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In 1913, Cambridge mathematician G. F. Hardy received an unusual letter from a young scholar in India named Srinivasa Ramanujan. In this letter Ramanujan introduced himself as a student of mathematics with little money and without a formal education. As evidence of his ability, he included in the letter over one hundred mathematical theorems that he had discovered, without justification.
The letter sparked Hardy’s curiosity, and he replied asking for proofs of some of the theorems, stating, “there are some results which appear to be new and important.”
Ramanujan replied with the requested work, and asked Hardy to write a letter of recommendation for him, so he could obtain a scholarship for university studies. He explained that he hardly had enough money to eat, let alone attend school. “To preserve my brains I want food,” he wrote, “and this is my first consideration.”
On Hardy’s recommendation, the University of Madras accepted Ramanujan as a student, and a year later, he was offered to come to Cambridge to work directly with Hardy.
Ramanujan accepted the offer, and in England, collaborating with Hardy, he produced important work in a number of areas in pure mathematics: he wrote papers on integer partitions, elliptic functions, continued fractions and hypergeometric functions.
As they began working together, Ramanujan’s lack of formal training became apparent to Hardy. Ramanujan had a masterful sense of intuition, and produced many wonderful conjectures, but he sometimes could not verify them. He had little experience in writing formal mathematical proofs, was not well-read in the journals of the day, and was relatively ignorant in many important fields, like analysis and algebra. Hardy would later remark of Ramanujan, “The limitations of his knowledge were as striking as its profundity.”
It was Littlewood who undertook the task of giving Ramanujan a rigorous, formal mathematics education. Littlewood found Ramanujan a sometimes exasperating student. “Every time some matter was mentioned,” Littlewood remarked once, “Ramanujan’s response was an avalanche of original ideas.”
During World War I, Littlewood was called away to war duty and Hardy continued working with Ramanujan. Ramanujan’s Ph.D. dissertation was written on highly composite integers – integers that have more divisors than any integer smaller than them – and his degree was awarded in 1916.
Ramanujan, an observant Brahmin, was a vegetarian. In England, he could not obtain the diet he had been used to in India. What was worse, the food that was available to him came only in meager quantities, due to war rationing. Hence, Ramanujan’s health quickly declined. He was often bedridden, but never stopped writing down new ideas.
His amazing mathematical insight and creativity earned him fame in the few years he worked at Cambridge. In February 1918, Ramanujan was elected a fellow of the Cambridge Philosophical Society. Only three days after his induction there, he was proposed for membership and then elected into the Royal Society of London. His name had been suggested by the most prominent English mathematicians of the time, including Whitehead, Whittaker, Forsyth and of course Hardy and Littlewood.
In 1919, his education complete, it came time for Ramanujan to return home to India. Hardy foresaw a great future for the prodigious young man, stating “I am confident that India will regard him as the treasure he is.”
However, Ramanujan’s health failed to improve on returning to India. Just a year later he died in bed, only thirty-two years old. His medical doctors diagnosed tuberculosis. Up until the end, Ramanujan engaged himself in his favorite activity: devising new and ingenious mathematical results.
Although his life was short, and his mathematical career spanned only a few years, Ramanujan left behind enough unfinished work to keep other mathematicians busy for decades. He had written thousands of his unpublished results in his notebooks; these were mostly in rough form, with only the result quoted and none of the establishing work done - seemingly pulled from the air. Most of these results have by now been established (and a few incorrect assertions disproved, as well) but they have often been quite deep. Ramanujan’s notebooks, for the past eighty years, have provided material for intense research, and have given many others inspiration for groundbreaking work in pure mathematics. Even now, this self-taught wonder and his unusual and surprising results continue to astound us, like he astounded the mathematical community in Cambridge almost ninety years ago.