MATHEMATICIANS CONTRIBUTIONS: MODULE 4 - SRINIVASA RAMANUJAN (1887 AD - 1920 AD)

THURSDAY, JULY 26, 2012

MODULE 4 - SRINIVASA RAMANUJAN (1887 AD - 1920 AD)

SRINIVASA RAMANUJAN (1887 AD - 1920 AD)

Born Died Residence Nationality Fields Institutions Friend 22nd December 1887 AD 1920 AD Erode , Kumbakonam Indian Mathematics, Astronomy Cambridge university, madras university Hardy

Srinivasa Ramanujan, one of India’s greatest mathematical geniuses, was born in his grandmother’s house in Erode, a small village about 400 km southwest of Madras, on 22nd December 1887. His father worked in kumbakonam as a clerk in a cloth merchant’s shop. In 1917 he was hospitalized, his doctors fearing for his life. By late 1918 his health had improved; he returned to India in 1919. But his health failed again, and he died the next year. Ø Five years old – primary school Ø Jan 1898 – town high school in Kumbakonam Ø 1904 – he got scholarship Ø 1906 – he entered in to Pachaiyappa’s college Ø 14th July 1909 – he married ten year old girl S.Janaki Ammal Ø 1911 – His first paper published, 17 page works on Bernoulli numbers - journal of the Indian Mathematical Society. Ø Ramanujan was appointed to the post of clerk and began his duties on 1stMarch 1912. Ø 1914 – he went England Ø 1916 – Cambridge university granted him a bachelor of science degree Ø 1919 – he returned India Contributions · 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 = ¡ is true for all values of gamma. thiyagusurimathematicians.blogspot.in/2012/07/module-4-srinivasa-ramanujan-1887-ad.html

(gamma)

1/3

5/27/13

MATHEMATICIANS CONTRIBUTIONS: MODULE 4 - SRINIVASA RAMANUJAN (1887 AD - 1920 AD)

·

·

·

·

·

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. 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 Cubic Equations and Quadratic Equation: Ramanujam 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....