Srinivasa Ramanujan

(Dec. 22, 1887 -- April 26, 1920)

K. Srinivasa Rao

The Institute of Mathematical Sciences, Madras-600 113.

Srinivasa Ramanujan (1887-1920) hailed as an all-time great mathematician, like Euler, Gauss or Jacobi, for his natural genius, has left behind 4000 original theorems, despite his lack of formal education and a short life-span. In his formative years, after having failed in his F.A. (First examination in Arts) class at College, he ran from pillar to post in search of a benefactor. It is during this period, 1903-1914, he kept a record of the final results of his original research work in the form of entries in two large-sized Note Books. These were the ones which he showed to Dewan Bahadur Ramachandra Rao (Collector of Nellore), V. Ramaswamy Iyer (Founder of Indian Mathematical Society), R. Narayana Iyer (Treasurer of IMS and Manager, Madras Port Trust), and to several others to convince them of his abilities as a Mathematician. The orchestrated efforts of his admirers, culminated in the encouragement he received from Prof. G.H. Hardy of Trinity College, Cambridge, whose warm response to the historic letter of Ramanujan which contained about 100 theorems, resulted in inducing the Madras University, to its lasting credit, to rise to the occasion thrice - in offering him the first research scholarship of the University in May 1913 ; then in offering him a scholarship of 250 pounds a year for five years with 100 pounds for passage by ship and for initial outfit to go to England in 1914 ; and finally, by granting Ramanujan 250 pounds a year as an allowance for 5 years commencing from April 1919 soon after his triumphant return from Cambridge ``with a scientific standing and reputation such as no Indian has enjoyed before''.

Ramanujan was awarded in 1916 the B.A. Degree by research of the Cambridge University. He was elected a Fellow of the Royal Society of London in Feb. 1918 being a ``Research student in Mathematics Distinguished as a pure mathematician particularly for his investigations in elliptic functions and the theory of numbers'' and he was elected to a Trinity College Fellowship, in Oct. 1918 (- a prize fellowship worth 250 pounds a year for six years with no duties or condition, which he was not destined to avail of). The ``Collected Papers of Ramanujan'' was edited by Profs. G.H.Hardy, P.V. Seshu Aiyar and B.M. Wilson and first published by Cambridge University Press in 1927 (later by Chelsea, 1962 ; and by Narosa, 1987), seven years after his death. His `Lost' Notebook found in the estate of Prof. G.N. Watson in the spring of 1976 by Prof. George Andrews of Pennsylvania State University, and its facsimile edition was brought out by Narosa Publishing House in 1987, on the occasion of Ramanujan's birth centenary. His bust was commissioned by Professors R. Askey, S. Chandrasekhar, G.E. Andrews, Bruce C. Berndt (`the gang of four'!) and `more than one hundred mathematicians and scientists who contributed money for the bust' sculpted by Paul Granlund in 1984 and another was commissioned for the Ramanujan Institute of the University of Madras, by Mr. Masilamani in 1994. His original Note Books have been edited in a series of five volumes by Bruce C. Berndt (``Ramanujan Note Books'', Springer, Parts I to V, 1985 onwards), who devoted his attention to each and every one of the three to four thousand theorems. Robert Kanigel recently wrote a delightfully readable biography entitled : ``The Man who knew Infinity : a life of the Genius Ramanujan'' (Scribners 1991; Rupa & Co. 1993). Truly, the life of Ramanujan in the words of C.P. Snow: ``is an admirable story and one which showers credit on nearly everyone''.

During his five year stay in Cambridge, which unfortunately overlapped with the first World War years, he published 21 papers, five of which were in collaboration with Prof. G.H. Hardy and these as well as his earlier publications before he set sail to England are all contained in the ``Collected Papers of Srinivasa Ramanujan'', referred earlier. It is important to note that though Ramanujan took his ``Note Books'' with him he had no time to delve deep into them. The 600 formulae he jotted down on loose sheets of paper during the one year he was in India, after his meritorious stay at Cambridge, are the contents of the `Lost' Note Book found by Andrews in 1976. He was ailing throughout that one year after his return from England (March 1919 - April 26, 1920). The last and only letter he wrote to Hardy, from India, after his return, in Jan. 1920, four months before his demise, contained no news about his declining health but only information about his latest work : ``I discovered very interesting functions recently which I call `Mock' theta-functions. Unlike the `False' theta-functions (studied partially by Prof. Rogers in his interesting paper) they enter into mathematics as beautifully as ordinary theta-functions. I am sending you with this letter some examples ... ''. The following observation of Richard Askey is noteworthy: ``Try to imagine the quality of Ramanujan's mind, one which drove him to work unceasingly while deathly ill, and one great enough to grow deeper while his body became weaker. I stand in awe of his accomplishments; understanding is beyond me. We would admire any mathematician whose life's work was half of what Ramanujan found in the last year of his life while he was dying''.

