Indian Postage Stamp on Ramanujan
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India Post issues second time, a postage stamp on S. RamanujanDate of Issue : 26 December 2011 |
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| India Post issued a commemorative postage stamp on 75th Birth Anniversary of Srinivasa Ramanujan on 22 December 1962. |
रविवार, 19 फ़रवरी 2012
Indian Postage Stamp on Ramanujan
गुरुवार, 16 फ़रवरी 2012
Rediscovering Ramanujan : Interview with Prof. Bruce C. Berndt.
Rediscovering Ramanujan
Interview with Prof. Bruce C. Berndt.
Interviewer : Asha Krishnakumar
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."
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 Ramanujan'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 Ramanujan'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.
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 know 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 in 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 more 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.
गुरुवार, 9 फ़रवरी 2012
Srinivasa Ramanujan: A Remarkable Mathematical Genius
| Srinivasa Ramanujan A Remarkable Mathematical Genius |
By: Dr
Subodh Mahanti
|
|
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Ramanujan’s brief life and death are symbolic
of conditions in India. Of our millions how few get any education
at all; how many live on the verge of starvation.
Jawaharlal Nehru in his Discovery of India
Sheer intuitive brilliance coupled to long, hard hours on his slate made up for most of his educational lapse. This ‘poor and solitary Hindu pitting his brains against the accumulated wisdom of Europe’ as Hardy called him, had rediscovered a century of mathematics and made new discoveries that would captivate mathematicians for next century.
Robert Kanigel in The Man who Knew Infinity
: A Life of the Genius Ramanujan
|
I still say to myself when I am depressed and find myself forced
to listen to pompous and tiresome people, ‘Well, I have done
one thing you could never have done, and that is to have collaborated
with both Littlewood and Ramanujan on something like equal terms’.
Godfrey Harold Hardy
“Ramanujan’s life”, as Robert
Kanigel, the author of a marvellous biography of Ramanujan, wrote,
“can be made to serve as parable for almost any lesson you
want to draw from it.” Ramanujan’s example stirred the
imagination of many–particularly that of mathematicians. Thus,
Subrahmanyan Chandrasekhar (1910-95), the Indian born astrophysicist,
who got Nobel Prize in 1983, said : “I think it is fair to
say that almost all the mathematicians who reached distinction during
the three or four decades following Ramanujan were directly or indirectly
inspired by his example.” Even those who do not know about
Ramanujan’s work are bound to be fascinated by his life. As
Kanigel wrote: “Few can say much about his work, and yet something
in the story of his struggle for the chance to pursue his work on
his own terms compels the imagination, leaving Ramanujan a symbol
for genius, for the obstacles it faces, for the burden it bears,
for the pleasure it takes in its own existence.”
Ramanujan’s life is full of strange contrasts.
He had no formal training in mathematics but yet “he was a
natural mathematical genius, in the class of Gauss and Euler.”
Probably Ramanujan’s life has no parallel in the history of
human thought. Godfrey Harold Hardy, (1877-1947), who made it possible
for Ramanujan to go to Cambridge and give formal shape to his works,
said in one of his lectures given at Harvard Universty (which later
came out as a book entitled Ramanujan: Twelve Lectures on Subjects
Suggested by His Life and Work): “I have to form myself, as
I have never really formed before, and try to help you to form,
some of the 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 all cannons 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.”
Srinivasa Ramanujan Iyengar (best known as Srinivasa
Ramanujan) was born on December 22, 1887, in Erode about 400 km
from Chennai, formerly known as Madras where his mother’s
parents lived. After one year he was brought to his father’s
town, Kumbakonam. His parents were K. Srinivasa Iyengar and Komalatammal.
He passed his primary examination in 1897, scoring first in the
district and then he joined the Town High School. In 1904 he entered
Kumbakonam’s Government College as F.A. student. He was awarded
a scholarship. However, after school, Ramanujan’s total concentration
was focussed on mathematics. The result was that his formal education
did not continue for long. He first failed in Kumbakonam’s
Government College. He tried once again in Madras from Pachaiyappa’s
College but he failed again.
