Srinivasa Ramanujan

"If n is any positive quantity shew that \frac 1{n} > \frac 1{n+1} + \frac 1{{(n+2)}^2} + \frac 3{{(n+3)}^3} + \frac {4^2}{{(n+4)}^4} + \frac {5^3}{{(n+5)}^5} + \dots Find the difference approximately when n is great. Hence shew that \frac 1{1001} + \frac 1{1002^2} + \frac 3{1003^3} + \frac {4^2}{1004^4} + \frac {5^3}{1005^5} + \dots < \frac 1{1000} by 10^{-440} nearly."

"Sir, an equation has no meaning for me unless it expresses a thought of GOD."

"I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras... I have 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". ...Very recently I came across a tract published by you styled Orders of Infinity in page 36 of which I find a statement that no definite expression has been as yet found for the number of prime numbers less than any given number. I have found an expression which very nearly approximates to the real result, the error being negligible. I would request that you go through the enclosed papers. Being poor, if you are convinced that there is anything of value I would like to have my theorems published. I have not given the actual investigations nor the expressons that I get but I have indicated the lines on which I proceed. Being inexperienced I would very highly value any advice you give me. Requesting to be excused for the trouble I give you. I remain, Dear Sir, Yours truly..."

"Ramanujan lived in a tiny hut in India. No formal education, no access to other works. But he came across an old math book and from this basic text he was able to extrapolate theories that had baffled mathematicians for years. … Ramanujan's genius was unparalleled."

"Paul Erdős has passed on to us Hardy's personal ratings of mathematicians. Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100, Hardy gave himself a score of 25, Littlewood 30, Hilbert 80 and Ramanujan 100."

"He began to focus on mathematics at an early age, and, at the age of about fifteen, borrowed a copy of G. S. Carr's Synopsis of Pure and Applied Mathematics, which served as his primary source for learning mathematics. Carr was a tutor and compiled this compendium of approximately 4000-5000 results (with very few proofs) to facilitate his tutoring."

"At about the time Ramanujan entered college, he began to record his mathematical discoveries in notebooks... Ramanujan devoted all of his efforts to mathematics and continued to record his discoveries without proofs in notebooks for the next six years."

"After Ramanujan died, Hardy strongly urged that Ramanujan's notebooks be edited and published. By "editing," Hardy meant that each claim made by Ramanujan in his notebooks should be examined. If a theorem is known, sources providing proofs should be provided; if an entry is known, then an attempt should be made to prove it."

"Ramanujan's approach to the theory of theta functions does not appear to have been influenced by any other writer."

"He was sent at seven to the High School at , and remained there nine years. ...His biographers say ...that soon after he had begun the study of , he discovered for himself "Euler's theorems for the sine and cosine (by which I understand the relations between the circular and exponential functions), and was very disappointed when he found later, apparently from the second volume of Loney's Trigonometry that they were known already. Until he was sixteen he had never seen a mathematical book of higher class. Whittaker's Modern Analysis had not yet spread so far, and Bromwich's Infinite Series did not exist. ...[E]ither of these books would have made a tremendous difference ..."

"Ramanujan did not seem to have any definite occupation, except mathematics, until 1912. In 1909 he married, and it became necessary for him to have some regular employment, but he had great difficulty in finding any because of his unfortunate college career. About 1910 he began to find more influential Indian friends, Ramaswami Aiyar and his two biographers, but all their efforts to find a tolerable position for him failed, and in 1912 he became a clerk in the office of the Port Trust of Madras, at a salary of about £30 per year. He was nearly twenty-five. The years between eighteen and twenty-five are the critical years in a mathematician's career, and the damage had been done. Ramanujan's genius never had again its chance of full development."

"It has not the simplicity and the inevitableness of the very greatest work; it would be greater if it were less strange. One gift it shows... profound and invincible originality. He would probably been a greater mathematician if he could have been caught and tamed a little in his youth; he would have discovered more that was new, and... of greater importance. On the other hand he would have been less of a Ramanujan, and more of a European professor, and the loss might have been greater than the gain... the last sentence is... ridiculous sentimentalism. There was no gain at all when the College at Kumbakonam rejected the one great man they had ever possessed, and the loss was irreparable..."

"The formulae... defeated me completely; I had never seen anything in the least like them before. A single look at them is enough to show that they could only have been written by a mathematician of the highest class. They must be true because, if they were not true, no one would have the imagination to invent them."

"I hardly asked him a single question of this kind; I never even asked him whether (as I think he must have done) he had seen Cayley's or Greenhill's Elliptic Functions. ... he was a mathematician anxious to get on with the job. And after all I too was a mathematician, and a mathematician meeting Ramanujan had more interesting things to think about than historical research. It seemed ridiculous to worry him about how he had found this or that known theorem, when he was showing me half a dozen new ones almost every day."

