GREATEST MATHEMATICIAN :: S. RAMANUJAN

If we talk about the greatest scientfic minds

i can think of one person in our recent past who has given a

"NEW DIMENSION IN MATHS"

YES I am talking of SRINIVASA RAMANUJAN

 

Ramanujan was born on 22 December 1887 in Erode, Tamil Nadu, India, at the place of residence of his maternal grandparents. His father, K. Srinivasa Iyengar worked as a clerk in a sari shop and hailed from the district of thanjavur. His mother, Komalatammal was a house wife and also a singer at a local temple. They lived in Sarangapani Street in a south-Indian-style home (now a museum) in the town of Kumbakonam.

 

By age eleven, he had exhausted the mathematical knowledge of two college students, who were lodgers at his home. He was later lent books on advanced trigonometry written by S.L. Loney. He completely mastered this book by the age of thirteen and he discovered sophisticated theorems on his own. By fourteen, he achieved merit certificates and academic awards throughout his school career and also assisted the school in the logistics of assigning its 1200 students (each with their own needs) to its 35-odd teachers. He completed mathematical exams in half the allotted time, and showed a familiarity with infinite series. When he was sixteen, Ramanujan came across the book, A synopsis of elementary results in pure and applied mathematics written by George S. Carr. This book was a collection of 5000 theorems, and it introduced Ramanujan to the world of mathematics. The next year, he had independently developed and investigated the Bernoulli numbers and had calculated Euler's constant up to 15 decimal places. His peers of the time commented that they "rarely understood him" and "stood in respectful awe" of him.

 

 

He received a scholarship to study at Government College in kumbakonam, known as the "Cambridge of South India." However, Ramanujan was so intent on studying mathematics that he could not focus on any other subjects and failed most of them, losing his scholarship in the process.

 

 

In August 1905, he ran away from home, heading towards Visakhapatnam.

 He later enrolled at Pachaiyappa's College in Madras. He again excelled in mathematics, but performed poorly in other subjects such as physiology. Ramanujan failed his F. A degree exam in December 1906 and again a year later. Without a degree, he left college and continued to pursue independent research in mathematics. At this point in his life, he lived in extreme poverty and was often near the point of starvation

 

He met deputy collector V. Ramaswami Iyer who had recently founded the Indian Mathematical Society. Ramanujan, wishing for a job at the revenue department where Iyer worked, showed him his mathematics notebooks. As Iyer later recalled:

 

 

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

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Iyer sent Ramanujan, with introduction letters, to his mathematical friends in Madras. Some of these friends 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. Ramachandra Rao was impressed by Ramanujan's work, but was doubtful that it was actually his own work. Ramanujan mentioned a correspondence he had with Professor Saldhana, a notable Bombay (now Mumbai) mathematician, in which Saldhana expressed a lack of understanding for his work, but concluded that he was not a phony. Ramanujan's friend, C. V. Rajagopalachari, persisted with Ramachandra Rao and tried to quell any doubts over Ramanujan's academic morality. Rao agreed to give him another chance, and he listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his theory of divergent series which Rao said ultimately "converted me" to believe Ramanujan's mathematical brilliance.Rao: "ask him what he wanted", and Ramanujan replied that he needed some work and financial support. Rao consented and sent him to Madras. He continued his mathematical research with Rao's financial aid supporting his daily needs. Ramanujan, with the help of Ramaswami Iyer, had his work published in the Journal of Indian Mathematical Society.

 

 

 

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

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To supplement Hardy's endorsement, a former mathematical lecturer at Trinity College in Cambridge, Gilbert Walker, looked at Ramanujan's work and expressed amazement and urged him to spend time at Cambridge. As a result of Walker's endorsement, B. Hanumantha Rao, a mathematics professor at an engineering college, invited Ramanujan's colleague Narayana Iyer to a meeting of the Board of Studies in Mathematics to discuss "what we can do for S. Ramanujan." The board met and agreed to grant Ramanujan a research scholarship of 75 rupees per month for the next two years at the University of Madras. While he was engaged as a research student, Ramanujan continued to submit papers to the Journal of the Indian Mathematical Society. In one paper, Ramanujan anticipated the work of a Polish mathematician who had published his work shortly after. In his quarterly papers, Ramanujan drew up theorems to make definite integrals more easily solvable. Working off Giuliano Frullani's 1821 integral theorem, Ramanujan formulated generalizations that could be made to evaluate formerly unyielding integrals.

Hardy's correspondence with Ramanujan soured after Ramanujan refused to come to England. Hardy enlisted a colleague lecturing in Madras, E. H. Neville, to mentor and bring Ramanujan to England. Neville asked Ramanujan why he was not coming to Cambridge. Ramanujan apparently had now accepted the proposal, as Neville put it, "Ramanujan needed no converting and that his parents' opposition had been withdrawn." Apparently, Ramanujan's friends convinced his mother to accept the journey to Cambridge. Ramanujan was personally convinced by a vivid dream his mother had, in which the family goddess Namagiri commanded her "to stand no longer between her son and the fullfilment of his life's purpose."

 

Ramanujan went aboard the S. S. Nevasa on 17 March 1913

Hardy and Ramanujan began to take a look at Ramanujan's work in his notebooks. Hardy had already received 120 theorems from Ramanujan in the first two letters, but there were many more results and theorems to be found in the notebooks. Hardy saw that some were wrong, some were already discovered and the rest were new breakthroughs.

 

 

While the master was still in India, Ramanujan recorded the bulk of his results in four notebooks of loose leaf paper. These results were mostly written up without any derivations. This is probably the origin of the misperception that Ramanujan was unable to prove his results and simply thought up the final result directly. Mathematician Bruce C. Berndt, in his review of these notebooks and Ramanujan's work, says that Ramanujan most certainly was able to make the proofs of most of his results, but chose not to.

This style of working may have been for several reasons. Since paper was very expensive, Ramanujan would do most of his work and perhaps his proofs on slate, and then transfer just the results to paper. Using a slate was common for mathematics students in India at the time. He was also quite likely to have been influenced by the style of G. S. Carr's book, which stated results without proofs. Finally, it is possible that Ramanujan considered his workings to be for his personal interest alone; and therefore only recorded the results.^[81]

The first notebook has 351 pages with 16 somewhat organized chapters and some unorganized material. The second notebook has 256 pages in 21 chapters and 100 unorganized pages, with the third notebook containing 33 unorganized pages. The results in his notebooks inspired numerous papers by later mathematicians trying to prove what he had found. Hardy himself created papers exploring material from Ramanujan's work as did G. N. Watson, B. M. Wilson, and Bruce Berndt. A fourth notebook, the so-called "lost notebook", was rediscovered in 1976 by George Andrews.

 

 

Plagued by health problems all through his life, living in a country far away from home, and obsessively involved with his mathematics, Ramanujan's health worsened in England, perhaps exacerbated by stress, and by the scarcity of vegetarian food during the First World War. He was diagnosed with tuberculosis and a severe vitamin deficiency and was confined to a sanatorium. Ramanujan returned to Kumbakonam, India in 1919 and died soon thereafter at the age of 32. His wife, S. Janaki Ammal, lived in Chennai (formerly Madras) until her death in 1994.