Srinivasa Ramanujan was a renowned Indian mathematician who made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions despite his lack of formal training. Some of his key achievements include formulating the Ramanujan prime and discovering the Ramanujan theta function. He worked closely with his mentor, G.H. Hardy, at Cambridge University where he was elected Fellow of the Royal Society. Unfortunately, Ramanujan's life was cut short at the young age of 32, but he left a lasting legacy as one of the greatest mathematicians of all time.

1. Srinivasa Ramanujan Iyengar (Best known as S. Ramanujan)(22 Dec
1887 - 26 April 1920)
The Man Who
Knew Infinity

2. About His History
 Born - 22 December 1887
Place - Kumbakonam, Madras Presidency
 Died - 26 April 1920
Place - Chetput, Madras, British India
 College - Government Arts College
Pachaiyappa’s College
Cambridge University
 Academic Advisors - G.H.Hardy
J.E.Littlewood

3. Example of His Brain
Teacher: n / n =1, for every integer n.
Ramanujan: “Is zero divided by zero is also one?”
Ramanujan’s Answer: “Zero divided by zero maybe anything. The zero of the
numerator may be several times the zero of the denominator advice versa”.(7 year
old prodigy was thinking of limits and limiting processes.)

4. The Genius Mathematician
 S. Ramanujan 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.
 Probably Ramanujan’s life has no parallel in the history of human thought. He
was a autodidact mathematical genius not only of the twentieth century but for
all time to come.
 Discovered theorems of his own

5. His Ups & Downs
 After high school, Ramanujan passed a competitive examination in English and Mathematics
and secured the Scholarship. He joined the Government Arts college, in the F.A.(First
Examination in Arts) class at the age of 18 years.
 Due to his love & preoccupation with Mathematics, he could not pass in English and Sanskrit
and hence was not promoted to the Senior F.A. Class.
 He lost his Scholarship & forced to discontinue his studies due to this and the poverty of his
family.
 However, he appeared privately for the F.A.examination but failed to pass the university
examinations and this marked the end of his formal education.
 After having failed in his F.A. class at College, he ran from pillar to post in search of a
benefactor for 6 (from 18 years of age to 24 years) years.

6. His Ups & Downs
 He tried for tutoring assignments to earn livelihood but failed. His love and dedication for
math continued.
 He lived in absolute poverty. 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.”
 With the help of a recommendation letter from a mathematician, Ramanujan secured a
clerical post in the Accounts Section of the Madras Port Trust at an age of 24 years.
 Meanwhile in 1911 at an age of 23 years, the first full length (15 page) research paper of
Ramanujan, entitled: “Some properties of Bernoulli Numbers”, appeared in the Journal of the
Indian Mathematical Society. This brought him to the forefront of international mathematics.
 After going through a paper by British mathematician Hardy, about some theorems on prime
numbers, where he claimed that no definite expression has been found as yet, Ramanujan
derived the expression which very nearly approximated to the real result.

7. His Ups & Down
 The orchestrated efforts of his admirers, resulted in inducing the Madras University, 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 &100 pounds for passage by ship and for initial outfit to go to England in
1914 at an age of 26 years.
 Ramanujan initially refused to go to England possibly due to his caste prejudice. But after mentoring
from Neville, Ramanujan finally set sail for England at a age of 26 years in 1914.
 Prior to Ramanujan departure to England, Prof. Littlehailes & Mr. Arthur Davies arranged with the
University for £ 60 (out of Scholarship amount of £ 250 ) per year to be sent to his parents in India.
Thus, he fulfilled his responsibilities as the eldest son of the family.
 At Cambridge, Ramanujan worked with Littlewood and Hardy, mathematicians of repute of that time.
 Ramanujan was awarded the B.A. degree by research (this degree was later renamed as PhD)in
March 1916 at an age of 28 years, for his work on Highly Composite Numbers.
 He was elected a Fellow of the Royal Society of London in February 1918 at an age of 30 years. He
was the second Indian to become a Fellow of the Royal Society (1stone was in 1841, around 80 years
prior) & one of the youngest Fellows in the entire history of the Royal Society.

8. Role of Prof. G.H. Hardy
The role of G.H. Hardy (1877 - 1947), Professor of Mathematics at
Trinity College, Cambridge, in the life and career of Ramanujan is
immeasurable, peerless and beyond praise.
 Convinced that Ramanujan was a natural genius, Hardy made up his
mind that Ramanujan should be brought to Cambridge.
The eccentric British mathematician G.H. Hardy is known for his
achievements in number theory and mathematical analysis. But he is
perhaps even better known for his adoption and mentoring of the self-
taught mathematical genius, Ramanujan.
Before writing to Hardy, Ramanujan had written to two well-known
Cambridge mathematicians. But both of them had expressed their
inability to help Ramanujan.

