from 'Icons from the world of science', by S.Ananthanarayanan - home page

The phenomenon that was Ramanujan

Srinivasa Ramanujan was perhaps the first of the true ‘moderns’ of Indian science. He quietly took for India a place of honour in the stream of discovery in the sciences that swept through the western world since the 17th century. In mathematics, India did claim the development of the decimal system, the discovery of ‘zero’ and boasted of greats like Bhaskara and Brahmagupta. There were also clever results and quick ways to work out problems in arithmetic. But abstract and analytical work that was systematic and gave importance to proving things, progressing from particular results to general principles, was first discovered by the Greeks and then continued in Western Mathematics. It is true that India had attained, since ancient times, proficiency in architecture and metallurgy, but in the 19th century, Europe was the leader in a multitude of fields, of engineering, ship-building, chemical technology, spinning and weaving,… and also the sciences!

The education system introduced in India by the British was based upon the system in England and basic, formal mathematics was also taught. The sharpest of Indians excelled in law and in commerce, in engineering and in mathematics, they even established a ‘mathematical society’. This society tried to make available to its members the expensive books and journals from America and England. But still, the purpose of the educational system was only to supply the British regime with literate and competent workers. It was not to create philosophers and thinkers, which was the objective of the mother universities in England.

It was in this unlikely setting that Srinivasa Ramanujan astonished the western world by what he achieved. He looked deep into the simple, practical mathematics that was taught to him by the existing system and went on to rediscover much of what the greatest of European mathematicians had done in the last three centuries. And he went further, to add to mathematics whole new areas of research and marvels that are studied and analysed to this day.

Ramanujan was born on 22 Dec 1987, in Erode, on the banks of the Cauvery, 245 miles south-west of Chennai, in the south of India. His father was an accountant in a firm that dealt in saris and textiles, in Kumbakonam, and does not appear to have had any noticeable influence on Ramanujan. Ramanujan’s mother was a much stronger person, took most of the decisions in the household and was of some refinement. She was a singer of Bhajans in the local temple, to earn a few rupees to supplement the family income.

And so, in the placid, temple town of Kumbakonam, Ramanujan spent his childhood, reared as a traditional Hindu boy, full of scriptures, religion, the temple, and rituals unchanged for centuries

Till Ramanujan was ten years of age, he attended Kangayan Primary School. In 1897, he stood first in the district in English, Tamil, Arithmetic and Geography in the primary school examination. And the following year he enrolled the English language high school, Town High.

Town High was an institution of some tradition, with revered and wizened teachers and some record of fine students. Ramanujan made a mark early, and in just a year after he had enrolled, had become something of a legend in mathematics. Very soon, he became the one his companions turned to for help, and in a year, he began to challenge his teachers. Just about this time, Ramanujan came across Trigonometry, by S L Loney, the classic English High School text book on the subject, which does have some advanced material too. By the time Ramanujan was 13, he had mastered his Loney, an achievement that would do an undergraduate proud!

Ramanujan soon learnt many difficult things, like ‘cubic equations’, a topic that any but professional mathematicians would keep at a safe distance. He was at home with the mysteries about pi and e. Pi, or the ratio of the circumference of a circle to its diameter, and usually shown as 22/7, is a number that is remarkable in that it can never, even in theory, be exactly evaluated. ‘e’ is another such number, connected with how fast a thing, for example a bank deposit, grows if the compound interest is added not once a year, or even once a month, but continuously. Pi and e are also related, in a surprising way, involving the use of imaginary numbers, or quantities that involve the square root of –1! Subjects like these were Ramanujan’s close companions. And students two or three years senior came to him with problems that had evaded them for weeks and saw him solve them at a glance!

Trigonometry itself, as the name suggests, is a study of triangles, and associates the ratios of the sides of triangles with the angle between the sides. A far more sophisticated way to look at these ratios is to trace a link between angles, triangles and circles and to work in terms of infinitely long series of numbers. While still a fledgling of mathematics, Ramanujan developed, on his own, this approach to trigonometric ratios. In fact, he was rather mortified when he learnt, a little while later, that the method had been discovered by Euler in the 18th century!

By the time Ramanujan finished school he had become a celebrity for his talent in Mathematics. He won prizes and accolades and joined Government College, also in Kumbakonam, as an F.A. or ‘First Arts’ student, in 1904, with a scholarship.

So far, despite all the notoriety that comes of academic brilliance, Ramanujan had been a conventionally well-behaved lad and quite in his mother’s control. He was phenomenal in mathematics, it was true, but he did reasonably well in other subjects too. But in university, mathematics seemed to take hold of Ramanujan’s very being and brought to the surface a willfulness, an almost irrational, eccentric aspect that resulted in his having to drop out of regular formal, university education, for what it was worth, for many years.

