The greatest mathematical genius produced in India in the modern times, Srinivasa Ramanujan
made extraordinary contributions to mathematical analysis, number theory, infinite series,
and continued fractions though he had almost no formal training in pure mathematics. In this
article the life and work of Ramanujan have been discussed.
each student still get a banana?”3 Ramanujan has been
quoted to have said:
“Zero divided by zero may be
he year 2012 has been designated as National
anything. The zero of the numerator may be several times
Mathematics Year by the Prime Minister of India.
the zero of the denominator and vice versa”. This anecdote
He also announced that December 22 would be
reveals that Ramanujan was thinking of limits and limiting
celebrated as National Mathematics Day from 2012 on.
processes. The teacher was astonished observing the talent
The greatest mathematical genius produced in India
of Ramanujan that even as a school student it was apparent
in the modern times, Srinivasa Ramanujan made
to Ramanujan that zero divided by zero is undefined.4 His
extraordinary contributions to mathematical analysis,
devotion to research in mathematics in spite of various odds
number theory, infinite series, and continued fractions
is inspiring to us even today.
though he had almost no formal training in pure
mathematics. Without any access to the larger mathematical
Early Life and Education
community, which was centered in Europe , he developed
He was born in a poor Brahmin family on 22nd
his own mathematical research independently. As a result,
December, 1887 in Erode, Madras Presidency, at the
he sometimes rediscovered known theorems in addition to
residence of his maternal grandparents. His father was K.
producing new work. According to the famous English
Srinivasa Iyengar and his mother was Komalatammal. They
mathematician G.H. Hardy Ramanujan was a natural genius
lived in Kumbakonam. In December 1889 Ramanujan had
like Euler and Gauss.1 Another British mathematician
smallpox and recovered, unlike thousands in the Thanjavur
Littlewood commented, “I can believe that he is at least a
District who died from the disease that year. He moved with
Jacobi”. He showed his talent in mathematics when he was
his mother to her parents’ house in Kanchipuram, near
a school student. In a class, the teacher was explaining
Madras (now Chennai). On 1 October 1892, Ramanujan
with examples that if any number is divided by the same
was enrolled at the local school. In March 1894, he was
number the quotient will be one. If three mangoes are
moved to a Telugu medium school. After his maternal
equally divided among three persons each will get one
grandfather lost his job as a court official in
mango. Ramanujan is said to have surprised everyone by
Kanchipuram, Ramanujan and his mother moved back to
asking: “Sir, if no banana is distributed to no student, will
Kumbakonam and he was enrolled in the Kangayan
Primary School, where Ramanujan performed well. Just
* Retired Reader and Head, Department of Physics, Presidency
before the age of 10, in November 1897, he passed his
College, Kolkata-73. e-mail
primary examinations in English, Tamil, geography and
arithmetic. He stood first in the district. He then entered
was often on the brink of starvation. In January 1910, a
Town Higher Secondary School. He mastered a book
doctor volunteered to do the surgery without any
on advanced trigonometry written by S. L. Loney by the
remuneration. After his successful surgery, Ramanujan
age of 13. He even discovered theorems of his own, and
searched for a job. He stayed at friends’ houses while he
re-discovered Euler’s identity independently. He
went door to door around the city of Madras looking for
demonstrated unusual mathematical skills at school, winning
a clerical position. To make some money, he tutored some
accolades and awards. By 14, he was receiving merit
students at Madras Presidency College who were preparing
certificates and academic awards which continued
for their F.A. examination. In late 1910, Ramanujan was
throughout his school career. He completed mathematical
sick again. After recovery he continued his research in
examinations in half the allotted time, and showed a
mathematics and at the same time searched for a job.
familiarity with infinite series. Ramanujan was shown how
to solve cubic equations in 1902 and he went on to find
his own method to solve the quartic.
Ramanujan, wishing for a job, met deputy collector V.
Once a student, two years senior to Ramanujan, could
Ramaswamy Aiyer, founder of the Indian Mathematical
not solve a problem and requested Ramanujan if he could
Society. Ramanujan showed him his mathematics notebooks
solve it. The problem was to find the values of x and y if
which were sufficient to convince him that Ramanujan was
x + y =7 and x + y = 11. Almost immediately Ramanujan
a gifted mathematician .He advised Ramanujan to publish
solved the problem, putting x
= p2 and y = q2. So,
his results of research in the
“Journal of Indian
p + q2 = 7 and p2 + q = 11. Subtracting the 1st equation
Mathematical Society”. Ramaswamy Aiyer sent Ramanujan
from the 2nd we get, p2 - q2 - ( p - q) = 4, or (p-q)( p+q-
back to Madras with a letter of introduction to P. V. Sheshu
1) =4. Its only solution is p= 3 and q = 2, i.e., x=9 and
Aiyer of Madras Presidency College and a distinguished
y=4. How genius he was.
