Next Time, Remember Ramanujan

One hundred years ago on Decem ber 22, a most extraordinary mathematician was born in the town of Erode, 160 miles from Madras in Southern India. Srinivasa Ramanujan Aiyangar was the son of a petty accountant and a bailiff’s daughter. He grew up in Kumbakonam, where his father worked. At the age of 15 he borrowed a copy of A Synopsis of Elementary Results in Pure Mathematics by G.S. Carr, which lists some 6,000 theorems but gives no proofs. Captivated, Ramanujan set about finding the proofs for himself, and progressed to independent research. Too poor to afford paper, he did his calculations on a slate and jotted down the results in a series of notebooks. In 1913 Godfrey H. Hardy, an outstanding mathematician at Cambridge University, England, received a letter. “Dear Sir, I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras.” Ramanujan enclosed some of his discoveries. “This man is either a crank or a genius,” said Hardy. A few hours later Hardy and his colleague J.E. Littlewood emerged from the Trinity College chess room with the verdict: genius. With some difficulty, Hardy persuaded Ramanujan to leave India for Cambridge, and in March 1914 he set sail for England. But after three years his health deteriorated severely, and in 1919 he returned to India. In the meantime he had been elected a Fellow of the Royal Society, the first Indian to receive this high scientific honor. He died on April 26, 1920, leaving a mathematical legacy that still holds many deep secrets. Where did he get his remarkable ideas? It often is said that Ramanujan’s formulas were inspired in dreams by the goddess Namagiri. However, according to his widow, S. Janaki Ammal Ramanujan, her husband “never had time to go to the temple because he was constantly obsessed with mathematics.” Hardy thought that “all mathematicians think, at bottom, in the same way, and Ramanujan was no exception,” but added, “He combined a power of generalisation, a feeling for form, and a capacity for rapid modification of his hypotheses, that were often really startling.” Ramanujan’s genius was in a style of mathematics that is no longer fashionable: the manipulation and invention of formulas. “It was his insight into algebraical formulae, transformations of infinite series, and so forth, that was most amazing,” Hardy writes. In this respect, “most certainly I have never met his equal, and I can compare him only to Euler or Jacobi.” There always seemed to be some special quirk, some unexpected twist, to Ramanujan’s formulas that made them very much his own. Some of his best work is in the theory of partitions. In how many ways can a given number be written as a sum of smaller ones? Ramanujan noticed that whenever the original number leaves remainder 4 on division by 5, then the number of its partitions is divisible by 5. There are similar results for the numbers 7 and 11 in place of 5, and no others. The proof is tricky in the extreme, and before Ramanujan, nobody had suspected that partitions might have arithmetical properties of this kind. In analytic number theory, approximate formulas are sought for quantities such as the number of primes less than a given value. In collaboration with Hardy, Rarnanujan applied these methods to the number of partitions. Not only did they find an approximate formula: they found an exact one! Ramanujan left three notebooks, a fourth “lost” notebook consisting of loose sheets that was redisoovered in 1976 by George Andrews of Pennsylvania State University, and various manuscripts, some of which are still missing. Bruce Berndt, at the University of Illinois, recently published the first part of a three-volume work entitled Ramanujan’s Notebooks (Springer-Verlag, New York, 1985), aiming to supply proofs of all his formulas. Berndt thinks that Ramanujan, far from being old-fashioned, was ahead of his time. “I may be able to prove [a formula], but I don’t know where it comes from and where it fits into the rest of mathematics,” Berndt said. Ramanujan left unsolved problems that still tax the power of mathematics. One, the Ramanujan Conjecture, was proved in 1975. It was a spinoff from one of the major mathematical events of this century—a dramatic breakthrough in algebraic geometry by the Belgian mathematician Pierre Deligne. Genius can flower in the most unlikely habitat. Before Hardy, two other prominent British mathematicians had received letters from Ramanujan, and had ignored them. The next time you get an unsolicited manuscript from a “crank,” think of them, of Hardy, and of Ramanujan. Ian Stewart is a reader in mathematics at the Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK.

Next Time, Remember Ramanujan

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November 1987

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