906 Ramanujan
went to Madras where he entered Pachaiyappa's College. His aim was to
pass the First Arts examination which would allow him to be admitted
to the University of Madras. He attended lectures at Pachaiyappa's
College but became ill after three months study. He took the First
Arts examination after having left the course. He passed in
mathematics but failed all his other subjects and therefore failed
the examination. This meant that he could not enter the University of
Madras. In the following years he worked on mathematics developing
his own ideas without any help and without any real idea of the then
current research topics other than that provided by Carr's book.
Continuing his mathematical work Ramanujan studied
continued fractions and divergent series in 1908. At this stage he
became seriously ill again and underwent an operation in April 1909
after which he took him some considerable time to recover. He married
on 14 July 1909 when his mother arranged for him to marry a ten year
old girl S Janaki Ammal. Ramanujan did not live with his wife,
however, until she was twelve years old. Ramanujan
continued to develop his mathematical ideas and began to pose
problems and solve problems in the Journal of the Indian Mathematical
Society. He devoloped relations between elliptic modular equations in
1910. After publication of a brilliant research paper on Bernoulli
numbers in 1911 in the Journal of the Indian Mathematical Society he
gained recognition for his work. Despite his lack of a university
education, he was becoming well known in the Madras area as a
mathematical genius. In 1911 Ramanujan approached
the founder of the Indian Mathematical Society for advice on a job.
After this he was appointed to his first job, a temporary post in the
Accountant General's Office in Madras. It was then suggested that he
approach Ramachandra Rao who was a Collector at Nellore. Ramachandra
Rao was a founder member of the Indian Mathematical Society who had
helped start the mathematics library. He writes in [30]:
A short uncouth figure, stout, unshaven, not over clean, with one
conspicuous featureshining eyes walked in with a frayed notebook
under his arm. He was miserably poor. ... He opened his book and
began to explain some of his discoveries. I saw quite at once that
there was something out of the way; but my knowledge did not permit
me to judge whether he talked sense or nonsense. ... I asked him what
he wanted. He said he wanted a pittance to live on so that he might
pursue his researches. Ramachandra Rao told him to
return to Madras and he tried, unsuccessfully, to arrange a
scholarship for Ramanujan. In 1912 Ramanujan applied for the post of
clerk in the accounts section of the Madras Port Trust. In his letter
of application he wrote [3]: I have passed the
Matriculation Examination and studied up to the First Arts but was
prevented from pursuing my studies further owing to several untoward
circumstances. I have, however, been devoting all my time to
Mathematics and developing the subject. Despite
the fact that he had no university education, Ramanujan was clearly
well known to the university mathematicians in Madras for, with his
letter of application, Ramanujan included a reference from E W
Middlemast who was the Professor of Mathematics at The Presidency
College in Madras. Middlemast, a graduate of St John's College,
Cambridge, wrote [3]: I can strongly recommend
the applicant. He is a young man of quite exceptional capacity in
mathematics and especially in work relating to numbers. He has a
natural aptitude for computation and is very quick at figure work.
On the strength of the recommendation Ramanujan was
appointed to the post of clerk and began his duties on 1 March 1912.
Ramanujan was quite lucky to have a number of people working round
him with a training in mathematics. In fact the Chief Accountant for
the Madras Port Trust, S N Aiyar, was trained as a mathematician and
published a paper On the distribution of primes in 1913 on
Ramanujan's work. The professor of civil engineering at the Madras
Engineering College C L T Griffith was also interested in Ramanujan's
abilities and, having been educated at University College London,
knew the professor of mathematics there, namely M J M Hill. He wrote
to Hill on 12 November 1912 sending some of Ramanujan's work and a
copy of his 1911 paper on Bernoulli numbers. Hill
replied in a fairly encouraging way but showed that he had failed to
understand Ramanujan's results on divergent series. The
recommendation to Ramanujan that he read Bromwich's Theory of
infinite series did not please Ramanujan much. Ramanujan wrote to E W
Hobson and H F Baker trying to interest them in his results but
neither replied. In January 1913 Ramanujan wrote to G H Hardy having
seen a copy of his 1910 book Orders of infinity. In Ramanujan's
letter to Hardy he introduced himself and his work [10]:
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'. Hardy, together with Littlewood,
studied the long list of unproved theorems which Ramanujan enclosed
with his letter. On 8 February he replied to Ramanujan [3], the
letter beginning: 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. Your results seem to me to fall into roughly three
classes: (1) there are a number of results that are already
known, or easily deducible from known theorems; (2) there
are results which, so far as I know, are new and interesting, but
interesting rather from their curiosity and apparent difficulty than
their importance; (3) there are results which appear to be
new and important... Ramanujan was delighted with
Hardy's reply and when he wrote again he said [8]:
I have found a friend in you who views my labours sympathetically.
... I am already a half starving man. To preserve my brains I want
food and this is my first consideration. Any sympathetic letter from
you will be helpful to me here to get a scholarship either from the
university of from the government. Indeed the
University of Madras did give Ramanujan a scholarship in May 1913 for
two years and, in 1914, Hardy brought Ramanujan to Trinity College,
Cambridge, to begin an extraordinary collaboration. Setting this up
was not an easy matter. Ramanujan was an orthodox Brahmin and so was
a strict vegetarian. His religion should have prevented him from
travelling but this difficulty was overcome, partly by the work of E
H Neville who was a colleague of Hardy's at Trinity College and who
met with Ramanujan while lecturing in India.
