Shrinivas Ramanujan: Contribution to mathematics
Srinivasa Ramanujan, an Indian mathematician was born in 22nd December, 1887 in Madras, India. Like Sophie Germain, he received no formal education in mathematics but made important contributions to advancement of mathematics.
His chief contribution in mathematics lies mainly in analysis, game theory and infinite series. He made in depth analysis in order to solve various mathematical problems by bringing to light new and novel ideas that gave impetus to progress of game theory. Such was his mathematical genius that he discovered his own theorems. It was because of his keen insight and natural intelligence that he came up with infinite series for π.
This series made up the basis of certain algorithms that are used today. One such remarkable instance is when he solved the bivariate problem of his roommate at spur of moment with a novel answer that solved the whole class of problems through continued fraction. Besides that he also led to draw some formerly unknown identities such as by linking coefficients of and providing identities for hyperbolic secant.
He also described in detail the mock theta function, a concept of mock modular form in mathematics. Initially, this concept remained an enigma but now it has been identified as holomorphic parts of maass forms. His numerous assertions in mathematics or concepts opened up new vistas of mathematical research for instance his conjecture of size of tau function that has distinct modular form in theory of modular forms. His papers became an inspiration with later mathematicians such as G. N. Watson, B. M. Wilson and Bruce Berndt to explore what Ramanujan discovered and to refine his work. His contribution towards development of mathematics particularly game theory remains unrivaled as it was based upon pure natural talent and enthusiasm. In recognition of his achievements, his birth date 22 December is celebrated in India as Mathematics Day. It would not be wrong to assume that he was first Indian mathematician who gained acknowledgment only because of his innate genius and talent.
Continued fraction
Continued fraction, expression of a number as the sum of an integer and a quotient , the denominator of which is the sum of an integer and a quotient, and so on. In general,
where a0, a1, a2, … and b0, b1, b2, … are all integers.
In a simple continued fraction (SCF), all the bi are equal to 1 and all the ai are positive integers. An SCF is written, in the compact form, [a0; a1, a2, a3, …]. If the number of terms ai is finite, the SCF is said to terminate, and it represents a rational number; for example, 802/251 = [3; 5, 8, 6]. If the number of these terms is infinite, the SCF does not terminate, and it represents an irrational number; for example, Square root of√23 = [4; 1, 3, 1, 8], in which the bar spans a sequence of terms that repeats indefinitely. A nonterminating SCF in which a sequence of terms recurs represents an irrational number that is a root of a quadratic equation with rational coefficients. Nonterminating SCFs 64S117.2 64 74.6 75.5c-23.5 6.3-42 24.9-48.3 48.6-11.4 42.9-11.4 132.3-11.4 132.3s0 89.4 11.4 132.3c6.3 23.7 24.8 41.5 48.3 47.8C117.2 448 288 448 288 448s170.8 0 213.4-11.5c23.5-6.3 42-24.2 48.3-47.8 11.4-42.9 11.4-132.3 11.4-132.3s0-89.4-11.4-132.3zm-317.5 213.5V175.2l142.7 81.2-142.7 81.2z"/> Subscribe on YouTube