S.RAMANUJAN HAD CONTRIBUTED A LOT TO THE MATHEMATICS AS HE IS THE FRIEND OF NUMBER

1. CONTRIBUTION OF SRINIVASA
RAMANUJAN TO MATHEMATICS
By S.P. SRIVATSAN XI B

2. His famous history was :- One day a primary School teacher of 3"
form was telling to his students 'If three fruits are divided among
three persons, each would get one, even would get one , even if
1000 fruits are divided among 1000 persons each would get one .
Thus, generalized that any number divided by itself was unity.
This Made a child of that class jump and ask- is zero divided by
zero also unity?" If no fruits are divided nobody, will each get
one? This little boy was none other than RAMANUJAN

3. Euatio
“ An equation means nothing to me unless is
expresses a thought of god “

4. He was an Indian mathematician
who lived during the British Rule in
India. Though he had almost no
formal training in pure mathematics,
he made substantial contributions to
mathematics
He was Borm on December 22, 1887.
In a village in Madras State, at Erode,
in Tanjore District, In a poor hindu
brahmin family.

5. At Kangayan Primary School Ramanujan
performed well. Just before turning 10, in
November 1897, he passed his primary
examinations in English, Tamil, geography
and arithmetic with the best scores in the
district. That year Ramanujan entered Town
Higher Secondary School, where he
encountered formal mathematics for the first
time.

6. When he graduated from Town Higher
Secondary School in 1904, Ramanujan was
awarded the K. Ranganatha Rao prize for
mathematics by the school's headmaster,
Krishnaswami Iyer. Iyer introduced
Ramanujan as an outstanding student who
deserved scores higher than the maximum.
He received a scholarship to study at
Government Arts College, Kumbakonam,
but was so intent on mathematics that he
could not focus on any other subjects and
failed most of them, losing his scholarship in
the process.

7. England honoured him by Royal
Society and Trinity fellowship.
But he Did not receive any
honour from India. In spring of
1917, he first appeared tobe
unwell,Active work for Royal
Society and Trinity Fellowship.
Due to TB, he left for India and
died in chetpet Madras, On April
26, 1920 at the age of 33.

8. HIS
ACHIEVEMENTS

9. Ramanujan summation is a
technique invented by the
mathematician Srinivasa
Ramanujan for assigning a value
to divergent infinite series.
Although the Ramanujan
summation of a divergent series
is not a sum in the traditional
sense, it has properties that make
it mathematically useful in the
study of divergent infinite series,
for which conventional
summation is undefined
1. RAMANUJAN
SUMMATION

10. The Rogers–Ramanujan
continued fraction is a continued
fraction discovered by Rogers
and independently by Srinivasa
Ramanujan, and closely related
to the Rogers–Ramanujan
identities. It can be evaluated
explicitly for a broad class of
values of its argument.
2. Rogers - ramanujan
continued fraction

11. Ramanujan loved infinite series
and integrals. Of course, it is
impossible to adequately cover
what Ramanujan accomplished
inhis devotion to integrals. Many
of Ramanujan’s theorems and
examples of integrals have
inspired countless
mathematicians to take
Ramanujan’s thoughts and
proceed further.
3. Definite integral

12. Once Hardy visited to Putney were
Ramanujan hospitalized, He visited
there in a taxi cab having 1729.
Hardy was very superstitious due to
his such nature when he entered into
Ramanujan's room, he quoted that he
had just came in a taxi cab having
number 1729 which seemed to him an
unlucky number, but at that time,
Ramanujan replied that this was a very
interesting number a
s it is the smallest number can be expressed as
the sum of cubes of two numbers. Later some
theorems were established in theory of elliptic
curves which involves this fascinating number.
4. Hardy - Ramanujan
number

13. Sreenivasa Ramanujan also
discovered some remarkable
infinite series of pi around
1910.The series, Computes a
further eight decimal places of a
with each term in the series.
Later on, a number of efficient
algorithms have been developed
by number theorists using the
infinite series of given by
Ramanujan.
5. Infinite seried for pi

14. Goldbach’s conjecture is one of
the important illustrations of
Ramanujan contribution towards
the proof of the conjecture. The
statement is every even integer
> 2 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).
6. Goldbatch’s conjecture

15. Ramanujan was shown how to
solve cubic equations in 1902
and he went on to find his own
method to solve the quadratic. He
derived the formula to solve
biquadratic equations.The
following year, he tried to
provide the formula for solving
quintic but he couldn’t as he was
not aware of the fact that quintic
could not be solved by radicals.
7. Theory of equations

16. Ramanujan’s one of the major
work was in the partition of
numbers. By using partition
function 𝑝(𝑛),
he derived a number of formulae
in order to calculate the partition
of numbers. In a joint paper with
Hardy,
Ramanujan gave an asymptotic
formulas for𝑝(𝑛).In fact, a
careful analysis of the generating
function for 𝑝 𝑛
leads to the Hardy-Ramanujan
asumptotic.
8. Hardy - ramanujan
asympotic formula

17. Ramanujan’s congruences are
some remarkable congruences for
the partition function. He
discovered the congruencesIn his
1919 paper, he gave proof for
the first two congruencesusing
the following identities using
Pochhammer symbol notation.
After the death of Ramanujan, in
1920, the proof of all above
congruences extracted from his
unpublished work.
9. Ramanujan’s congruences

18. A natural number n is said to
behighly composite number if it
has more divisors than any
smaller natural number. If we
denote the number of divisors of
n by d(n), then we say 𝑛 є N is
called a highly composite
10. Highly composite number
if 𝑑 𝑚 < 𝑑(𝑛) ∀𝑚 < 𝑛 where𝑚 є N.For
example, 𝑛 = 36is highly composite
because it has 𝑑 36 = 9 and
smaller natural numbers have less
number of divisors.If
𝑛 =2k2 3k3 …… pkp
(by Fundamental theorem of Arithmetic

19. Some other contribution of S. Ramanujan
1. Apart from the contributions mentioned above, he worked in some other
areas of mathematics such as hypogeometric series, Bernoulli numbers,
Fermat’s lasttheorem.
2. He focused mainly on developing the relationship between partial sums
and products of hyper-geometric series.
3. He independently discovered Bernoulli numbers and using these
numbers, he formulated the value of Euler’s constant up to 15 decimal
places.
4. He nearly verified Fermat’s last theorem which states that no three
natural number 𝑥, 𝑦 and 𝑧 satisfy the equation x n + yn = zn for any
integer 𝑛 > 2.