ARCHIMEDIS [287BC-212BC]

Archimedis is considered to be one of the greatest mathematicians of all time. His father was a greek astronomer. He studied in Alexandria. Archimedis contributed to both pure and applied mathematics. He calculated the value of pie with a remarkable accuracy between 3.1408 and 3.1429.he also did a great amount of work on Analytical geometry. He wrote many books on mathematics such as Measurement of circles,On floating bodies.

M.S.NARASIMHAN ANA C.P.RAMANUJAN

              M.S. NARASIMHAN

 Seshachalu Narasimhan (born 1932) is an eminent Indian mathematician. He is well known along with C S Seshadri for their proof of the Narasimhan–Seshadri theorem, and both were elected as FRS.

Education.

Narasimhan did his undergraduate studies at Loyola College, Chennai, where he was taught by Fr Racine. Fr Racine had studied with the famous French mathematicians Élie Cartan and Jacques Hadamard, and connected his students with the latest developments in modern mathematics. Among Racine's other students who achieved eminence, Mudumbaiwe may count Minakshisundaram, K. G. Ramanathan, C S Seshadri, Raghavan Narasimhan, and C. P. Ramanujam.

Narasimhan went to the Tata Institute of Fundamental Research (TIFR), Bombay, for his graduate studies. He obtained his Ph.D. from University of Mumbai in 1960; his advisor was K. Chandrasekharan. Among Narasimhan's distinguished students is M. S. Raghunathan who followed in this footsteps to bag the Shanti Swarup Bhatnagar Prize as well as become FRS. Two other students who made a mark as top-notch mathematicians are S. Ramanan and V. K. Patodi.

Degrees and posts held
              Awards and felicitations

•  Visiting Scholar, Institute for Advanced Study 

   (1968-1969)

•  Fellow of the Royal Society, London

•  Head, Mathematics Group of the Abdus Salam International Centre for Theoretical Physic

   (1992–1999)

•  Honorary Fellow, Tata Institute of Fundamental Research, Bangalore Centre.

•  Shanti Swarup Bhatnagar Prize (1975)

•  Third World Academy Award for Mathematics (1987)

•  Padma Bhushan (1990)

•  King Faisal International Prize for Science, 2006 (jointly with Simon Donaldson, Imperial College)

            

CP RAMANUJAM

Chakravarthi Padmanabhan Ramanujam (January 9, 1938 – October 27, 1974) worked in the fields of number theory and algebraic geometry. He was elected a Fellow of the Indian Academy of Sciences in 1973.

Like his namesake Srinivasa Ramanujan, Ramanujam also had a very short life.

As David Mumford put it, Ramanujam felt that the spirit of mathematics demanded of him not merely routine developments but the right theorem on any given topic. "He wanted mathematics to be beautiful and to be clear and simple. He was sometimes tormented by the difficulty of these high standards, but in retrospect, it is clear to us how often he succeeded in adding to our knowledge, results both new, beautiful and with a genuinely original stamp".

Career

Ramanujam set out for Mumbai at the age of eighteen to pursue his interest in mathematics. He and his friend and schoolmate Raghavan Narasimhan, and S. Ramanan joined TIFR together in 1957. At the Tata Institute there was a stream of first rate visiting mathematicians from all over the world. It was a tradition for some graduate student to write up the notes of each course of lectures. Accordingly, Ramanujam wrote up in his first year, the notes of Max Deuring's lectures on Algebraic functions of one variable. It was a nontrivial effort and the notes were written clearly and were well received. The analytical mind was much in evidence in this effort as he could simplify and extend the notes within a short time period. "He could reduce difficult solutions to be simple and elegant due to his deep knowledge of the subject matter" states Ramanan. "Max Deuring's lectures gave him a taste for Algebraic Number Theory. He studied not only algebraic geometry and analytic number theory of which he displayed a deep knowledge but he became an expert in several other allied subjects as well".

On the suggestion of his doctoral advisor, K. G. Ramanathan, he began working on a problem relating to the work of the German number theorist Carl Ludwig Siegel. In the course of proving the main result to the effect that every cubic form in 54 variables over any algebraic number field K had a non-trivial zero over that field, he had also simplified the earlier method of Siegel. He took up Waring's problem in algebraic number fields and got interesting results. In recognition of his work and his contribution to Number Theory, the Institute promoted him as Associate Professor. He protested against this promotion as 'undeserved', and had to be persuaded to accept the position. He proceeded to write his thesis in 1966 and took his Doctoral examination in 1967. Dr. Siegel who was one of the examiners was highly impressed with the young man's depth of knowledge and his great mathematical abilities.

