Srinivasa Ramanujan

And Now a Movie about Srinivasa Ramanujam

Born: 22 Dec 1887 in Erode, Tamil Nadu state, India
Died: 26 April 1920 in Madras, Tamil Nadu state, India

The Man who knew Infinity , so hailed because of his tremendous panache for working with infinite series

Ramanujan was one of India's greatest mathematical geniuses. He made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinite series.

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.

References:

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

References for Srinivasa Ramanujan

Biography in Dictionary of Scientific Biography (New York 1970-1990).
Biography in Encyclopaedia Britannica.

Books:

B C Berndt and R A Rankin, Ramanujan : Letters and commentary (Providence, Rhode Island, 1995).
G H Hardy, Ramanujan (Cambridge, 1940).
R Kanigel, The man who knew infinity : A life of the genius Ramanujan (New York, 1991).
J N Kapur (ed.), Some eminent Indian mathematicians of the twentieth century (Kapur, 1989).
S Ram, Srinivasa Ramanujan (New Delhi, 1979).
S Ramanujan, Collected Papers (Cambridge, 1927).
S R Ranganathan, Ramanujan : the man and the mathematician (London, 1967).
P K Srinivasan, Ramanujan : Am inspiration 2 Vols. (Madras, 1968).

Articles:

P V Seshu Aiyar, The late Mr S Ramanujan, B.A., F.R.S., J. Indian Math. Soc. 12 (1920), 81-86.
G E Andrews, An introduction to Ramanujan's 'lost' notebook, Amer. Math. Monthly 86 (1979), 89-108.
B Berndt, Srinivasa Ramanujan, The American Scholar 58 (1989), 234-244.
B Berndt and S Bhargava, Ramanujan - For lowbrows, Amer. Math. Monthly 100 (1993), 644-656.
B Bollobas, Ramanujan - a glimpse of his life and his mathematics, The Cambridge Review (1988), 76-80.
B Bollobas, Ramanujan - a glimpse of his life and his mathematics, Eureka 48 (1988), 81-98.
J M Borwein and P B Borwein, Ramanujan and pi, Scientific American 258 (2) (1988), 66-73.
S Chandrasekhar, On Ramanujan, in Ramanujan Revisited (Boston, 1988), 1-6.
L Debnath, Srinivasa Ramanujan (1887-1920) : a centennial tribute, International journal of mathematical education in science and technology 18 (1987), 821-861.
G H Hardy, The Indian mathematician Ramanujan, Amer. Math. Monthly 44 (3) (1937), 137-155.
G H Hardy, Srinivasa Ramanujan, Proc. London Math, Soc. 19 (1921), xl-lviii.
E H Neville, Srinivasa Ramanujan, Nature 149 (1942), 292-294.
C T Rajagopal, Stray thoughts on Srinivasa Ramanujan, Math. Teacher (India) 11A (1975), 119-122, and 12 (1976), 138-139.
K Ramachandra, Srinivasa Ramanujan (the inventor of the circle method), J. Math. Phys. Sci. 21 (1987), 545-564.
K Ramachandra, Srinivasa Ramanujan (the inventor of the circle method), Hardy-Ramanujan J. 10 (1987), 9-24.
R A Rankin, Ramanujan's manuscripts and notebooks, Bull. London Math. Soc. 14 (1982), 81-97.
R A Rankin, Ramanujan's manuscripts and notebooks II, Bull. London Math. Soc. 21 (1989), 351-365.
R A Rankin, Srinivasa Ramanujan (1887- 1920), International journal of mathematical education in science and technology 18 (1987), 861-.
R A Rankin, Ramanujan as a patient, Proc. Indian Ac. Sci. 93 (1984), 79-100.
R Ramachandra Rao, In memoriam S Ramanujan, B.A., F.R.S., J. Indian Math. Soc. 12 (1920), 87-90.
E Shils, Reflections on tradition, centre and periphery and the universal validity of science : the significance of the life of S Ramanujan, Minerva 29 (1991), 393-419.
D A B Young, Ramanujan's illness, Notes and Records of the Royal Society of London 48 (1994), 107-119.