"He could remember the idiosyncrasies of numbers in an almost uncanny way. It was Littlewood who said that every positive integer was one of Ramanujan's personal friends. I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.""

"The years between 18 and 25 are the critical years in a mathematician's career, and the damage had been done. Ramanujan's genius never had again its chance of full development. ... a mathematician is often comparatively old at 30, and his death may be less of a catastrophe than it seems. Abel died at 26 and, although he would no doubt have added a great deal more to mathematics, he could hardly have become a greater man. The tragedy of Ramanujan was not that he died young, but that, during his five unfortunate years, his genius was misdirected, side-tracked, and to a certain extent distorted."

"In his insight into algebraical formulae, transformation of infinite series, and so forth, that was most amazing. On this side most certainly I have never met his equal, and I can compare him only with Euler or Jacobi."

"The formulae (1.10) - (1.13) are on a different level and obviously both difficult and deep... (1.10) - (1.12) defeated me completely; I had never seen anything in the least like them before. A single look at them is enough to show that they could only be written by a mathematician of the highest class. They must be true because, if they were not true, no one would have the imagination to invent them."

"His death is the saddest event in my professional career. It is not for me to assess Ramanujan's mathematical genius. But at the human level, he was one of the noblest men I have met in my life-shy, reserved and endowed with an infinite capacity to bear the agonies of the mind and spirit with fortitude."

"Srinivasa Ramanujan was the strangest man in all of mathematics, probably in the entire history of science. He has been compared to a bursting supernova, illuminating the darkest, most profound corners of mathematics, before being tragically struck down by tuberculosis at the age of 33, like Riemann before him."

"The number 24 appearing in Ramanujan's function is also the origin of the miraculous cancellations occurring in string theory. ...each of the 24 modes in the Ramanujan function corresponds to a physical vibration of a string. Whenever the string executes its complex motions in space-time by splitting and recombining, a large number of highly sophisticated mathematical identities must be satisfied. These are precisely the mathematical identities discovered by Ramanujan. ...The string vibrates in ten dimensions because it requires... generalized Ramanujan functions in order to remain self-consistent."

"A more recent example of anomalous creativity is the work of the great mathematician Srinivasa Ramanujan, who died in 1920 at the age of 32. His notebook, which was lost and forgotten for about 50 years and published only in 1988, contains several thousand formulas -- without proof, in different areas of mathematics -- that were well ahead of their time, and the methods by which he found the formulas remain elusive. Ramanujan himself claimed that the formulas were revealed to him in his sleep. Is it possible that the creative process draws upon the unconscious in a manner that cannot be explained in rational terms?"

"Ramanujan learned from an older boy how to solve cubic equations. He came to understand trigonometric functions not as the ratios of the sides in a right triangle, as usually taught in school, but as far more sophisticated concepts involving infinite series. He'd rattle off the numerical values of π and e, "transcendental" numbers appearing frequently in higher mathematics, to any number of decimal places. He'd take exams and finish in half the allotted time. Classmates two years ahead would hand him problems they thought difficult, only to watch him solve them at a glance. … By the time he was fourteen and in the fourth form, some of his classmates had begun to write Ramanujan off as someone off in the clouds with whom they could scarcely hope to communicate. "We, including teachers, rarely understood him," remembered one of his contemporaries half a century later. Some of his teachers may already have felt uncomfortable in the face of his powers. But most of the school apparently stood in something like respectful awe of him, whether they knew what he was talking about or not. He became something of a minor celebrity. All through his school years, he walked off with merit certificates and volumes of English poetry as scholastic prizes. Finally, at a ceremony in 1904, when Ramanujan was being awarded the K. Ranganatha Rao prize for mathematics, headmaster Krishnaswami Iyer introduced him to the audience as a student who, were it possible, deserved higher than the maximum possible marks. An A-plus, or 100 percent, wouldn't do to rate him. Ramanujan, he was saying, was off-scale."

"Ramanujan was an artist. And numbers — and the mathematical language expressing their relationships — were his medium. Ramanujan's notebooks formed a distinctly idiosyncratic record. In them even widely standardized terms sometimes acquired new meaning. Thus, an "example" — normally, as in everyday usage, an illustration of a general principle — was for Ramanujan often a wholly new theorem. A "corollary" — a theorem flowing naturally from another theorem and so requiring no separate proof — was for him sometimes a generalization, which did require its own proof. As for his mathematical notation, it sometimes bore scant resemblance to anyone else's."

"Ramanujan was a man for whom, as Littlewood put it, "the clear-cut idea of what is meant by proof ... he perhaps did not possess at all"; once he had become satisfied of a theorem's truth, he had scant interest in proving it to others. The word proof, here, applies in its mathematical sense. And yet, construed more loosely, Ramanujan truly had nothing to prove. He was his own man. He made himself. "I did not invent him," Hardy once said of Ramanujan. "Like other great men he invented himself." He was svayambhu."

