Biography hardy ramanujan partition function

National Academy of Sciences. June Retrieved Advances in Mathematics. Bibcode : arXiv The Ramanujan Journal. External links [ edit ]. Categories : Theorems in number theory Srinivasa Ramanujan Equivalence mathematics. Hidden categories: Articles with short description Short description matches Wikidata. Toggle the table of contents. Ramanujan's congruences.

And to my mind, the most remarkable thing about Ramanujan is that he could define something as seemingly arbitrary as this, and have it turn out to be useful a century later. But to us today, this just seems like a random fact of mathematics, not of any particular significance. Instead, more and more of them are being found to be connected to deep, elegant mathematical principles.

To enunciate these principles in a direct and formal way requires layers of abstract mathematical concepts and language which have taken decades to develop. But somehow, through his experiments and intuition, Ramanujan managed to find concrete examples of these principles. Often his examples look quite arbitrary—full of seemingly random definitions and numbers.

And part of the reason seems to be that to do so—and to create the kind of narrative needed for a good proof—one actually has no choice but to build up much more abstract and conceptually complex structures, often in many steps. So how is it that Ramanujan managed in effect to predict all these deep principles of later mathematics?

I think there are two basic logical possibilities. The first is that if one drills down from any sufficiently surprising result, say in number theory, one will eventually reach a deep principle in the effort to explain it. And the second possibility is that while Ramanujan did not have the wherewithal to express it directly, he had what amounts to an aesthetic sense of which seemingly random facts would turn out to fit together and have deeper significance.

But to understand this a little more, we should talk about the overall structure of mathematics. At an underlying level, mathematics is based on simple axioms. So the question then is, why should the truth of what seem like random facts of number theory even be decidable? Conceivably the Goldbach conjecture will turn out to be an example.

Ramanujan no doubt convinced himself of many of his results by what amount to empirical methods—and often it worked well. In the case of the counting of primes, however, as Hardy pointed out, things turn out to be more subtle, and results that might work up to very large numbers can eventually fail. Now the next question: are these statements connected in any way?

Imagine one could find proofs of the statements that are true. These proofs effectively correspond to paths through a directed graph that starts with the axioms, and leads to the true results. One possibility is then that the graph is like a star—with every result being independently proved from the axioms. I have to say that these considerations lead to an important question for me.

I have spent many years studying what amounts to a generalization of mathematics: the behavior of arbitrary simple programs in the computational universe. But I have also found evidence—not least through my Principle of Computational Equivalence —that undecidability is rife there. But now the question is, when one looks at all that rich and complex behavior, are there in effect Ramanujan-like facts to be found there?

But perhaps there are networks of facts that can be reasoned about—and that all connect to deeper principles of some kind. Repetitive behavior and nested behavior are two almost trivial examples. But now the question is whether among all the specific details of particular programs there are other general forms of organization to be found. Of course, whereas repetition and nesting are seen in a great many systems, it could be that another form of organization would be seen only much more narrowly.

Will there ever be another Ramanujan? Are these numerical facts significant? Number theory is a common topic. So are relativity and gravitation theory. And particularly in recent years, AI and consciousness have been popular too. The pictures I got certainly seemed visually interesting. And one consequence of this is that I frequently get letters that show remarkable behavior in some particular cellular automaton or other simple program.

Many years later we would collect as many of these as we could to build them into the algorithms and knowledgebase of Mathematica and the Wolfram Language. But at the time probably the most significant aspect of their publication was the proofs that were given: the stories that explained why the results were true. Because in these proofs, there was at least the potential that concepts were introduced that could be reused elsewhere, and build up part of the fabric of mathematics.

It would take us too far afield to discuss this at length here, but there is a kind of analog in the study of the computational universe: the methodology for computer experiments. Just as a proof can contain elements that define a general methodology for getting a mathematical result, so the particular methods of search, visualization or analysis can define something in computer experiments that is general and reusable, and can potentially give an indication of some underlying idea or principle.

When a letter one receives contains definite mathematics, in mathematical notation, there is at least something concrete one can understand in it. I have no story yet as dramatic as Hardy and Ramanujan. Ramanujan did his calculations by hand—with chalk on slate, or later pencil on paper. Today with Mathematica and the Wolfram Language we have immensely more powerful tools with which to do experiments and make discoveries in mathematics not to mention the computational universe in general.