While at school he came across a book entitled
A Synopsis of Elementary Results in Pure and Applied Mathematics
by George Shoobridge Carr. The title of the book does not reflect
its contents. It was a compilation of about 5000 equations in algebra,
calculus, trigonometry and analytical geometry with abridged demonstrations
of the propositions. Carr had compressed a huge mass of mathematics
that was known in the late nineteenth century within two volumes.
Ramanujan had the first one. It was certainly not a classic. But
it had its positive features. According to Kanigel, “one strength
of Carr’s book was a movement, a flow to the formulas seemingly
laid down one after another in artless profusion that gave the book
a sly seductive logic of its own.” Thisbook had a great influence
on Ramanujan’s career. However, the book itself was not very
great. Thus Hardy wrote about the book: “He (Carr) is now
completely forgotten, even in his college, except in so far as Ramanujan
kept his name alive”. He further continued, “The book
is not in any sense a great one, but Ramanujan made it famous and
there is no doubt it influenced him (Ramanujan) profoundly”.
We do not know how exactly Carr’s book influenced Ramanujan
but it certainly gave him a direction. `It had ignited a burst of
fiercely single-minded intellectual activity’. Carr did not
provide elaborate demonstration or step by step proofs. He simply
gave some hints to proceed in the right way. Ramanujan took it upon
himself to solve all the problems in Carr’s Synopsis. And
as E. H. Neville, an English mathematician, wrote
: “In proving one formula, as he worked through Carr’s
synopsis, he discovered many others, and he began the practice of
compiling a notebook.” Between 1903 and 1914 he had three
notebooks.
While Ramanujan made up his mind to pursue mathematics
forgetting everything else but then he had to work under extreme
hardship. He could not even buy enough paper to record the proofs
of his results. Once he said to one of his friends, “when
food is problem, how can I find money for paper? I may require four
reams of paper every month.” In fact Ramanujan was in a very
precarious situation. He had lost his scholarship. He had failed
in examination. What is more, he failed to prove a good tutor in
the subject which he loved most.
At this juncture, Ramanujan was helped by R. Ramachandra
Rao, then Collector of Nellore. Ramchandra Rao was educated at Madras
Presidency College and had joined the Provincial Civil Service in
1890. He also served as Secretary of the Indian Mathematical Society
and even contributed solution to problem posed in its Journal. The
Indian Mathematical Society was founded by V. Ramaswami Iyer, a
middle-level Government servant, in 1906. Its Journal put Ramanujan
on the world’s mathematical map. Ramaswami Iyer met Ramanujan
sometime late in 1910. Ramaswami Iyer gave Ramanujan notes of introduction
to his mathematical friends in Chennai (then Madras). One of them
was P.V. Seshu Iyer, who earlier taught Ramanujan at the Government
College. For a short period (14 months) Ramanujan worked as clerk
in the Madras Port Trust which he joined on March 1, 1912. This
job he got with the help of S. Narayana Iyer.
Ramanujan’s name will always be linked to
Godfrey Harold Hardy, a British mathematician.
It is not because Ramanujan worked with Hardy at Cambridge but it
was Hardy who made it possible for Ramanujan to go to Cambridge.
Hardy, widely recognised as the leading mathematician of his time,
championed pure mathematics and had no interest in applied aspects.
He discovered one of the fundamental results in population genetics
which explains the properties of dominant, and recessive genes in
large mixed population, but he regarded the work as unimportant.
Encouraged by his well-wishers, Ramanujan, then
25 years old and had no formal education, wrote a letter to Hardy
on January 16, 1913. The letter ran into eleven pages and it was
filled with theorems in divergent series. Ramanujan did not send
proofs for his theorems. He requested Hardy for his advice and to
help getting his results published. Ramanujan wrote : “I beg
to introduce myself to you as a clerk in the Accounts Department
of the Port Trust Office at Madras on a salary of only £ 20
per annum. I have had no university education but I have undergone
the ordinary school course. After leaving school I have been employing
the spare time at my disposal to work at mathematics. I have not
trodden through the conventional regular course which is followed
in a university course, but I am striking out a new path for myself.
