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Born 1350
Kerala, India
Died 1425

Madhavan (മാധവന്) of Sangamagramam (13501425) was a prominent mathematician-astronomer from Kerala, India. He was the founder of the Kerala School of Mathematics and is considered the founder of mathematical analysis for having taken the decisive step from the finite procedures of ancient mathematics to treat their limit-passage to infinity, which is the kernel of modern classical analysis.[1] He is considered as one of the greatest mathematician-astronomers of the Middle Ages due to his important contributions to the fields of mathematical analysis, infinite series, calculus, trigonometry, geometry and algebra. Events 29 August - An English fleet personally commanded by King Edward III defeats a Spanish fleet in the battle of Les Espagnols sur Mer. ...   (IPA: ; , Written as àµ‡à´•à´°à´³à´‚ in the native language Malayalam) is a state on the Western Coast of south-western India. ... Events Foundation of the Katholieke Universiteit Leuven, Belgium Births John II, Duke of Lorraine (died 1470) Edmund Sutton, English nobleman (died 1483) Deaths January 18 - Edmund Mortimer, 5th Earl of March, English politician (born 1391) March 17 - Ashikaga Yoshikazu, Japanese shogun (born 1407) May 24 - Murdoch Stewart, 2nd Duke of... Events 29 August - An English fleet personally commanded by King Edward III defeats a Spanish fleet in the battle of Les Espagnols sur Mer. ... Events Foundation of the Katholieke Universiteit Leuven, Belgium Births John II, Duke of Lorraine (died 1470) Edmund Sutton, English nobleman (died 1483) Deaths January 18 - Edmund Mortimer, 5th Earl of March, English politician (born 1391) March 17 - Ashikaga Yoshikazu, Japanese shogun (born 1407) May 24 - Murdoch Stewart, 2nd Duke of... Leonhard Euler is considered by many to be one of the greatest mathematicians of all time A mathematician is the person whose primary area of study and research is the field of mathematics. ... An astronomer or astrophysicist is a person whose area of interest is astronomy or astrophysics. ...   (IPA: ; , Written as àµ‡à´•à´°à´³à´‚ in the native language Malayalam) is a state on the Western Coast of south-western India. ... The Kerala School was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, South India which included as its prominent members Parameshvara, Nilakantha Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ... Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ... In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ... This article or section is not written in the formal tone expected of an encyclopedia article. ... The Middle Ages formed the middle period in a traditional schematic division of European history into three ages: the classical civilization of Antiquity, the Middle Ages, and modern times, beginning with the Renaissance. ... In mathematics, a series is a sum of a sequence of terms. ... Calculus [from Latin, literally chalk pebble (used in reckoning)] is a major area in mathematics, with applications in science, engineering, business, and medicine. ... Wikibooks has a book on the topic of Trigonometry Trigonometry (from the Greek Trigona = three angles and metron = measure[1]) is a branch of mathematics which deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees (right angled triangles). ... Table of Geometry, from the 1728 Cyclopaedia. ... Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ...

Unfortunately, most of Madhava's original works have been lost in course of time, as they were written primarily on perishable material. However his works have been detailed by later scholars of the Kerala School, primarily Nilakantha Somayaji and Jyesthadeva.[2] Nilakantha Somayaji (à¤¨à¥€à¤²à¤•à¤£à¥à¤  à¤¸à¥‹à¤®à¤¯à¤¾à¤œà¤¿) (1444-1544), from Kerala, was a major mathematician and astronomer. ... Jyesthadeva (1500-1575), born in Kerala, was a major mathematician, and author of the 1501 Yukti-bhasa, which was a survey of Kerala mathematics and astronomy that was unique at the time for its exacting proofs of the theorems it presented. ...

Perhaps Madhava's most significant contribution was in moving on from the finite procedures of ancient mathematics to 'treat their limit passage to infinity', which is considered to be the essence of modern classical analysis, and thus he is considered the founder of mathematical analysis. In particular, Madhava invented the fundamental ideas of: Image File history File links Circle-question-red. ... Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ...

