**SRINIVASA RAMANUJAN :**

**Srinivasa Ramanujan** was an Indian mathematician who lived during the British Rule in India. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems considered to be unsolvable. Srinivasa Ramanujan initially developed his own mathematical research in isolation; it was quickly recognized by Indian mathematicians. Seeking mathematicians who could better understand his work. In 1913 he began a postal partnership with the English mathematician G. H. Hardy at the University of Cambridge, England. Recognizing the extraordinary work sent to him as samples, Hardy arranged travel for Ramanujan to Cambridge. In his notes, Ramanujan had produced groundbreaking new theorems, including some that Hardy stated had “defeated [him and his colleagues] completely”, in addition to rediscovering recently proven but highly advanced results.

**ABOUT HIS LIFE:**

Ramanujan independently compiled nearly 3,900 results (mostly identities and equations). His original and highly unconventional results, such as the Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions, have opened entirely new areas of work and inspired a vast amount of further research. As late as 2011 and again in 2012, researchers continued to discover that mere comments in his writings about “simple properties” and “similar outputs” for certain findings were themselves profound and subtle number theory results that remained unsuspected until nearly a century after his death and which relied on work published in 2006. He became one of the youngest Fellows of the Royal Society and only the second Indian member, and the first Indian to be elected a Fellow of Trinity College, Cambridge.

**ACHIEVEMENTS:**

**Ramanujan** met deputy collector V. RamaswamyAiyer, who had founded the Indian Mathematical Society, wishing for a job at the revenue department where Aiyer worked, Ramanujan showed him his mathematics notebooks.

As Aiyer later recalled:

I was struck by the extraordinary mathematical results contained in [the notebooks]. I had no mind to smother his genius by an appointment in the lowest rungs of the revenue department.

Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends in Madras. Some of them looked at his work and gave him letters of introduction to R. Ramachandra Rao, the district collector for Nellore and the secretary of the Indian Mathematical Society. Rao was impressed by Ramanujan’s research but doubted that it was his own work. Ramanujan’s friend C. V. Rajagopalachari tried to quell Rao’s doubts about Ramanujan’s academic integrity. Rao agreed to give him another chance and listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his theory of divergent series, which Rao said ultimately converted him to a belief in Ramanujan’s brilliance. When Rao asked him what he wanted, Ramanujan replied that he needed work and financial support. Rao consented and sent him to Madras. With Aiyer’s help, Ramanujan had his work published in the Journal of the Indian Mathematical Society.

**HIS WORKS:**

He waited for a solution to be offered in three issues, over six months but failed to receive any. In the end, Ramanujan supplied the solution to the problem himself. On page 105 of his first notebook, he formulated an equation that could be used to solve the infinitely nested radicals problem.

Using this equation, the answer to the question posted in the Journal was simply 3, obtained by setting x = 2, n = 1, and a = 0. Ramanujan wrote his first formal paper for the Journal on the properties of Bernoulli numbers. One property he discovered was that the denominators (sequence A027642 in the OEIS) of the fractions of Bernoulli numbers were always divisible by six. He also devised a method of calculating Bn based on previous Bernoulli numbers. One of these methods follows:

It will be observed that if n is even but not equal to zero,

- Bn is a fraction and the numerator of Bn/n in its lowest terms is a prime number,
- the denominator of Bn contains each of the factors 2 and 3 once and only once,
- 2n(2n − 1)Bn/n is an integer and 2(2n − 1)Bn consequently is an odd integer.

In his 17-page paper, “Some Properties of Bernoulli’s Numbers” (1911), Ramanujan gave three proofs, two corollaries and three conjectures. Ramanujan’s writing initially had many flaws. As Journal editor M. T. Narayana Iyengar noted:

**Mr Ramanujan’s** methods were so terse and novel and his presentation so lacking in clearness and precision, that the ordinary [mathematical reader], unaccustomed to such intellectual gymnastics, could hardly follow him.

