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Self-taught genius: Srinivasa Ramanujan (right) played by Dev Patel in "The Man Who Knew Infinity" (IFC Films)

This month a new movie about a mathematical genius appears. The Man Who Knew Infinity, starring Dev Patel and Jeremy Irons, deals with Srinivasa Ramanujan (1887–1920), brought from India to England by the eminent Cambridge mathematician G.H. Hardy. Ramanujan’s story is an extraordinary one because he had no very suitable books to help him in India and lacked a strong formal education.

Such education is always a problem with teenagers of startling creativity and mathematical ability who cast aside the usual classroom studies as mere obstacles to their passion. After starting original research at 14 he was already dealing with deep matters by the age of 16. He was a fine scholar but serious illness caused him to miss school, and after failing all subjects except mathematics he was unable to enter the University of Madras. Undeterred, he continued his research, and his letters to various mathematicians in England finally found a response from Hardy. By the time he reached Cambridge in 1914, a few months before the start of the Great War, he had filled his famous notebooks with hundreds of startling results, some of which have still not been properly investigated. How he did these things is often a mystery, and the meaning of his results was sometimes lost on other mathematicians, but here is one that everybody can appreciate.

The old Greek problem of squaring the circle — constructing a square having the same area as a given circle using only a straight edge and a pair of compasses — was finally shown to be impossible in 1882, using a new theorem about π. Yet this still raised the question of finding good approximations, and although geometry was not one of Ramanujan’s main concerns he gave a simple construction yielding the side of a square that he rightly claimed “is one hundredth of an inch greater than the true length if the given circle is 14 square miles in area”. He also produced several formulae approximating π to numerous decimal places, including a series for 1/π that he asserted to be “rapidly convergent”. Indeed, the first term of this series was already correct to eight decimal places, and in 1986 two computer scientists used a version of his formula to calculate π to 17 million decimal places, finding it converges far more rapidly than any previous method.

To say a series converges means that the more terms you add the closer you get to a finite number. For example, 1/2 + 1/4 + 1/8 + 1/16 + . . . converges fairly rapidly to 1 — think of slicing up a cake. Series that do not converge are said to diverge, and Ramanujan discovered numerous intriguing results on divergent series. One he sent to Hardy stated that:

1 + 2 + 3 + 4 + 5 + . . .  = −1/12

On the face of it this is nonsense, but Hardy did not dismiss such things, because Ramanujan’s work contained other extraordinary statements that he judged to be the product of a first-rate mind:

A single look at them is enough to show that they could only be written by a mathematician of the highest class. They must be true because, if they were not true, no one would have the imagination to invent them.