"An equation has no meaning for me unless it expresses a thought of GOD."
Srinivasa Ramanujan
--The Man who knew Infinity
22 December 1887 – 26 April 1920
Life facts
¼ Srinivasa Ramanujan was born on December 22, 1887 in the town of Erode,
in Tamil Nadu, in the south east of India. His father was K. Srinivasa Iyengar,
an accounting clerk for clothing merchant. His mother was Komalatammal, who
earned a small amount of money each month as a singer at the local temple.
¼ As Ramanujan’s mathematical knowledge developed, his main source of
inspiration and expertise became Synopsis of elementary results in
pure mathematics by George S.
Carr.
¼ He could afford only
a small amount of paper, doing most of his work on slate with chalk,
transferring a minimal amount of his working and his results to paper.
¼
His memory for
mathematical formulas and constants seems to have been boundless: he amazed
classmates with his ability to recite the values of irrational numbers like π,
e, and √2 to as many decimal places as they asked for.
¼
Ramanujan failed his
non-mathematical exams and lost his scholarship. In 1905 he traveled to Madras
and enrolled at Pachaiyappa’s College, but again failed his non-mathematical
exams.
¼
Ramanujan tried to
find work at the government revenue department, and there he met an official
whose name was Ramaswamy Aiyer. Ramanujan did not have a resume to show
Ramaswamy Aiyer; all he had was his notebooks – the results of his mathematical
work.
¼
Ramaswamy Aiyer
contacted the secretary of the Indian Mathematical Society, R. Ramachandra Rao,
suggesting he provide financial support for Ramanujan.
¼
A meeting with
Ramanujan, however, convinced Rao that he was dealing with a genuine
mathematical genius. He agreed to provide support for Ramanujan, and Ramaswamy
Aiyer began publishing Ramanujan’s work in the Journal
of the Indian Mathematical Society.
¼ Sir Francis Spring, an engineer, who was Chairman of the
Madras Port Trust, began pressing for Ramanujan’s mathematical work to be
supported by the government and for him to be appointed to a research position
at one of the great British universities.
¼ Ramanujan and his supporters contacted a
number of British professors, but only one was receptive – an eminent pure
mathematician at the University of Cambridge – Godfrey Harold Hardy, known to
everyone as G. H. Hardy, who received a letter from Ramanujan in January 1913.
By this time, Ramanujan had reached the age of 25.
¼
Hardy was eager for Ramanujan to move to Cambridge, but in accordance
with his Brahmin beliefs, Ramanujan refused to travel overseas. Instead, an
arrangement was made to fund two years of work at the University of Madras.
During this time, Ramanujan’s mother had a dream in which the goddess Namagiri
told her she should give her son permission to go to Cambridge, and this she
did.
¼ Ramanujan arrived in Cambridge in April
1914, three months before the outbreak of World War 1. Within days he had begun
work with Hardy and Littlewood. Two years later, he was awarded the equivalent
of a Ph.D. for his work – a mere formality.
¼ In 1917 he was diagnosed with tuberculosis and worryingly
low vitamin levels. He spent months being cared for in sanitariums and nursing
homes.
¼ In February 1919 his health seemed to have recovered
sufficiently for him to return to India, but sadly he would only live for about
a year on his return.
¼ Srinivasa Ramanujan died aged 32 in Madras on April 26,
1920.
Inventions & Discoveries in Mathematics
"No, it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways, the two ways being 13 + 123 and 93 + 103.”
― Srinivasa Ramanujan
Ramanujan number: 1729 is a famous ramanujan number. It is the smaller number which
can be expressed as the sum of two cubes in two different ways- 1729 = 13 +
123 = 93 + 103
Ramanujam made substantial contributions to
the analytical theory of numbers and worked on elliptic
functions, continued fractions and infinite 1900 he began to work on his own on
mathematics summing geometric and arithmetic series.
He worked on Divergent series. He
sent 120 theorems on imply divisibility properties of the partition function.
He gave a meaning to Eulerian second
integral for all values of n (negative, positive and fractional). He
proved that the integral of xn-1 e-7 =¡ (gamma)
is true for all values of gamma.
Goldbach’s conjecture: Goldbach’s conjecture is one of the
important illustrations of ramanujan contribution towards the proof of the
conjecture. The statement is every even integer greater that two is the sum of
two primes, that is, 6=3+3 : Ramanujan and his associates had shown that every
large integer could be written as the sum of at most four (Example:
43=2+5+17+19).
Partition of whole numbers: Partition of whole numbers is another
similar problem that captured ramanujan attention. Subsequently ramanujan
developed a formula for the partition of any number, which can be made to yield
the required result by a series of successive approximation. Example
3=3+0=1+2=1+1+1;
Numbers: Ramanujan studied the highly composite numbers also which are
recognized as the opposite of prime numbers. He studies their structure,
distribution and special forms.
Fermat Theorem: He also did considerable work on the unresolved Fermat theorem,
which states that a prime number of the form 4m+1 is the sum of two squares.
Cubic Equations and Quadratic Equation: Ramanujan was shown how to solve cubic
equations in 1902 and he went on to find his own method to solve the quadratic.
