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 ",  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 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 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.
The paper was more than 50 pages and proved various properties of such numbers. Hardy remarked that it was one of the most unusual papers seen in mathematical research at that time and that Ramanujan showed extraordinary ingenuity in handling it. 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.
Throughout his life, Ramanujan was plagued by health problems. 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 in England and wartime rationing during — He was diagnosed with tuberculosis and a severe vitamin deficiency at the time, and was confined to a sanatorium. In he returned to Kumbakonam , Madras Presidency , and soon thereafter, in , died at the age of 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.
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 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. She died at her Triplicane residence in A analysis of Ramanujan's medical records and symptoms by Dr. 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. Ramanujan has been described as a person of a somewhat shy and quiet disposition, a dignified man with pleasant manners. He credited his acumen to his family goddess , Mahalakshmi of Namakkal. He looked to her for inspiration in his work : Afterward he would receive visions of scrolls of complex mathematical content unfolding before his eyes.
Hardy cites Ramanujan as remarking that all religions seemed equally true to him. At the same time, he remarked on Ramanujan's strict vegetarianism. 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.
See also the more general Ramanujan—Sato series. 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. Hans Rademacher , in , was able to refine 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, there is one that was highly influential on 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 which "explains" these congruences and more generally all modular forms. Pierre 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 loose-leaf 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.
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. This may have been for any number of 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 the Madras Presidency at the time. He was also quite likely to have been influenced by the style of G.
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 recorded only the results. The first notebook has pages with 16 somewhat organised chapters and some unorganised material. The second notebook has pages in 21 chapters and unorganised pages, with the third notebook containing 33 unorganised 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. Wilson , and Bruce Berndt.
The number is known as the Hardy—Ramanujan number after a famous visit by Hardy to see Ramanujan at a hospital. 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.
Generalizations of this idea have created the notion of " taxicab numbers ". In his obituary of Ramanujan, which he wrote for Nature in , Hardy observed Ramanujan's work primarily involved fields less known even amongst 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.
It is not extravagant to suppose that he might have become the greatest mathematician of his time. What he actually did is wonderful enough He combined a power of generalization, a feeling for form, and a capacity for rapid modification of his hypotheses, that were often really startling, and made him, in his own peculiar field, without a rival in his day.
The limitations of his knowledge were as startling as its profundity. Here was a man who could work out modular equations and theorems When asked about the methods Ramanujan employed to arrive at his solutions, Hardy said that they were "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account.
Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to , Hardy gave himself a score of 25, J. Littlewood 30, David Hilbert 80 and Ramanujan In his book Scientific Edge , the physicist Jayant Narlikar spoke of "Srinivasa Ramanujan, discovered by the Cambridge mathematician Hardy, whose great mathematical findings were beginning to be appreciated from to His achievements were to be fully understood much later, well after his untimely death in For example, his work on the highly composite numbers numbers with a large number of factors started a whole new line of investigations in the theory of such numbers.
The year after his death, Nature listed Ramanujan among other distinguished scientists and mathematicians on a "Calendar of Scientific Pioneers," who had achieved eminence. A stamp picturing Ramanujan was released by the Government of India in — the 75th anniversary of Ramanujan's birth — commemorating his achievements in the field of number theory,  and a new design was issued on 26 December , by the India Post.
A prize for young mathematicians from developing countries has been created in Ramanujan's name by the International Centre for Theoretical Physics ICTP in cooperation with the International Mathematical Union , which nominate members of the prize committee.
House of Ramanujan Mathematics, a museum on life and works of the Mathematical prodigy, Srinivasa Ramanujan, also exists on this campus. In , on the th anniversary of his birth, the Indian Government declared that 22 December will be celebrated every year as National Mathematics Day. From Wikipedia, the free encyclopedia. This is the latest accepted revision , reviewed on 11 September For other uses, see Ramanujan disambiguation.
In this Indian name , the name Srinivasa is a patronymic , not a family name , and the person should be referred to by the given name , Ramanujan. List of things named after Srinivasa Ramanujan. Charlemagne and His Heritage: Number Theory in the Spirit of Ramanujan. The American Mathematical Monthly. Facts on File Inc. The Man Who Knew Infinity: Kolata, Gina 19 June Not much is known about his early life and schooling except that he was a solitary child by nature.
It is believed that he was born as a result of ardent prayers to the goddess Namgiri. Later Ramanujan attributed his mathematical power to this goddess of creation and wisdom. For him nothing was useful unless it expressed the essence of spirituality.
Ramanujan found mathematics as a profound manifestation of the Reality. He was such a great mathematician and genius as transcends all thoughts and imagination. He was an expert in the interpretation of dreams and astrology. These qualities he had inherited from his mother. His interest and devotion to mathematics was to the point of obsession. He ignored everything else and would play with numbers day and night on a slate and in his mind.
One day he came to possess G. Ramanujan made it his constant companion and improved it further on his own.
His obsession and preoccupation with mathematics did not allow him to pass his intermediate examination in spite of three attempts. He could not get even the minimum pass marks in other subjects. Ramanujan was married to a nine year old girl called lauaki and it added more to his family responsibilities. Janaki and her mother joined Ramanujan in after he secured a job. This was a stable job that allowed him plenty of time to continue his research in mathematics.
He earned 20 rupees per month. During this period Ramanujan was completely focused on Mathematics. A friend who knew him from college noted that when Ramanujan was concentrating on some mental tasks, the pupils of his eyes disappear.
His former Professor Seshu Aiyer was so impressed with his work after reviewing the publications that he had requested Ramanujan to write a letter to then famous mathematician G. Hardy, a fellow of trinity college, Cambridge.
In Ramanujan wrote a letter to Dr. Hardy that was 11 pages long with divergent series without proofs. 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 not given the actual investigations nor the expressions that I get but I have indicated the lines on which I proceed. Page 1 of Read Full Essay Save.
ADVERTISEMENTS: Read this Essay on Srinivasa Ramanujan ( A.D. – A.D.)! One of the greatest mathematicians of India, Ramanujan’s contribution to the theory of numbers has been profound. He was indeed a mathematical phenomenon of the twentieth century. This legendary genius of India ranks among the all time greats like Euler and .
Srinivasa Ramanujan (22 December 26 April ) was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions.
Srinivasa Ramanujan Ramanujan was born in India to a poor family in Erode, a city in Madras state. His father was a clerk and his mother a deeply religious housewife. His father was a clerk and his mother a deeply religious housewife. Srinivasa Ramanujan: Srinivasa Ramanujan, Indian mathematician who made pioneering contributions to number theory.
Read Srinivasa Ramanujan free essay and over 88, other research documents. Srinivasa Ramanujan. It is one of the most romantic stories in the history of mathematics: in , the English mathematician G. H/5(1). In ramanujan journal was launched to publish work in areas mathematics influenced by Ramanujan”. thiyagusurimathematicians. blogspot. in//07/modulesrinivasa-ramanujanad. html 3/3 Popular Essays.