A tribute to the legend of the legends: Srinivas Ramanujan

 

(The picture above is taken from a stamp issued by the Indian Post Office to celebrate the 75th anniversary of his birth.)

     

Srinivasa Ramanujan (श्रीनिवास रामानुजन), the man who knew infinity, an exceptionally gifted young Indian mathematician , who shook the whole world with his theories that still are mystery to present day mathematicians.

He was born in a Brahmin family in Erode,Madras (now Chennai) on the date 22 December 1887.

His father worked as a clerk in a cloth merchant's shop.

Ramanujan entered the Town High School in Kumbakonam in January 1898.

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,he tried to solve the quintic that could not be solved even by radicals.

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 mathematic.

By 1904 Ramanujan had begun to undertake deep research. He investigated the series summation of 1by 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.

Nevertheless of all his primary  achivements , he faced many challenges in his journey of life.

Without money he was soon in difficulties and, without telling his parents, he ran away to the town of vishakhapatnam about 650 km north of Madras. He continued his mathematical work and at this time he worked on hypergeometric series and investigated relations between integrals and series.

The problems faced were so massive and his family was such poverty strucken that he ran out of pages even for making his new theorems.

In 1906 Ramanujan went to Madras where he entered Pachaiyappa's College. 

Continuing his mathematical work Ramanujan studied continued fractions and divergent series in 1908.

He married on 14 July 1909 with Janaki Ammal.

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. 

Job:-

He was appointed to the post of clerk on 1 March 1912. 

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.

Ramanujan 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. 

 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 

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.

Now 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.

Firstly, Ramanujan was an Brahmin and so was a strict vegetarian. 

Also Hindu Shastras don't give permission to travel through to go to foreign Island nations.

But he did it with a penance as described in shastras for cleansing after returning.

Ramanujan sailed from India on 17 March 1914. 

He arrived in London on 14 April 1914 and was met by Neville 

Right from the beginning, however, he had problems with his diet. The outbreak of World War I made obtaining special items of food harder .

The  English students at the University misbehaved with the Indian students but however the two professors - Hardy and Little wood sympathetically helped him.

On 16 March 1916 Ramanujan graduated from Cambridge with a Bachelor of Arts 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 and spent most of his time in various nursing homes.

On 18 February 1918 Ramanujan was elected a fellow of the Cambridge Philosophical Society and then three days later, the greatest honour that any great mathematician would receive, his name appeared on the list for election as a fellow of the Royal Society of London.

On 18 February 1918 Ramanujan was elected a fellow of the Cambridge Philosophical Society and then three days later, the greatest honour that any great mathematician would receive, his name appeared on the list for election as a fellow of the Royal Society of London.

But gradually on seeing the declining health of  Ramanujan.

Hardy wrote in a letter 

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. 

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 and how's that we lost our never ending precious diamond.

Date of death:26 April 1920

Place:Kumbakonam, Tamil Nadu state, India

Some of the theories:–

Ramanujan worked out the Riemann series, the elliptic integrals, hypergeometric series and functional equations of the zeta function. 

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)p(n) for small numbers nn, 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)p(n). It had the remarkable property that it appeared to give the correct value of p(n)p(n).

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

 

At the last I want to tell an interesting incidence from the life of Srinivas Ramanujan ji:-

Once when he was doing early morning puja in his hostel room in England, an English student mocked by saying that I think every mathematicians is an atheist.

Than Ramanujan ji remarked by saying that.

"An equation for me has no meaning, unless it expresses a thought of God."

"मेरे लिए एक समीकरण का कोई मतलब नहीं है, जब तक कि यह भगवान के बारे में एक विचार व्यक्त नहीं करता है।"