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Biography of the Mathematician Madhava of Sangamagramma (1350 - 1425) is a well-known Indian mathematician who made great impact on the development of Calculus. He was born in 1350 in Sangamagramma in Kerala state in the southwestern part of India. Madhava invented the ideas of Taylor series, Maclaurin series, Power series, trigonometric series, infinite series, and gave formulae for? that leads to the approximation 3. 14159265359. He is also considered the founder of mathematical analysis because he undertook a decisive step from the finite procedures of ancient mathematics to treat their limit-passage to infinity, which is the kernel of modern classical analysis (Biography of Madhava, n. p. ).
The mathematician also laid the basis for further development of calculus, including integral calculus and differential calculus that were further examined and developed in his Kerala School. Madhava's innovations brought a completely new perspective on mathematics. As far as he was much more innovative than any other Indian mathematician, he is considered the greatest mathematician-astronomer of medieval India. It is accepted as correct that Madhava of Sangamagramma made the following discoveries: = tan - (tan 3) / 3 + (tan 5) / 5 -... , (analogous to Gregory series) (9 III. Madhava of Sangamagramma); r = {r (rain) / 1 (rco's) }-{r (rain) 3 / 3 (rco's) 3 }+{r (rain) 5 / 5 (rco's) 5 }-...
sin = - 3 / 3! + 5 / 5! -... , (Madhava Newton power series); cos = 1 - 2 / 2! + 4 / 4! -... , - it is necessary to remember that Indian cos = rco's, and Indian sin = rain. However, both results are often attributed to Maclaurin. ? / 4-... 1 /n (-fi (n+ 1) ), i = 1, 2, 3, and where f 1 = n/ 2, f 2 = (n/ 2) / (n 2 + 1) and f 3 = ( (n/ 2) 2 + 1) / ( (n/ 2) (n 2 + 4 + 1) ) 2 - these are the infinite series expansion of? ? This formula () was obtained from the power series expansion of the arc-tangent function. Madhava used a particular rational approximation of this series? ? d 2 d + 4 d/ (22 - 1) - 4 d/ (42 - 1) +... 4 d/ (n 2 + 1) etc. it allowed improving the approximations of?
and allowed the mathematician calculating? ? to? ? ? decimal places. Thus, Madhava calculated the value? ? as? 3. 14159265359. This value is unique to Kerala School.
In such a way, he gave two methods of calculating? ? ? The second method involved adding a remainder term to the original series of? ? ? The remainder term? was also used on the infinite series expansion of? ? ? ? to improve the approximation of? to 13 decimal places of accuracy when n = 76 (Madhava of Sangamagramma).
The next idea is attributed to Eulers series, where? +... 1 /n {-f (n+ 1) } One of his most significant approximations are: sin (x + h) sin x + (h / r ) cos x - (h 2 / 2 r 2) sin x and cos (x + h) cos x - (h / r ) sin x - (h 2 / 2 r 2) cos x. Both of them are special cases of Taylor series. Madhava of Sangamagramma was known not only as a mathematician. He was a talented astronomer and a founder of Kerala School of Astronomy and Mathematics. He was a prominent figure and a creative personality. Madhava made plenty of important discoveries in trigonometry (the analysis of trigonometric functions, sine and cosine tables, etc), in mathematical analysis (additional Taylor series approximations of cosine and sine functions, trigonometric series for arc tangent and tangent functions, methods of polynomial expansion, tests of convergence for infinite series, etc), in geometry (the analysis of the circle, computation of tt, etc), in algebra and in calculus (integration, differentiation, etc).
In such a way, we can conclude that Madhava's innovations brought a completely new perspective on mathematics. Bibliography 9 III. Madhava of Sangamagramma. 16 February 2007 < web >. Joseph, George Gheverghese. The Crest of the Peacock: Non-European Roots of Mathematics, 2 nd Edition. Penguin Books, 2000.
Madhava of Sangamagramma. 2007 February 16 < web >. Madhava. Biography of Madhava. School of Mathematics and Statistics University of St Andrews, Scotland... 16 February 2007 < web >. O'Connor, John J. , and Edmund F.
Robertson. "Madhava of Sangamagramma. " Mac Tutor History of Mathematics archive, St Andrews University, 2000.
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