But without the intuitive genius of Ramanujan, Hardy would never have formulated such an astonishing result; and without Hardy, Ramanujan would never have been able to prove it. Littlewood said [9, p. 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 December , followed by fellowships of the Royal Society in May and Trinity College in October The war had enforced a prolonged stay in England, but by November , his 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 February , he 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 atheist , there 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 to , he 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 to , for 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. Your email address will not be published. This site uses Akismet to reduce spam. Thus was Srinivasa Ramanujan introduced to the mathematical world. Born in South India, Ramanujan was a promising student, winning academic prizes in high school.
But at age 16 his life took a decisive turn after he obtained a book titled A Synopsis of Elementary Results in Pure and Applied Mathematics. The book was simply a compilation of thousands of mathematical results, most set down with little or no indication of proof. It was in no sense a mathematical classic; rather, it was written as an aid to coaching English mathematics students facing the notoriously difficult Tripos examination, which involved a great deal of wholesale memorization.
But in Ramanujan it inspired a burst of feverish mathematical activity, as he worked through the book's results and beyond. Unfortunately, his total immersion in mathematics was disastrous for Ramanujan's academic career: ignoring all his other subjects, he repeatedly failed his college exams. As a college dropout from a poor family, Ramanujan's position was precarious. He lived off the charity of friends, filling notebooks with mathematical discoveries and seeking patrons to support his work.
Finally he met with modest success when the Indian mathematician Ramachandra Rao provided him with first a modest subsidy, and later a clerkship at the Madras Port Trust. During this period Ramanujan had his first paper published, a page work on Bernoulli numbers that appeared in in the Journal of the Indian Mathematical Society.
Still no one was quite sure if Ramanujan was a real genius or a crank. He said he wanted a pittance to live on so that he might pursue his researches. Ramachandra Rao told him to return to Madras and he tried, unsuccessfully, to arrange a scholarship for Ramanujan. In Ramanujan applied for the post of clerk in the accounts section of the Madras Port Trust. In his letter of application he wrote [ 3 ] :- I have passed the Matriculation Examination and studied up to the First Arts but was prevented from pursuing my studies further owing to several untoward circumstances.
I have, however, been devoting all my time to Mathematics and developing the subject. Despite the fact that he had no university education, Ramanujan was clearly well known to the university mathematicians in Madras for, with his letter of application, Ramanujan included a reference from E W Middlemast who was the Professor of Mathematics at The Presidency College in Madras.
Middlemast, a graduate of St John's College, Cambridge, wrote [ 3 ] :- I can strongly recommend the applicant. He is a young man of quite exceptional capacity in mathematics and especially in work relating to numbers. He has a natural aptitude for computation and is very quick at figure work. On the strength of the recommendation Ramanujan was appointed to the post of clerk and began his duties on 1 March Ramanujan was quite lucky to have a number of people working round him with a training in mathematics.
He wrote to Hill on 12 November sending some of Ramanujan's work and a copy of his paper on Bernoulli numbers. Hill replied in a fairly encouraging way but showed that he had failed to understand Ramanujan's results on divergent series. The recommendation to Ramanujan that he read Bromwich 's Theory of infinite series did not please Ramanujan much. In Ramanujan's letter to Hardy he introduced himself and his work [ 10 ] :- 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 made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as 'startling'. Hardy , together with Littlewood , studied the long list of unproved theorems which Ramanujan enclosed with his letter.
On 8 February he replied to Ramanujan [ 3 ] , the letter beginning:- I was exceedingly interested by your letter and by the theorems which you state. You will however understand that, before I can judge properly of the value of what you have done, it is essential that I should see proofs of some of your assertions.
Your results seem to me to fall into roughly three classes: 1 there are a number of results that are already known, or easily deducible from known theorems; 2 there are results which, so far as I know, are new and interesting, but interesting rather from their curiosity and apparent difficulty than their importance; 3 there are results which appear to be new and important Ramanujan was delighted with Hardy 's reply and when he wrote again he said [ 8 ] :- I have found a friend in you who views my labours sympathetically.
I am already a half starving man. To preserve my brains I want food and this is my first consideration. Any sympathetic letter from you will be helpful to me here to get a scholarship either from the university of from the government. Indeed the University of Madras did give Ramanujan a scholarship in May for two years and, in , Hardy brought Ramanujan to Trinity College, Cambridge, to begin an extraordinary collaboration.
Setting this up was not an easy matter. Ramanujan was an orthodox Brahmin and so was a strict vegetarian. His religion should have prevented him from travelling but this difficulty was overcome, partly by the work of E H Neville who was a colleague of Hardy 's at Trinity College and who met with Ramanujan while lecturing in India.
Ramanujan sailed from India on 17 March It was a calm voyage except for three days on which Ramanujan was seasick. He arrived in London on 14 April and was met by Neville. After four days in London they went to Cambridge and Ramanujan spent a couple of weeks in Neville 's home before moving into rooms in Trinity College on 30 th April. Right from the beginning, however, he had problems with his diet.
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.
Since his death at age 32 mathematicians have analyzed his notebooks pdf , which are full of formulas but light on justification. Most of the formulas have turned out to be correct, and researchers continue to learn from his work while trying to understand and prove them. India's mathematical heritage extends far beyond Ramanujan's time. The nation is considered home of the concept of zero.
Babylonians had used a space as a placeholder similar to the role of "0" in the number , but this space could not stand alone or at the end of a number. In our number system, as in theirs, this could be problematic; imagine trying to tell the difference between the numbers 1 and 10 by context alone. In India, however, zero was treated as a number like any other.
India is also the home of our decimal numeral system. Indian government and mathematical societies pursued several projects to celebrate their year of mathematics, from enrichment programs for students and teachers to the "Mathematical Panorama Lectures" that occurred around the country. This series of 20 short lecture courses, which will continue into , brings prominent mathematicians from different fields to Indian universities to deliver five or six lectures.
0コメント