June 1998

MacTutor History of Mathematics
[http://www-history.mcs.st-andrews.ac.uk/References/Ramanujan.html]

MATHEMATICS

http://www.hinduonnet.com/fline/fl1617/16170810.htm

Rediscovering Ramanujan

Interview with Prof. Bruce C. Berndt.

The academic lineage of most eminent scholars can be traced to famous centres of learning, inspiring teachers or an intellectual milieu, but Srinivasa Ramanujan, perhaps the greatest of Indian mathematicians, had none of these advantages. He had just one year of education in a small college; he was basically self-taught. Working in isolation for most of his short life of 32 years, he had little contact with other mathematicians.

"Many people falsely promulgate mystical powers to Ramanujan's mathematical thinking. It is not true. He has meticulously recorded every result in his three notebooks," says Dr. Bruce C. Berndt, Professor of Mathematics at the University of Illino is, whose 20 years of research on the three notebooks has been compiled into five volumes.

Between 1903 and 1914, before Ramanujan went to Cambridge, he compiled 3,542 theorems in the notebooks. Most of the time Ramanujan provided only the results and not the proof. Berndt says: "This is perhaps because for him paper was unaffordable and so he worked on a slate and recorded the results in his notebooks without the proofs, and not because he got the results in a flash."

K. GAJENDRAN

Berndt is the only person who has proved each of the 3,542 theorems. He is convinced that nothing "came to" Ramanujan but every step was thought or worked out and could in all probability be found in the notebooks. Berndt recalls Ramanujan's well-known i nteraction with G.H. Hardy. Visiting Ramanujan in a Cambridge hospital where he was being treated for tuberculosis, Hardy said: "I rode here today in a taxicab whose number was 1729. This is a dull number." Ramanujan replied: "No, it is a very interestin g number; it is the smallest number expressible as a sum of two cubes in two different ways." Berndt believes that this was no flash of insight, as is commonly thought. He says that Ramanujan had recorded this result in one of his notebooks before he cam e to Cambridge. He says that this instance demonstrated Ramanujan's love for numbers and their properties.

Although Ramanujan's mathematics may seem archaic by today's standards, in many respects he was far ahead of his time. While the thrust of 20th century mathematics has been on building general theories, Ramanujan was a master in finding particular result s which are now recognised as providing the core for the theories. His results opened up vistas for further research not only in mathematics but in other disciplines such as physics, computer science and statistics.

After Ramanujan's death in 1920, the three notebooks and a sheaf of papers that he left behind were handed over to the University of Madras. They were sent to G.N. Watson who, along with B.M. Wilson, edited sections of the notebooks. After Watson's death in 1965, the papers, which contained results compiled by Ramanujan after his return to India from Cambridge in 1914, were handed over to Trinity College, Cambridge. In 1976, G. E. Andrews of Pennsylvania State University rediscovered the papers at the T rinity College Library. Since then these papers have been called Ramanujan's "lost notebook". According to Berndt, the lost notebook caused as much stir in the mathematical world as Beethoven's Tenth Symphony did in the world of Western classical music.

Berndt says that the "unique circumstances surrounding Ramanujan and his mathematics" make it very difficult to assess his greatness among such mathematical giants as Newton, Gauss, Euler and Reimann. According to Berndt, Hardy had provided the following assessment of his contemporary mathematicians on a scale of 0 to 100: "On the basis of pure talent he gave himself a rating of 25, his collaborator J. E. Littlewood 30, German mathematician D. Hilbert 80, and Ramanujan 100." Berndt says that it is not R amanujan's greatness but only its measure that is in doubt.

Besides the five volumes, Berndt has written over 100 papers on Ramanujan's works. He has guided a number of research students in this area. He now works on Ramanujan's "lost notebook" and on some other manuscripts and fragments of notes. Recently in Che nnai to give lectures on Ramanujan's works at the Indian Institute of Technology, the Institute of Mathematical Sciences and the Ramanujan Museum and Mathematical Centre, Berndt spoke to Asha Krishnakumar on his work on Ramanujan's notebooks, the broad areas in mathematics that Ramanujan had covered, the vistas his work has opened up and the application of his work in physics, statistics and communication.

