Indian mathematician ramanujan

After that, he enrolled in Town High School and spent six years there. In nature, he was peaceful, kind, and emotional. He would take a close look at everything and begin to consider it. Ramanujan was an insatiable questioner. His professors found his queries to be a little odd at times. For example, who was the first guy in the world? How far is the earth from the clouds?

His talent began to influence other students and professors at school. During his school years, he not only studied college-level mathematics but also guided college students in trigonometry. He received a Subramaniam scholarship for good grades in math and English after passing the high school examination, and he was also recognized for further college education.

The principal of his school had even found that the school's examinations were meaningless to Ramanujan. However, a problem occurred. His love for mathematics grew to the point where he ignored other disciplines. Math was the only subject solved in history, biology, and English classrooms. Consequently, he received a perfect score in mathematics, but he failed all of his other subjects and lost his scholarship.

He also attempted to attend school for a time, but Ramanujan ran away from home at seventeen when he no longer felt like it. Even Ramanujan's teachers were stumped by some of the queries they couldn't answer. His mathematics teacher was shocked when he saw his notepad. He began to spend more time teaching Ramanujan to solve math problems. Ramanujan's teacher would solve the problem in 12 stages, but he would do it in three.

Littlewood, his collaborator, concluded that this one was different. The letter contained statements on theorems related to the infinite series, improper integrals, continued fractions and the number theory. Hardy wrote back to Ramanujan and his acknowledgement changed everything for the young mathematician. He became a research scholar at the University of Madras earning almost double what his job as a clerk was paying him.

However, Hardy wanted him to come over to England. Ramanujan worked with Hardy for five years. Hardy was astonished by the genius of the young mathematician and said that he had never met anyone like him. His years at England were very decisive. Discussed below is the history, achievements, contributions, etc. Candidates preparing for the upcoming civil services exam must analyse this information carefully.

The intention behind encouraging the significance of mathematics was mainly to boost youngsters who are the future of the country and influence them to have a keen interest in analysing the scope of this subject. Also, aspirants appearing in the civil services exam can choose mathematics as an optional and the success stories of IAS Toppers from the past have shown the scope of this subject.

To get details of UPSC , candidates can visit the linked article. For any further information about the upcoming civil services examination , study material, preparation tips and strategy, candidates can visit the linked article. Your Mobile number and Email id will not be published. Littlewood , to take a look at the papers. Littlewood was amazed by Ramanujan's genius.

After discussing the papers with Littlewood, Hardy concluded that the letters were "certainly the most remarkable I have received" and that Ramanujan was "a mathematician of the highest quality, a man of altogether exceptional originality and power". Neville , later remarked that "No one who was in the mathematical circles in Cambridge at that time can forget the sensation caused by this letter On 8 February , Hardy wrote Ramanujan a letter expressing interest in his work, adding that it was "essential that I should see proofs of some of your assertions".

To supplement Hardy's endorsement, Gilbert Walker , a former mathematical lecturer at Trinity College, Cambridge , looked at Ramanujan's work and expressed amazement, urging the young man to spend time at Cambridge. 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.

While he was engaged as a research student, Ramanujan continued to submit papers to the Journal of the Indian Mathematical Society. In one instance, Iyer submitted some of Ramanujan's theorems on summation of series to the journal, adding, "The following theorem is due to S. Ramanujan, the mathematics student of Madras University. Ross of Madras Christian College , whom Ramanujan had met a few years before, stormed into his class one day with his eyes glowing, asking his students, "Does Ramanujan know Polish?

Working off Giuliano Frullani's integral theorem, Ramanujan formulated generalisations 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. Neville, to mentor and bring Ramanujan to England.

Ramanujan apparently had now accepted the proposal; Neville said, "Ramanujan needed no converting" and "his parents' opposition had been withdrawn". Ramanujan departed from Madras aboard the S. Nevasa on 17 March 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, a five-minute walk from Hardy's room. Hardy and Littlewood began to look at Ramanujan's notebooks. Hardy had already received 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.

Littlewood commented, "I can believe that he's at least a Jacobi ", [ 95 ] 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. Hardy and Ramanujan had highly contrasting personalities. 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 was recognised. 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. 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 Arts by Research degree [ 97 ] [ 98 ] the predecessor of the PhD degree in March for his work on highly composite numbers , sections of the first part of which had been published the preceding year in the Proceedings of the London Mathematical Society. The paper was more than 50 pages long and proved various properties of such numbers.

Hardy disliked this topic area but remarked that though it engaged with what he called the 'backwater of mathematics', in it Ramanujan displayed 'extraordinary mastery over the algebra of inequalities'. At age 31, Ramanujan was one of the youngest Fellows in the Royal Society's history. He was elected "for his investigation in elliptic functions and the Theory of Numbers.

Ramanujan had numerous health problems throughout his life. His health worsened in England; possibly he was also less resilient due to the difficulty of keeping to the strict dietary requirements of his religion there and because of wartime rationing in — He was diagnosed with tuberculosis and a severe vitamin deficiency, and confined to a sanatorium.

He attempted suicide in late or early by jumping on the tracks of a London underground station. Scotland Yard arrested him for attempting suicide which was a crime , but released him after Hardy intervened. After his death, his brother Tirunarayanan compiled Ramanujan's remaining handwritten notes, consisting of formulae on singular moduli, hypergeometric series and continued fractions.

Ramanujan's widow, Smt. Janaki Ammal, moved to Bombay. In , she returned to Madras and settled in Triplicane , where she supported herself on a pension from Madras University and income from tailoring. In , she adopted a son, W. Narayanan, who eventually became an officer of the State Bank of India and raised a family. In her later years, she was granted a lifetime pension from Ramanujan's former employer, the Madras Port Trust, and 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 to increase his public recognition; prominent mathematicians, including George Andrews, Bruce C. She died at her Triplicane residence in A analysis of Ramanujan's medical records and symptoms by D. 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, than tuberculosis.

