Not Pi In The Sky After All
An Indian genius had little formal education but his findings astounded Cambridge mathematicians
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.
The seemingly ludicrous series above featured in a 2007 play about Ramanujan by Simon McBurney and Théâtre de Complicité — A Disappearing Number — the apparent nonsense coming from something called the Riemann Zeta Function. This is named after the German mathematician Bernhard Riemann, who investigated it in the mid-19th century, and whose work on multi-dimensional curved geometry laid the foundation for Einstein’s General Theory of Relativity.
Here is the idea. Take the series:
1/1s + 1/2s + 1/3s + 1/4s + . . .
When s = 2 the sum converges to π2/6, and as s varies the sum varies, yielding what is called the Zeta Function. It features in the most famous unsolved conjecture of mathematics — the so-called Riemann Hypothesis — which if true would have huge implications for the distribution of prime numbers.
The connection to Ramanujan’s extraordinary statement is that when s = −1 you get precisely the series 1 + 2 + 3 + 4 + . . . , and by allowing s to move, like a tiny beetle, from the number 2 to the number −1 on a circuitous route avoiding zero, the value of the Zeta Function changes from π2/6 to −1/12. That circuitous route takes you through a field called the complex numbers, taught in advanced high school courses, though the Zeta Function itself lies at a far more advanced university level, and it is hard to know just how Ramanujan obtained his insights.
In his early teens he acquired a book summarising results in elementary pure mathematics, and its very concise style was the only model he had for how to write mathematical arguments. Thank goodness he found a welcome in Cambridge, and although his notebooks from previous years accompanied him he left them “sleeping in a corner”. His life moved on, and many of the proofs he had in mind for results in the notebooks remain elusive a century later. The tragedy of Ramanujan’s life was compounded by ill-health and a vegetarian diet not adequately catered for in England.
When Hardy once went to see him in hospital he noted the number of the taxi that had taken him there, knowing Ramanujan’s fascination with numbers. It was 1729, which Hardy took to be a dull number, hoping it was not an ill omen. On the contrary, said Ramanujan, “it is a very interesting number; it is the first number expressible as the sum of two cubes in two different ways” — it equals 13 + 123 and 93 + 103.
In February 1919 after the Great War was over Ramanujan sailed for India, and never returned to Britain. This great self-taught genius died at 32 the following year, a loss of such incalculable magnitude that the movie industry has chosen some of the world’s finest actors to bring his story once again to life.