The stage for this transformation was probably set by the influence of a mathematical work that Ramanujan had come upon just before he left school. The book in question was Carr’s Synopsis of Elementary Results in Pure Mathematics, a collection of important mathematical results, just results stated, without proof or explanation. Although not a work of any importance, by itself, the book had the effect of introducing the young and brilliant Ramanujan to the wonders of mathematics as developed by greats like Newton, Euler, Laplace, ever since the 17th century.

George Shoobridge Carr, the author, in fact, was a mathematician of average ability who had compiled his synopsis mainly as a volume of reference for students. During the eighteenth and early nineteenth century, mathematics students in England were preoccupied, almost insanely obsessed, with the challenge of the Tripos, an infamously testing examination of the University of Cambridge

The Tripos demanded incredible facility with the most demanding topics in mathematics and consisted of a set of problems, any of which would do a student credit to have solved at all. Success in the Tripos demanded that students solved many of them and a good rank in the Tripos usually assured for a student a bright career in any line of work he chose

Tutoring students for the Tripos had hence become a paying occupation. Carr, who had an MA in mathematics from Cambridge University, was an enthusiastic teacher. After years of painstaking work in helping students master the vast areas that the Tripos required them to, Carr compiled his Synopsis, a listing of important results, seemingly just a vast catalogue of theorems and formulas, perhaps not of much utility except in conjunction with all the material that went before.

But to curious and mathematically almost intuitive Ramanujan, Carr’s Synopsis proved a wonderland of challenges that took him swiftly through what may have been the course of many years of conventional training. To be fair to the Synopsis, although it states results with just, occasionally, a hint of a proof, it does have a ‘progression’, of results leading from one to another. It is no guide through the history of mathematics but it is at least a list of the highlights in nearly the correct order!

The lack of detailed proofs forced Ramanujan to work them out himself, thereby developing insight and discovering new results and extensions along the way! One of the definite results of this apprenticeship to Carr was certainly that Ramanujan did not learn the rigorous method of western mathematics, to clearly state the assumptions and then to systematically arrive at conclusions, with proofs. Following this example of the Synopsis, much of Ramanujan’s original work consists of brilliant results simply stated, with generations of researchers and students who came after him having to plod to find the proof.

The main result of this exposure to Carr, however, was that by the time Ramanujan was in college, he worked on mathematics and little else. When he finished his FA, his performance in mathematics, surely enough, was brilliant, but he failed in English and did dismally in other subjects. His scholarship was withdrawn and though Ramanujan tried to make ends meet by giving tuitions, he was soon a drop-out of Government College, Kumbakonam.

The months that followed were tense and troubled, with Ramanujan a drain on his family and far from prepared, academically or temperamentally for any kind of paying work. And still driven by a passion for mathematics! The effect was that Ramanujan ran away from home, first at the age of 17, when he went as far as Vishakhapatnam, on the eastern coast of Andhra Pradesh. What really he did at Vishakhapatnam, and how he supported himself is not very clear, except that in a few months, he was traced by his family and coaxed back home. He obviously found the pressure of being expected to come to some good too hard to bear, because he ran away from home more than once, till, at the age of 19, he was found a place in Pachaiyappa’s College, Chennai, to have another go at the First Arts course.

Pachaiyappa’s was an institution with a fine academic tradition and boasted an enlightened mathematics faculty. In mathematics, Ramanujan thrived. But as before, he neglected all else. Of his reaction to the class on the digestive system of the rabbit, which was taught as part of the course on physiology, Ramanujan’s biographer, Robert Kanigel reports that Ramanujan wrote in his exam paper, “Sir, this is my undigested product of the chapter on the digestive system”.

Ramanujan worked incessantly and crammed his now celebrated notebooks with theorems and discoveries. The notebooks have survived and are a record of the vast canvas of pure mathematics that Ramanujan worked with in those years. He had set out first, to put down proofs for the results in Carr’s Synopsis, but soon went much further. Each theorem brought out new and unimagined refinements and discoveries. The notebooks are a meandering record of fevered creativity, in many directions and in notation that is difficult for any but a talented mathematician to comprehend. The later notebooks were edited versions, where the material has been composed and annotated, for publication, are again full of intuitive leaps and steps encapsulating whole flights in one, whose unraveling has formed for many a student the work of a lifetime!

During the years 1908 to 1912 Ramanujan worked as well as his straightened circumstances would allow. He moved from Kumbakonam to Chennai, to Villipuram, near Pondicherry, or wherever a sponsor or kindly soul would keep him, and he constantly worked at his mathematics. The notebooks were edited and put in order, as a kind of visiting card, to seek whatever employment he could get, to keep him in two meals a day. He was also able to get his first path breaking papers published in the Journal of the Indian Mathematical society.

These first papers dealt, as much of his work, with series of numbers extending to Infinity.
An instance is a number like this:

or a square root which contains a square root, which contains a square root, and so on. When this was published in the Journal as a problem to challenge readers, no solution was forthcoming for months! Ramanujan’s notebooks and papers contained general theorems and stratagems to solve mysteries as this one as well as studies of the properties of the maze of mathematical constants and functions, known and newly discovered.