He had two episodes of dysentery before he left India. When not properly treated, amoebic dysentery can lie dormant for years and lead to hepatic amoebiasis, whose diagnosis was not then well established. While asleep, I had an unusual experience. There was a red screen formed by flowing blood, as it were. I was observing it. Suddenly a hand began to write on the screen.

I became all attention. That hand wrote a number of elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing. Ramanujan has been described as a person of a somewhat shy and quiet disposition, a dignified man with pleasant manners. He looked to her for inspiration in his work [ ] and said he dreamed of blood drops that symbolised her consort, Narasimha.

Later he had visions of scrolls of complex mathematical content unfolding before his eyes.

Indian mathematician ramanujan

Hardy cites Ramanujan as remarking that all religions seemed equally true to him. At the same time, he remarked on Ramanujan's strict vegetarianism. Similarly, in an interview with Frontline, Berndt said, "Many people falsely promulgate mystical powers to Ramanujan's mathematical thinking. It is not true. He has meticulously recorded every result in his three notebooks," further speculating that Ramanujan worked out intermediate results on slate that he could not afford the paper to record more permanently.

Berndt reported that Janaki said in that Ramanujan spent so much of his time on mathematics that he did not go to the temple, that she and her mother often fed him because he had no time to eat, and that most of the religious stories attributed to him originated with others. However, his orthopraxy was not in doubt. In mathematics, there is a distinction between insight and formulating or working through a proof.

Ramanujan proposed an abundance of formulae that could be investigated later in depth. Hardy said that Ramanujan's discoveries are unusually rich and that there is often more to them than initially meets the eye. As a byproduct of his work, new directions of research were opened up. This might be compared to Heegner numbers , which have class number 1 and yield similar formulae.

One of Ramanujan's remarkable capabilities was the rapid solution of problems, illustrated by the following anecdote about an incident in which P. Mahalanobis posed a problem:. Imagine that you are on a street with houses marked 1 through n. There is a house in between x such that the sum of the house numbers to the left of it equals the sum of the house numbers to its right.

If n is between 50 and , what are n and x? Ramanujan thought about it and gave the answer with a twist: He gave a continued fraction. The unusual part was that it was the solution to the whole class of problems. Mahalanobis was astounded and asked how he did it. The minute I heard the problem, I knew that the answer was a continued fraction. Which continued fraction, I asked myself.

Then the answer came to my mind', Ramanujan replied. His intuition also led him to derive some previously unknown identities , such as. In , Hardy and Ramanujan studied the partition function P n extensively. They gave a non-convergent asymptotic series that permits exact computation of the number of partitions of an integer. In , Hans Rademacher refined their formula to find an exact convergent series solution to this problem.

Ramanujan and Hardy's work in this area gave rise to a powerful new method for finding asymptotic formulae called the circle method. In the last year of his life, Ramanujan discovered mock theta functions. Although there are numerous statements that could have borne the name Ramanujan conjecture, one was highly influential in later work. It was finally proven in , as a consequence of Pierre Deligne 's proof of the Weil conjectures.

The reduction step involved is complicated. Deligne won a Fields Medal in for that work. This congruence and others like it that Ramanujan proved inspired Jean-Pierre Serre Fields Medalist to conjecture that there is a theory of Galois representations that "explains" these congruences and more generally all modular forms. Deligne in his Fields Medal-winning work proved Serre's conjecture.

The proof of Fermat's Last Theorem proceeds by first reinterpreting elliptic curves and modular forms in terms of these Galois representations. Without this theory, there would be no proof of Fermat's Last Theorem. While still in Madras, Ramanujan recorded the bulk of his results in four notebooks of looseleaf paper. They were mostly written up without any derivations.

This is probably the origin of the misapprehension 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 prove most of his results, but chose not to record the proofs in his notes.

This may have been for any number of reasons. Since paper was very expensive, Ramanujan did most of his work and perhaps his proofs on slate , after which he transferred the final results to paper. At the time, slates were commonly used by mathematics students in the Madras Presidency. He was also quite likely to have been influenced by the style of G.

Carr 's book, which stated results without proofs. It is also possible that Ramanujan considered his work to be for his personal interest alone and therefore recorded only the results. The first notebook has pages with 16 somewhat organised chapters and some unorganised material. The second has pages in 21 chapters and unorganised pages, and the third 33 unorganised pages.

The results in his notebooks inspired numerous papers by later mathematicians trying to prove what he had found. Hardy himself wrote papers exploring material from Ramanujan's work, as did G. Watson , B. Wilson , and Bruce Berndt. In , George Andrews rediscovered a fourth notebook with 87 unorganised pages, the so-called "lost notebook". The number is known as the Hardy—Ramanujan number after a famous visit by Hardy to see Ramanujan at a hospital.

In Hardy's words: [ ]. I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number and remarked that the number seemed to me rather a dull one , and that I hoped it was not an unfavorable omen. Immediately before this anecdote, Hardy quoted Littlewood as saying, "Every positive integer was one of [Ramanujan's] personal friends.

Generalisations of this idea have created the notion of " taxicab numbers ". That's one reason I always read letters that come in from obscure places and are written in an illegible scrawl. I always hope it might be from another Ramanujan. In his obituary of Ramanujan, written for Nature in , Hardy observed that Ramanujan's work primarily involved fields less known even among other pure mathematicians, concluding:.

His insight into formulae was quite amazing, and altogether beyond anything I have met with in any European mathematician. It is perhaps useless to speculate as to his history had he been introduced to modern ideas and methods at sixteen instead of at twenty-six.