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, formulas
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well out of date by the time Ramanujan used it.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.
The
picture above is taken from a stamp issued by the Indian
Post Office to celebrate the 75th anniversary of his birth.
