SRINIVASA
RAMANUJAN -A MATHEMATICIAN
INTRODUCTION
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.
EARLY LIFE:
He was
born in his grandmother's house in Erode, a small village about 400 km
southwest of Madras. When he 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.
CONTRIBUTION:
Ramanujan
was shown how to solve cubic
equations in 1902 and he went on to find his own method to solve the quadratic. The following year, not knowing that the quintic(Polynomial
of degree five) could not be solved by radicals, he 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. The book contained theorems, formulae and short proofs.
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. He worked on hypergeometric
series and investigated relations between integrals and series. He
was to discover later that he had been studying elliptic functions.
Continuing
his mathematical work Ramanujan studied continued fractions and divergent
series in 1908. 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.
RAMANUJAN-HARDY
In January 1913 Ramanujan
wrote to G H Hardy having seen a
copy of his 1910 book Orders
of infinity. They had communicated through letters. Hardy, together with Littlewood, studied the long
list of unproved theorems which Ramanujan enclosed with his letter.
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.
Ramanujan sailed from
India on 17 March 1914. He arrived in London on 14 April 1914. The outbreak of World War I made obtaining
special items of food harder and it was not long before Ramanujan had health
problems.
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). 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.
HARDY- RAMANUJAN NUMBER:
The number 1729 is known as the Hardy–Ramanujan
number after a famous anecdote of the British mathematician G. H. Hardy regarding a visit
to the hospital to see Ramanujan. In Hardy's words
“I
remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked
that the number seemed to me rather a dull one, and that I hoped it was not an
unfavorable omen. “No”, he replied, “it
is a very interesting number; it is the smallest number expressible as the sum
of two cubes in two different ways”.
The
two different ways are 1729 = 13 + 123 = 93+103
Generalizations
of the idea have created the notion of “ taxicab numbers”. Coincidentally, 1729 is also a Carmichael
number.
ACHIEVEMENT:
On 18
February 1918 Ramanujan was elected a fellow of the Cambridge Philosophical
Society and then three days later, the greatest honor that he would receive;
his name appeared on the list for election as a fellow of the Royal Society of London. 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 honors
which were bestowed on Ramanujan seemed to help his health improve a little and
he renewed his efforts at producing mathematics. By the end of November 1918
Ramanujan's health had greatly improved.
BACK TO INDIA:
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.
Ramanujan independently
discovered results of Gauss, Kummer and others on
hyper geometric series. Ramanujan's own work on partial sums and products of
hyper geometric series have led to major development in the topic. 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.
RECOGNITION:
1.
Tamil Nadu Celebrates Ramanujan’s Birthday, 22
December, as 'State IT Day'.
3.
The 75th
anniversary of Ramanujan's birth – commemorating his achievements in the field
of number theory, and a new design was issued on December 26, 2011, by the India Post.
4.
A prize for young mathematicians from developing
countries has been created in the name of Ramanujan by the International
Centre for Theoretical Physics (ICTP).
5.
On the 125th anniversary of
his birth, India declared the birthday of Ramanujan, December 22, as 'National
Mathematics Day.'
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