**Srinivasa
Ramanujan - His life and his genius**

By

Prof.

(Expository address delivered on Sep.16, 1987
at Visvesvarayya Auditorium as part of the celebrations of Ramanujan Centenary
by the IISC, Bangalore)

Srinivasa Ramanujan
was a mathematical genius not only of the twentieth century but for all time.
He was an enigma to his teachers even at school because of his prodigious
memory and unusual mathematical talent which began to show even before he was
ten years old. That was the age when he topped the whole district at the
Primary examination and this procured him a half-fee concession at school
namely,

He passed the
matriculation examination in first class and this earned him a scholarship in
the FA class at

In 1910 Ramanujan, now
23, heard of the Indian Mathematical
Society and its founder Prof. V. Ramaswami Iyer, a Deputy Collector. To
Ramaswamy Iyer goes the credit of being the first among those who recognised the mathematical
genius in Ramanujan and also acted on that recognition. Ramanujan was
introduced to Prof. Seshu Iyer, who introduced him to Dr. Ramachandra Rao,
Collector of Nellore. Ramachandra Rao
looked into his notebooks, and
also listened to him and decided that
Ramanujan was a remarkable man. Ramachandra Rao supported him for some time.
After a few months he accepted a clerk's job in the office of the Madras Port
Trust in March 1912. By this time the Chairman of the Madras Port Trust,

Ramanujan thus became
a professional mathematician and
remained as such for the rest of his life, which was not much, unfortunately, -
it was only seven years. Upon the suggestion of Prof. Seshu Iyer and others he
had then already started a correspondence with Prof. G.H. Hardy, a Fellow of
the Royal Society and Cayley lecturer in Mathematics at

The Hindu of Madras in its edition of May 13, 1914,
reported as follows: Mr. S. Ramanujan of

The common man who
knows very little of mathematics may not be able to grasp the full impact of
Ramanujan's genius and intuition for discovery and creativity except through
statements of appreciation from others. Instead of quoting these appreciations
and encomia we give below a few anecdotes and make them simple enough to be
understood (at least partially) by the layman so that the latter can have a
first-hand knowledge of the super-genius that Ramanujan was.

Dr. P. C. Mahalanobis, who later in Free India became
Nehru's right hand man of National Planning, was a student at

Here the reader must know what a continued fraction
is. A simple infinite continued fraction, for example, is

1 + 1/(2 +1/(3 + 1/(4 +1/(5 + … ))) ….)

If you truncate it
at each step, you get what are called the successive convergents of the
continued fraction, thus:

1 ; 1 + 1/2 = 3/2 ;
1 + 1/(2 + 1/3) = 1 + 3/7 = 10/7 ;

1 + 1/(2 + 1/(3 + 1/4)) = 1 +
1 /(2 + 4/13) = 1 + 13/30 = 43/30 ;

and so on,
endlessly.^{}

Now let us try
to understand Ramanujan's flash of an
answer to the problem in Strand Magazine, though we do not have the exact
problem which was the subject of conversation between Mahalonobis and
Ramanujan, *let us concoct a simplified
version of it, for the lay reader to understand it. * The problem is to find the door numbers of the
two houses, call these numbers x and y,
such that a mathematical relation, say

x^{2} - 10 y^{2}
= +1 or -1

is satisfied. Mahalanobis tries it and comes out with the
answer x = 3 and y = 1 in no time. Now Ramanujan, as soon as he hears the
statement of the problem, dictates the following continued fraction:

3 + 1/(6 + 1/(6 + 1/(6 + 1/(6
+ …
))) … )

Here 3/1 is the first
convergent. The numbers 3 and 1 are the first answers to x and y which
Mahalanobis has himself discovered by trial and error. Now Ramanujan's continued fraction answer not
only gives this first elementary answer but gives an infinity of answers on the
assumption that the street may have an infinite number of houses in it! Thus
the second convergent, 3 + 1/6 = 19/6 gives the pair, 19 and 6 for x and y.
This can be verified to be a true answer.

19^{2}
- 10 x 6^{2} = + 1.

The third convergent, 3 + 1/(6+1/6) = 3 + 6/37 = 117 / 37 gives the answer x = 117 and y
= 37 which also satisfies the relation specified by the problem. The fourth convergent can be calculated to be
721/228 and this pair x = 721 and y = 228 can also be verified to satisfy the
relation. And so on it goes. Every convergent gives a pair of numbers, which
constitute an answer. The greatness of
Ramanujan was that as soon as he heard of the problem, without effort he
decided that the answer could be given in the form of a continued fraction and
immediately gave the continued fraction, which not only solved the problem but
gave an infinity of solutions to it!

