Contribution of S. Ramanujan .pptx

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 ACHI EVEMENTS

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 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 =2 k2 3 k3 …… p kp (by Fundamental theorem of Arithmetic

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