Highly composite numbers ramanujan biography

As Aiyer later recalled:. I was struck by the extraordinary mathematical results contained in [the notebooks]. I had no mind to smother his genius by an appointment in the lowest rungs of the revenue department. Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends in Madras. Ramachandra Raothe district collector for Nellore and the secretary of the Indian Mathematical Society.

Ramanujan mentioned a correspondence he had with Professor Saldhana, a notable Bombay mathematician, in which Saldhana expressed a lack of understanding of his work but concluded that he was not a fraud. Rajagopalachari tried to quell Rao's doubts about Ramanujan's academic integrity. Rao agreed to give him another chance, and listened as Ramanujan discussed elliptic integralshypergeometric seriesand his theory of divergent serieswhich Rao said ultimately convinced him of Ramanujan's brilliance.

Rao consented and sent him to Madras. He continued his research with Rao's financial aid. One of the first problems he posed in the journal [ 30 ] was to find the value of:. He waited for a solution to be offered in three issues, over six months, but failed to receive any. At the end, Ramanujan supplied an incomplete [ 59 ] solution to the problem himself.

On page of his first notebook, he formulated an equation that could be used to solve the infinitely nested radicals problem. One property he discovered was that the denominators of the fractions of Bernoulli numbers sequence A in the OEIS are always divisible by six. He also devised a method of calculating B n based on previous Bernoulli numbers.

One of these methods follows:.

Highly composite numbers ramanujan biography: The mathematician Jean-Pierre Kahane suggested

In his page paper "Some Properties of Bernoulli's Numbers"Ramanujan gave three proofs, two corollaries and three conjectures. As Journal editor M. Narayana Iyengar noted:. Ramanujan's methods were so terse and novel and his presentation so lacking in clearness and precision, that the ordinary [mathematical reader], unaccustomed to such intellectual gymnastics, could hardly follow him.

Ramanujan later wrote another paper and also continued to provide problems in the Journal. He lasted only a few weeks. Sir, I understand there is a clerkship vacant in your office, and I beg to apply for the same. I have passed the Matriculation Examination and studied up to the F. I have, however, been devoting all my time to Mathematics and developing the subject.

I can say I am quite confident I can do justice to my work if I am appointed to the post. I therefore beg to request that you will be good enough to confer the appointment on me. Attached to his application was a recommendation from E. Middlemasta mathematics professor at the Presidency Collegewho wrote that Ramanujan was "a young man of quite exceptional capacity in Mathematics".

Ramanujan's boss, Sir Francis Springand S. Narayana Iyer, a colleague who was also treasurer of the Indian Mathematical Society, encouraged Ramanujan in his mathematical pursuits. Middlemast tried to present Ramanujan's work to British mathematicians. Hill of University College London commented that Ramanujan's papers were riddled with holes. With the help of friends, Ramanujan drafted letters to leading mathematicians at Cambridge University.

The first two professors, H. Baker and E. Hobsonhighly composite numbers ramanujan biography Ramanujan's papers without comment. Hardywhom he knew from studying Orders of Infinity The first result had already been determined by G. Bauer in The second was new to Hardy, and was derived from a class of functions called hypergeometric serieswhich had first been researched by Euler and Gauss.

Hardy found these results "much more intriguing" than Gauss's work on integrals. Littlewoodto 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".

Nevillelater 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 FebruaryHardy 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 Walkera former mathematical lecturer at Trinity College, Cambridgelooked 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 Collegewhom 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".

Highly composite numbers ramanujan biography: Ramanujan was awarded a Bachelor

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 numberssections 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 crimebut 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. Inshe returned to Madras and settled in Triplicanewhere she supported herself on a pension from Madras University and income from tailoring. Inshe 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 NaduAndhra 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 amoebiasisan 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. 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 highly composite numbers ramanujan biography, "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.

Highly composite numbers ramanujan biography: In this paper, Ramanujan

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 numberswhich 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 andwhat 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 identitiessuch as. InHardy 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. InHans 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 inas 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. Berndtin 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 slateafter 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 highly composite numbers ramanujan biography 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. WatsonB. Wilsonand Bruce Berndt. InGeorge 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 oneand 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 inHardy 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.

We owe the theorem, to a singularly happy collaboration of two men, of quite unlike gifts, in which each contributed the best, most characteristic, and most fortunate work that was in him. He was diagnosed with tuberculosis and severe vitamin deficiency, although recent analysis has concluded that he may have been suffering from hepatic amoebiasis, a complication arising from previous attacks of dysentery.