Ramanujan sailed from India on 17 March 1914. It was a calm voyage
except for three days on which Ramanujan was seasick. He arrived in
London on 14 April 1914 and was met by Neville. After four days in
London they went to Cambridge and Ramanujan spent a couple of weeks
in Neville's home before moving into rooms in Trinity College on 30th
April. Right from the beginning, however, he had problems with his
diet. The outbreak of World War I made obtaining special items of
food harder and it was not long before Ramanujan had health problems.
Right from the start Ramanujan's collaboration with Hardy
led to important results. Hardy was, however, unsure how to approach
the problem of Ramanujan's lack of formal education. He wrote [1]:
What was to be done in the way of teaching him modern
mathematics? The limitations of his knowledge were as startling as
its profundity. Littlewood was asked to help teach
Ramanujan rigorous mathematical methods. However he said ([31]):
... that it was extremely difficult because every time some
matter, which it was thought that Ramanujan needed to know, was
mentioned, Ramanujan's response was an avalanche of original ideas
which made it almost impossible for Littlewood to persist in his
original intention. The war soon took Littlewood
away on war duty but Hardy remained in Cambridge to work with
Ramanujan. Even in his first winter in England, Ramanujan was ill and
he wrote in March 1915 that he had been ill due to the winter weather
and had not been able to publish anything for five months. What he
did publish was the work he did in England, the decision having been
made that the results he had obtained while in India, many of which
he had communicated to Hardy in his letters, would not be published
until the war had ended. On 16 March 1916
Ramanujan graduated from Cambridge with a Bachelor of Science by
Research (the degree was called a Ph.D. from 1920). He had been
allowed to enrol in June 1914 despite not having the proper
qualifications. Ramanujan's dissertation was on Highly composite
numbers and consisted of seven of his papers published in England.
Ramanujan fell seriously ill in 1917 and his doctors feared
that he would die. He did improve a little by September but spent
most of his time in various nursing homes. In February 1918 Hardy
wrote (see [3]): Batty Shaw found out, what other
doctors did not know, that he had undergone an operation about four
years ago. His worst theory was that this had really been for the
removal of a malignant growth, wrongly diagnosed. In view of the fact
that Ramanujan is no worse than six months ago, he has now abandoned
this theory  the other doctors never gave it any support. Tubercle
has been the provisionally accepted theory, apart from this, since
the original idea of gastric ulcer was given up. ... Like all Indians
he is fatalistic, and it is terribly hard to get him to take care of
himself. On 18 February 1918 Ramanujan was elected
a fellow of the Cambridge Philosophical Society and then three days
later, the greatest honour that he would receive, his name appeared
on the list for election as a fellow of the Royal Society of London.
He had been proposed by an impressive list of mathematicians, namely
Hardy, MacMahon, Grace, Larmor, Bromwich, Hobson, Baker, Littlewood,
Nicholson, Young, Whittaker, Forsyth and Whitehead. His election as a
fellow of the Royal Society was confirmed on 2 May 1918, then on 10
October 1918 he was elected a Fellow of Trinity College Cambridge,
the fellowship to run for six years. The honours
which were bestowed on Ramanujan seemed to help his health improve a
little and he renewed his effors at producing mathematics. By the end
of November 1918 Ramanujan's health had greatly improved. Hardy wrote
in a letter [3]: I think we may now hope that he
has turned to corner, and is on the road to a real recovery. His
temperature has ceased to be irregular, and he has gained nearly a
stone in weight. ... There has never been any sign of any diminuation
in his extraordinary mathematical talents. He has produced less,
naturally, during his illness but the quality has been the same. ....
He will return to India with a scientific standing and
reputation such as no Indian has enjoyed before, and I am confident
that India will regard him as the treasure he is. His natural
simplicity and modesty has never been affected in the least by
success  indeed all that is wanted is to get him to realise that he
really is a success. Ramanujan sailed to India on
27 February 1919 arriving on 13 March. However his health was very
poor and, despite medical treatment, he died there the following
year. The letters Ramanujan wrote to Hardy in 1913
had contained many fascinating results. Ramanujan worked out the
Riemann series, the elliptic integrals, hypergeometric series and
functional equations of the zeta function. On the other hand he had
only a vague idea of what constitutes a mathematical proof. Despite
many brilliant results, some of his theorems on prime numbers were
completely wrong. Ramanujan independently
discovered results of Gauss, Kummer and others on hypergeometric
series. Ramanujan's own work on partial sums and products of
hypergeometric series have led to major development in the topic.
Perhaps his most famous work was on the number p(n) of partitions of
an integer n into summands. MacMahon had produced tables of the value
of p(n) for small numbers n, and Ramanujan used this numerical data
to conjecture some remarkable properties some of which he proved
using elliptic functions. Other were only proved after Ramanujan's
death. In a joint paper with Hardy, Ramanujan gave
an asymptotic formula for p(n). It had the remarkable property that
it appeared to give the correct value of p(n), and this was later
proved by Rademacher. Ramanujan left a number of
unpublished notebooks filled with theorems that mathematicians have
continued to study. G N Watson, Mason Professor of Pure Mathematics
at Birmingham from 1918 to 1951 published 14 papers under the general
title Theorems stated by Ramanujan and in all he published nearly 30
papers which were inspired by Ramanujan's work. Hardy passed on to
Watson the large number of manuscripts of Ramanujan that he had, both
written before 1914 and some written in Ramanujan's last year in
India before his death. Article by: J J O'Connor
and E F Robertson Source:
www.history.mcs.standrews.ac.uk/Mathematicians



