“In small proportions we just beauties see; And in short measures life may perfect be” - ‘The Noble Nature,’ Ben Jonson
In April 1984, when he read the letter thanking him for contributing to a bronze bust in memory of the mathematician Srinivasa Ramanujan, Takashi Ono was moved to tears. The letter was from Janaki Ammal, Ramanujan’s wife. From first hearing about Ramanujan in Tokyo in 1955 from the mathematician Andre Weil and having Ramanujan as an inspiration and then to being one of the few mathematicians in the world to fulfil the wish of Janaki Ammal that a statue of her husband be made, life had come a full circle for Takashi.
There was one more connection that Takashi had with Ramanujan: Takashi’s son Ken Ono made key contributions to ‘mock theta functions’ in early 2000s, a topic that Ramanujan had worked on in the last few months before his death in April 1920. In the mathematical community, it was no surprise that Ramanujan’s work was being expanded actively eight decades after his demise: many of Ramanujan’s findings anticipated research areas by many years.
Research at a feverish pitch
Ramanujan, born on December 22, 1887 was an autodidact who specialised in pure mathematics. While he excelled in mathematics, he neglected other subjects and could not complete his pre-university course. By 1908 he gave up studies, but not his research in mathematics. He struggled in poverty until in 1910, a benefactor, Ramachandra Rao, district collector of Nellore, provided him monthly allowance from his own pocket so that Ramanujan could pursue research. This would continue for a couple of years until Ramanujan managed to become a clerk at Madras Port Trust. He initiated contact with the British mathematician G.H. Hardy under whose insistence Ramanujan travelled to England in early 1914. His partnering with Hardy was productive: Ramanujan published more than 20 research papers between 1914 and 1919. During his stay, he was awarded a doctorate and made Fellow of Royal Society. When he returned to India in 1919, he was “...with a scientific standing and reputation such as no Indian has enjoyed before”. Unfortunately he lived only a year after his return succumbing to illness which was diagnosed then as tuberculosis but now revised as hepatic amoebiasis. However, in that one year, he continued his research at a feverish pitch.
Until he left for England in 1914, Ramanujan recorded his mathematical results, mostly equations, in his notebooks. There were three such notebooks (preserved now). One more was added when Ramanujan returned to India. Together there were about 4,000 results. The results were the culmination of research backed by deep intuition and insights. However, Ramanujan did not record proofs of his results: that work would be taken up by future generations of mathematicians.
Ramanujan’s work was in number theory, infinite series, analysis (theoretical underpinnings of calculus) and a few other areas in pure mathematics. Specifically, as G.H. Hardy wrote, these subjects were “...the applications of elliptic functions to the theory of numbers, the theory of continued fractions and... the theory of partitions”.
A few significant contributions were multiple formulae to calculate pi with great accuracy to billions of digits (22/7 is only an approximation to pi), partition functions (a partition is a way to represent a positive integer — for example, 1+1+1+1 is a partition of 4, 1+3 is another partition of 4, and so on), modular forms and hypergeometric series (the terms in every consecutive pair in the series form rational functions).
The importance of many of his works became apparent much later. One such was ‘Ramanujan conjecture’ which he published in 1916 and was proved in 1973 by Pierre Deligne. The conjecture inspired the development of theory of Galois representation that was employed in Andrew Wiles’ proof of Fermat’s last theorem published in 1995. In recent years, Ramanujan’s works and their extensions have found applications in signal processing to identify periodic information, akin to Fourier analysis. Mock theta functions have found applications in the study of black holes in astrophysics.
But to look for applications of his works is exactly how not to appreciate Ramanujan. As G.N. Watson, a contemporary mathematician of Ramanujan in Cambridge said, “The study of Ramanujan’s works gives me a thrill which is indistinguishable from the thrill which I feel... when I see before me the... beauty of the four statues... which Michelangelo has set over the tombs of Guiliano de Medici and Lorenzo de Medici”. For Ramanujan, mathematics was art.
Robert Kanigel, the celebrated biographer of Ramanujan, noted: “People will try to explain it in an easy way but I think they are unjustified in doing it. I think some people really are a few steps beyond where the rest of us live. We are forced to view those intellects, those artistic sensibilities, as a little bit mysterious or a little beyond what is the common realm.”
For Ramanujan, his mathematics was an end in itself. In this he thought like G.H. Hardy who claimed in A Mathematician’s Apology , “A mathematician, like a painter or a poet, is a maker of patterns... The ‘seriousness’ of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is ‘significant’ if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas.”
How did Ramanujan develop his mathematical abilities? Here again we have to look to G.H. Hardy: “...with his memory, his patience and his power of calculation, he combined a power of generalisation, a feeling for form, a capacity for rapid modification of his hypothesis, that were often really startling, and made him, in his own peculiar field, without a rival in his day”.
Power of intuition and insight
Let’s remember that Ramanujan was always precocious in his mathematical talent. And by virtue of working alone on problems and theorems that were advanced for his age during adolescence, day after day, hour after hour, he developed an incredible power of intuition and insight. The problems he worked on were from a book by one Carr. This book was “a spark which`ignited the flame... [but] as the depth of Ramanujan’s discoveries deepened, Carr’s influence certainly waned,” the mathematician Bruce Berndt said.
Ramanujan is remembered in many ways. His birthday (today is his 133rd birth anniversary) is celebrated as National Mathematics Day in India. The Ramanujan Journal publishes advancements in the areas that Ramanujan contributed to. His home in Kumbakonam in Tamil Nadu has been converted to a museum by SASTRA University.
Ramanujan’s greatness is not that he was a poor man who travelled to England and did research. It is that he was prolific and that his works were diverse, original and transcended time; and all this while he surmounted many odds coming from indigence. Ramanujan’s works, especially to mathematicians, are of enduring elegance. The mathematician and physicist Freeman Dyson said, “Whenever I am angry or depressed, I pull down the Collected papers [of Ramanujan] from the shelf and take a quiet stroll in Ramanujan’s garden”.
Varahasimhan is a history of science enthusiast based in Chennai