Indian Mathematicians

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10 Indian Mathematicians and their contributions.

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Indian Mathematicians

  1. 1. Indian Mathematicians Math Project for 2013-2014
  2. 2. Introduction There is no doubt that the world today is indebted to the contributions made by Indian mathematicians. Some of the most important contributions made by Indian mathematicians were the introduction of decimal system as well as the invention of zero. Here are some the famous Indian mathematicians, along with their contributions.
  3. 3. Aryabhata • Place value system and zero The place-value system, was clearly in place in Aryabhata’s work. While he did not use a symbol for zero, the French mathematician Georges Ifrah argues that knowledge of zero was suggested in Aryabhata's place-value system as a place holder for the powers of ten with null coefficients. • Approximation of π Aryabhata worked on the approximation for pi (π), and may have come to the conclusion that it is irrational. • Trigonometry Aryabhata gave the area of a triangle as "for a triangle, the result of a perpendicular with the half-side is the area.”
  4. 4. Brahmagupta •Zero Brahmagupta's Brahmasphuṭasiddhanta is the first book that mentions zero as a number, hence Brahmagupta is considered the first to formulate the concept of zero. He gave rules of using zero with negative and positive numbers. Zero plus a positive number is the positive number and negative number plus zero is a negative number etc. The Brahmasphutasiddhanta is the earliest known text to treat zero as a number in its own right. •Brahmagupta's formula Brahmagupta's most famous geometrical work is his formula for cyclic quadrilaterals. Given the lengths of the sides of any cyclic quadrilateral, Brahmagupta gave an approximate and an exact formula for the figure's area, i.e., “the area is the square root from the product of the halves of the sums of the sides diminished by each side of the quadrilateral.”
  5. 5. Srinivasa Ramanujan In mathematics, there is a distinction between having an insight and having a proof. Ramanujan's talent suggested a plethora of formulae that could then be investigated in depth later. It is said that Ramanujan's discoveries are unusually rich and that there is often more to them than initially meets the eye. As a byproduct, new directions of research were opened up. Examples of the most interesting of these formulae include the intriguing infinite series for π, one of which is given below
  6. 6. P.C. Mahalanobis •Mahalanobis Distance P.C. Mahalnobis asked Nelson Annandale the questions on what factors influence the formation of European and Indian marriages. He wanted to examine if the Indian side came from any specific castes. He used the data collected by Annandale and the caste specific measurements made by Herbert Risley to come up with the conclusion that the sample represented a mix of Europeans mainly with people from Bengal and Punjab but not with those from the Northwest Frontier Provinces or from Chhota Nagpur. He also concluded that the intermixture more frequently involved the higher castes than the lower ones. This analysis was described by his first scientific paper in 1922. During the course of these studies he found a way of comparing and grouping populations using a multivariate distance measure. This measure, denoted "D2" and now named Mahalanobis distance, is independent of measurement scale.
  7. 7. C.R. Rao •Estimation theory Estimation theory is a branch of statistics that deals with estimating the values of parameters based on measured/empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the measured data. An estimator attempts to approximate the unknown parameters using the measurements. In estimation theory, two approaches are generally considered. The probabilistic approach assumes that the measured data is random with probability distribution dependent on the parameters of interest The set-membership approach assumes that the measured data vector belongs to a set which depends on the parameter vector.
  8. 8. D.R. Kaprekar D.R. Kaprekar discovered a number of results in number theory and described various properties of numbers including Kaprekar Constant, Kaprekar Numbers, self numbers, Harshad numbers and Demlo numbers. •Kaprekar Constant Kaprekar discovered the Kaprekar constant or 6174 in 1949. He showed that 6174 is reached in the limit as one repeatedly subtracts the highest and lowest numbers that can be constructed from a set of four digits that are not all identical. Thus, starting with 1234, we have 4321 − 1234 = 3087, then 8730 − 0378 = 8352, and 8532 − 2358 = 6174. Repeating from this point onward leaves the same number (7641 − 1467 = 6174). In general, when the operation converges it does so in at most seven iteration.
  9. 9. Harish Chandra He was influenced by the mathematicians Hermann Weyl and Claude Chevalley. From 1950 to 1963 he was at the Columbia University and worked on representations of semisimple Lie groups. During this period he established as his special area the study of the discrete series representations of semisimple Lie groups, which are analogues of the Peter–Weyl theory in the non-compact case. •Discrete Series Representation In mathematics, a discrete series representation an irreducible unitary representation of a locally compact topological group G that is a subrepresentation of the left regular representation of G on L²(G). In the Plancherel measure, such representations have positive measure. The name comes from the fact that they are exactly the representations that occur discretely in the decomposition of the regular representation.
  10. 10. Satyendra Nath Bose •Bose-Einstein Statistics/Condensate Bose showed that the contemporary theory was inadequate, because it predicted results not in accordance with experimental results. In the process of describing this discrepancy, Bose for the first time took the position that the Maxwell–Boltzmann distribution would not be true for microscopic particles, where fluctuations due to Heisenberg's uncertainty principle will be significant. Thus he stressed the probability of finding particles in the phase space, each state having volume h3, and discarding the distinct position and momentum of the particles. Bose's interpretation is now called Bose–Einstein statistics. Einstein adopted the idea and extended it to atoms. This led to the prediction of the existence of phenomena which became known as Bose–Einstein condensate, a dense collection of bosons, which was demonstrated to exist by experiment in 1995.
  11. 11. Bhāskara II Some of Bhāskara II’s contributions are: •A proof of the Pythagorean theorem by calculating the same area in two different ways and then canceling out terms to get a2 + b2 = c2. •Solutions of quadratic, cubic and quartic indeterminate equations are explained. •Solutions of indeterminate quadratic equations (of the type ax2 + b = y2). •Integer solutions of linear and quadratic indeterminate equations. •Solved quadratic equations with more than one unknown, and found negative and irrational solutions. •In Siddhanta Shiromani, Bhaskara developed spherical trigonometry along with a number of other trigonometric results.
  12. 12. Narendra Karmarkar •Karmarkar's algorithm Karmarkar's algorithm solves linear programming problems in polynomial time. These problems are represented by "n" variables and "m" constraints. The previous method of solving these problems consisted of problem representation by an "x" sided solid with "y" vertices, where the solution was approached by traversing from vertex to vertex. Karmarkar's novel method approaches the solution by cutting through the above solid in its traversal. Consequently, complex optimization problems are solved much faster using the Karmarkar algorithm. His algorithm thus enables faster business and policy decisions. Karmarkar's algorithm has stimulated the development of several other interior point methods, some of which are used in current codes for solving linear programs.
  13. 13. Done by: Jerome George, IX-C DPS, Riyadh

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