Ramanujan (1887-1920)

Ramanujan was an Indian mathematician who was self-taught and had an uncanny mathematical manipulative ability.

He was unable to pass his school examinations in India, and could only obtain a clerk's position in the city of Madras. However, he continued to pursue his own mathematics, and sent letters to three mathematicians in England.

G. H. Hardy recognized Ramanujan's intrinsic mathematical ability and arranged for him to come to Cambridge. Because of his lack of formal training, Ramanujan sometimes did not differentiate between formal proof and apparent truth based on intuitive or numerical evidence. Although his intuition and computational ability allowed him to determine and state highly original and unconventional results which continued to defy formal proof until recently (Berndt 1985-1997), Ramanujan did occasionally state incorrect results.

Ramanujan had an intimate familiarity with numbers, and excelled especially in number theory and modular function theory. His familiarity with numbers were demonstrated by the following incident. During an illness in England, Hardy visited Ramanujan in the hospital. When Hardy remarked that he had taken taxi number 1729, a singularly unexceptional number, Ramanujan immediately responded that this number was actually quite remarkable: it is the smallest integer that can be represented in two ways by the sum of two cubes:

1729= 1 raised to the power 3 + 12 raised to the power 3 =

9 raised to the power 3 + 10 raised to the power 3.

Unfortunately, Ramanujan's health deteriorated rapidly in England, due perhaps to the unfamiliar climate, food, and to the isolation which Ramanujan felt as the sole Indian in a culture which was largely foreign to him. Ramanujan was sent home to recuperate in 1919, but tragically died the next year at the very young age of 32.

One remarkable result of the Hardy-Ramanujan collaboration was a formula for the number p(n) of partitions of a number n. A partition of a positive integer n is just an expression for n as a sum of positive integers, regardless of order. Thus p(4) = 5 because 4 can be written as 1+1+1+1, 1+1+2, 2+2, 1+3, or 4.

The problem of finding p(n) was studied by Euler, who found a formula for the generating function of p(n) ( that is, for the infinite series whose nth term is p(n)x n).

While this allows one to calculate p(n) recursively, it doesn't lead to an explicit formula. Hardy and Ramanujan came up with such a formula (though they only proved it works asymptotically; Rademacher proved it gives the exact value of p(n)).

Part of this page has been created based on informatrion on the following web page:

http://scienceworld.wolfram.com/biography/Ramanujan.html