CONTRIBUTIONS OF SANGAMGRAMA MADHAVA.pptx

Contributions of Sangamagrama Madhava

1. HARITHA BS MATHEMATICS SEMESTER 1 MOUNT TABOR TRAINING COLLEGE ,PATHANAPURAM CONTRIBUTIONS OF SANGAMGRAMA MADHAVA

2. INTRODUCTION Irinnattapilli Madhavan Nampudiri known as Madhava of Sangamagrama . [c.1340-c.1425] born near Cochin on the coast in Kerala. Mathematician and Astronomer. Started a school in Kerala called “The Kerala School of Mathematics And Astronomy.” Made significant contributions in Calculus, geometry, infinite series, algebra and trigonometry.

3. First Mathematician who has applied the endless series in trigonometric functions like sin, cos and tan. Most of his books are lost, the books which are found have been used by today’s mathematicians and researchers to research Mathematics Worked as an Astronomer-Mathematician till his last breadth.

4. Aganita-grahacara Chandravakyani Golavada Lagnaprakarana Madhyamanayanaprakara Mahajyanayanaprakara Sphutacandrapti Venvaroka BOOKS

5. Infinite Series Trigonometry The Value of ∏ Calculus CONTRIBUTIONS

6. Among his many contributions, he discovered infinite series for the trigonometric functions of sine, cosine, arctangent, and many methods for calculating the circumference of a circle. In the text ‘ Yuktibhasa ’, ‘ Jyesthadeva ’ describes the series in the following manner: “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 a 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 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.” INFINITE SERIES

7. This yeilds Or equivalently, This series is known as ‘ Gregory’s series ’. It’s has been attributed to Gottfried Wilhelm Leibniz [1646-1716] and James Gregory [1638-1675]. The series was known in Kerala more than two centuries before its European discoveries were born.

8. Madhava composed an accurate table of sines . Marking a quarter circle at twenty-four equal intervals, he gave the lengths of the half-chord ( sines ) corresponding to each of them. It is believed that he may have computed these values based on the series expansions: TRIGONOMETRY

9. Madhava’s work on the value of the mathematical constant pi is cited in the ‘ Mahajyanayanaprakara ’ (“Methods for the great sines ”). While some scholars such as Sarma feel that this book may have been composed by Madhava himself, it is more likely the work of a 16 th -century successor. This text attributes most of the expansions to Madhava , and gives the following infinite series expansions of ∏, now known as the Madhava -Leibniz series: which he obtained from the power series expansion of the arctangent function. THE VALUE OF ∏ (pi)

10. He also gave a correction term R n for the error after computing the sum up to n terms, namely; where the third correction leads to highly accurate computation of ∏. It has long been speculated how Madhava found these correction terms.

11. They are the first three convergents of a finite continued fraction, which when combined with the original Madhava’s series evaluated to n terms, yields about 3n/2 correct digits:

12. The absolute value of the correction term in next higher order is He also gave a more rapidly converging series by transforming the original infinite series of ∏, obtaining the infinte series, By using the first 21 terms to compute an approximation of ∏, he obtains a value correct to 11 decimal places [3.14159265359]. The value of 3.1415926535898, correct to 13 decimals, is sometimes attributed to Madhava , but may be due to one of his followers. These were the most accurate approximations of ∏ given, since the 5 th century

13. Madhava developed the power series expansion for some trigonometry functions which were further developed by his successors at the Kerala school of astronomy and mathematics. Madhava also extended some results found in earlier works, including those of Bhaskara II. However, they did not combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, or turn calculus into the powerful problem-solving tool we have today. CALCULUS

14. The Kerala school of astronomy and mathematics flourished for at least two centuries beyond Madhava . In Jyesthadeva we find the notion of integration, termed sankalitam , (lit. collection), as in the statement: ‘ ekadyekothara pada sankalitam samam padavargathinte pakuti ’. which translates as the integral of a variable ( pada ) equals half that variable squared ( varga ), i.e. The integral of x dx is equal to x 2 /2. This is clearly a start to the process of integral calculus. A related result states that the area under a curve is its integral. Most of these results pre-date similar results in Europe by several centuries. In many senses, Jyeshthadeva’s Yuktibhasa may be considered the world’s first calculus text. KERALA SCHOOL OF ASTRONOMY AND MATHEMATICS

15. The group also did much other work in astronomy; indeed many more pages are developed to astronomical computations than are for discussing analysis related results. The Kerala school also contributed much to linguistics (the relation between language and mathematics is an ancient Indian tradition). The ayurvedic and poetic traditions of Kerala can also be traced back to this school. The famous poem, Narayaneeyam , was composed by Narayana Bhattathiri , a prominent scholar of this school.