"Graduating from high school in 1904, he entered the University of Madras on a scholarship. However, his excessive neglect of all subjects except mathematics caused him to lose the scholarship after a year, and Ramanujan dropped out of college. He returned to the university after some traveling through the countryside, but never graduated. ...His marriage in 1909 compelled him to earn a living. Three years later, he secured a low-paying clerk's job with the Madras Port Trust."

"Every positive integer is one of Ramanujan's personal friends."

"I read in the proof-sheets of Hardy on Ramanujan: 'As someone said, each of the positive integers was one of his personal friends.' My reaction was, 'I wonder who said that; I wish I had.' In the next proof- sheets I read (what now stands), 'It was Littlewood who said... '"

"Ramanujan's great gift is a 'formal' one; he dealt in 'formulae'. To be quite clear what is meant, I give two examples (the second is at random, the first is one of supreme beauty):p(4)+p(9) x+p(14) x^2+\ldots=5 \frac{\left\{\left(1-x^5\right)\left(1-x^{10}\right)\left(1-x^{15}\right) \ldots\right\}^5}{\left\{(1-x)\left(1-x^2\right)\left(1-x^3\right) \ldots\right\}^6} where p(n) is the number of partitions of n; ... But the great day of formulae seems to be over. No one, if we are again to take the highest standpoint, seems able to discover a radically new type, though Ramanujan comes near it in his work on partition series; it is futile to multiply examples in the spheres of Cauchy's theorem and elliptic function theory, and some general theory dominates, if in a less degree, every other field. A hundred years or so ago his powers would have had ample scope... The beauty and singularity of his results is entirely uncanny... the reader at any rate experiences perpetual shocks of delighted surprise. And if he will sit down to an unproved result taken at random, he will find, if he can prove it at all, that there is at lowest some 'point', some odd or unexpected twist. ... His intuition worked in analogies, sometimes remote, and to an astonishing extent by empirical induction from particular numerical cases... his most important weapon seems to have been a highly elaborate technique of transformation by means of divergent series and integrals. (Though methods of this kind are of course known, it seems certain that his discovery was quite independent.) He had no strict logical justification for his operations. He was not interested in rigour, which for that matter is not of first-rate importance in analysis beyond the undergraduate stage, and can be supplied, given a real idea, by any competent professional."

"He was eager to work out a theory of reality which would be based on the fundamental concept of "zero", "infinity" and the set of finite numbers … He sometimes spoke of "zero" as the symbol of the absolute (Nirguna Brahman) of the extreme monistic school of Hindu philosophy, that is, the reality to which no qualities can be attributed, which cannot be defined or described by words and which is completely beyond the reach of the human mind. According to Ramanuja the appropriate symbol was the number "zero" which is the absolute negation of all attributes."

"Srinivasa Ramanujan, discovered by the Cambridge mathematician G. H. Hardy, whose great mathematical findings were beginning to be appreciated from 1915 to 1919. His achievements were to be fully understood much later, well after his untimely death in 1920. 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."

"Ramanujam used to show his notes to me, but I was rarely able to make head or tail of at least some of the things he had written. One day he was explaining a relation to me; then he suddenly turned round and said, "Sir, an equation has no meaning for me unless it expresses a thought of GOD." I was simply stunned. Since then I had meditated over this remark times without number. To me, that single remark was the essence of Truth about God, Man and the Universe. In that statement, I saw the real Ramanujam, the philosopher mystic-mathematician."

"The manuscript of Ramanujan contained theorems and propositions that Hardy classified in three categories: 1) important results already known or demonstrable, through theorems which Ramanujan was certainly not acquainted with; 2) false results (few in number) or results concerning marginal curiosities; 3) important theorems not demonstrated, but formulated in such a manner that presupposed views... which only a genius could have."

"Hardy... in vain, tried to convince him to learn classical foundations of mathematics and, in particular, the rigorous expositive method of mathematical demonstrations. Every time Hardy introduced a problem, Ramanujan considered it ex novo [new] applying unconventional reasoning which was sometimes incomprehensible to his fellow colleagues."

"The great advances in mathematics have not been made by logic but by creative imagination. The title of mathematician can scarcely be denied to Ramanajan who hardly gave any proofs of the many theorems which he enumerated."

"That Ramanujan conceived these problems, sometimes before anyone else had done so, with no contact with the European mathematical community, and that he correctly obtained the dominant terms in asymptotic formulas are astounding achievements that should not be denigrated because of his unrigorous, but clever, arguments."

"Ramanujan proved many theorems for products of hypergeometric functions and stimulated much research by W. N. Bailey and others on this topic."