Four years at

A *prime number*
is a number p which is exactly divisible by no number
except 1 and p. So 1 and p are the only (two) divisors of p.
1,2,3,5,7,11,13,17,19,23, … are prime numbers.
A *composite number* is one
which has not only 1 and itself as a divisor, but also at least one more
divisor. The number 4 is divisible by 1, 2 and 4. It has three divisors. So 4 is a composite number. 4, 6,
8,9,10,12,14,15,16,18,20, … are all composite numbers. A *highly composite number* is a composite number which has more divisors than all its
predecessors. 4 is one such. Because, the number of divisors of 4 is 3 and no number less than 4 has so many
divisors. The list of highly composite
numbers goes as follows:

4, 6, 12, 24, 36, 60, 120, …

Ramanujan has analysed the properties of highly
composite numbers very deeply and produced this paper. When he was writing this
paper for publication, that was in 1914, he showed the manuscript to some of
the mathematics seniors in

In 1918 Ramanujan was elected Fellow of the Royal Society and
in the same year was also elected Fellow of Trinity College, both honours
coming as a first to any Indian. The

Ramanujan's innovations have not been
surpassed either before or after him. His results continue to have unforeseen
impact in fields of mathematics which were not even born in his time. Here are
two unparalleled anecdotes in this
context:

One of the creations of Ramanujan, which goes back to
his notebooks is the tau-function. The
function is so important it is nowadays called Ramanujan's tau-function. It is a function of the variable *n*,
but the definition is rather complicated for a non-mathematician; so we will
not attempt it here. But what is relevant to our story is the question of how
big the function grows for larger and larger values of *n*. Ramanujan himself conjectured that

Tau(n) £ K n^{5 1/2} ^{+ }^{e}

But he offered no proof. So mathematician after
mathematician tried to prove it. Hardy could prove the result with 8 in the
place of 5 1/2. That was certainly a
weaker result than Ramanujan's, but at least it had a proof. So other people
went on trying to bridge the gap between 8 and 5 1/2. The problem was to
estimate the growth of the function more correctly and bring it down to 5 1/2 from 8. Ramanujan himself now
offered a proof to bring it down to 7. Hardy and Littlewood brought it down to
6 in 1918. Each such effort was a major effort and resulted in an important
technical paper. Kloosterman in 1927 brought it down to 5 7/8. In 1933, ^{th} century armoury of modern algebraic geometry, a subject which
was not even born in Ramanujan's time?

Here is another which is more astounding. The
reduction process of reducing all matter to some primitives started in the 17^{th}
century and is still going on. The list of elementary and sub-elementary
particles seems to be a receding horizon. We have photons for conveying
electro-magnetic force, gravitons whose exchange is what causes gravitation,
quarks which combine in threes to make protons and neutrons and gluons which
are messenger particles. Further research has led scientists to believe that
the ultimate is what they call a string as tiny and probably tinier than a
length of 1/(10 raised to the power 31) millimetres with no mass but with
intrinsic tension and the capacity to oscillate as well as vibrate. The
electron is one mode of vibration of the string, the quark is another mode of
vibration and graviton, still another. Interaction between particles is a
matter of strings breaking or joining together or an open string forming a
loop. The latest in this list is a superstring. The deeper the physicists go
into these the more they find many advanced tools of mathematics are needed, first
for the formulation of these objects, and secondly for the understanding of
their behaviour. It appears the strings
live in a world of ten dimensions. The
physicists are comfortable with four dimensions - three of space and one of
time. But the extra dimensions that the strings need seem to be lost in a
peculiar curling up which is where they need all the mathematics in the world.
Much of this mathematics is awaiting the
21^{st} century for its development.
There are 26-dimensional models of string theory and we are told
Ramanujan's work is exactly what is needed here!

It is no surprise therefore that as years pass by, his fame increases and
the influence of his work spreads into new areas. Valmiki was eulogised by Brahma the Creator on his famous epic with
the words: So long as mountains and rivers survive in this world, that long
this Ramayana will survive. We can also
say of Ramanujan: So long as there is mathematical science and scientific activity
in the world, that long we will hear of Ramanujan and his genius. When comes
such another?

***************************

**Copyright © V. Krishnamurthy
1 Jan. 2007 Homepage**