Excerpts from the interview:

How did you get interested in Ramanujan's notebooks?

After my Ph.D. at the University of Wisconsin, I took my first position at the University of Glasgow (Scotland) in 1966-67. Prof. R. A. Rankin was a leader in number theory at that time. I remember being in Rankin's office in 1967 when he told me about R amanujan's notebooks for the first time. He said: "I have a copy of the notebooks published by the Tata Institute of Fundamental Research, Bombay (Mumbai). Would you be interested in looking at it?" I said, "No, I am not interested in it."

I did not think about the notebooks for some years until early 1974 when I was on leave at the Institute for Advanced Studies in Princeton, U.S. In February that year, I was reading two papers of Emil Grosswald in which he proves some formulae from Raman ujan's notebooks. I realised I could prove these formulae as well by using a theorem I proved two years ago. I did that and then I was curious to find out whether there were other formulae in the notebooks that I could prove using my methods. So, I went to the Princeton University library and got hold of Ramanujan's notebooks published by the TIFR. I was thrilled to find out that I could actually prove some more formulae. But there were a few thousand others I could not.

I was fascinated with the notebooks and in the next few years I wrote papers around the formulae I had proved from the notebooks. The first was a repository paper on Ramanujan's theta 2n+1 formula, for which I did a lot of historical research on other pr oofs of the formula. This I wrote for a special volume called Srinivasa Ramanujan's Memorial Volume, published by Jupiter Press in Madras (Chennai) in 1974. After that, wherever I went, I was all the time working on, and proving, the various formulae of Ramanujan's - to be precise - from Chapter 14 of the second notebook. Then I wrote a sequel to this.

Let me jump ahead to May 1977, when I decided to try and prove all the formulae in Chapter 14. I took this on as a challenge. There were in all 87 results in this chapter. I worked on this for the next one year. I took the help of my first Ph.D. student, Ron Evans.

After about a year of working on this, the famous mathematician George E. Andrews visited Illinois and told me that he discovered in the spring of 1976 Ramanujan's "lost notebook" along with G. N. Watson and B. M. Wilson's edited volumes on Ramanujan's t hree notebooks and some of their unpublished notes in the Trinity College Library. I then got photocopies of Ramanujan's lost notebook and all the notes of Watson and Wilson. And so I went to the beginning of the second notebook.

What does the second notebook contain?

This is the main notebook because it is the revised and enlarged version of the first. I went back to the beginning and went about working my way through it using Watson and Wilson's notes when necessary.

How long did you work on the second notebook?

I really do not know how many years exactly. But some time in the early 1980s Walter Kaufmann-Buhler, the mathematics editor of Springer Verlag in New York, showed interest in my work and decided to publish it. That had not occurred to me till the n. I agreed and signed a contract with Springers.

That was when I started preparing the results with a view to publishing them. I finally came out with five volumes; I had thought it would be three. It also took a much longer time than I had anticipated.

After I completed 21 chapters of the second notebook, the 100 pages of unorganised material in the second notebook and the 33 pages in the third had a lot more material. I also found more material in the first which was not there in the second. So, I fou nd a lot of new material. It was 20 years before I eventually completed all the three notebooks.

Why did you start with the second notebook and not the first?

I knew that the second was the revised and enlarged edition of the first. The first was in a rough form and the second, I was relatively certain, had most of the things that were there in the first and a lot more.

What did each notebook contain?

The new results that were in the second notebook were generally among the unorganised pages of the first. And the third notebook was all unorganised. A higher percentage of the results in the unorganised parts of the second and the third were new. In oth er words, you got a higher percentage of new results as you went into the unorganised material.

What do you mean by new results?

Results that have not been got earlier.

What is the percentage of new results in the notebooks?