I have made a special investigation of divergent series in general
and the results I get are termed by the local mathematicians as
“startling“… I would request you to go through
the enclosed papers. Being poor, if you are convinced that there
is anything of value I would like to have my theorems published.
I have not given the actual investigations nor the expressions that
I get but I have indicated the lines on which I proceed. Being inexperienced
I would very highly value any advice you give me “. The letter
has become an important historical document. In fact, ‘this
letter is one of the most important and exciting mathematical letters
ever written’. At the first glance Hardy was not impressed
with the contents of the letter. So Hardy left it aside and got
himself engaged in his daily routine work. But then he could not
forget about it. In the evening Hardy again started examining the
theorems sent by Ramanujan. He also requested his colleague and
a distinguished mathematician, John Edensor Littlewood (1885-1977)
to come and examine the theorems. After examining closely they realized
the importance of Ramanujan’s work. As C.P. Snow recounted,
‘before mid-night they knew and knew for certain’ that
the writer of the manuscripts was a man of genius’. Everyone
in Cambridge concerned with mathematics came to know about the letter.
Many of them thought `at least another Jacobi in
making had been found out’. Bertrand Arthur William
Russell (1872-1970) wrote to Lady Ottoline Morell. “I
found Hardy and Littlewood in a state of wild excitement because
they believe, they have discovered a second Newton, a Hindu Clerk
in Madras … He wrote to Hardy telling of some results he has
got, which Hardy thinks quite wonderful.”
Fortunately for Ramanujan, Hardy realised that
the letter was the work of a genius. In the next three months Ramanujan
received another three letters from Hardy. However, in the beginning
Hardy responded cautiously. He wrote on 8 February 1913. To quote
from the letter. “I was exceedingly interested by your letter
and by the theorems which you state. You will however understand
that, before I can judge properly of the value of what you have
done it is essential that I should see proofs of some of your assertions
… I hope very much that you will send me as quickly as possible
at any rate a few of your proofs, and follow this more at your leisure
by more detailed account of your work on primer and divergent series.
It seems to me quite likely that you have done a good deal of work
worth publication; and if you can produce satisfactory demonstration
I should be very glad to do what I can to secure it” .
In the meantime Hardy started taking steps for
bringing Ramanujan to England. He contacted the Indian Office in
London to this effect. Ramanujan was awarded the first research
scholarship by the Madras University. This was possible by the recommendation
of Gilbert Walker, then Head of the Indian Meteorological Department
in Simla. Gilbert was not a pure mathematician but he was a former
Fellow and mathematical lecturer at Trinity College, Cambridge.
Walker, who was prevailed upon by Francis Spring to look through
Ramanujan’s notebooks wrote to the Registrar of the Madras
University : “The character of the work that I saw impressed
me as comparable in originality with that of a Mathematical Fellow
in a Cambridge College; it appears to lack, however, as might be
expected in the circumstances, the completeness and precision necessary
before the universal validity of the results could be accepted.
I have not specialised in the branches of pure mathematics at which
he worked, and could not therefore form a reliable estimate of his
abilities, which might be of an order to bring him a European reputation.
But it was perfectly clear to me that the University would be justified
in enabling S. Ramanujan for a few years at least to spend the whole
of his time on mathematics without any anxiety as to his livelihood.”
Ramanujan was not very eager to travel abroad.
In fact he was quite apprehensive. However, many of his well-wishers
prevailed upon him and finally Ramanujan left Madras by S.S. Navesa
on March 17, 1914. Ramanujan reached Cambridge on April 18, 1914.