I rather think he would have been quite an adventurer—going out into the mathematical universe and finding all sorts of strange and wonderful things, then using his intuition and aesthetic sense to see what fits together and what to study further. Ramanujan unquestionably had remarkable skills. Posted in: Historical Perspectives , Mathematics.

Please enter your comment at least 5 characters. Please enter a valid email address. The casual reader may find it a bit lengthy, but your personal inputs and speculations are amazing. The unique perspective of not being a mathematician yourself but being extensively involved with the subject really makes this a unique read. Thank you, sir!

Interesting, I had no idea this was such a big deal. Made no sense to me. If they are potentially interesting, I could raid some attics. Very nice article. I look forward to seeing this movie. Thank you! If experimental mathematics becomes more default way of exploring first , then I am sure we will make faster progress. Wow great stuff. My Math teacher in college was Indian and boy did he know his stuff.

Fascinating and brilliant article, carefully constructed and very well documented in its essence. Thank you. Nice read sir. I am just a simple Dutch math-teacher, but I have read your article with great pleasure. It is very inspiring and I hope to share it with some of my best pupils. Mr Mark Littlewood will do a great service if he could raid the attics and unearth whatever papers of his own great uncle and Ramanujan and he could lay his hands on.

The world of mathematics and science will be grateful. Fascinating read! This should be a two day paper-and pencil job for a grad student in I actually went to college because of Ramanujan. This article is wonderful and thank you for adding your thoughts on arguably one of the greatest mathematicians the wold has ever seen. Thank you for this very impressive article, Sir!

I know next to nothing about mathematics but I was intrigued by the movie and wanted to find out more about Ramanujan. I really enjoyed the article even though a lot of it was way over my head! Your writing certainly kept my interest. Thanks sir for this post. It is very very interesting. I like it. And perhaps first post in my internet-life, I read a post all at a time.

This article would have been so much better with no mentions of Hardy. Ramanujan is one of the, sorry the best Mathematician of the entire world, no western mathematician can even touch the dust of his feet, such was his skills. Apparently India failed to properly recognize Ramanujan as they are caught up in their own turmoil by using English as their language for education.

I did empirical studies like Ramujan to fill in time while studying at university. As stated, most findings are mere curiosities. The real highly-advanced skill is to recognize the useful ones. It is pretty hot here in India and therefore we tend to avoid physical work. We like to sit in the shade and brood about the nature of things. Sometimes we do accomplish much with a paper and pencil and of course knowledge and ideas.

He was nearly zeroised by Eddington. But we are a patient breed. We still learn our multiplication tables. And we do respect scholarship. Do read how the Russian Perelman did it solo. Hats off to all the mathematicians who struggling it alone. In , S. Barnard and J. Child stated that the different types of partitions of n in symbolic form.

In this paper, different types of partitions of n are also explained with symbolic form.

Biography hardy ramanujan partition function

In , E. Grosswald quoted that the linear Diophantine equation has distinct solutions; the set of solution is the number of partitions of n. This paper proves theorem 1 with the help of certain restrictions. In , Godfrey Harold Hardy and E. Theorem 2 has been proved here with easier mathematical calculations. In , British mathematician Norman Macleod Ferrers explained a partition graphically by an array of dots or nodes.

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. It is not extravagant to suppose that he might have become the greatest mathematician of his time.

What he actually did is wonderful enough… when the researches which his work has suggested have been completed, it will probably seem a good deal more wonderful than it does to-day. Hardy further said: [ ]. He combined a power of generalisation, 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 As an example, Hardy commented on 15 theorems in the first letter. Of those, the first 13 are correct and insightful, the 14th is incorrect but insightful, and the 15th is correct but misleading. When asked about the methods Ramanujan used to arrive at his solutions, Hardy said they were "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account.

Hardy thought his achievements were greatest in algebra, especially hypergeometric series and continued fractions. It is possible that the great days of formulas are finished, and that Ramanujan ought to have been born years ago; but he was by far the greatest formalist of his time. There have been a good many more important, and I suppose one must say greater, mathematicians than Ramanujan during the last 50 years, but not one who could stand up to him on his own ground.

Playing the game of which he knew the rules, he could give any mathematician in the world fifteen. He discovered fewer new things in analysis, possibly because he lacked the formal education and did not find books to learn it from, but rediscovered many results, including the prime number theorem. In analysis, he worked on the elliptic functions and the analytic theory of numbers.

