Who are We?
What do we do?
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
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.
Ramanujan (Dec. 22, 1887 -- April 26, 1920)
of Mathematical Sciences, Madras-600 113.
(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''.
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
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.
Dictionary of Scientific Biography
G H Hardy,
Ramanujan (Cambridge, 1940).
R A Rankin,
Ramanujan's manuscripts and notebooks, Bull.
London Math. Soc. 14
R A Rankin,
Ramanujan's manuscripts and notebooks II, Bull.
London Math. Soc. 21
The man who knew infinity : A life of the genius
Ramanujan (New York, 1991).
Srinivasa Ramanujan, The American Scholar
58 (1989), 234-244.
B Berndt and S
Bhargava, Ramanujan - For lowbrows, Amer. Math.
Monthly 100 (1993),
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
Srinivasa Ramanujan (New Delhi, 1979).
Srinivasa Ramanujan (1887-1920) : a centennial tribute,
International journal of mathematical education in
science and technology 18
R A Rankin,
Srinivasa Ramanujan (1887- 1920), International
journal of mathematical education in science and
technology 18 (1987),
Rao, Srinivasa Ramanujan: a Mathematical Genius
(EastWest Books (Madras) Pvt. Ltd., 1998).
References for Srinivasa
- Biography in Dictionary
of Scientific Biography (New York 1970-1990).
- Biography in
- B C Berndt and R A Rankin,
Ramanujan : Letters and commentary (Providence,
Rhode Island, 1995).
- G H Hardy, Ramanujan
- R Kanigel, The man who
knew infinity : A life of the genius Ramanujan (New
- J N Kapur (ed.), Some
eminent Indian mathematicians of the twentieth century
- S Ram, Srinivasa
Ramanujan (New Delhi, 1979).
- S Ramanujan, Collected
Papers (Cambridge, 1927).
- S R Ranganathan,
Ramanujan : the man and the mathematician (London,
- P K Srinivasan,
Ramanujan : Am inspiration 2 Vols. (Madras, 1968).
- 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),
- B Berndt and S Bhargava,
Ramanujan - For lowbrows, Amer. Math. Monthly 100
- 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)
- S Chandrasekhar, On
Ramanujan, in Ramanujan Revisited (Boston, 1988),
- 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
- 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.
MacTutor History of Mathematics
Interview with Prof. Bruce
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
"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
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.
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
Excerpts from the
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
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
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 that have not been
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
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
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
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
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
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
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
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
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
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
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
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
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 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
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
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".
Last modified:11/19/2002 14:51:09