As for his place in the world of Mathematics, we quote Bruce C Berndt: ``Paul Erdos 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, Littlewood 30, Hilbert 80 and Ramanujan 100''. G.H.Hardy, in 1923, edited Chapter XII of Ramanujan's second Notebook on Hypergeometric series which contained 47 main theorems, many of them followed by a number of corollaries and particular cases. This work had taken him so many weeks that he felt that if he were to edit the entire Notebooks ``it will take the whole of my lifetime. I cannot do my own work. This would not be proper.'' He urged Indian authorities and G.N.Watson and B.M. Wilson to edit the Notebooks. Watson and Wilson divided the task of editing the Notebooks - Chapters 2 to 13 were to be edited by Wilson and Chapters 14 to 21 by Watson. Unfortunately, the premature death of Wilson, in 1935, at the age of 38, aborted this effort. In 1957, with monetary assistance from Sir Dadabai Naoroji Trust, at the instance of Professors Homi J Bhabha and K. Chandrasekaran, the Tata institute of Fundamental Research published a facsimile edition of the Notebooks of Ramanujan in two volumes, with just an introductory para about them. The formidable task of truly editing the Notebooks was taken up in right earnest by Professor Bruce C. Berndt of the University of Illinois, in May 1977 and his dedicated efforts for nearly two decades has resulted in the Ramanujan's Notebooks published by Springer-Verlag in five Parts, the first of which appeared in 1985. The three original Ramanujan Notebooks are with the Library of the University of Madras, some of the correspondence, papers/letters on or about Ramanujan are with the National Archives at New Delhi and the Tamil Nadu Archives, and a large number of his letters and connected papers/correspondence and notes by Hardy, Watson, Wilson are with the Wren Library of Trinity College, Cambridge. ``Ramanujan : Letters and Commentary'', by Bruce C. Berndt and Robert A. Rankin (published jointly by the American Mathematical Society and London Math. Society, 1995) is a recent publication. The Ramanujan Institute for Advanced Study in Mathematics of the University of Madras is situated at a short distance from the famed Marina Beach and is close to the Administrative Buildings of the University and its Library. The bust of Ramanujan made by Mr. Masilamani is housed in the Ramanujan Institute. In 1992, the Ramanujan Museum was started in the Avvai Kalai Kazhagam in Royapuram. Mrs. Janakiammal Ramanujan, the widow of Ramanujan, lived for several decades in Triplicane, close to the University's Marina Campus and died on April 13, 1994. A bust of Ramanujan, sculpted by Paul Granlund was presented to her and it is now with her adopted son Mr. W. Narayanan, living in Triplicane.


  1. Dictionary of Scientific Biography
  2. Biography in Encyclopaedia Britannica
  3. G H Hardy, Ramanujan (Cambridge, 1940).
  4. R A Rankin, Ramanujan's manuscripts and notebooks, Bull. London Math. Soc. 14 (1982), 81-97.
  5. R A Rankin, Ramanujan's manuscripts and notebooks II, Bull. London Math. Soc. 21 (1989), 351-365.
  6. R Kanigel, The man who knew infinity : A life of the genius Ramanujan (New York, 1991).
  7. B Berndt, Srinivasa Ramanujan, The American Scholar 58 (1989), 234-244.
  8. B Berndt and S Bhargava, Ramanujan - For lowbrows, Amer. Math. Monthly 100 (1993), 644-656.
  9. J M Borwein and P B Borwein, Ramanujan and pi, Scientific American 258 (2) (1988), 66-73.
  10. S R Ranganathan, Ramanujan : the man and the mathematician (London, 1967).
  11. S Ram, Srinivasa Ramanujan (New Delhi, 1979).
  12. L Debnath, Srinivasa Ramanujan (1887-1920) : a centennial tribute, International journal of mathematical education in science and technology 18 (1987), 821-861.
  13. R A Rankin, Srinivasa Ramanujan (1887- 1920), International journal of mathematical education in science and technology18 (1987), 861- .
  14. K. Srinivasa Rao, Srinivasa Ramanujan: a Mathematical Genius (EastWest Books (Madras) Pvt. Ltd., 1998).


by G.H.Hardy

Godfrey Hardy was the Cambridge mathematician who `discovered' the great Indian mathematician Ramanujan. This is a condensed version of the (20 page!) first chapter in "Ramanujan: 12 lectures on subjects suggested by his life and work" by Hardy. It was not possible to get Hardy's approval for this due to technical reasons.


I have set myself a task that is genuinely difficult, even impossible --- to form some sort of reasoned estimate of the most romantic figure in the recent history of mathematics; a man whose career seems full of paradoxes and contradictions, who defies almost all the canons by which we are accustomed to judge one another, and about whom all of us will probably agree in one judgement only, that he was in some sense a very great mathematician.

The difficulties in judging Ramanujan are clear --- he was an Indian, I am an Englishman, and the two parties have always found it hard to understand one another. He was at best, a half-educated Indian, since he never could rise to be even a "failed B.A.". He worked for most of his life ignorant of modern European maths, and died when he was thirty and when his mathematical education had in some ways hardly begun. He published abundantly (at least 400 pages worth) but left behind even more unpublished stuff. While this work includes much that is new, about two-thirds is rediscovery, that too usually imperfect rediscovery.