member of the Mathematical Society. Sheshu Aiyer was
greatly impressed with the contents of the notebooks. He
In 1903 at the age of 16, Ramanujan obtained a book
gave Ramanujan a note of recommendation to the lover of
by G. S. Carr titled “A Synopsis of Elementary Results in
mathematics, Dewan Bahadur R. Ramachandra Rao, who
Pure and Applied Mathematics”. It was a collection of 5000
was then the District Collector of Nellore. Ramachandra
theorems. Ramanujan studied the book in detail. The next
Rao was also highly impressed by the original results
year, he had independently developed and investigated
proved by Ramanujan in various topics of mathematics.
the Bernoulli numbers and had calculated Euler’s
The best description of Ramanujan’s first meeting with
constant up to 15 decimal places. His peers at the time
Diwan Bahadur is in his own words: “ A short uncouth
commented that they “rarely understood him” and “stood
figure, stout, unshaved, not over-clean, with one
in respectful awe” of him. In 1904 when he graduated
conspicuous feature - shining eyes - walked in with a
from Town Higher Secondary School , Ramanujan was
frayed notebook under his arm. He was miserably poor. I
awarded the K. Ranganatha Rao prize for mathematics by
asked him what he wanted. He said he wanted a pittance
Krishnaswami Iyer, the headmaster of the school. Iyer
to live on so that he might pursue his researches.” During
introduced Ramanujan as an outstanding student who
this period Ramanujan had his first paper published, a 17-
deserved scores higher than the maximum possible
page work on Bernoulli numbers, that appeared in 1911 in
marks. He received a scholarship to study at Government
the ”Journal of the Indian Mathematical Society”.
Arts College, Kumbakonam, However, Ramanujan was so
Ramachandra Rao supported him for some time, but
intent on studying mathematics that neglected other subjects
Ramanujan was not willing to live on somebody’s help
and failed most of them. As a result, lost his scholarship.
indefinitely. He also tried to obtain some scholarship but
He was later enrolled at Pachaiyappa’s College in Madras.
failed. In early 1912 Ramanujan got a temporary service
He again excelled in mathematics but performed poorly in
at the Office of the Accountant General of Madras
other subjects. He failed his Fine Arts degree examination
Presidency at a monthly salary of Rs.20/- . He worked there
in December 1906 and again next year. He left college
for a few weeks. In March, 1912 Ramanujan joined as a
without a degree and continued to pursue independent
Class III, Grade IV accounting clerk at Madras Port Trust
research in mathematics.
at a monthly salary of Rs.30/- .At his office, Ramanujan
On 14 July 1909, Ramanujan was married to a nine-
easily and quickly completed the work given to him and
year old bride, Janaki Ammal. After the marriage,
spent his spare time doing mathematical research. He
Ramanujan fell seriously ill and required a surgical
received encouragement from S. Narayana Iyer , the
operation. At that time he lived in extreme poverty and
manager of the Port Trust (also the Treasurer of the Indian
VOL. 79, NOS. 3-4
Mathematical Society) and Sir Francis Spring, Chairman
you give me. Requesting to be excused for the trouble I
of the Trust. Sir Gilbert Walker, Director General of
give you”. Alongwith this letter he enclosed about 120
Observatories in Simla, who was on a visit to Madras, was
statements of theorems on infinite series, improper integrals,
also highly impressed by Ramanujan’s capabilities. He
continued fractions, and number theory. Coming from an
wrote a letter to Francis Didsbury, Registrar of the
unknown mathematician, the nine pages of mathematics
University of Madras “that the university would be justified
made Hardy initially view Ramanujan’s manuscripts as a
in enabling S. Ramanujan for a few years at least to spend
possible “fraud”. Hardy recognised some of Ramanujan’s
the whole of his time on mathematics without any anxiety
formulae but others “seemed scarcely possible to believe”.
as to his livelihood.” Dr. Walker’s recommendation was
After he saw Ramanujan’s theorems on continued fractions
accepted by the University of Madras, with the express
on the last page of the manuscripts, Hardy commented that
consent of the Governor of Madras. The special scholarship
the “[theorems] defeated me completely; I had never seen
amount was Rs. 75 per month, for two years.2 Ramanujan
anything in the least like them before”. He figured that
accepted this first research scholarship of the University
Ramanujan’s theorems “must be true, because, if they were
of Madras, from May 1913.
not true, no one would have the imagination to invent
them”. Ramanujan deduced 4000 theorems with proofs at
Contacting British Mathematicians
the end without showing steps. Hardy asked his colleague, J.
E. Littlewood, to take a look at the papers. Littlewood was
On the spring of 1913, Narayana Iyer, Ramachandra
amazed by the mathematical genius of Ramanujan. After
Rao and E. W. Middlemast tried to present Ramanujan’s
discussing the papers with Littlewood, Hardy concluded
work to British mathematicians. One mathematician, M. J.
that the letters were “certainly the most remarkable I have
M. Hill of University College London, commented that
received” and commented that Ramanujan was
Ramanujan’s papers were ‘riddled with holes’. He said that
mathematician of the highest quality, a man of altogether
although Ramanujan had “a taste for mathematics, and
exceptional originality and power”. One colleague, E. H.
some ability”, he lacked the educational background and
Neville, later commented that “not one [theorem] could
foundation needed to be accepted by mathematicians. Two
have been set in the most advanced mathematical
professors, H. F. Baker and E. W. Hobson, returned
examination in the world”. It may be mentioned in this
Ramanujan’s papers without comment. Without being
connection the assertions made by an eminent Indian
disheartened, on the suggestion of Sheshu Aiyer, Ramanujan
mathematician Prof. Hans Raj Gupta which are given here
wrote to Professor G. H. Hardy of Trinity College,
- “A man may become famous if he can either prove or
Cambridge. His letter to Hardy dated January 16, 1913
disprove any one of Ramanujan’s conjectures.” On 8
ran as follows 3.
1913, Hardy wrote a letter to Ramanujan,
“Dear Sir,
expressing his interest for his work. Hardy also added that
I beg to introduce myself to you as a clerk in the
it was “essential that I should see proofs of some of your
Accounts Department of the Port Trust office at Madras
assertions”. Ramanujan was delighted with Hardy’s reply
on a salary of only Rs. 20 per annum. I am now about 23
and sent a letter packed with theorems to Hardy, writing,
years of age. I have had no university education but I have
‘I have found a friend in you who views my labours
undergone the ordinary school course. After leaving school
I am already a half starving man. To
I have been employing the spare time at my disposal to
preserve my brains I want food and this is my first
work at Mathematics. I have not trodden through the
consideration. Any sympathetic letter from you will be
conventional regular course which is followed in a
helpful to me here to get a scholarship either from the
University course but I am striking out a new path for
university or from the government.’ This reminds us of
myself. I have made a special investigation of divergent
another genius of the
17th century, Johanness Kepler,
series in general and the results I get are termed by the
through whose lips came in utter despair a bunch of painful
local mathematicians as ‘startling’.
words, “My hungry stomach looks up like a little dog to
its master who used to feed it.” Hardy sent him an
I would request you to go through the enclosed papers.
invitation to come to England. In accordance with his
Being poor, if you are convinced that there is anything of
Brahmin upbringing, Ramanujan refused to go to a foreign
value, I would like to have my theorems published. I have
land. Hardy requested his colleague lecturing in Madras,
not given the actual investigations nor the expressions that
E. H. Neville, to mentor and bring Ramanujan to
I get but I have indicated the lines on which I proceed.
England. Neville asked Ramanujan why he would not go
Being inexperienced, I would very highly value any advice
to Cambridge. Ramanujan apparently had now accepted the
proposal; as Neville put it,
“Ramanujan needed no
Ramanujan was honoured in several ways. On 6
converting and that his parents’ opposition had been
1917, he was elected to the London
withdrawn”. Apparently, Ramanujan’s mother had a vivid
Mathematical Society. He was the first mathematician who
dream in which the family Goddess Namagiri comanded
became a Fellow of the Royal Society in 1918. He was
her “to stand no longer between her son and the fulfillment
the second Indian and one of the youngest Fellows in the
of his life’s purpose”. Neville, wrote on January 28, 1914
history of the Royal Society. He was elected “for his
to the Registrar of the University of Madras about “the
investigation in Elliptic functions and the Theory of
importance of securing to Ramanujan a training in the
Numbers.” On 13 October 1918, he became the first Indian
refinements of modern methods and a contact with men
to be elected a Fellow of Trinity College, Cambridge.
who knew what range of ideas have been explored and
During his five year stay in Cambridge, Ramanujan
what have not”. He prophesied that Ramanujan would
21 research papers containing theorems on
respond to such a stimulus and that “his name will become
definite integral, modular equations, Riemann’s zeta
one of the greatest in the history of mathematics and the
function, infinite series, summation of series, analytic
University and the city of Madras will be proud to have
number theory, asymptotic formulae, modular functions,
assisted in his passage from obscurity to fame.” Within a
partitions and combinatorial analysis. Five of these
week, the University of Madras decided to set aside
papers were in collaboration with Hardy. His paper entitled
Rs. 10,000 to offer Ramanujan a scholarship of £ 100 for
“Highly Composite Numbers” which appeared in the
a passage by ship and for his initial outfit. On the 17th
Journal of the London Mathematical Society, in 1915, is
March 1914, Ramanujan sailed for England. On 14th April
62 pages long and contains 269 equations. Ramanujan also
he reached England. On 18 April, 1914 he joined as a
published five short notes in the Records of Proceedings
research scholar on an annual scholarship of 250 pounds.
at meetings of the London Mathematical Society and six
more in the journal of the Indian Mathematical Society.
Life in England
Illness and Return to India
Ramanujan’s arrival at Cambridge was the beginning
of a very successful five-year collaboration with Hardy.
The alien climate and culture took a toll on his health.
He immediately began his work with Littlewood and Hardy.
Ramanujan faced the English winter, and he had to do all
his own cooking to adhere to his caste’s strict dietary rules.
Hardy had already received 120 theorems from Ramanujan
in the first two letters, but there were many more results
Ramanujan was purely vegetarian and there was scarcity
and theorems to be found in the notebooks. Hardy saw
of vegetarian food during the First World War. In 1917 he
that some were wrong, others had already been discovered,
fell ill and was hospitalized. He was diagnosed
with tuberculosis and a severe vitamin deficiency. The
while the rest were new breakthroughs. Ramanujan left a
illness grew from bad to worse and his doctors feared for
deep impression on Hardy and Littlewood. Hardy did his
his life. By late 1918 his health had improved; and it was
best to fill in the gaps in Ramanujan’s education. He was
decided in 1919 that he should be sent back to India where
amazed by Ramanujan’s uncanny formal intuition in
the warm climate might help in his recovery. He returned
manipulating infinite series, continued fractions, and the
to India on the 2nd April, 1919. But his health failed again.
like: “I have never met his equal, and can compare him
Ramanujan was brought to Madras for expert medical
only with Euler or Jacobi.” Towards the end of 1914 the
treatment, in January 1920. Despite all the tender attention
First World War started. Littlewood joined war duty
he could get from his wife who nursed him throughout
but Hardy remained in Cambridge to work with Ramanujan.
this period, and the best medical attention from the doctors,
In 1916 Ramanujan was awarded the B. A. degree of
his untimely end came on 26th April 1920, at Chetput,
the University of Cambridge (the degree was called a Ph.D.
Madras, when Ramanujan was 32 years old. Even on his
from 1920) on the basis of his research work on highly
death bed he devoted himself completely to Mathematics
composite numbers, which was published as a paper in
and produced research work of the highest order, a glimpse
the Journal of the London Mathematical Society. The paper
of which he communicated in his last letter to Hardy, but
was over 50 pages with different properties of such
most of it remained unnoticed till during the summer of
numbers proven. Hardy remarked that this was one of the
1976, Andrews accidentally came across some 140 sheets
most unusual papers seen in mathematical research at that
in Ramanujan’s own handwriting and containing some 600
time and that Ramanujan showed extraordinary ingenuity
formulae. This was the work that Ramanujan did in the
in handling it.
last year of his life after returning to India.
VOL. 79, NOS. 3-4
any means of support, and more so because he continued
to produce work of the highest order even in the face of
The author expresses his sincere thanks to the
death. But he did not get due recognition in India until he
Reviewer for his valuable comments, some of which has
got recognition from abroad. Unless his mathematical
been incorporated in this revised paper.
talents were detected and recognized by an eminent
mathematician like Prof. Hardy, this prodigious son of India
could have been lost into oblivion. According to J. B. S.
Robert Kanigel. The Man Who Knew Infinity: A life of the
Haldane, “Today in India Ramanujan could not get even a
Genius Ramanujan, Charles Scribner’s Sons, New York (1991);
lectureship in a rural college because he had no degree.
(Indian Edition by Rupa & Co.) (1993).
Much less could he get a post through the Union Public
S R Ranganathan. Ramanujan. The Man and the Mathematician.
(1967) (Asia Publishing House).
Service Commission? This fact is a disgrace to India. I
Bruce C Berndt and Robert A Rankin. Ramanujan: Letters and
am aware that he was offered a chair in India after
(1995) Amer. Math. Soc. and London Math. Soc.
becoming a Fellow of the Royal Society. But it is
Indian Edition by Affiliated East West Press Pvt. Ltd. (1997).
scandalous that India’s great mean should have to wait for
4. Life
foreign recognition. If Ramanujan’s work had been
and Work of Ramanujan
recognized in India as early as it was in England, he might
Some Relevant Books
never have emigrated and might be alive today. We can
Collected works of Ramanujan : Edited by Prof. G.H.Hardy with
cast the blame for Ramanujan’s non-recognition on the
an introduction by him
British Raj. We cannot do so when similar cases occur
Mathematician’s Apology by Prof. G.H.Hardy
Ramanujan’s Note Books in 5 volumes : Edited by Bruce Brendt.
VOL. 79, NOS. 3-4