Contributions Of Great Mathematician S.RAMANUJAN

نجنامار سيا

History Of S.RAMANUJAN-

 Born on December 22 , 1887.

 In a village in Madras State, at Erode, in Tanjore District.

  In a poor HINDU BRAHMIN family.

Full name is “ SRINIVAS RAMANUJAN AYYANGER ”.

 Son of Srinivas Iyenger.

  Accountant to a cloth merchant at KUMBHAKONAM. Daughter of petty official ( Amin ) in District Munsif’s court at Erode.

Daughter of petty official ( Amin ) in District Munsif’s court at Erode.

 First went to school at the age of 7 .

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His famous history was :-

One day a primary School teacher of 3 rd 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 .

 So intelligent that as students of class 3 rd or primary school.

 Solved all problems of Looney’s Trigonometry meant for degree classes.

 At the age of seven , he was transferred to Town High School at Kumbhakonam.

 He held scholarship.

 Stood first in class.

 Popular in mathematics.

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 At the age of 12, he was declared “ CHILD MATHEMATICIAN” by his teachers.

 Entertain his friends with theorem and formulas.

 Recitation of complete list of Sanskrit roots and repeating value of ∏ and square root of 2, to any number of decimal places.

 In 1903 , at the age of 15, in VI form he got a book , “Carr’s Synopsis”.

 “Pure and Applied Mathematics”

 Gained first class in matriculation in December 1903.

 Secured Subramanian’s scholarship.

 Joined first examination in Arts (F.A).

 Tried thrice for F.A.

 In 1909, he got married to Janaki ammal.

 Got job as clerk.

 Office of Madras port trust.

Born

4 November 1897 Tellicherry,Kerala

Died

February 1984 (aged 87)

Nationality Fields Institutions Alma mater

Indian Botany, Cytology University Botany,,Laboratory Madras University of Michigan

 Published his work in “Journal of Indian Mathematical Society”.

 In 1911, at 23 , wrote a long article on some properties of “Bernoullis Numbers”.  Correspondence with Prof.J.H Hardy.

 Attached 120 theorems to the first letter.

 In 1912, Mr. Walker, held high post under the Government.

 Obtained, scholarship of Rs. 75/- per month.

 In 1914, invited to Cambridge University, and in 1916, got Hon. B.A. Degree of University of Cambridge

His 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

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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−ا e −ͯ = Gamma is true for all Gamma.

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.

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He detected congruence, symmetries and relationships and different wonderful properties.Taxi cab Nowas an interesting number to him.

1729 = 1³+12³ = 9³ + 10³ 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.

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”.

 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.

 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.