He spent much of the year in various nursing homes, but his mathematical output, although reduced, remained as remarkable as ever. His spirits were raised by his election to membership of the London Mathematical Society in Decemberfollowed by fellowships of the Royal Society in May and Trinity College in October The war had enforced a prolonged stay in England, but by Novemberhis health had improved sufficiently for Hardy to write about a return to his homeland [5, p.

He will return to India with a scientific standing and reputation such as no Indian has enjoyed before, and I am confident that India will regard him as the treasure he is. His natural simplicity and modesty has never been affected in the least by success — indeed all that is wanted is to get him to realise that he really is a success. On 27 Februaryhe embarked for India, arriving in Kumbakonam two weeks later, but his health deteriorated again despite medical treatment.

He died on 26 April at the age of Ramanujan was described as being enthusiastic and eager, with a good-natured personality, although somewhat shy and quiet in official settings. Not particularly introspective, he was never able to give a completely coherent account of how he came up with his ideas — indeed, although his religious beliefs were later downplayed by Hardy an atheistthere is evidence that Ramanujan believed that some form of divine inspiration was involved.

In any case, he seems to have been quite modest about his own abilities and scrupulously keen to acknowledge help from any other sources. Littlewood famously remarked [2, p. I remember once going to see him when he was lying ill at Putney. I had ridden in taxi-cab No. With regard to his mathematics, its prime characteristic is its overwhelming wealth of algebraic formulae and vast computational complexity.

Ramanujan was gifted with a power of calculation and symbolic dexterity unavailable to most mathematicians prior to the computer age. He also had an uncanny ability to spot patterns that nobody knew existed. For example, from the list of partition numbers from 1 tohe deduced a number of attractive, but hitherto unknown, congruences, including.

Most mathematicians would be satisfied with the mere discovery of relationships such as these, but in order to prove them Ramanujan was led to an even more stunning result. His ability to conjure up a myriad of bizarre yet almost supernaturally accurate approximations was another overwhelming feature of his mathematics. From his work on elliptic and modular functions came irrational expressions surprisingly close to integer values, such as.

It also yielded a series of tremendously accurate approximations tofor example. The Hardy—Ramanujan partition formula itself was refined and improved by Hans Rademacher in the s and, as well as its utility in mathematics, now serves as a useful function in superstring theory in physics and the study of phase transitions in chemistry.

Hardy was kind in giving opportunity for an Indian and to share his work. There was a French Mathematician prodigy who died young. This Indian Mathematician is framed in that manner. All Rights Reserved. Site by Measured Designs. Home » Publications » Mathematics Today » Srinivasa Ramanujan — : The Centenary of a Remarkable Mathematician This month marks the centenary of the death of one of the most remarkable mathematicians of the 20th century.

Figure 1: Srinivasa Ramanujan — Upon graduation from secondary school inhe was awarded a scholarship to study at the Government College in Kumbakonam, also in Tamil Nadu. Firstly, there were theorems that, unbeknown to Ramanujan, were already known, such as the integral formula: Secondly, there were results that, while new, were interesting rather than important, for example: where and is the gamma function.

And finally, there were entirely original results that were simply astonishing, such as: These formulae, Hardy later wrote [4, part II, p. Figure 2: G. Hardy — Adapting to life in Britain was equally challenging for Ramanujan. Figure 3: Extract from a postcard sent by Ramanujan to Hardy as their work on partitions neared completion [5, p.

So for instance, the integer can be written in five different ways, namely, meaning that. If and where,and is a -th root of unity, then, for someThey then proceeded to show that their formula was capable of producing results of unprecedented accuracy. For example, from the list of partition numbers from 1 tohe deduced a number of attractive, but hitherto unknown, congruences, including and Most mathematicians would be satisfied with the mere discovery of relationships such as these, but in order to prove them Ramanujan was led to an even more stunning result which is one of the most beautiful formulae he ever produced.

From his work on elliptic and modular functions came irrational expressions surprisingly close to integer values, such as It also yielded a series of tremendously accurate approximations tofor example, which is correct to 8 decimal places. The outbreak of World War I made obtaining special items of food harder and it was not long before Ramanujan had health problems.