Ramanujan later wrote another paper and also continued to provide problems in the Journal. In early 1912, he got a temporary job in the Madras Accountant General’s office, with a salary of 20 rupees per month. He lasted only a few weeks. Toward the end of that assignment, he applied for a position under the Chief Accountant of the Madras Port Trust.

**LIFE IN ENGLAND:**

**Ramanujan** departed from Madras aboard the S.S. Nevada on 17 March 1914. When he disembarked in London on 14 April, Neville was waiting for him with a car. Four days later, Neville took him to his house on Chesterton Road in Cambridge. Ramanujan immediately began his work with Littlewood and Hardy. After six weeks, Ramanujan moved out of Neville’s house and took up residence on Whewell’s Court. Hardy and Littlewood began to look at Ramanujan’s notebooks. Hardy had already received 120 theorems from Ramanujan in the first two letters, but there were many more results and theorems in the notebooks. Hardy saw that some were wrong, others had already been discovered, and the rest were new breakthroughs. Ramanujan left a deep impression on Hardy and Littlewood. Littlewood commented, “I can believe that he’s at least a Jacobi”, while Hardy said he “can compare him only with Euler or Jacobi.”

**Ramanujan** spent nearly five years in Cambridge collaborating with Hardy and Littlewood and published part of his findings there. Their collaboration was a clash of different cultures, beliefs, and working styles. In the previous few decades, the foundations of mathematics had come into question and the need for mathematically rigorous proofs recognized. Hardy was an atheist and an apostle of proof and mathematical rigour, whereas Ramanujan was a deeply religious man who relied very strongly on his intuition and insights. While in England, Hardy tried his best to fill the gaps in Ramanujan’s education and to mentor him in the need for formal proofs to support his results, without hindering his inspiration – a conflict that neither found easy.

Ramanujan was awarded a Bachelor of Science degree by research (this degree was later renamed PhD) in March 1916 for his work on highly composite numbers, the first part of which was published as a paper in the Proceedings of the London Mathematical Society. Hardy remarked that it was one of the most unusual papers seen in mathematical research at that time. On 6 December 1917, he was elected to the London Mathematical Society. In 1918 he was elected a Fellow of the Royal Society, the second Indian admitted to the Royal Society, following ArdaseerCursetjee in 1841. At age 31 Ramanujan was one of the youngest Fellows in the history of the Royal Society. He was elected “for his investigation in Elliptic functions and the Theory of Numbers.” On 13 October 1918, he was the first Indian to be elected a Fellow of Trinity College, Cambridge.

**LAST DAYS OF RAMANUJAN : **

**Srinivasa Ramanujan** was diagnosed with tuberculosis and a severe vitamin deficiency at the time and was confined to a sanatorium.

In 1919 he returned to Kumbakonam, Madras Presidency, and soon thereafter, in 1920, died at the age of 32. After his death, his brother Tirunarayanan chronicled Ramanujan’s remaining handwritten notes consisting of formulae on singular moduli, hypergeometric series and continued fractions and compiled them.

Ramanujan’s widow, **Smt. Janaki Ammal** moved to Bombay. In 1931 she returned to Madras and settled in Triplicane, where she supported herself on a pension from Madras University and income from tailoring. In her later years, she was granted a lifetime pension from Ramanujan’s former employer, the Madras Port Trust, and was also granted pensions from, among others, the Indian National Science Academy and the state governments of Tamil Nadu, Andhra Pradesh and West Bengal. She continued to cherish Ramanujan’s memory, and was active in efforts towards increasing his public recognition; prominent mathematicians, including George Andrews, Bruce C. Berndt and BélaBollobás made it a point to visit her while in India. She died at her Triplicane residence in 1994.

A 1994 analysis of **Ramanujan’s** medical records and symptoms by Dr D. A. B. Young concluded that his medical symptoms—including his past relapses, fevers, and hepatic conditions—were much closer to those resulting from hepatic amoebiasis, an illness then widespread in Madras, rather than tuberculosis. He had two episodes of dysentery before he left India. When not properly treated, dysentery can lie dormant for years and lead to hepatic amoebiasis, whose diagnosis was not then well established.

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