Srinivasa Ramanujan · Mathematics
The 1913 letter that stunned Cambridge, and the five-year collaboration between G. H. Hardy - apostle of rigorous proof - and Ramanujan - intuition incarnate - that produced results neither could have reached alone, including the great asymptotic formula for the partition function and the circle method.
In January 1913 a large envelope from Madras landed on the desk of G. H. Hardy at Trinity College, Cambridge, then the leading pure mathematician in England. Inside was a letter from an unknown Indian clerk, introducing himself and enclosing page after page of mathematical results - about a hundred and twenty theorems on series, integrals, continued fractions, and the distribution of prime numbers, stated without proof. Hardy’s first reaction was suspicion; the letter might be a fraud or the work of a crank. But as he and his colleague J. E. Littlewood studied the strangest formulae that evening, they realised no crank could have written them. Some results were already known, some were false or imprecise, but others were so bizarre and so profound that, as Hardy said, they must be true, because no one would have had the imagination to invent them. By the end of the night Hardy was convinced he was reading the work of a mathematician of the very first rank.
Hardy arranged for Ramanujan to come to Cambridge, and in March 1914 the young man sailed for England, beginning one of the most remarkable collaborations in the history of mathematics. Hardy came to feel that his discovery of Ramanujan was the one genuinely romantic incident of his life, and when he later amused himself by rating mathematicians for sheer natural talent on a scale of one hundred, the numbers told the story. He gave himself a 25 and his brilliant collaborator Littlewood a 30. To David Hilbert, the towering figure of the age, he gave an 80. To Ramanujan he gave a 100. It was Hardy’s way of saying that in raw mathematical power he had never met, and never expected to meet, Ramanujan’s equal. Yet he also saw, clearly, what Ramanujan lacked, and set himself the delicate task of supplying it without extinguishing the gift.
The greatest fruit of the partnership grew from a deceptively simple question. A partition of a whole number is a way of writing it as a sum of positive whole numbers, order not mattering. The number 4 has five partitions: 4, then 3+1, then 2+2, then 2+1+1, then 1+1+1+1. The partition function p(n) counts them, so p(4) equals 5. The counts start gently but explode: p(10) is 42, p(100) is already more than 190 million. No simple formula gives p(n) exactly, and its wild growth had defeated everyone. Hardy and Ramanujan attacked it together, combining Ramanujan’s intuition for the right shape of the answer with Hardy’s command of rigorous analysis, and in 1918 they produced an astonishing asymptotic formula that predicts p(n) with almost unbelievable accuracy. It was the perfect collaboration: a problem where seeing the answer and proving the answer both took genius.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned that Ramanujan’s 1913 letter of unproved theorems stunned Hardy, who brought him to Cambridge; that Hardy supplied the rigour Ramanujan lacked; and that together they found an asymptotic formula for the partition function using the ‘circle method.’ Explain how their two gifts combined.
Leads to G. H. Hardy.
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