Hardy estimated that over two-thirds of the work Ramanujan did in India was rediscovered. That is much too high. I found that well over half is new. It is difficult to say precisely. I would say that most results were new because we also have to consider that in the meantime, from 1920 until I started doing this work, other people discovered these things. So, I would say that at least two-thirds of the material was really new when Ramanujan died.

THE HINDU PHOTO LIBRARY
Srinivasa Ramanujam.

Ramanujan is popularly known as a number theorist. Would you give a broad idea about the results in his notebooks? What areas of mathematics do they cover?

You are right. To much of the mathematical world and to the public in general, Ramanujan is known as a number theorist. Hardy was a number theorist but he was also into analysis. When Ramanujan was at Cambridge with Hardy, he was naturally influenced by him (Hardy). And so most of the papers he published while he was in England were in number theory. His real great discoveries are in partition functions.

Along with Hardy, he found a new area in mathematics called probabilistic number theory, which is still expanding. Ramanujan also wrote sequels in highly composite numbers and arithmetical functions. There are half a dozen or more of these papers that ma de Ramanujan very famous. They are still very important papers in number theory.

However, the notebooks do not contain much of number theory. It is, broadly speaking, in analysis. I will try and break that down a little bit. I would say that the area in which Ramanujan spent most of his time, more than any other, is in elliptic funct ions (theta functions), which have strong connections with number theory. In particular, Chapters 16 to 21 of the second notebook and most of the unorganised portions of the notebooks are on theta functions. There is a certain type of theta functions ide ntity which has applications in other areas of mathematics, particularly in number theory, called modular equations. Ramanujan devoted an enormous amount of effort on refining modular equations.

Ramanujan is also popular for his approximations to pie. Many of his approximations came with his work on elliptic functions. Ramanujan computed what are called class invariants. Even as he discovered them, they were computed by a German mathematician, H . Weber, in the late 19th and early 20th centuries. But Ramanujan was unaware of this. He computed 116 of these invariants which are much more complicated. These have applications not only in approximations to pie but in many other areas as well.

Have you gone through every one of the 3,254 entries in the three notebooks and proved each of them, including in the unorganised material?

I have gone through every entry in the notebooks. If a result has already been proved in the literature, then I just wrote the entry down and said that proofs can be found in this literature and so on. But I will also discuss the relevance in history of the entry.

What are the applications of Ramanujan's discoveries in areas such as physics, communications and computer science?

This is a very difficult question to answer because of the way mathematics and science work. Mathematics is discovered and it is then there for others to use. And you do not always know who uses it. But I have regular contact with some physicists who I k now use Ramanujan's work. They find the results very useful in their own application.

What are the areas in physics in which Ramanujan's work is used?

The most famous application in physics is in the area of statistical mechanics. Among those who I know have used Ramanujan's mathematics extensively is W. Backster, the well-known physicist from Australia. He used the famous Rogers-Ramanujan identities i n what is called the hard hexagon model to describe the molecular structure of a thin film.

Many of Ramanujan's works are used but his asymptotic formulae have found the most important application; I first wrote this in 1974 from his notebook.

Then there is a particular formula of Ramanujan's involving the exponential function which has been used many times in statistics and probability. Ramanujan had a number of conjectures in regard to this formula and one is still unproven. He made this con jecture in a problem he submitted to the Indian Mathematical Society. The asymptotic formula is used, for instance, in the popular problem: What is the minimum number of people you can have in a room so that the probability that two share a common birthd ay is more than half? I think it is 21, 22 or 23. Anyway, this problem can be generalised to many other types of similar problems.

Have you looked at the lost notebook?

That is what I am working on now with Andrews. It contains about 630 results. About 60 per cent of these are of interest to Andrews. He has proved most of these results. The other 40 per cent are of great interest to me as most of them were a continuatio n of what Ramanujan considered in his other notebooks. So, I began working on them.

What are your experiences of working on Ramanujan's notebooks? Do you think Ramanujan was a freak or a genius or he had the necessary motivation to write the notebooks?