When Ramanujan reached England he was fully abreast of the recent
developments in his field. This was described by J. R. Newman in
1968: “Ramanujan arrived in England abreast and often ahead
of contemporary mathematical knowledge. Thus, in a lone mighty sweep,
he had succeeded in recreating in his field, through his own unaided
powers, a rich half century of European mathematics. One may doubt
whether so prodigious a feat had ever been accomplished in the history
of thought.”
Today it is simply futile to speculate about what
would have happened if Ramanujan had not come in contact with Hardy.
It could happen either way. But then Hardy should be given due credit
for recognizing Ramanujan’s originality and helping him to
carry out his work. Hardy himself was very clear about his role.
“Ramanujan was”, Hardy wrote, “my discovery. I
did not invent him — like other great men, he invented himself
— but I was the first really competent person who had the
chance to see some of his work, and I can still remember with satisfaction
that I could recognize at once what I treasure I had found.”
It may be noted that before writing to Hardy, Ramanujan
had written to two well-known Cambridge mathematicians viz., H.F.
Baker and E.W. Hobson. But both of them had expressed their inability
to help Ramanujan.
Ramanujan was awarded the B.A. degree in March
1916 for his work on ‘Highly composite Numbers’ which
was published as a paper in the Journal of the London Mathematical
Society. He was the second Indian to become a Fellow of the Royal
Society in 1918 and he became one of the youngest Fellows in the
entire history of the Royal Society. He was elected “for his
investigation in Elliptic Functions and the Theory of Numbers.”
On 13 October 1918 he was the first Indian to be elected a Fellow
of Trinity College, Cambridge.
Much of Ramanujan’s mathematics comes under
the heading of number theory — a purest realm of mathematics.
The number theory is the abstract study of the structure of number
systems and properties of positive integers. It includes various
theorems about prime numbers (a prime number is an integer greater
than one that has not integral factor). Number theory includes analytic
number theory, originated by Leonhard Euler (1707-89);
geometric theory - which uses such geometrical methods of analysis
as Cartesian co-ordinates, vectors and matrices; and probabilistic
number theory based on probability theory. What Ramanujan did will
be fully understood by a very few. In this connection it is worthwhile
to note what Hardy had to say of the work of pure mathematicians:
“What we do may be small, but it has certain character of
permanence and to have produced anything of the slightest permanent
interest, whether it be a copy of verses or a geometrical theorem,
is to have done something beyond the powers of the vast majority
of men.” In spite of abstract nature of his work Ramanujan
is widely known.
Ramanujan was a mathematical genius in his own
right on the basis of his work alone. He worked hard like any other
great mathematician. He had no special, unexplained power. As Hardy,
wrote: “I have often been asked whether Ramanujan had any
special secret; whether his methods differed in kind from those
of other mathematicians; whether there was anything really abnormal
in his mode of thought. I cannot answer these questions with any
confidence or conviction; but I do not believe it. My belief that
all mathematicians think, at bottom, in the same kind of way, and
that Ramanujan was no exception.”
Of course, as Hardy observed Ramanujan “combined
a power of generalization, a feeling for form and a capacity for
rapid modification of his hypotheses, that were often really startling,
and made him, in his peculiar field, without a rival in his day.
Here we do not attempt to describe what Ramanujan
achieved. But let us note what Hardy had to say about the importance
of Ramanujan’s work. “Opinions may differ as to the
importance of Ramanujan’s work, the kind of standard by which
it should be judged and the influence which it is likely to have
on the mathematics of the future. It has not the simplicity and
the inevitableness of the greatest work; it would be greater if
it were less strange. One gift it shows which no one will deny—profound
and invincible originality.”
The Norwegian mathematician Atle Selberg, one of
the great number theorists of this century wrote : “Ramanujan’s
recognition of the multiplicative properties of the coefficients
of modular forms that we now refer to as cusp forms and his conjectures
formulated in this connection and their later generalization, have
come to play a more central role in the mathematics of today, serving
as a kind of focus for the attention of quite a large group of the
best mathematicians of our time. Other discoveries like the mock-theta
functions are only in the very early stages of being understood
and no one can yet assess their real importance. So the final verdict
is certainly not in, and it may not be in for a long time, but the
estimates of Ramanujan’s nature in mathematics certainly have
been growing over the years. There is doubt no about that.”