The following year, not knowing that the quintic could not be solved by
radicals, he tried (and of course failed) to solve the quintic.
Euler’s constant : By 1904 Ramanujam had began to
undertake deep research. He investigated the series (1/n) and calculated Euler’s
constant to 15 decimal places.
Hypo geometric series: He worked hypo geometric series, and
investigated relations between integrals and series. He was to discover later
that he had been studying elliptic functions. Ramanujan’s own works on partial
sums and products of hyper-geometric series have led to major development in
the topic.
Bernoulli
numbers: He
published a brilliant research paper on Bernoulli numbers in 1911 in the
journal of the Indian mathematical society and gained recognition for his work.
Despite his lack of a university education, he was becoming well known in the
madras area as a mathematical genius. He began to study the Bernoulli numbers,
although this was entirely his own independent discovery.
Ramanujan
worked out the Riemann series, the elliptic integrals hyper geometric series
and functions equations of the zeta functions on the other hand he had only a
vague idea of what constitutes a mathematical proof. Despite many brilliant
results, some of his theorems on prime numbers were completely wrong.
Ramanujan
independently discovered results of Gauss, Kummar and others on
hyper-geometric series.
Perhaps
has most famous work was on the number p(n) for small numbers n, and Ramanujan
used this numerical data to conjecture some remarkable properties some of which
he proved using elliptic functions. others were only proved after Ramanujan’s
death. In a joint paper with hardly,Ramanujan gave an asymptotic
formulas for p(n). It had the remarkable property that it appeared to give the
correct value of p(n), and this was later proved by Rademacher.
Ramanujan
discovered a number of remarkable identities that imply Divisibility properties
of the partition function. He also produced quite a number of results in
definite integrals in the form of general formulate.
Besides his published work, Ramanujan left
behind several notebooks filled with theorems that mathematicians have continued
to study. The English Mathematician G.N Watson, from 1918 to 1951,
published 14 papers under the general title theorems stated by Ramanujan and in
all he published nearly 30 papers which were inspired by Ramanujan work. In 1997
Ramanujan journal was launched to publish work in areas mathematics influenced
by Ramanujan”.
Number Theory
and String Theory: In 1918 Ramanujan became the first Indian Mathematician to be elected a Fellow
of the British Royal Society:
“Distinguished
as a pure mathematician particularly for his investigation in elliptic
functions and the theory of numbers.”In his
short lifetime he produced almost 4000 proofs, identities, conjectures and
equations in pure mathematics.His theta
function lies at the heart of string theory in physics.
Others about Ramanujan
"Every positive integer is one of Ramanujan's personal friends."
JOHN LITTLEWOOD-Mathematician
"For my part, it is difficult for me to say what I
owe to Ramanujan – his originality has been a constant source of suggestion to
me ever since I knew him, and his death is one of the worst blows I have ever
had.”
"It
was his insight into algebraical formulae, transformations of infinite series,
and so forth that was most amazing. On this side most certainly I have never
met his equal, and I can compare him only with Euler or Jacobi.”
"I
had never seen anything in the least like them before. A single look at them is
enough to show that they could only be written by a mathematician of the
highest class. They must be true because, if they were not true, no one would
have the imagination to invent them.”
G. H. HARDY, 1877 – 1947-Mathematician
“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.”
PAUL ERDŐS, 1913 – 1996-Mathematician
“That was the wonderful thing about Ramanujan. He
discovered so much, and yet he left so much more in his garden for other people
to discover.”
FREEMAN DYSON, BORN 1923-Mathematician and Physicist
"… each of the 24 modes in the Ramanujan function
corresponds to a physical vibration of a string. Whenever the string executes
its complex motions in space-time by splitting and recombining, a large number
of highly sophisticated mathematical identities must be satisfied. These are
precisely the mathematical identities discovered by Ramanujan.”
MICHIO KAKU, BORN 1947-Theoretical Physicist
Recognition
Ramanujan's home state of Tamil Nadu celebrates 22 December
(Ramanujan's birthday) as 'State IT Day'.
A stamp picturing Ramanujan was released by the Government
of India in 1962 – the 75th anniversary of Ramanujan's birth – commemorating
his achievements in the field of number theory,and a new design was issued on
26 December 2011, by the India Post.
Since Ramanujan's centennial year, his birthday, 22
December, has been annually celebrated as Ramanujan Day by the Government Arts
College, Kumbakonam where he studied and at the IIT Madras in Chennai.
A prize for young
mathematicians from developing countries has been created in Ramanujan's name
by the International Centre for Theoretical Physics (ICTP) in cooperation with
the International Mathematical Union, which nominate members of the prize
committee.
The SASTRA University, based in the state of Tamil Nadu in
South India, has instituted the SASTRA Ramanujan Prize of US$10,000 to be given
annually to a mathematician not exceeding the age of 32 for outstanding
contributions in an area of mathematics influenced by Ramanujan.
Vasavi College of
Engineering named its Department of Computer Science and Information Technology
"Ramanujan Block".
In 2011, on the 125th anniversary of his birth, the Indian
Government declared that 22 December will be celebrated every year as "National
Mathematics Day". Then Indian Prime Minister Manmohan Singh also declared that
the year 2012 would be celebrated as the National Mathematics Year.