Explanation of the sine rule in Yuktibhasa

 “ 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 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. ”

This yields $rtheta={frac {r sin theta }{cos theta }}-(1/3),r,{frac { left(sin theta right) ^ {3}}{ left(cos theta right) ^{3}}}+(1/5),r,{frac { left(sin theta right) ^{5}}{ left(cos theta right) ^{5}}}-(1/7),r,{frac { left(sin theta right) ^{7}}{ left(cos theta right) ^{ 7}}} + ...$

which further yields the theorem

$theta = tan theta - (1/3) tan^3 theta + (1/5) tan^5 theta - ldots$

popularly attributed to James Gregory, who discovered it three centuries after Madhava. This series was traditionally known as the Gregory series but scholars have recently begun referring to it as the Madhava-Gregory series, in recognition of Madhava's work.[4] James Gregory James Gregory (November 1638 â€“ October 1675), was a Scottish mathematician and astronomer. ...

Madhava also found the infinite series expansion of π: In mathematics, a series is often represented as the sum of a sequence of terms. ... Lower-case pi The mathematical constant Ï€ is a real number which may be defined as the ratio of a circles circumference (Greek Ï€ÎµÏÎ¹Ï†Î­ÏÎµÎ¹Î±, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. ...

$frac{pi}{4} = 1 - frac{1}{3} + frac{1}{5} - frac{1}{7} + cdots + frac{(-1)^n}{2n + 1} + cdots$

which he obtained from the power series expansion of the arc-tangent function.

Using a rational approximation of this series, he gave values of the number π as 3.14159265359 - correct to 11 decimals; and as 3.1415926535898 - correct to 13 decimals. These were the most accurate approximations of π given since the 5th century.(see History of Pi). Lower-case pi The mathematical constant Ï€ is a real number which may be defined as the ratio of a circles circumference (Greek Ï€ÎµÏÎ¹Ï†Î­ÏÎµÎ¹Î±, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. ... Europe in 450 The 5th century is the period from 401 - 500 in accordance with the Julian calendar in the Christian Era. ... When a circles diameter is 1, its circumference is Ï€. The mathematical constant Ï€ is an irrational real number, approximately equal to 3. ...

He gave two methods for computing the value of π.

• One of these methods is to obtain a rapidly converging series by transforming the original infinite series of π. By doing so, he obtained the infinite series
$pi = sqrt{12}left(1-{1over 3cdot3}+{1over5cdot 3^2}-{1over7cdot 3^3}+cdotsright)$

and used the first 21 terms to compute an approximation of π correct to 11 decimal places as 3.14159265359.

• The other method was to add a remainder term to the original series of π. The remainder term was used
$frac{n^2 + 1}{4n^3 + 5n}$

in the infinite series expansion of $frac{pi}{4}$ to improve the approximation of π to 13 decimal places of accuracy when n = 76.

Madhava was also responsible for many other original discoveries, including:

Mathematical analysis

• Trigonometric series for tangent and arctangent functions
• Additional Taylor series approximations of sine and cosine functions
• Investigations into other series for arclengths and the associated approximations to rational fractions of π
• Methods of polynomial expansion.
• Tests of convergence of infinite series.
• Analysis of infinite continued fractions.

Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ... In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. ... In mathematics, the integral test for convergence is a method used to test infinite series of non-negative terms for convergence. ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ...

Trigonometry

• The analysis of trigonometric functions (as described above).
• Sine table to 12 decimal places of accuracy.
• Cosine table to 9 decimal places of accuracy.

Wikibooks has a book on the topic of Trigonometry Trigonometry (from the Greek Trigona = three angles and metron = measure[1]) is a branch of mathematics which deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees (right angled triangles). ...

Geometry

• The analysis of the circle (as described above).
• Many methods for calculating the circumference of a circle.
• Computation of π correct to 13.00 decimal places.

Table of Geometry, from the 1728 Cyclopaedia. ...

Algebra

Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ... A transcendental function is a function which does not satisfy a polynomial equation whose coefficients are themselves polynomials. ... The word iteration is sometimes used in everyday English with a meaning virtually identical to repetition. ... In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ...

Calculus

Some scholars have suggested that Madhava's work was transmitted to Europe via traders and Jesuit missionaries, and as a result, had an influence on later European developments in analysis and calculus. (See Possible transmission of Kerala mathematics to Europe for further information.) The Society of Jesus (Latin: Societas Iesu), commonly known as the Jesuits, is a Roman Catholic religious order. ... The Kerala School was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, South India which included as its prominent members Parameshvara, Nilakantha Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ...

