Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work
I suppose that the book is substantially a summary of Carr's coaching notes. If you were a pupil of Carr, you worked through the appropriate sections of the Synopsis. It covers roughly the subjects of Schedule A of the present Tripos as these subjects were understood in Cambridge in , and is effectively the " synopsis " it professes to be.
It contains the enunciations of theorems, systematically and quite scientifically arranged, with proofs which are often little more than cross-references and are decidedly the least interesting part of the book. All this is exaggerated in Ramanujan's famous notebooks which contain practically no proofs at all , and any student of the notebooks can see that Ramanujan's ideal of presentation had been copied from Carr's. In December he passed the Matriculation Examination of the University of Madras, and in the January of the succeeding year he joined the Junior First in Arts class of the Government College, Kumbakonam, and won the Subrahmanyam scholarship, which is generally awarded for proficiency in English and Mathematics He had an impossible handicap -- a poor and solitary Hindu pitting his brains against the accumulated wisdom of Europe.
There is not much to say about the rest of Ramanujan's life. His first substantial paper had been published in , and in his exceptional powers began to be understood. It is significant that, though Indians could befriend him, it was only the English who could get anything effective done. Hardy Hardy, G. The British were the colonial rulers of India, justifying their rule in terms of their manifest destiny to improve the globe.
Under these circumstances, an unknown Indian writes a letter to two mathematicians at Cambridge - E. Neville and G. A long list of results are attached. They see a spark somewhere, that they cannot explain. Hardy noted later that it could not have been a charlatan, since "great mathematicians are commoner than humbugs of such incredible skill. They bring Ramanujan to Cambridge, and the clash of cultures continues.
He is a strict vegetarian, a devotee of goddess Namakkal - the deity Namagiri Lakshmi at the district town of Namakkal, not far from Trichy. At Cambridge, he cooks his own food. Hardy tells us that "he never cooked without first changing into pyjamas. The mystery of a colonial encounter Given this lifestyle, Hardy is surprised one day, when ramanujan tells him "that all religions seemed to him more or less equally true.
Therefore religion, concludes Hardy, except in a strictly material sense, played no important part in his life. Steeped in his own religious culture, Hardy is unable to understand a tradition that accepts multiplicity of religious thought. You can have your own gods, and I can have mine.
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Hardy appears to interpret this statement as that of holding no belief: if a strict Brahmin like Ramanujan told me, as he certainly did, that he had no definite beliefs, then it is to 1 that he meant what he said. So we find that "all religions are equally true" is taken to be equivalent form for the statement "he had no definite beliefs". Based on this, Hardy is convinced that Ramanujan was merely an "observing Hindu", not a true believer He was not a reasoned infidel, but an "agnostic " in its strict sense, who saw no particular good, and no particular harm, in Hinduism or in any other religion.
In this manner Hardy seems to be reconstructing Ramanujan into the straitjacket of his own cultural milieu. On the basis of this understanding he challenges the views of Seshu Aiyar and Ramachaundra Rao, who had known Ramanujan in Madras: Ramanujan had definite religious views.
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He had a special veneration for the Namakkal goddess He believed in the existence of a Supreme Being and in the attainment of Godhead by men He had settled convictions about the problem of life and after For my part I have no doubt at all; I am quite certain that I am This type of cultural misunderstanding apart, the mathematics of the interaction speaks in its own language, where too there are many misunderstandings, owing to the very different background that Ramanujan had.
But here, Hardy is on much more solid ground in terms of interpreting the provenance of his find. Ramanujan's mathematics: The Partition function Most of us, even those who have had a reasonably good mathematical education, are quite lost when it comes to judge Ramanujan's quality. Here is a simple example, that most people can grasp. The partition function grows very rapidly. Earlier, Euler and others had proposed complex recurrence relations where you do long infinite sums to get the value of P n. But Ramanujan came up with the approximation: With some help from Hardy, they were able to show that this result follows from some theorems in modular functions.