Ramanujam was a scribe for Igor Shafarevich's course of lectures in 1965 on minimal models and birational transformation of two dimensional schemes. Professor Shafarevich subsequently wrote to say that Ramanujam not only corrected his mistakes but complemented the proofs of many results. The same was the case with Mumford's lectures on abelian varieties which was delivered at TIFR around 1967. Mumford wrote in the preface to his book that the notes improved upon his work and that his current work on abelian varieties was a joint effort between him and Ramanujam. A little known fact is that during this time he started teaching himself German, Italian, Russian and French so that he could study mathematical works in their original form. His personal library contained quite a few non-English mathematical works.

                                               DONE BY

                                                         SUCHITRA

BHRMAGUPTHA.........

                   BRAHMAGUPTHA

BRAHMAGUPTA -Indian Mathematicians And Their Contributions * Brahma Gupta was born in 598A.D in Pakistan. * He gave four methods of multiplication. * He gave the following formula, used in G.P series a + ar + ar2 + ar3 +……….. + arn-1 = (arn-1) ÷ (r – 1) * He gave the following formulae : Area of a cyclic quadrilateral with side a, b, c, d= √(s -a)(s- b)(s -c)(s- d) where 2s = a + b + c + d Length of its diagonals = bio1 2 Indian Mathematicians And Their Contributions

ARCHIMEDES............

 ARCHIMEDES...........

 Archimedes of Syracuse (Greek: Ἀρχιμήδης; c. 287 BC – c. 212 BC) was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an explanation of the principle of the lever. He is credited with designing innovative machines, including siege engines and the screw pump that bears his name. Modern experiments have tested claims that Archimedes designed machines capable of lifting attacking ships out of the water and setting ships on fire using an array of mirrors.

Archimedes is generally considered to be the greatest mathematician of antiquity and one of the greatest of all time. He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of pi. He also defined thArchimedes died during the Siege of Syracuse when he was killed by a Roman soldier despite orders that he should not be harmed. Cicero describes visiting the tomb of Archimedes, which was surmounted by a sphere inscribed within a cylinder. Archimedes had proven that the sphere has two thirds of the volume and surface area of the cylinder (including the bases of the latter), and regarded this as the greatest of his mathematical achievements.

 Archimedes' principle

 The most widely known anecdote about Archimedes tells of how he invented a method for determining the volume of an object with an irregular shape. According to Vitruvius, a votive crown for a temple had been made for King Hiero II, who had supplied the pure gold to be used, and Archimedes was asked to determine whether some silver had been substituted by the dishonest goldsmith. Archimedes had to solve the problem without damaging the crown, so he could not melt it down into a regularly shaped body in order to calculate its density. While taking a bath, he noticed that the level of the water in the tub rose as he got in, and realized that this effect could be used to determine the volume of the crown. For practical purposes water is incompressible, so the submerged crown would displace an amount of water equal to its own volume. By dividing the mass of the crown by the volume of water displaced, the density of the crown could be obtained. This density would be lower than that of gold if cheaper and less dense metals had been added. Archimedes then took to the streets naked, so excited by his discovery that he had forgotten to dress, crying "Eureka!" (Greek: "εὕρηκα!" meaning "I have found it!"). The test was conducted successfully, proving that silver had indeed been mixed in.

The story of the golden crown does not appear in the known works of Archimedes. Moreover, the practicality of the method it describes has been called into question, due to the extreme accuracy with which one would have to measure the water displacement. Archimedes may have instead sought a solution that applied the principle known in hydrostatics as Archimedes' principle, which he describes in his treatise On Floating Bodies. This principle states that a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces. Using this principle, it would have been possible to compare the density of the golden crown to that of solid gold by balancing the crown on a scale with a gold reference sample, then immersing the apparatus in water. The difference in density between the two samples would cause the scale to tip accordingly. Galileo considered it "probable that this method is the same that Archimedes followed, since, besides being very accurate, it is based on demonstrations found by Archimedes himself." In a 12th-century text titled Mappae clavicula there are instructions on how to perform the weighing in the water in order to calculate the percentage of silver used, and thus solve the problem. The Latin poem Carmen de ponderibus et mensuris of the 4th or 5th century describes the use of a hydrostatic balance to solve the problem of the crown, and attributes the method to Archimedes.e spiral bearing his name, formulae for the volumes of surfaces of revolution and an ingenious system for expressing very large numbers.

BY-ASISH E.S