I think one has to be really motivated to do the kind of mathematics he was doing, through either teachers or books. We understand from Ramanujan's biographers that he was motivated in particular by two books: S. L. Loney's Plane Trigonometry and Carr's Synopsis of Elementary Results in Pure Mathematics (which was a compilation of 5,000 theorems with a few proofs) at the age of 12. How much his teachers motivated him, we really do not know as nothing about it has been recorded. Reading these book s and going through the problems must have aroused the curiosity that he had and inspired him.

He is particularly amazing because he took off from the little bit he knew and extended it so much in so many directions, leading to so many new and beautiful results.

Did you find any results difficult to decipher in any of Ramanujan's notbooks?

Oh yes. I get stuck all the time. At times I have no idea where these formulae are coming from. Earlier, Ron Evans, whom I have already mentioned as having worked on Chapter 14, helped me out a number of times. There are times I would think of a formula over for about six months or even a year, not getting anywhere. Even now there are times when we wonder how Ramanujan was ever led to the formulae. There has to be some chain of reasoning to lead him to think that there might be a theorem there. But ofte n this is missing. To begin with, the formulae look strange but over time we understand where they fit in and how important they are than they were previously thought to be.

Did you find any serious errors in Ramanujan's notebooks?

There are a number of misprints. I did not count the number of serious mistakes but it is an extremely small number - maybe five or ten out of over 3,000 results. Considering that Ramanujan did not have any rigorous training, it is really amazing that he made so few mistakes.

Are the methods of mathematics teaching today motivating enough to produce geniuses like Ramanujan?

Some like G. E. Andrews think that much of the reforms have come about because students do not study as much. This, along with the advent of computers, has changed things. A lot of mathematics which can be done by computations, manipulations and by doing exercises in high school are now being done using calculators and computers. And the computer, I do not think, gives any motivation.

The books on calculus reform (that is now introduced in the U.S.) include sections on using a computer. To calculate the limit of a sequence given by a formula, the book says press these numbers, x, y and z... Then there appears a string of numbers that get smaller and smaller and then you can see that is tends to zero. But that does not lead to any understanding as to why they are tending to zero. So, this reasoning, motivation and understanding of why the sequence tends to zero is not being taught. I think that is wrong.

There seem to be two schools of thought: one which thinks that the development of concepts and ideas is important and the other, like that in India, which thinks that development of skills is important in teaching mathematics. Which do you think is m ore important?

I think you cannot have one without the other. Both must be taught. The tendency in the U.S. is to move away from skills and rely on computers. I do not think this is correct because if you have the skills and understanding, then you can see if you have made an error in punching in the computers. Andrews and I have the experience of students putting down results that are totally ridiculous because they have not understood what is going on. They do not even realise that they made mistakes while punching in the computers. So, developing skills is absolutely necessary. But on the other hand if you just go on with the skills and have no understanding of why you are doing this, you lose the motivation and it becomes just a mechanical exercise.

However, even now there is a possibility that geniuses like Ramanujan will emerge. It is important that once you identify such children, books and material should be found for them specially. The greatest thing about number theory in which Ramanujan work ed is that you can give it to people of all ages to stimulate them. Number theory has problems that are challenging, that are not too easy, but yet they are durable and motivating. A foremost mathematician (Atle Selberg) and a great physicist (Freeman Dy son) of this century have said that they were motivated by Ramanujan's number theory when they were in their early teens.

Srinivasa Ramanujan

It is one of the most romantic stories in the history of mathematics: in 1913, the English mathematician G. H. Hardy received a strange letter from an unknown clerk in Madras, India. The ten-page letter contained about 120 statements of theorems on infinite series, improper integrals, continued fractions, and number theory (Here is a .dvi file with a sample of these results). Every prominent mathematician gets letters from cranks, and at first glance Hardy no doubt put this letter in that class. But something about the formulas made him take a second look, and show it to his collaborator J. E. Littlewood. After a few hours, they concluded that the results "must be true because, if they were not true, no one would have had the imagination to invent them".

