Madhavan (മാധവന്) of Sangamagramam (1350–1425) was a prominent mathematicianastronomer 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 limitpassage to infinity, which is the kernel of modern classical analysis.^{[1]} He is considered as one of the greatest mathematicianastronomers 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 southwestern 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 southwestern 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¤¿) (14441544), from Kerala, was a major mathematician and astronomer. ...
Jyesthadeva (15001575), born in Kerala, was a major mathematician, and author of the 1501 Yuktibhasa, which was a survey of Kerala mathematics and astronomy that was unique at the time for its exacting proofs of the theorems it presented. ...
Contributions
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 Circlequestionred. ...
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. ...
Among his many contributions, he discovered the infinite series for the trigonometric functions of sine, cosine, tangent and arctangent, and many methods for calculating the circumference of a circle. One of Madhava's series is known from the text Yuktibhasa, which contains the derivation and proof of the power series for inverse tangent, discovered by Madhava.^{[3]} In the text, Jyesthadeva describes Madhava's series in the following manner: Image File history File links No higher resolution available. ...
Image File history File links No higher resolution available. ...
In trigonometry, the law of sines (or sine law) is a statement about arbitrary triangles in the plane. ...
Yuktibhasa (Malayalam:à´¯àµà´•àµà´¤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. ...
In mathematics, a series is a sum of a sequence of terms. ...
In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...
As the degree of the Taylor series rises, it approaches the correct function. ...
As the degree of the taylor series rises, it approaches the correct function. ...
In mathematics, a Fourier series of a periodic function, named in honor of Joseph Fourier (17681830), represents the function as a sum of periodic functions of the form where e is Eulers number and i the imaginary unit. ...
All of the trigonometric functions of an angle Î¸ can be constructed geometrically in terms of a unit circle centered at O. Trigonometric functions: , , , , , In mathematics, the trigonometric functions (also called circular functions) are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other...
In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...
In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...
In mathematics, the word tangent has two distinct but etymologicallyrelated meanings: one in geometry and one in trigonometry. ...
In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...
The circumference is the distance around a closed curve. ...
Circle illustration This article is about the shape and mathematical concept of circle. ...
Yuktibhasa (Malayalam:à´¯àµà´•àµà´¤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. ...
In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...
In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. ...
Jyesthadeva (15001575), born in Kerala, was a major mathematician, and author of the 1501 Yuktibhasa, which was a survey of Kerala mathematics and astronomy that was unique at the time for its exacting proofs of the theorems it presented. ...
“  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 which further yields the theorem 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 MadhavaGregory 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. ...
Lowercase pi The mathematical constant Ï€ is a real number which may be defined as the ratio of a circles circumference (Greek Ï€ÎµÏÎ¹Ï†ÎÏÎµÎ¹Î±, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. ...
which he obtained from the power series expansion of the arctangent 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). Lowercase pi The mathematical constant Ï€ is a real number which may be defined as the ratio of a circles circumference (Greek Ï€ÎµÏÎ¹Ï†ÎÏÎµÎ¹Î±, 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
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

in the infinite series expansion of 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:
 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 nonnegative 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. ...
 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). ...
 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 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 nonzero polynomial equation with integer (or, equivalently, rational) coefficients. ...
Madhava laid the foundations for the development of calculus, including differential calculus and integral calculus, which were further developed by his successors at the Kerala School.^{[5]}^{[6]}^{[4]}^{[7]} (It should be noted that Archimedes also contributed to integral calculus, though not to differential calculus.) Madhava also extended some results found in earlier works, including those of Bhaskara. Calculus [from Latin, literally chalk pebble (used in reckoning)] is a major area in mathematics, with applications in science, engineering, business, and medicine. ...
In mathematics, a derivative is the rate of change of a quantity. ...
In calculus, the integral of a function is an extension of the concept of a sum. ...
Calculus [from Latin, literally chalk pebble (used in reckoning)] is a major area in mathematics, with applications in science, engineering, business, and medicine. ...
Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ...
This article deals with the concept of an integral in calculus. ...
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. ...
Archimedes (Greek: ; c. ...
Bhaskara (11141185), also known as Bhaskara II and Bhaskara AchÄrya (Bhaskara the teacher), was an Indian mathematicianastronomer. ...
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 
The Kerala School was a school of mathematics and astronomy founded by Madhava in Kerala (in South India) which included as its prominent members Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. It flourished between the 14th and 16th centuries and has its intellectual roots with Aryabhatta who lived in the 5th century. The lineage continues down to modern times but the original research seems to have ended with Narayana Bhattathiri (15591632). These astronomers, in attempting to solve problems, invented revolutionary ideas of calculus. These discoveries included the theory of infinite series, tests of convergence (often attributed to Cauchy), differentiation, term by term integration, iterative methods for solutions of nonlinear equations, and the theory that the area under a curve is its integral. They achieved most of these results up to several centuries before European mathematicians. 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. ...
(IPA: ; , Written as àµ‡à´•à´°à´³à´‚ in the native language Malayalam) is a state on the Western Coast of southwestern India. ...
South India is a linguisticcultural region of India that comprises the four Indian states of Andhra Pradesh, Karnataka, Kerala and Tamil Nadu and the Union Territory of Pondicherry, whose inhabitants are collectively referred to as South Indians. ...