These publications soon started getting Ramanujan recognition and there began to develop around him a group of friends and admirers. Finally, in 1912, with the help of some of these, Ramanujan got a sinecure of a job as a clerk in the Madras Port Trust. With this backing of a regular income, he was able, at last to devote himself seriously to mathematics. He was already 25 years of age, an age by which mathematicians have generally done their greatest work, and Ramanujan had to hurry to catch up!

It was now that Ramanujan’s work came to the notice of educated Englishmen working in Madras and they sent portions of Ramanujan’s work to abler mathematicians in England for an opinion. The response was hesitant and cautious, but of a quality that convinced Ramanujan that it was in England that he should seek appreciation and educated critique of his work.

So Ramanujan began writing to leading mathematicians in England, with samples of his work. One after another, the letters seemed to arouse no interest, till, providentially, a brilliant, young mathematician, at Cambridge, G.H.Hardy, FRS, took enthusiastic notice and was thunderstruck by the incredible originality of the material sent to him by the remarkable Indian clerk.

What followed is historic. Even before Hardy wrote back to Ramanujan, he had set in motion the process of inviting Ramanujan to Cambridge.

Ramanujan baulked at first. Orthodox Hindus believed crossing the ocean resulted in losing caste. But Hardy’s letter infused him with renewed confidence and the desire for recognition. Hardy and Ramanujan began an exchange of letters on mathematics, and while, on the strength of this communication, Ramanujan got a scholarship from the Madras University, in March 1914. Ramanujan actually set out for England.

At Cambridge Hardy got to work at first hand with the remarkable notebooks, of which some 120 theorems he had seen so far, by post. Soon their character became clearer. A good portion was rediscovery of work that had gone before, even during the 40 years that had passed after the publication of Carr’s Synopsis. Some parts were even incorrect. But a substantial portion, over a third, Hardy could see, was brilliant and mint new! Other commentators have put this portion to almost half.

Hardy undertook in full earnest the task of editing Ramanujan’s notebooks for publication. Ramanujan also set to work in academically electrifying surroundings of Cambridge. Ramanujan knew now that a proper university education, like in England, would have trained him in much mathematics that had been discovered by talented men. This would not only have saved him the trouble of discovering them again, to no credit, but would have launched him onto greater things. 1915, the year after Ramanujan’s arrival in England thus saw a flurry of papers published, much being new work.

These papers and the work of subsequent years covered novel approaches to evaluating the random and infinite digits in the expansion of the number , modular forms, divergent series, elliptic integrals, number theory, discoveries in prime numbers, partitions, round numbers…. These were subjects that gripped contemporary mathematicians and Ramanujan’s novel approach and tremendous intuitive feel for numbers made deep impact and provoked great interest.

Such was the stature of the work being turned out that Hardy soon felt obliged to propose Ramanujan to be elected as a Fellow of Trinity College. Some university politics and perhaps a touch of racism actually kept Ramanujan from being elected for two years running. This did affect Ramanujan in spirit. In body, he was already in poor shape, the winters in England not having been comfortable and now he contracted tuberculosis.

Soon after the Trinity Fellowship disappointment, he entered Matlock, a TB sanatorium. But the sanatorium could hardly help him, as he was resolutely vegetarian and now began to crave for south Indian food. He was too sick to be mathematically productive and this rankled too.

In the meantime, undeterred by the failure at Trinity, Hardy went on with trying to get Ramanujan the recognition he deserved. In December 1917, Ramanujan was elected to the London Mathematical society. And then, just two weeks later, Hardy and eleven other mathematical heavyweights of England nominated Ramanujan for election as a Fellow of the Royal Society!

“Distinguished as a pure mathematician, particularly for his investigation of elliptic functions and the theory of numbers’ was how he was described in their nomination. In January 1918, the Royal Society published a list of 103, Ramanujan included, for election.

On the face of it, Ramanujan was against difficult odds. His election would also have been premature, at the age of just 31. Still, Hardy lobbied for him, “his claims are such as, in the long run, could not be ignored… There is an absolute gulf between him and other mathematical candidates…”.

In February that year, Ramanujan was elected to the Cambridge Philosophical Society. And then, ten days later, Ramanujan could hardly believe the telegram he received from Hardy, he was elected a Fellow of the Royal Society! And a few months later, also as a Fellow of Trinity College.

The effect on Ramanujan’s spirits was, as expected, “a brief period of brilliant invention”, with more papers and publications. His health too, seemed to get a little better. With the end of the war and the sea-lanes being open once more, it was time now for Ramanujan to return to India, which he did in March 1919.

He arrived in India, as the Journal of the Indian Mathematical Society announce, ‘in indifferent health’. For a year after this, Ramanujan did correspond a little with Hardy about “theta function’, an area of complex mathematics, involving infinites and elliptic ratios, just what was needed to cheer his troubled spirits. But his health deteriorated continuously, till, on the 26th April, 1920, he died, one of the most original mathematicians of the century.