But, to check the theorem a fellow Cambridge mathematician tallied by hand the partitions for Many have commented that Ramanujan's numerical accuracy was unprecedented - matched perhaps only by Euler or Jacobi. One can never cease to wonder how such insights could have come to Ramanujan. Reading his biography, together with other calculating prodigies and other gifted individuals, sometimes one thinks that for some types of minds or maybe all? Hardy noted that he was in the best position to resolve this mystery, but admitted not even once asking Ramanujan about the source of his inspiration.
The Hardy—Ramanujan asymptotic series for the partition function is the main focus of the mathematical discussions in The Man Who Knew Infinity. The Hardy—Ramanujan formula for partitions is also one of the greatest achievements in number theory and the crowning glory of their collaboration.
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Additionally, this discovery illustrates beautifully how the intuition of Ramanujan and the sophistication of Hardy combined to establish a truly remarkable result. A partition of a positive integer n is a representation of n as a sum of positive integers, two such representations being considered the same if they differ only in the order of the summands, or parts. The number of partitions of n is denoted by by p n. For a small number like 4, it is easy enough to work out all its partitions.
How then do we know the value of p ?
In the mid-eighteenth century, Leonhard Euler, founder of the theory of partitions, discovered the following remarkable recurrence relation:. This seemed impossible because partitions represent a discrete process. But his remark conveys the near impossibility of such a representation using continuous functions without resorting to technical jargon that would be indecipherable for the public. Euler was not only interested in p n , but in other partition functions as well. Many of these have generating functions with elegant infinite product or series representations.
These generating function evaluations yield beautiful relations among various partition functions. From the fundamental Cauchy residue theorem in complex variable theory—formulated a few decades after the time of Euler—it follows that for each positive integer n ,. The difficulty here is in determining the contour that would lead to the evaluation of p n.
Notice that the product part in 5 indicates that F z would become large when z is near a root of unity. This is very deep, difficult, and sophisticated, and is at the heart of the powerful circle method that they introduced in their famous paper of The final result they proved is that. There is a wonderful convention in mathematics that for a joint paper, all authors are equal. Hardy was a firm believer in this practice and did not elaborate on the respective contributions of the pair. At this point Hardy asked P.
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MacMahon, a noted combinatorialist, to check the formula for numerical accuracy. This was established by Derrick Henry Lehmer in In 2 , the expression on the right is equal to.
Thus 2 and 9 are generating functions of a weighted version of p n. In 2 Ramanujan was using hyperbolic functions, just as Rademacher did for p n. While 2 is not correct as stated by Ramanujan in his first letter to Hardy, it does yield a genuine approximation. During their meeting, MacMahon challenges Ramanujan to compute the square root of a particular number. Ramanujan responds instantly.
MacMahon then invites Ramanujan to pose a computational problem for him to solve. MacMahon provides the answer, just as quickly. From time to time, MacMahon and Ramanujan engaged in friendly computational contests. Despite Ramanujan having found several spectacular results, Hardy needed something totally unexpected and staggering, such as the asymptotic series for p n , to obtain these honors. The scenes depicting the tremendous efforts made by Hardy attempting to convince the academic aristocracy that Ramanujan should be honored are deeply moving.
He was, however, successful in getting Ramanujan elected a fellow of the Royal Society. S hortly after Ramanujan arrives in Cambridge, he is shown walking across a lawn at Trinity College only to be stopped by a guard. Ramanujan is ordered to stay off the grass and follow the path running around the edge of the lawn. Unbeknownst to Ramanujan, only a privileged few, such as fellows of the college, are entitled to traverse the sacrosanct and immaculate lawns. After Ramanujan is elected a fellow of Trinity, Hardy informs him that he may now walk across the lawn with confidence. Similarly, while Hardy insisted on proofs and conveyed their importance to Ramanujan, he did not chastise Ramanujan to the extent shown in the film.
In one scene Hardy tells Ramanujan that he will not see him again unless he provides proofs for certain results. Hardy was never this harsh with Ramanujan. In reality, Hardy helped Ramanujan write up his results and filled in missing steps so that his work could be published.
One scene depicts Ramanujan jumping with joy when Hardy hands him a reprint of his paper on highly composite numbers. By contrast, the depictions of discrimination and racial prejudice against Ramanujan in England are overdone.