9. Story of Hardy – Ramanujan “Taxicab numbers”
A common anecdote about Ramanujan relates how Hardy arrived at Ramanujan’s
house in a cab numbered 1729, a number he claimed to be totally uninteresting.
Ramanujan is said to have stated on the spot that, on the contrary, it was actually a
very interesting number mathematically, being the smallest number representable
in two different ways as a sum of two cubes. Such numbers are now sometimes
referred to as "taxicab numbers".

10. Taxicab – number
1,729 is the smallest number which can be represented in two different ways as the sum
of two cubes:
1729 = 13 + 123
= 93 + 103
It is also incidentally the product of 3 prime numbers:
1729 = 7 X 13 X 19
The largest known similar number is :
885623890831 = 75113 + 77303
= 87593 + 59783
= 3943 X 14737 X 15241

11. Ramanujan’s Magic Square
22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11
This square looks like
any other normal magic
square. But this is
formed by great
mathematician of our
country – Srinivasa
Ramanujan.
What is so great in it?

12. 22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11
Sum of numbers of
any row is 139.
What is so great in it.?

13. Sum of numbers of
any column is also 139.
Oh, this will be there in
any magic square.
What is so great in it..?
22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11

14. Sum of numbers of
any diagonal is also
139.
Oh, this also will be there
in any magic square.
What is so great in it…?
22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11

15. Sum of corner
numbers is also 139.
Interesting?
22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11

16. Look at these
possibilities. Sum of
identical coloured
Numbers is also 139.
Interesting..?
22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11

17. Look at these
possibilities. Sum of
identical coloured
Numbers is also 139.
Interesting..?
22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11

18. Look at these central
numbers.
Interesting…?
22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11

19. Can you try these
combinations?
Interesting…..?
22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11

20. Try these
combinations also?
Interesting.…..?
22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11

21. NOW
LETS FACE
THE
CLIMAX
22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11

22. Do you know date of
birth of Srinivasa
Ramanujan?
22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11

23. It is 22nd Dec 1887.
22.12.1887
22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11

24. Achievements
1) Divergent Series:- When
Dr. Hardy examined his
investigation – “I had never
seen anything the least
like them before. A single
look at them is enough to
show that this could only be
written by
Mathematician of highest
class”.
2) Hyper Geometric series
and continued Fraction:
He was compared with Euler
and Jacobi.
3) Definite Integrals
4)Elliptic Functions
5)Partition functions
6)Fractional Differentiation:
He gave a meaning to
Eulerian second integral for
all values of n .He proved
x ⁿ− ‫ا‬e −ͯ = Gamma is true for
all Gamma.

25. 7)Theory of Numbers: The modern theory of
numbers is
most difficult branch of mathematician .It has
many
unsolved problems. Good Example is of Gold
Bach’s
Theorem which states that every even number is
sum
of two prime numbers. Ramanujan discovered
Reimann’s series , concerning prime numbers .
For him
every integer was one of his personal friend.
He detected congruence, symmetries
and
relationships and different wonderful
properties.
8. Partition of whole numbers: Take case of 3. It
can
be written as…
3+0,1+2,1+1+1
He developed a formula
, for partition of any number
which can be made to yield the
required result by a series of
successive approximation.
”.

26. 9). Highly Composite Numbers : Highly
composite
number is opposite of prime numbers.
Prime number has
two divisions, itself and unity . A highly
composite
number has more divisions than any
preceding number
like: 2,4,6,12,24,36,48,60,120,etc.He
studied the
structure ,distribution and special forms of
highly
composite numbers. Hardy says –
“Elementary analysis
of highly composite numbers is most
remarkable and
shows very clearly Ramanujan’s extra-
ordinary mastery
over algebra of inequalities

27. Greatest masters in the field of higher geometric theories and
continued fractions.
He could work out modular equation, work out theorems of complex
multiplication, mastery of continued fraction.
Found for him self functional equation of zeta function.
Mathematician whom only first class mathematicians follow.
England honoured by Royal Society and Trinity Fellowship.

28. Ramanujan is eponym of mathematical topics listed below
 Ramanujan Magic Square
 Brocard-Ramanujan Diophantine equation
 Dougall-Ramanujan identity
 Hardy-Ramanujan number
 Landau-Ramanujan constant
 Ramanujan’s congruences
Ramanujan-Nagell equation
 Ramanujan-Peterssen conjecture
 Ramanujan-Skolems theorem
 Ramanujan-Soldner constant
 Ramanujan theta function
 Ramanujan graph
 Ramanujan’s tau function
 Ramanujan’s ternary quadratic form
 Ramanujan prime
 Ramanujan’s constant
 Ramanujan’s sum
 Rogers-Ramanujan identity

29. MADE BY
ROHAN KUMAR SAHU
CLASS
VIII - A