Right from the start Ramanujan's collaboration with Hardy led to important results. Hardy was, however, unsure how to approach the problem of Ramanujan's lack of formal education. He wrote [ 1 ] :- What was to be done in the way of teaching him modern mathematics? The limitations of his knowledge were as startling as its profundity. Littlewood was asked to help teach Ramanujan rigorous mathematical methods.

However he said [ 31 ] The war soon took Littlewood away on war duty but Hardy remained in Cambridge to work with Ramanujan. Even in his first winter in England, Ramanujan was ill and he wrote in March that he had been ill due to the winter weather and had not been able to publish anything for five months. What he did publish was the work he did in England, the decision having been made that the results he had obtained while in India, many of which he had communicated to Hardy in his letters, would not be published until the war had ended.

He had been allowed to enrol in June despite not having the proper qualifications. Ramanujan's dissertation was on Highly composite numbers and consisted of seven of his papers published in England. Ramanujan fell seriously ill in and his doctors feared that he would die. He did improve a little by September but spent most of his time in various nursing homes.

In February Hardy wrote see [ 3 ] :- Batty Shaw highly composite numbers ramanujan biography out, what other doctors did not know, that he had undergone an operation about four years ago. His highly composite numbers ramanujan biography theory was that this had really been for the removal of a malignant growth, wrongly diagnosed.

In view of the fact that Ramanujan is no worse than six months ago, he has now abandoned this theory - the other doctors never gave it any support. Tubercle has been the provisionally accepted theory, apart from this, since the original idea of gastric ulcer was given up. Like all Indians he is fatalistic, and it is terribly hard to get him to take care of himself.

On 18 February Ramanujan was elected a fellow of the Cambridge Philosophical Society and then three days later, the greatest honour that he would receive, his name appeared on the list for election as a fellow of the Royal Society of London. His election as a fellow of the Royal Society was confirmed on 2 Maythen on 10 October he was elected a Fellow of Trinity College Cambridge, the fellowship to run for six years.

The honours which were bestowed on Ramanujan seemed to help his health improve a little and he renewed his effors at producing mathematics. By the end of November Ramanujan's health had greatly improved. Hardy wrote in a letter [ 3 ] :- I think we may now hope that he has turned to corner, and is on the road to a real recovery. His temperature has ceased to be irregular, and he has gained nearly a stone in weight.

There has never been any sign of any diminuation in his extraordinary mathematical talents. He has produced less, naturally, during his illness but the quality has been the same. He will return to India with a scientific standing and reputation such as no Indian has enjoyed before, and I am confident that India will regard him as the treasure he is.

His natural simplicity and modesty has never been affected in the least by success - indeed all that is wanted is to get him to realise that he really is a success. Ramanujan sailed to India on 27 February arriving on 13 March. However his health was very poor and, despite medical treatment, he died there the following year. The letters Ramanujan wrote to Hardy in had contained many fascinating results.

Ramanujan worked out the Riemann series, the elliptic integrals, hypergeometric series and functional equations of the zeta function. On the other hand he had only a vague idea of what constitutes a mathematical proof. Despite many brilliant results, some of his theorems on prime numbers were completely wrong. Ramanujan independently discovered results of GaussKummer and others on hypergeometric series.

Ramanujan's own work on partial sums and products of hypergeometric series have led to major development in the topic. Perhaps his most famous work was on the number p n of partitions of an integer n n n into summands. MacMahon had produced tables of the value of p n p n p n for small numbers n n nand Ramanujan used this numerical data to conjecture some remarkable properties some of which he proved using elliptic functions.

Other were only proved after Ramanujan's death. In a joint paper with HardyRamanujan gave an asymptotic formula for p n p n p n. It had the remarkable property that it appeared to give the correct value of p n p n p nand this was later proved by Rademacher. Ramanujan left a number of unpublished notebooks filled with theorems that mathematicians have continued to study.

G N WatsonMason Professor of Pure Mathematics at Birmingham from to published 14 papers under the general title Theorems stated by Ramanujan and in all he published nearly 30 papers which were inspired by Ramanujan's work. Hardy passed on to Watson the large number of manuscripts of Ramanujan that he had, both written before and some written in Ramanujan's last year in India before his death.

The picture above is taken from a stamp issued by the Indian Post Office to celebrate the 75 th anniversary of his birth. References show. Biography in Encyclopaedia Britannica. G H Hardy, Ramanujan Cambridge, J N Kapur ed. S Ramanujan, Collected Papers Cambridge,