Parameshvara (परमेश्वर) (13601425) was a major mathematician of the Kerala school. ...
Neelakanta Somayaji (14441545) was a member of the Kerala school of astronomy and mathematics and was a student of Damodara. ...
Jyestadeva (15001610), was an astronomer of the Kerala school founded by Madhava of Sangamagrama and a student of Damodara. ...
Achyuta Pisharati (1550â€“1621) was a renowned Sanskrit grammarian, astrologer and mathematician of his time. ...
Melpathur Narayana Bhattathiri (15591632), third student of Achyuta Pisharati, was a member of Madhava of Sangamagramas school of Astronomy and Mathematics. ...
This 14thcentury statue from south India depicts the gods Shiva (on the left) and Uma (on the right). ...
(15th century  16th century  17th century  more centuries) As a means of recording the passage of time, the 16th century was that century which lasted from 1501 to 1600. ...
Aryabhata (आर्यभट) (Āryabhaṭa) is the first of the great astronomers of the classical age of India. ...
Europe in 450 The 5th century is the period from 401  500 in accordance with the Julian calendar in the Christian Era. ...
Melpathur Narayana Bhattathiri (15591632), third student of Achyuta Pisharati, was a member of Madhava of Sangamagramas school of Astronomy and Mathematics. ...
January 15  Elizabeth I of England is crowned in Westminster Abbey. ...
See also: 1632 (novel) Events February 22  Galileos Dialogue Concerning the Two Chief World Systems is published July 23  300 colonists for New France depart Dieppe November 8  Wladyslaw IV Waza elected king of the PolishLithuanian Commonwealth after Zygmunt III Waza death November 16  Battle of LÃ¼tzen...
Calculus [from Latin, literally chalk pebble (used in reckoning)] is a major area in mathematics, with applications in science, engineering, business, and medicine. ...
As the degree of the Taylor series rises, it approaches the correct function. ...
In the absence of a more specific context, convergence denotes the approach toward a definite value, as time goes on; or to a definite point, a common view or opinion, or toward a fixed or equilibrium state. ...
Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ...
Differentiation can mean the following: In biology: cellular differentiation; evolutionary differentiation; In mathematics: see: derivative In cosmogony: planetary differentiation Differentiation (geology); Differentiation (logic); Differentiation (marketing). ...
In calculus, the integral of a function is an extension of the concept of a sum. ...
An iterative method attempts to solve a problem (for example an equation or system of equations) by finding successive approximations to the solution starting from an initial guess. ...
To do: 20th century mathematics chaos theory, fractals Lyapunov stability and nonlinear control systems nonlinear video editing See also: Aleksandr Mikhailovich Lyapunov Dynamical system External links http://www. ...
In calculus, the integral of a function is an extension of the concept of a sum. ...
Jyeshtadeva consolidated the Kerala School's discoveries in the Yuktibhasa, the world's first calculus text.^{[5]}^{[6]}^{[4]}^{[7]} Yuktibhasa (Malayalam:à´¯àµà´•àµà´¤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 southwestern 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 (15591632), third student of Achyuta Pisharati, was a member of Madhava of Sangamagramas school of Astronomy and Mathematics. ...
References For the Manfred Mann album, see 2006 (album). ...
August 12 is the 224th day of the year (225th in leap years) in the Gregorian Calendar. ...
For the Manfred Mann album, see 2006 (album). ...
July 9 is the 190th day of the year (191st in leap years) in the Gregorian Calendar, with 175 days remaining. ...
For the Manfred Mann album, see 2006 (album). ...
July 9 is the 190th day of the year (191st in leap years) in the Gregorian Calendar, with 175 days remaining. ...
For the Manfred Mann album, see 2006 (album). ...
July 9 is the 190th day of the year (191st in leap years) in the Gregorian Calendar, with 175 days remaining. ...
For the Manfred Mann album, see 2006 (album). ...
July 9 is the 190th day of the year (191st in leap years) in the Gregorian Calendar, with 175 days remaining. ...
For the Manfred Mann album, see 2006 (album). ...
July 7 is the 188th day of the year (189th in leap years) in the Gregorian Calendar, with 177 days remaining. ...
Bibliography  George Gheverghese Joseph. The Crest of the Peacock: NonEuropean Roots of Mathematics, 2nd Edition. Penguin Books, 2000.
 T. Hayashi, T. Kusuba and M. Yano. 'The correction of the Madhava series for the circumference of a circle', Centaurus 33 (pages 149174). 1990.
 C. T. Rajagopal and M. S. Rangachari. 'On an untapped source of medieval Keralese mathematics', Archive for History of Exact Sciences 18 (pages 89102). 1978.
 O'Connor, John J., and Edmund F. Robertson. "Madhava of Sangamagrama". MacTutor History of Mathematics archive. St Andrews University, 2000.
 Ian Pearce. Madhava of Sangamagrama at the MacTutor archive. St Andrews University, 2002.
 Dr. Sarada Rajeev. Neither Newton nor Leibnitz  The PreHistory of Calculus and Celestial Mechanics in Medieval Kerala. University of Rochester, 2005
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 14101413 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 14101413 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. ...
See also 