Madhava's Approximation of π

Madhava the great Mathematician!

Madhava, an Indian mathematician and astronomer from Thrissur District, Kerala, India was born in c.1340 BC. He did his basic schooling from Kerala school of mathematics and astronomy. His major contributions are in calculus, geometry, infinite series, algebra, and trigonometry. He was the first mathematician who has applied the endless infinite series in trigonometric functions like sin, cosine, tangent even before Leibnitz did.

From a very young age, he was interested in mathematics. He wrote more than 10 books on mathematics covering multiple mathematical areas. Though most of his books are lost, the books which are found have been used by today’s mathematicians and researchers to research mathematics.

Madhava of Sangamagrama Books

Most of Madhava Mathematical works have been lost, but whatever remained changed the phase of mathematics, Madhava of has written the following books:

• Aganita-grahacara
• Chandravakyani
• Golavada
• Lagnaprakarana
• Madhyamanayanaprakara

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Madhava's Approximation of Pi

He realized that, by successively adding and subtracting different odd number fractions to infinity, he could home in on an exact formula for π (this was two centuries before Lebnitz was to come to the same conclusion in Europe). Through his application of this series, Madhava obtained a value for π correct to an astonishing 13 decimal places.

He went on to use the same mathematics to obtain infinite series expressions for the sine formula, which could then be used to calculate the sine of any angle to any degree of accuracy, as well as for other trigonometric functions like cosine, tangent and arctangent. Perhaps even more remarkable, though, is that he also gave estimates of the error term or correction term, implying that he quite understood the limit nature of the infinite series.

Madhava’s use of infinite series to approximate the trigonometric functions, were further developed by his students and effectively laid the foundations of calculus and analysis, and further developed an early form of integration for simple functions.

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Here is one of the famous quote by Madhava:

The first term is the product of the given sine and radius of the desired arc divided by the cosine of the arc. The succeeding terms are obtained by the process of iteration when the first term is repeatedly multiplied by the square of the sine and divided by the square of the cosine. All the terms are then divided by the odd numbers 1, 3, 5 … The arc is obtained by adding and subtracting, respectively, the terms of odd rank and those of even rank. It is laid down that the sine of the arc or that of its complement whichever is the smaller should be taken here as the given sine. Otherwise, the terms obtained by this above iteration will not tend to the vanishing magnitude.        

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