Kerala School of Astronomy and Mathematics

Main article: Kerala School

Jyeshtadeva consolidated the Kerala School's discoveries in the Yuktibhasa, the world's first calculus text.[5][6][4][7] Yuktibhasa (Malayalam:à´¯àµà´•àµà´¤à´¿à´­à´¾à´· ; meaning â€” rationale language ) also known as Ganita Yuktibhasa (compendium of astronomical rationale) is a major treatise on Mathematics and Astronomy, written by Indian astronomer Jyesthadeva of the Kerala School of Mathematics in AD 1530. ... Calculus [from Latin, literally chalk pebble (used in reckoning)] is a major area in mathematics, with applications in science, engineering, business, and medicine. ...

The Kerala School also contributed much to linguistics. The ayurvedic and poetic traditions of Kerala were founded by this school. The famous poem, Narayaneeyam, was composed by Narayana Bhattathiri. Shirodhara, one of the techniques of Ayurveda Ayurveda (Devanagari: ) or Ayurvedic medicine is a practice in use primarily in the Indian subcontinent, which advocates argue assists with health and healing. ...   (IPA: ; , Written as àµ‡à´•à´°à´³à´‚ in the native language Malayalam) is a state on the Western Coast of south-western India. ... Narayaneeyam is a devotional Sanskrit work, in the form of a poetical hymn, consisting of 1034 verses (called slokas in Sanskrit). ... Melpathur Narayana Bhattathiri (1559-1632), third student of Achyuta Pisharati, was a member of Madhava of Sangamagramas school of Astronomy and Mathematics. ...

References

1. ^ Madhava. Biography of Madhava. School of Mathematics and Statistics University of St Andrews, Scotland. Retrieved on 2006-08-12.
2. ^ A book on rationales in Indian Mathematics and Astronomy — An analytic appraisal. Yuktibhasa of Jyesthadeva. K V Sharma & S Hariharan. Retrieved on 2006-07-09.
3. ^ The Kerala School, European Mathematics and Navigation. Indian Mathemematics. D.P. Agrawal — Infinity Foundation. Retrieved on 2006-07-09.
4. ^ a b c Science and technology in free India. Government of Kerala — Kerala Call, September 2004. Prof.C.G.Ramachandran Nair. Retrieved on 2006-07-09.
5. ^ a b Neither Newton nor Leibniz - The Pre-History of Calculus and Celestial Mechanics in Medieval Kerala. MAT 314. Canisius College. Retrieved on 2006-07-09.
6. ^ a b An overview of Indian mathematics. Indian Maths. School of Mathematics and Statistics University of St Andrews, Scotland. Retrieved on 2006-07-07.
7. ^ a b Charles Whish (1835). Transactions of the Royal Asiatic Society of Great Britain and Ireland.

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Bibliography

Penguin Books is a British publisher founded in 1935 by Allen Lane. ... Centaurus, a scientific journal devoted to the history of mathematics, science, and technology. ... University of St Andrews The University of St Andrews was founded between 1410-1413 and is the oldest university in Scotland and the third oldest in the United Kingdom. ... The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ... University of St Andrews The University of St Andrews was founded between 1410-1413 and is the oldest university in Scotland and the third oldest in the United Kingdom. ... The University of Rochester is a private, coeducational and nonsectarian research institution located in Rochester, New York. ...

Results from FactBites:

 Madhava biography (894 words) Madhava of Sangamagramma was born near Cochin on the coast in the Kerala state in southwestern India. Madhava discovered the series equivalent to the Maclaurin expansions of sin x, cos x, and arctan x around 1400, which is over two hundred years before they were rediscovered in Europe. Madhava also gave a table of almost accurate values of half-sine chords for twenty-four arcs drawn at equal intervals in a quarter of a given circle.
 Melpathur Narayana Bhattathiri (1755 words) One of Madhava's series is known from the text Yuktibhasa which describes - 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. Madhava of Sangamagrama, Madhava of Sangamagrama - Contributions, Madhava of Sangamagrama - Kerala School of Astronomy and Mathematics
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