## The Man Who Knew Infinity

Way back in the 1914 Prof G.H.Hardy, a famous British mathematician, went to see his friend, who was recovering from an illness in a hospital, in London’s Putney district. He remarked to him, that he had come in a cab which had a number 1729, and some how that number seemed, dull and unremarkable to him. To which his friend immediately replied

“No Hardy, it’s a very interesting number, it is the smallest number that can be expressed as the sum of two cubes in two different ways”.

To the mathematically challenged, it means that 1729 can be expressed as

1729 = 1728+1 = 12^3 + 1^3
1729 = 1000+729 = 10^3 + 9^3

This was an anecdote, which we often read as school students, and even during our mathematics classes. The friend here is of course Srinivasa Ramanujan, one of India’s greatest scientists and one of the world’s greatest mathematicians.

The bonding between Hardy and Ramanujan was much earlier though. Exactly a year back, Hardy received a letter from Ramanujan,  then working as a clerk at Madras Port Trust.  It was a 10 page letter, which had around 120 statements on theorems, continued fractions and number theory.  Initially Hardy dismissed that letter as a crank, the ones mathematicians usually get, but a second look made him revise his opinion. He discussed it over with his colleague, J.E.Littlewood, and their conclusions “the results must be true because, if they were not true, no one would have had the imagination to invent them”.

The world of mathematics owes a great debt to India , because this is the land from which the basic theories of mathematics have originated. Aryabhatta who contributed the number-place value system and the concept of zero, as well as calculating the area of the triangle. Bhaskara who came up the concept of the decimal system. Halayudha who provided a clear description of the Pascal’s triangle.  And between 1300-1600 AD, we had the Kerala school of Mathematics and Astronomy founded by Madhava of Sangamagrama which made significant contributions to the fields of  infinite series and calculus.

Ramanujan was born into an Iyengar family, in the town of Erode, Tamil Nadu on Dec 22, 1887. His father was a clerk in a saree shop, while his mother was a devout Brahmin housewife. Theirs was a typical Tamil Brahmin household, pretty religious and orthodox. Growing up under his mother’s care and guidance, he learnt about Indian tradition and puranas. His first brush with mathematics was in 1898, in Higher Secondary School.  And that was when his prodigal talent came to being. At 11 years, he exhausted the mathematical knowledge of two college students, by 13 years he completely mastered the books on advanced trigonometry written by S.L.Loney. At 14 years he was assisting his school in the logistics of assigning its 1200 odd students to 35 teachers. When he was 16 years old in 1903, Ramanujan, got a book from the library, through his friend, by the British mathematician, George.S.Carr, titled  A Synopsis of Pure Mathematics, which was written in 1866.  The book a collection of around 5000 theorems,  stoked his curiosity, made him explore mathematics much more deeper.  The following year, Ramanujan, developed and investigated the Bernoulli numbers, calculated the Euler-Mascheroni constant up to 15 decimal places, winning him a new found respect among his peers.

Prof E.W.Middlemast and Sir Francis Spring.

However his obsession with mathematics, made him neglect other subjects, that he often ended up failing in most of them. He dropped out of college, and pursued maths research on his own. He made ends meet, by taking up tuition for school students. He later left for Villupuram, where he met the Dy. Collector, Mr Ramaswamy Iyer, who founded the Indian Mathematical Society, and applied for a clerical job, in the revenue department. However Iyer was wonder struck by Ramanujan’s genius at mathematics, and sent him to Madras, with letters of introduction. In Iyer’s own words.

I was struck by the extraordinary mathematical results contained in the notebooks. I had no mind to smother his genius by an appointment in the lowest rungs of the revenue department.

In Madras he got to meet the then Nellore district collector, R.Ramachandra Rao, who initially doubted that such a genius was original.  However with persistence from Rao’s friend, Rajagopalachari, he had a long discussion with Ramanujan over elliptic integrals and divergent series.  Rao, who was also the Secretary for the Indian Mathematical Society, was now convinced about Ramanujan, and supported him financially. With the help from Ramaswamy Aiyer, Ramanujan  managed to get his work published in Journal of the Indian Mathematical Society.  He however had a rather erratic style of writing, often not too clear enough,  and this  made his work quite hard to understand for the average reader. He had the most innovative methods to solve problems, yet his not too clear writing style and precision, could make it hard for others to understand.

Sir,
I understand there is a clerkship vacant in your office, and I beg to apply for the same. I have passed the Matriculation Examination and studied up to the F.A. but was prevented from pursuing my studies further owing to several untoward circumstances. I have, however, been devoting all my time to Mathematics and developing the subject. I can say I am quite confident I can do justice to my work if I am appointed to the post. I therefore beg to request that you will be good enough to confer the appointment on me.

The above application in 1912, along with a letter of recommendation from his Maths Professor, E.W.Middlemast at Presidency,  ensured he got a job as a clerk in the Accounts Department of the Madras Port Trust. During his spare time, he would still continue his research in mathematics. He was encouraged by his boss, Sir Francis Spring and his colleague, S.Narayana Iyer. Some of his friends like Spring, Narayana Iyer, Ramachandra Rao and Middlemast, helped him to send his work for Cambridge University, but he was rejected due to lack of formal educational qualifications. On 16 Jan, 1913, he wrote to Prof G.H. Hardy, who had the foresight to recognize Ramanujan’s skills. Hardy was especially impressed by Ramanujan’s work on continued fractions, claiming he had never seen such work before. One of Hardy’s colleagues Neville later remarked “not one of his theorems, could have been set in the most advanced mathematical examination in the world”.

Hardy requested Ramanujan to come to Cambridge, but he refused, due to the then prevailing sentiment of not going to a foreign land, strongly rooted in Indian society. For the time being, he was given a research grant at the University of Madras, where he did pioneering work on Frullani’s 1821 integral theorem. Hardy’s colleague Neville, again asked Ramanujan to come to Cambridge, and this time Ramanujan agreed. According to a popular anecdote, his mother had a dream, in which their family goddess Namagiri, commanded her not to stand in the way of her son’s progress.  From 1913-18, Ramanujan spent 5 fruitful years in Cambridge, collaborating with Hardy and Littlewood, on many research projects. While Littlewood said that “this man was at least a Carl Gustav Jacob”, Hardy remarked that he can be compared only with Euler or Jacobi.

In March 1916, Ramanujan was awarded the Doctorate for his research on highly composite numbers. Hardy later remarked that this was one of the most unusual papers he had ever seen. Both of them had contrasting ways of working, Hardy followed the Western model of proof and rigor, and was an atheist. Ramanujan was a devout believer and often relied on intuition and gut feeling. It was the perfect mix of East and West. On Oct 1918, he became the first Indian to be elected as Fellow of Trinity College, Cambridge. In 1917, he became a member of the London Mathematical Society. In 1918 he became the second Indian to gain fellowship of the Royal Society, and one of it’s youngest members. Unfortunately, his poor health, ensured he could not live long, and he died in 1919, at an age of 32. His home state of Tamil Nadu celebrates his birthday as State IT day, while Government Arts College, Kumbakonam remembers the day as Ramanujan Day.  It is fitting that his birthday should be declared as National Mathematics Day by Government, a truly great Indian.