On the 22nd of December, 1887, in what is now the state of Tamil Nadu
, Srinivasa Aiyangar Ramanujan was born. In his short lifetime, he would become known as one of the greatest mathematicians of India, certainly the greatest of his time; and quite possibly one of the most intelligent men ever.
Ramanujan was born in his grandmother's house in the town of Erode, about 400km from Madras. When he was a year old, he and his mother moved to the town of Kumbakonam, 160km away, to live with his father who worked there as a clerk in a cloth-merchant's shop.
When he was almost five years old, Ramanujan began to attend the primary school at Kumbakonam, and went on to attend many different primary schools before he entered the Town High School when he was 11. Here, Ramanujan did well in all his subjects, but found himself to be particularly adept at and interested in mathematics. When he was 13 he came across a book by G. S. Carr called Synopsis of Elementary Results in Pure Mathematics. This book allowed Ramanujan to study the subject on his own. Unfortunately, the book was well out of date by the time he got it, having been published almost half a century earlier in 1856.
He began by working with summing arithmetic and geometric series. When he was 15, he was shown how to solve cubic equations, and went on to solve quartic equations by himself. The next year, not knowing that quintic equations cannot be solved by radicals, he tried and (of course) failed. By the time he was 16, he had begun to undertake serious research, investigating the summa(1/n) series and calculating Euler's constant to 15 decimal places. He also discovered Bernoulli numbers independently.
At this time, because of his good work in school, Ramanujan got a scholarship to the Government College in Kumbakonam. However, Ramanujan devoted all his time to mathematics and neglected his other subjects. His scholarship was not renewed the next year, and he ran away to Vizagapatnam, about 1000km away. Here he continued his research and worked on hypergeometric series and their relationship with integrals. He later learned he had been studying elliptic functions.
When he was 17, he went to Madras and entered Pachaiyappa's College, with the aim of learning enough to pass the First Arts Examination and get into the University of Madras. Unfortunately, he fell ill three months into the course, dropped out, and failed all the subjects other than mathematics. He continued his research with no real idea of what was current in mathematics nor any help other than Carr's book. At 19, he began to study continued fractions and divergent series, soon after which he again fell seriously ill. WHen he was 20 he had to undergo an operation from which he took a long time to recover.
On July 14th, 1909, Ramanujan had an arranged marriage with nine year old Janaki Ammal. He did not live with her, however, until she was 12. Over the next few years, he began to publish some papers in the Journal of the Indian Mathematical Society and began to gain some recognition, particularly after a brilliant 1911 paper on Bernoulli numbers. Around this time he also got his first job, a temp in the Accountant General's office in Madras through the founder of the Indian Mathematical Society. After this, through recommendations from the mathematics professors at the University of Madras, he got a post as clerk in the accounts section of the Madras Port Trust on March 1st, 1912.
During this time, he also wrote to various mathematics professors in London showing them his work, including M. J. M. Hill, E. W. Hobson, and H. F. Baker; none of them showed any interest. Finally, in January 1913, he wrote to one G.H. Hardy, showing him a number of theorems that he had discovered, but did not in clude any proofs. Hardy was more sympathetic than the rest, and replied:
"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..."
Delighted, Ramanujan sent him proofs, and with a letter of recommendation obtained a two-year scholarship from the University of Madras in May 1913. In 1914, Hardy brought Ramanujan to Trinity College, Cabridge. Being an orthodox Brahmin he was a strict vegetarian, but a friend of Hardy's convinced Ramanujan while lecturing in India that he should come to the UK. On March 17, 1914, Ramanujan sailed to London, and arrived on April 14th. The outbreak of World War I, however, made obtaining special items of food harder, and before long Ramanujan had health problems from undernourishment. These problems would continue for the rest of his life.
The biggest problem facing Hardy was Ramanujan's almost complete lack of familiarity with current mathematics. For starters, he had almost no idea what a formal proof was, or any rigorous mathematical methods for that matter. What Ramanujan published while he was in England was work he did in England; he and Hardy agreed that the work he had done before would not be published until the war was over. Ramanujan, at this time, was also allowed to enrol in Trinity College despite his lack of formal qualifications, and on March 16th he graduated with a Bachelor of Science by Research degree, the equivalent of a Ph.D. His dissertation was on highly composite numbers and consisted of seven of his papers published in England.
In 1917, Ramanujan fell seriously ill with what has since been confirmed to have been tuberculosis, and his doctors feared he would die. It was during this time that the famous 1729 story happened. Hardy went to see Ramanujan during his sickness, and during the visit remarked that he had travelled in cab number 1729, a boring number; he hoped it wasn't a bad omen. Ramanujan instantly replied that 1729 is not a boring number at all as it is the smallest number that is expressible as the sum of two cubes in two different ways (10^3+9^3 = 1000+729 = 1729 ; 12^3+1^3 = 1728+1 = 1729). Just to give you an idea of his genius.
On February 18th, 1918, Ramanujan was elected a fellow of the Cambridge Philosophical Society, and three days later was nominated (by a long and impressive list of celebrated mathematicians) to be a fellow of the Royal Society of London. On May 2, 1918, he was elected a fellow, the greatest honor he received in his lifetime. On October 10, 1918, he was elected a fellow of Trinity College. These honors seemed to help his health, and by November of that year he had greatly improved, having regained a regular temperature and put on an extra stone (14lb). From this time untill February 1919, he once again worked with his original fervor, producing many important papers. Unfortunately, he had not much life remaining, and on April 26, 1920, he died in Kumbakonam, havin returned to India a year earlier.
Ramanujan independently discovered results of Gauss, Kummer and others on hypergeometric series. Ramanujan's own work on partial sums and products of hypergeometric series have led to major developments in the topic. Perhaps his most famous work was on the number p(n) of partitions of an integer n into summands. MacMahon had produced tables of the value of p(n) for small numbers n, and Ramanujan used this numerical data to conjecture some remarkable properties some of which he proved using elliptic functions. In a joint paper with Hardy, Ramanujan gave an asymptotic formula for p(n). It had the remarkable property that it appeared to give the correct value of p(n), later proved by Rademacher.1
Ramanujan's unpublished notebooks have been the subject of study since his death. Various mathematicians have published papers inspired by them. Most notably, G.N. Watson published 14 papers under the title Theorems stated by Ramanujan, besides 16 other papers also related to Ramanujan's work.
For further reading, read Robert Kanigel's excellent biography of Ramanujan, The Man Who Knew Infinity.
1: This paragraph c/p'd out of the source because I am not a mathematician and don't know what most of it means.