Often people tend to speculate what Ramanujan would
have achieved if he had not died or if his exceptional qualities
were recognised at the very beginning. There are many instances
of such untimely death of gifted persons, or rejection of gifted
persons by the society or the rigid educational system. In mathematics
we may cite the cases of Niels Henrik Abel (1809-29)
and Evarista Galois (1811-32). Abel solved one
of the great mathematical problems of his day - finding a general
solution for a class equations called quintiles. Abel solved the
problem by proving that such a solution was impossible. Galois pioneered
the branch of modern mathematics known as group theory. What is
important is that we should recognise the greatness of such people
and take inspiration from their work.
Even after more than 80 years of the death of Ramanujan
the situation is not very different as far the rigidity of the education
system. Today also a ‘Ramanujan’ is not likely to get
a chance to pursue his career. This situation remains very much
similar as described by JBS Haldane (1982-1964), a British born
geneticist and philosopher who spent last part of his life in India.
Haldane said : “Today in India Ramanujan could not get even
a lectureship in a rural college because he had no degree. Much
less could he get a post through the Union Public Service Commission.
This fact is a disgrace to India. I am aware that he was offered
a chair in India after becoming a Fellow of the Royal Society. But
it is scandalous that India’s great men should have to wait
for foreign recognition. If Ramanujan’s work had been recognised
in India as early it was in England, he might never have emigrated
and might be alive today. We can cast the blame for Ramanujan’s
non-recognition on the British Raj. We cannot do so when similar
cases occur today...”
Nehru’s statement given at the beginning
is very much valid even today. And for these very reasons the story
of Ramanujan should be told and retold to our younger people particularly
to those who aspire to do something extraordinary but feel dejected
under the prevailing circumstances. And in this connection it is
worthwhile to remember what Chandrasekhar had to
say: “I can recall the gladness I felt at the assurance that
one brought up under circumstances similar to my own could have
achieved what I could not grasp. The fact that Ramanujan’s
early years were spent in a scientifically sterile atmosphere, that
his life in India was not without hardships that under circumstances
that appeared to most Indians as nothing short of miraculous, he
had gone to Cambridge, supported by eminent mathematicians, and
had returned to India with very assurance that he would be considered,
in time as one of the most original mathematicians of the century
— these facts were enough, more than enough, for aspiring
young Indian students to break their bands of intellectual confinement
and perhaps soar the way what Ramanujan had.
"As someone has written “Ramanujan
did mathematics for its own sake, for thrill that he got in seeing
and discovering unusual relationships between various mathematical
objects.” Today Ramanujan’s work has some applications
in particle physics or in the calculation of pi up to a very large
number of decimal places. His work on Rieman’s Zeta Function
has been applied to the pyrometry, the investigations of the temperature
of furnaces. His work on the Partition Numbers resulted in two applications
— new fuels and fabrics like nylons. But then highlighting
the importance of the application side Ramanujan’s work is
really not very important.
Ramanujan died of tuberculosis in Kumbakonam on
April 26, 1920. He was only 32 years old. “It was always maths
... Four days before he died he was scribbling,” said Janaki,
his wife. The untimely death of Ramanujan was most unfortunate particularly
so when we take into account the circumstances under which he died.
As Times Magazine rightly wrote: “There is something peculiarly
sad in the spectacle of genius dying young, dying with the first
sweets of recognition and success tasted, but before the full recognition
of powers that lie within.
"The only Ramanujan Museum in the country,
founded by Shri P. K. Srinivasan, a mathematics teacher, operates
from March 1993 in the Avvai Academy, Royapuram, Madras. The achievement
of Ramanujan was so great that those who can really grasp the work
of Ramanujan ‘may doubt that so prodigious a feat had ever
been accomplished in the history of thought".
Further Reading
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