The Short Life of Srinivasa Ramanujan

Srinivasa Ramanujan (1887-1920)

Srinivasa Ramanujan (1887-1920)

On December 22, 1887, Indian mathematician and autodidact Srinivasa Ramanujan was born. Though he had almost no formal training in pure mathematics, he made major contributions to mathematical analysis, number theory, infinite series, and continued fractions. Supported by English mathematician G. H. Hardy from Cambridge, Ramanujan independently compiled nearly 3,900 results during his short life, which all have been proven correct.

Ramanujan was born into a Tamil Brahmin Iyengar family in Erode, Madras Presidency (now Tamil Nadu), at the residence of his maternal grandparents. His father, K. Srinivasa Iyengar, worked as a clerk in a sari shop and his mother, Komalatammal, was a housewife. When he was nearly five years old, Ramanujan entered the primary school in Kumbakonam although he would attend several different primary schools before entering the Town High School in Kumbakonam in January 1898. At the Town High School, Ramanujan was to do well in all his school subjects and showed himself an able all round scholar. In 1900 he began to work on his own on mathematics summing geometric and arithmetic series[2]. Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quartic. The following year, not knowing that the quintic could not be solved by radicals, he tried (and of course failed) to solve the quintic.[2]

It was in the Town High School that Ramanujan came across a mathematics book by G S Carr called Synopsis of elementary results in pure mathematics. This book, with its very concise style, allowed Ramanujan to teach himself mathematics, but the style of the book was to have a rather unfortunate effect on the way Ramanujan was later to write down mathematics since it provided the only model that he had of written mathematical arguments.[2] The next year, Ramanujan independently developed and investigated the Bernoulli numbers and calculated the Euler–Mascheroni constant up to 15 decimal places. His peers at the time commented that they “rarely understood him” and “stood in respectful awe” of him.

When he graduated from Town Higher Secondary School in 1904, Ramanujan was awarded the K. Ranganatha Rao prize for mathematics and received a scholarship to study at Government Arts College, Kumbakonam, but was so intent on mathematics that he could not focus on any other subjects and failed most of them, losing his scholarship in the process. He enrolled at Pachaiyappa’s College in Madras. There he passed in mathematics, choosing only to attempt questions that appealed to him and leaving the rest unanswered, but performed poorly in other subjects, such as English, physiology and Sanskrit. Ramanujan failed his Fellow of Arts exam in December 1906 and again a year later. He continued his mathematical work, however, and at this time he worked on hypergeometric series and investigated relations between integrals and series. He was to discover later that he had been studying elliptic functions. But, without a FA degree, he left college and continued to pursue independent research in mathematics, living in extreme poverty and often on the brink of starvation.

Ramanujan continued to develop his mathematical ideas and began to pose problems and solve problems in the Journal of the Indian Mathematical Society. He devoloped relations between elliptic modular equations in 1910. After publication of a brilliant research paper on Bernoulli numbers in 1911 in the Journal of the Indian Mathematical Society he gained recognition for his work. Despite his lack of a university education, he was becoming well known in the Madras area as a mathematical genius.

The professor of civil engineering at the Madras Engineering College C L T Griffith was also interested in Ramanujan’s abilities and, having been educated at University College London, knew the professor of mathematics there, namely M J M Hill. He wrote to Hill on in 1912 sending some of Ramanujan’s work and a copy of his 1911 paper on Bernoulli numbers. Hill replied in a fairly encouraging way but showed that he had failed to understand Ramanujan’s results on divergent series. The recommendation to Ramanujan that he read Bromwich’s Theory of infinite series did not please Ramanujan much. Ramanujan wrote to E W Hobson and H F Baker trying to interest them in his results but neither replied. In January 1913 Ramanujan wrote to G H Hardy having seen a copy of his 1910 book Orders of infinity. In Ramanujan’s letter to Hardy he introduced himself and his work:

‘I have had no university education but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at mathematics. I have not trodden through the conventional regular course which is followed in a university course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as ‘startling’.

Hardy initially view Ramanujan’s manuscripts as a possible fraud. Hardy recognised some of Ramanujan’s formulae but others “seemed scarcely possible to believe” [1]. After seeing Ramanujan’s theorems on continued fractions on the last page of the manuscripts, Hardy commented that “they [theorems] defeated me completely; I had never seen anything in the least like them before“. He figured that Ramanujan’s theorems “must be true, because, if they were not true, no one would have the imagination to invent them“. In 1914, Hardy brought Ramanujan to Trinity College, Cambridge, to begin an extraordinary collaboration. Setting this up was not an easy matter. Ramanujan was an orthodox Brahmin and so was a strict vegetarian. His religion should have prevented him from travelling but this difficulty was overcome, partly by the work of E H Neville who was a colleague of Hardy’s at Trinity College and who met with Ramanujan while lecturing in India. Right from the beginning, however, he had problems with his diet. The outbreak of World War I made obtaining special items of food harder and it was not long before Ramanujan had health problems.

Ramanujan spent nearly five years in Cambridge collaborating with Hardy and Littlewood, and published part of his findings there. Hardy and Ramanujan had highly contrasting personalities. Their collaboration was a clash of different cultures, beliefs, and working styles. Hardy was an atheist and an apostle of proof and mathematical rigour, whereas Ramanujan was a deeply religious man who relied very strongly on his intuition. While in England, Hardy tried his best to fill the gaps in Ramanujan’s education without interrupting his inspiration. Ramanujan was awarded a Bachelor of Science degree by research (this degree was later renamed PhD) in March 1916 for his work on highly composite numbers. On 6 December 1917, he was elected to the London Mathematical Society. In 1918 he was elected a Fellow of the Royal Society, at age 31 as one of the youngest Fellows in the history of the Royal Society.

Ramanujan sailed back to India in 1919. However his health was very poor and, despite medical treatment, he died there the following year. Bruce C. Berndt stated that over the last 40 years, as nearly all of Ramanujan’s theorems have been proven right: “Paul Erdős has passed on to us Hardy’s personal ratings of mathematicians. Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100, Hardy gave himself a score of 25, J. E. Littlewood 30, David Hilbert 80 and Ramanujan 100.’

At yovisto, you can learn more about Ramanujan’s work in the mathematics lecture of John D. Barrow from Gresham College, London, on Continuos Fractions.

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