Thus was Srinivasa Ramanujan (1887-1920) introduced to the mathematical world. Born in South India, Ramanujan was a promising student, winning academic prizes in high school. But at age 16 his life took a decisive turn after he obtained a book titled A Synopsis of Elementary Results in Pure and Applied Mathematics. The book was simply a compilation of thousands of mathematical results, most set down with little or no indication of proof. It was in no sense a mathematical classic; rather, it was written as an aid to coaching English mathematics students facing the notoriously difficult Tripos examination, which involved a great deal of wholesale memorization. But in Ramanujan it inspired a burst of feverish mathematical activity, as he worked through the book's results and beyond. Unfortunately, his total immersion in mathematics was disastrous for Ramanujan's academic career: ignoring all his other subjects, he repeatedly failed his college exams.

As a college dropout from a poor family, Ramanujan's position was precarious. He lived off the charity of friends, filling notebooks with mathematical discoveries and seeking patrons to support his work. Finally he met with modest success when the Indian mathematician Ramachandra Rao provided him with first a modest subsidy, and later a clerkship at the Madras Port Trust. During this period Ramanujan had his first paper published, a 17-page work on Bernoulli numbers that appeared in 1911 in the Journal of the Indian Mathematical Society. Still no one was quite sure if Ramanujan was a real genius or a crank. With the encouragement of friends, he wrote to mathematicians in Cambridge seeking validation of his work. Twice he wrote with no response; on the third try, he found Hardy.

Hardy wrote enthusiastically back to Ramanujan, and Hardy's stamp of approval improved Ramanujan's status almost immediately. Ramanujan was named a research scholar at the University of Madras, receiving double his clerk's salary and required only to submit quarterly reports on his work. But Hardy was determined that Ramanujan be brought to England. Ramanujan's mother resisted at first--high-caste Indians shunned travel to foreign lands--but finally gave in, ostensibly after a vision. In March 1914, Ramanujan boarded a steamer for England.

Ramanujan's arrival at Cambridge was the beginning of a very successful five-year collaboration with Hardy. In some ways the two made an odd pair: Hardy was a great exponent of rigor in analysis, while Ramanujan's results were (as Hardy put it) "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account". Hardy did his best to fill in the gaps in Ramanujan's education without discouraging him. He was amazed by Ramanujan's uncanny formal intuition in manipulating infinite series, continued fractions, and the like: "I have never met his equal, and can compare him only with Euler or Jacobi."

One remarkable result of the Hardy-Ramanujan collaboration was a formula for the number p(n) of partitions of a number n. A partition of a positive integer n is just an expression for n as a sum of positive integers, regardless of order. Thus p(4) = 5 because 4 can be written as 1+1+1+1, 1+1+2, 2+2, 1+3, or 4. The problem of finding p(n) was studied by Euler, who found a formula for the generating function of p(n) (that is, for the infinite series whose nth term is p(n)xⁿ). While this allows one to calculate p(n) recursively, it doesn't lead to an explicit formula. Hardy and Ramanujan came up with such a formula (though they only proved it works asymptotically; Rademacher proved it gives the exact value of p(n)).

Ramanujan's years in England were mathematically productive, and he gained the recognition he hoped for. Cambridge granted him a Bachelor of Science degree "by research" in 1916, and he was elected a Fellow of the Royal Society (the first Indian to be so honored) in 1918. But the alien climate and culture took a toll on his health. Ramanujan had always lived in a tropical climate and had his mother (later his wife) to cook for him: now he faced the English winter, and he had to do all his own cooking to adhere to his caste's strict dietary rules. Wartime shortages only made things worse. In 1917 he was hospitalized, his doctors fearing for his life. By late 1918 his health had improved; he returned to India in 1919. But his health failed again, and he died the next year.

Besides his published work, Ramanujan left behind several notebooks, which have been the object of much study. The English mathematician G. N. Watson wrote a long series of papers about them. More recently the American mathematician Bruce C. Berndt has written a multi-volume study of the notebooks. In 1997 The Ramanujan Journal was launched to publish work "in areas of mathematics influenced by Ramanujan".