Srinivasa Ramanujan was a genius. His approach to mathematics, his abilities, sense, insight, was unheard of. In his biographical notice for Ramanujan, G. H. Hardy wrote:
It was his insight into algebraical formulae, transformation of infinite series, and so forth, that was most amazing. On this side most certainly I have never met his equal, and I can compare him only with Euler or Jacobi. He worked, far more than the majority of modern mathematicians, by induction from numerical examples: all his congruence properties of partitions for example were discovered in this way. But with his memory, his patience, and his power of calculation he combined a power of generalisation, a feeling of form, and a capacity for rapid modification of his hypotheses, that were often really starling, and made him, in his own peculiar field, without a rival in his day.
“The answer came to my mind”
In December 1914, the popular English magazine Strand published a riddle about the imaginary town of Little Wurzelfold.
One Sunday morning soon after the December issue appeared, P. C. Mahalanobis sat with it at a table in Ramanujan’s rooms in Whewell’s Court. Now, with Ramanujan in the little back room stirring vegetables over the gas fire, Mahalanobis grew intrigued by the problem and figured he’d try it out on his friend.
“Now here’s a problem for you,” he yelled into the next room
“What problem? Tell me,” said Ramanujan, still stirring. And Mahalanobis read it to him.
“I was talking the other day,” said William Rogers to the other villagers gathered around the inn fire, “to a gentleman about the place called Louvain, what the Germans have burnt down. He said he knowed it well – used to visit a Belgian friend there. He said the house of his friend was in a long street, numbered on this side one, two, three, and so on, and that all the numbers on one side of him added up exactly the same as all the numbers on the other side of him. Funny thing that! He said he knew there was more than fifty houses on that side of the street, but not so many as five hundred. I made mention of the matter to our parson, and he took a pencil and worked out the number of the house where the Belgian lived. I don’t know how he done it.”
Perhaps the reader may like to discover the number of that house.
Through trial and error, Mahalanobis (who would go on to found the Indian Statistical Institute and become a Fellow of the Royal Society) had figured it out in a few minutes. Ramanujan figured it out, too, but with a twist. “Please take down the solution,” he said-and proceeded to dictate a continued fraction, a fraction whose denominator consists of a number plus a fraction, that fraction’s denominator consisting of a number plus a fraction, ad infinitum. This wasn’t just the solution to the problem, it was the solution to the whole class of problems implicit in the puzzle. As stated, the problem had but one solution – house no. 204 in a street of 288 houses; 1+2 + … 203 = 205 + 206 + … 288. But without the 50-to-500 house constraint, there were other solutions. For example, on an eight-house street, no. 6 would be the answer: 1 + 2 + 3 + 4 + 5 on its left equaled 7 + 8 on its right. Ramanujan’s continued fraction comprised within a single expression all the correct answers.
Mahalonobis was astounded. How, he asked Ramanujan, had he done it?
“Immediately I heard the problem it was clear that the solution should obviously be a continued fraction; I then thought, Which continued fraction? And the answer came to my mind.”
The answer came to my mind. That was the glory of Ramanujan – that so much came to him so readily, whether through the divine offices of the goddess Namagiri, as he sometimes said, or through what Westerners might ascribe, with equal imprecision, to “intuition.”
- Kanigel, Robert (1991). The Man Who Knew Infinity: a Life of the Genius Ramanujan. New York: Charles Scribner’s Sons.
“But why was Ramanujan so certain there was one?”
Preface: to understand the following story, no need to know much mathematics, and the technical details are not so important. The story discusses the function p(n) which gives for a positive number n the number of “partitions”, which is in how many ways we can write n as a sum of positive integers. For example, p(4) = 5 Because 4 is equal to: 4, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1. These are 5 different ways in total. The function grows very fast, and p(200), how many ways there are to write 200 as a sum of positive integers, is already equal to 3972999029388.
Mathematics looked for a formula to easily calculate p(n) for any n, or at least to find an approximation formula, that will give a “close enough” result. Hardy and Ramanujan found a formula that contained the following expression: eπ/q • sqrt(2n/3). However, the formula wasn’t close enough, and at some point, Ramanujan suggested to change the expression to eπ/q • sqrt(2(n – 1/24)/3). I.e., to replace n with n – 1/24. Why –1/24? How he came to this idea? This remains unclear, but since that moment, the formula started to behave well! Here is how Littlewood described these events:
The story of the theorem is a romantic one. (To do it justice I must infringe a little the rules about collaboration. I therefore add that Prof. Hardy confirms and permits my statements of bare fact.)
