Madhava (माधव) of Sangamagrama (1350-1425) was a major mathematician from Kerala, in South India. Madhava was the founder of the Kerala School, and is considered the father 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. He is also considered one of the greatest mathematician-astronomers of the Middle Ages, due to his important contributions to the fields of mathematical analysis, infinite series, calculus, and trignonometry. 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... Kerala (IPA: ; Malayalam: േകരളം — ) is a state on the tropical Malabar Coast of southwestern India. ... A map of Southern India, its rivers, regions and water bodies. ... Madhava founded the Kerala school, which included as its prominent members Parameswara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ... Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ... A limit can be: Limit (mathematics), including: Limit of a function Limit of a sequence Limit point Net (topology) Limit (category theory) A constraint (mathematical, physical, economical, legal, etc. ... Infinity refers to several distinct concepts which arise in theology, philosophy, mathematics and everyday life. ... 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. ...

Sadly, all of his mathematical works are lost, although it is possible that extant work may yet be 'unearthed'. It is vaguely possible that he may have written Karana Paddhati a work written sometime between 1375 and 1475, but this is only speculative. All we know of Madhava comes from works of later scholars, primarily Nilakantha Somayaji and Jyesthadeva. Events October 24 - Valdemar IV of Denmark dies and is succeeded by his grandson Olaf III of Denmark. ... Events August 29 - Treaty of Picquigny ends a brief war between France and England. ... Nilakantha Somayaji (नीलकण्ठ सोमयाजि) (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. ...

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: Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ...

• Infinite series expansions of functions.

• Power series.

• Taylor series.

• Maclaurin series.

• Rational approximations of infinite series.

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 describes - 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, 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 trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... This article is about the mathematical concept of tangent. For other meanings, see tangent (disambiguation). ... 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. ... A circle, in Euclidean geometry, is the set of all points at a fixed distance, called the radius, from a fixed point, the centre. ...

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, three centuries before him. 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. James Gregory James Gregory (November 1638 – October 1675), was a Scottish mathematician and astronomer. ...

Using a rational approximation of the Madhava-Gregory series, he gave a value of the number π as 3.14159265359 - correct to 11 decimals, the most accurate approximation of π after almost a thousand years. Lower-case pi The mathematical constant Ï€ is a real number which may be defined as the ratio of a circles circumference (Greek περιφέρεια, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. ...

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

• Taylor series of the sine and cosine functions (Madhava-Newton power series).

• Taylor series of the tangent function.

• Taylor series of the arctangent function (Madhava-Gregory series).

• Second-order Taylor series approximations of the sine and cosine functions.

• Third-order Taylor series approximation of the sine function.

• Power series of π (usually attributed to Leibniz).

• Power series of π/4 (Euler's series).

• Power series for any angle θ (equivalent to the Gregory series).

• Power series for the radius of a circle.

• Power series for the diameter of a circle.

• Power series for the circumference of a circle.

• Many methods for calculating the circumference of a circle.

• Infinite continued fractions.

• