ARTICLE

NATIONAL MATHEMATICS YEAR :

A TRIBUTE TO SRINIVASA RAMANUJAN

PRADIP K. DATTA*

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.

Introduction

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

T

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

: dattapradip@rediffmail.com,

dattapradip@gmail.com

primary examinations in English, Tamil, geography and

158

SCIENCE AND CULTURE, MARCH-APRIL, 2013

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

Career

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

159

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

“a

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.

February

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

sympathetically

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

160

SCIENCE AND CULTURE, MARCH-APRIL, 2013

proposal; as Neville put it,

“Ramanujan needed no

Ramanujan was honoured in several ways. On 6

converting and that his parents’ opposition had been

December

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

published

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

21

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

161

any means of support, and more so because he continued

Acknowlwdgement

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

References

could have been lost into oblivion. According to J. B. S.

1.

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

2.

S R Ranganathan. Ramanujan. The Man and the Mathematician.

(1967) (Asia Publishing House).

Service Commission? This fact is a disgrace to India. I

3.

Bruce C Berndt and Robert A Rankin. Ramanujan: Letters and

am aware that he was offered a chair in India after

Commentary,

(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.

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

1.

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

2.

Mathematician’s Apology by Prof. G.H.Hardy

today”.1

3.

Ramanujan’s Note Books in 5 volumes : Edited by Bruce Brendt.

VOL. 79, NOS. 3-4

163