In analytic number theory , he was as imaginative as usual, but much of what he imagined was wrong. Hardy blamed this on the inherent difficulty of analytic number theory, where imagination had led many great mathematicians astray. In analytic number theory, rigorous proof is more important than imagination, the opposite of Ramanujan's style.

His "one great failure" is that he knew "nothing at all about the theory of analytic functions ". Littlewood reportedly said that helping Ramanujan catch up with European mathematics beyond what was available in India was very difficult because each new point mentioned to Ramanujan caused him to produce original ideas that prevented Littlewood from continuing the lesson.

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 The year after his death, Nature listed Ramanujan among other distinguished scientists and mathematicians on a "Calendar of Scientific Pioneers" who had achieved eminence.

Stamps picturing Ramanujan were issued by the government of India in , , and The International Centre for Theoretical Physics ICTP has created a prize in Ramanujan's name for young mathematicians from developing countries in cooperation with the International Mathematical Union , which nominates members of the prize committee. House of Ramanujan Mathematics, a museum of Ramanujan's life and work, is also 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. Situated next to the Tidel Park , it includes 25 acres 10 ha with two zones, with a total area of 5. Commemorative stamps released by India Post by year :. Contents move to sidebar hide. Article Talk.

Read View source View history. Tools Tools. Download as PDF Printable version. In other projects. Wikimedia Commons Wikiquote Wikisource Wikidata item. Indian mathematician — For other uses, see Ramanujan disambiguation. In this Indian name , the name Srinivasa is a patronymic , and the person should be referred to by the given name , Ramanujan.

Ramanujan's sum Landau—Ramanujan constant Mock theta functions Ramanujan's congruences Ramanujan conjecture Ramanujan prime Ramanujan—Soldner constant Ramanujan theta function Rogers—Ramanujan identities Ramanujan's master theorem Hardy—Ramanujan asymptotic formula Ramanujan—Sato series. Hardy J. Pursuit of career in mathematics. Contacting British mathematicians.

Personality and spiritual life. Mathematical achievements. Main article: Ramanujan—Petersson conjecture. Ramanujan's notebooks. Further information: Ramanujan's lost notebook. Hardy—Ramanujan number Main article: number. Mathematicians' views of Ramanujan. Further information: List of things named after Srinivasa Ramanujan. Commemorative postal stamps.

Ramanujan, S. Messenger Math. Proceedings of the London Mathematical Society. The Journal of the Indian Mathematical Society. Mathematical Proceedings of the Cambridge Philosophical Society. Hardy, G. Ramanujan, Srinivasa S2CID Posthumously published extract of a longer, unpublished manuscript. Further works of Ramanujan's mathematics. Selected publications on Ramanujan and his work.

Berndt, Bruce C. Butzer, P. Turnhout, Belgium: Brepols Verlag. ISBN Archived PDF from the original on 9 September Ramanujan: Letters and Commentary. Ramanujan: Essays and Surveys. Number Theory in the Spirit of Ramanujan. Ramanujan's Notebooks: Part I. New York: Springer. Ramanujan's Notebooks: Part II. Ramanujan's Notebooks: Part IV. Ramanujan's Notebooks: Part V.

March The American Mathematical Monthly. JSTOR New York: Chelsea Pub. Henderson, Harry Modern Mathematicians. New York: Facts on File Inc. Kanigel, Robert New York: Charles Scribner's Sons. Leavitt, David The Indian Clerk paperback ed. London: Bloomsbury. Narlikar, Jayant V. New Delhi, India: Penguin Books. Ono, Ken ; Aczel, Amir D.

Sankaran, T. Selected publications on works of Ramanujan. Ramanujan, Srinivasa; Hardy, G. Collected Papers of Srinivasa Ramanujan. This book was originally published in [ ] after Ramanujan's death. It contains the 37 papers published in professional journals by Ramanujan during his lifetime. The third reprint contains additional commentary by Bruce C.

Ramanujan Notebooks 2 Volumes. Bombay: Tata Institute of Fundamental Research. These books contain photocopies of the original notebooks as written by Ramanujan. New Delhi: Narosa. This book contains photocopies of the pages of the "Lost Notebook". This was produced from scanned and microfilmed images of the original manuscripts by expert archivists of Roja Muthiah Research Library, Chennai.

Oxford University Press.