His early life

Srinivasa Aiyangar Ramanujan was born in 1887 in a poor Brahmin family at Erode near Kumbakonam, a fair sized town in the Tanjore district of Tamil Nadu. His father was a clerk in a cloth-merchant's office in Kumbakonam. He was sent at seven to the local high school and stayed there nine years. By the time he was in his early teens it was common knowledge that he was more than just a brilliant student, discovering for instance the relationship between circular and exponential functions (cos a + i sin a = e^ia). This of course had been discovered by Euler before, as he found out much to his chagrin later on.

When he was sixteen he came across "A synopsis of elementary results (actually, over 6000 theorems) in pure and applied mathematics" by George Carr, . This enthusiastic book served to introduce Ramanujan to the real world of mathematics, but in a highly personal style that relegated the proofs to mere footnotes. Ramanujan went through the entire book methodically and excitedly, proving its theorems by himself, often as he got up in the morn. He claimed that the goddess of Namakkal inspired him with formulae in dreams.

His religion

Was he religious? Certainly he observed his duties as a high-caste Hindu assiduously, like being a faultless vegetarian and cooking all his food himself (after changing into his pyjamas first). And while his excellent Indian biographers (Seshu Aiyar and Ramachandra Rao) say he believed in the existence of a Supreme Being, in Kharma, Nirvana and other Hindu tenets, I suspect he was not affected by religion any more than as a collection of rules to be followed. He told me once, to my surprise, that all religions seemed to him to be more or less equally true.

Some thought, and may still think, of Ramanujan as a unintelligible manifestation of the mystic East. Far from it! He had his oddities, no doubt mostly originating from his different culture, but he was as reasonable, sane and shrewd as anyone I've met. He was a man in whom society could take pleasure, with whom one could sip tea and discuss politics or mathematics. He was a normal human being who happened to be a great mathematician.

The rest of his life

Back to his early days. Thanks to his fine academic school record, he won a scholarship to university. But there he spent his time doing mathematics at the expense of his other subjects, which he consequently failed. His scholarship was not renewed. Further attempts to complete his degree failed. He married at 22 but could not find a university post, despite the fervent attempts of some influential Indians he had impressed with his results, Ramaswami Aiyar and his two biographers. Finally (at 25) in 1912 he found his first real job, a mundane clerical one in the Port Trust of Madras. But the damage had been done --- the years between 18 and 25 are the critical ones in a mathematician's life and his genius never again had the chance of full development. This, and not his early death, was the real tragedy, that his genius was misdirected, sidetracked and to some extent distorted by an inelastic and inefficient educational system.

But the foundations of at least a partial recovery had been laid. In 1911 he had published his first substantial paper and the following year two Britons, Sir Gilbert Walker and Sir Francis Spring secured for him a special scholarship (60 pounds a year) that was enough for a married man to live in tolerable comfort. He wrote to me in early 1913, and Professor Neville and myself got him to Britain after much difficulty in 1914. He then had three years of continuous work before falling ill in mid-1917. He was only able to work spasmodically (but as well as ever) after this, and died in 1920.

His letters to me

The stories, true and false, of what happened when I read the letters of an unknown Hindu clerk have been well spread --- like how I first stored them in my wastepaper basket before retrieving them for a second look, and so on. His letters contained the bare statement of about 120 theorems. Several of them were known already, others were not. Of these, some I could prove (after harder work than I had expected) while others fairly blew me away. I had never seen the like! Only a mathematician of the highest class could have written them. They had to be true, for if they were not, no one would have the imagination to invent them. A few were definitely wrong. But that only added credence to my feeling that the writer was totally honest, since great mathematicians are commoner than frauds of the incredible skill that would be needed to create such a letter.

My collaboration with him

While his mind had been hardened by the time I had access to him, Ramanujan could still learn new things, and learn them well. It was impossible to teach him systematically, but he gradually absorbed new points of view (like why proofs were important!). But there were theorems he should have revelled in, but never used, nor ever seemed to need! The line between what he learnt from books and learnt for himself was always very hazy. And here I shall have to apologize to the world for not asking him about such matters. For I could have easily asked him, seeing him daily, and he would have been perfectly willing to tell me. But I had no idea he was going to die so soon, and it seemed ridiculous to worry about how he had found this or that theorem when he was showing me half a dozen new ones almost every day.

How good was he?

In his favourite topics, like infinite series and continued fractions, he had no equal this century. His insight into algebraic formulae, often (and unusually) brought about by considering numerical examples, was truly amazing. But in analytic number theory, a subject he is often associated with, I do not believe he actually knew that much. He certainly contributed little of significance that was not known already. And in a subject that relied so much on proof, a subject where intuition had a bad habit of coming unstuck, he produced much that was false.

I have in the past tried to say things like "his failure was more wonderful than any of his triumphs", but that is absurd. It is no use trying to pretend that failure is something else. All we can say is that his failures give us additional, surprising evidence of his imagination and versatility. And we can respect him as one who let his mind run free, instead of keeping it under saddle and blinkers like so many others do.


But the reputation of a mathematician cannot be made by failures or by rediscoveries; it must rest primarily, and rightly, on actual and original achievement. And it is still possible to justify Ramanujan on these grounds.