Born 22 December 1887 in Erode, southern India, into a Brahmin family; his father kept the books at a sari shop. At fifteen he obtained Carr's A Synopsis of Elementary Results in Pure and Applied Mathematics — over five thousand formulas, almost none of them proved — and that book became his entire curriculum. He entered college twice, in 1904 and 1906, and dropped out both times after failing every subject except mathematics; he then lived on help from relatives until 1912, when he took a clerkship at the Madras Port Trust for 30 rupees a month. On 16 January 1913 he wrote to Hardy at Cambridge, reached England in April 1914, published more than twenty papers in five years, and was elected a Fellow of the Royal Society in 1918. He returned to India gravely ill in 1919 and died on 26 April 1920, aged 32.
First, in 1903, treating Carr's Synopsis as his entire mathematical education. It was a cram-book compilation: results listed, proofs almost never given. He had no other book, so he had to fill each one in himself. That shaped both his sides: unbound by any received framework, he grew conclusions straight out of form, at astonishing throughput; and he never acquired the habit of proof — in his system a result was true because it looked as though it had to be.
Second, on 16 January 1913, posting a third letter after two had vanished without trace. He wrote first to Hill in London and got a courteous but discouraging reply; then to Baker and Hobson at Cambridge, and heard nothing. When he rewrote it for Hardy, what he selected was not his safest results but his most impossible ones. Hardy said later that it was precisely those formulas — the ones nobody could have invented — that stopped him from throwing the letter away.
Third, in early 1914, deciding to cross the sea. For an orthodox Brahmin, an ocean voyage meant loss of caste (kala pani). Neville travelled from Cambridge to Madras to persuade him, and his mother finally relented on the strength of a dream from the goddess Namagiri. What he paid was not the fare, it was his social identity. And keeping a strict vegetarian diet in England meant cooking for himself — with wartime shortages, that discipline later took a direct hand in killing him.
Sources: Robert Kanigel, The Man Who Knew Infinity (1991); G. H. Hardy, Ramanujan: Twelve Lectures (1940); the letter to Hardy of 16 January 1913.One: Hardy's evening. In late January 1913 Hardy received over breakfast a sheaf of manuscript signed by a clerk at the Madras Port Trust, and his first reaction was that it was a hoax. At around nine that night he called in Littlewood, and the two read until midnight. Their method of judgment was peculiar: the formulas could not be checked on the spot, but "they must be true, because, if they were not true, no one would have had the imagination to invent them" — that is not a mathematical argument, it is an inference about a person.
Two: the Underground and the election. From 1917 he was in and out of hospital, diagnosed with tuberculosis (most researchers today lean toward hepatic amoebiasis). Around 1918 he lay down on the tracks in a London Underground station; the train stopped in time. Hardy went to the police station and got him released on the grounds that "he is a Fellow of the Royal Society" — which at that point he was not. Hardy then pressed the case hard, and on 2 May he was elected. The honour here was not a result; it was a rescue measure.
Sources: Hardy, Twelve Lectures (1940), Lecture I; C. P. Snow's foreword to A Mathematician's Apology; Kanigel (1991).He worked on a slate and wiped it with his elbow. Paper was too expensive, so the derivations went on slate and were rubbed out as he moved to the next step — only the result was copied into a notebook; the working was never kept. Most of the roughly 3,900 results in the three notebooks carry no proof, and supplying those proofs took later mathematicians close to a century.
Roughly thirty hours of work, then twenty hours of sleep. No schedule, only uninterrupted blocks. Cambridge read this as pathology; for him it was the only way to work.
He recorded conclusions, never the route. Asked "how did you get this," he often could not retrace the original path and would construct a different one on the spot. It took Hardy years to find the balance between teaching him to prove things and not damaging the intuition.
A strict vegetarian, cooking for himself in his Trinity rooms, often eating his first meal of the day late at night. Under wartime rationing from 1917 to 1919, with vegetables and fruit scarce, his nutrition deteriorated steadily and he refused to face it.
He credited the formulas to the goddess Namagiri. He said more than once that she wrote them for him in dreams. This was not a figure of speech: he sincerely took himself to be the copyist rather than the author — which also explains why he felt no obligation to prove anything.
Sources: Kanigel (1991); S. R. Ranganathan, Ramanujan: The Man and the Mathematician (1967).First, the theory he was proudest of was wrong. He believed he had found a nearly exact expression for the count of primes below x, and made it the centrepiece of what he sent Hardy. Hardy states flatly in Twelve Lectures that the theory was substantially wrong: not knowing about the non-real zeros of the zeta function, he misjudged the error term systematically. This was not bad luck but a necessity of the method — in a system that does not prove, brilliant results and wrong results leave the factory with exactly the same confidence, and cannot be told apart from inside.
Second, Janaki. In 1909, aged 21, he married Janaki Ammal, who was 10. By her own later account, during his five years in England her mother-in-law withheld the letters between them for long stretches and she heard almost nothing from her husband. After his death she lived by sewing, poor and unnoticed, for decades; she died in 1994. In the legend that gets retold over and over, she barely exists.
Third, "self-taught genius" is a harmful myth. The real chain runs: Ramaswamy Iyer of the Indian Mathematical Society, the administrator Ramachandra Rao, his Port Trust superior Narayana Iyer, the Madras University scholarship, Neville at Cambridge, plus five years of Hardy checking him line by line. The myth erases that network and leaves only "genius will out" — when in fact he sent three letters and two went unanswered; had Hardy not replied either, nobody would know his name today.
Sources: Hardy, Twelve Lectures (1940); Kanigel (1991); Janaki Ammal's interviews with the Indian press in the 1980s.The most direct lesson is to staff the generator and the verifier separately: he was a conjecture engine of enormous throughput with almost no self-checking, and Hardy was the slow, fastidious verification layer. The partition asymptotics, the circle method, the mock theta functions all came off that line, not from either end of it. The structure is identical to doing research with AI today: a high-output intuition source must be wired to a falsification layer that does not share its biases, or brilliant conclusions and systematic errors of the prime-theory kind get delivered with exactly the same confidence. Second, the arithmetic of distribution: three letters, two unanswered, the third changed the history of mathematics. One more letter costs almost nothing and the payoff is heavy-tailed — provided each one sends your most startling material, not your safest. Third, the bill he never settled: body, marriage and rest were all optimized away as residuals, and the machine halted at 32. A machine with no maintenance window isn't more productive; it is just drawing its lifetime output forward.