One of Ramanujan’s Indian conjectures was that the first term [of a series] was a very good approximation to p(n); this was established without great difficulty. At this stage the n – 1/24 was represented by a plain n – the distinction is irrelevant.
From this point the real attack begins. The next step in development, not a very great one, was to treat [the series] as an ‘asymptotic’ series, of which a fixed number of terms [for example, the first four] were to be taken, the error being of the order of the next term [that is, an error that grows with n].
But from now to the very end Ramanujan always insisted that much more was true than had been established: ‘there must be a formula with error O(1)’ [i.e., with a finite error for all n] This was his most important contribution; it was both absolutely essential and most extraordinary.
A severe numerical test was now made, which elicited the astonishing facts about p(100) and p(200). Then v was made a function of n [a mathematical detail, irrelevant for the story]; this was a very great step, and involved new and deep function-theory methods that Ramanujan obviously could not have discovered by himself.
The complete theorem thus emerged. But the solution of the final difficulty was probably impossible without one more contribution from Ramanujan, this time a perfectly characteristic one. As if its analytical difficulties were not enough, the theorem was entrenched also behind almost impregnable defences of a purely formal kind. The form of the function [which includes the above expression] is a kind of indivisible unit; among many asymptotically equivalent forms it is essential to select exactly the right one.
Unless this is done at the outset, and the –1/24 (to say nothing of the d/dn – in addition to replacing n with n – 1/24, Ramanujan also proposed to differentiate the key expression with respect to n) is an extraordinary stroke of formal genius, the complete result can never come into the picture at all.
There is, indeed, a touch of real mystery. If only we knew there was a formula with error O(1) [a finite error], we might be forced, by slow stages, to the correct form of the function. But why was Ramanujan so certain there was one? Theoretical insight, to be the explanation, had to be of an order hardly to be credited. Yet it is hard to see what numerical instances could have been available to suggest so strong a result. And unless the form of the function was known already, no numerical evidence could suggest anything of the kind – there seems no escape, at least, from the conclusion that the discovery of the correct form was a single stroke of insight.
We owe the theorem to a singularly happy collaboration of two men [Hardy and Ramanujan], of quite unlike gifts, in which each contributed the best, most characteristic, and most fortunate work that was in him. Ramanujan’s genius did have this one opportunity worthy of it.
- A Mathematician’s Miscellany, by Littlewood,J.E. Methuen And Company Limited.
“His personal friends”
Someone once remarked, that for Ramanujan, every positive integer was one of his personal friends (thanks to Hardy the phrase is often attributed to Littlewood). This must be true, as is demonstrated in the following short story:
It happened when Ramanujan was ill and Hardy went to visit him at Putney. Once, in the taxi from London, Hardy noticed its number, 1729. He must have thought about it a little because he entered the room where Ramanujan lay in bed and, with scarcely a hello, blurted out his disappointment with it. It was, he declared, “rather a dull number,” adding that he hoped that wasn’t a bad omen.
“No, Hardy,” said Ramanujan. “It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.” Finding numbers that were the sum of one pair of cubes was easy. For example, 2^3 + 3^3 = 35. But could you get to 35 by adding some other pair of cubes? You couldn’t. And as you tried the integers one by one, it was the same story. One pair sometimes, two pair never-never, that is, until you reached 1729, which was equal to 12^3 + 1^3, but also 10^3 + 9^3.
How did Ramanujan know? It was no sudden insight. Years before, he had observed this little arithmetic morsel, recorded it in his notebook and, with that easy intimacy with numbers that was his trademark, remembered it.
- Kanigel, Robert (1991). The Man Who Knew Infinity: a Life of the Genius Ramanujan. New York: Charles Scribner’s Sons.
Featured image of Ramanujan By Konrad Jacobs – Oberwolfach Photo Collection, original location, CC BY-SA 2.0 de, https://commons.wikimedia.org/w/index.php?curid=3911526
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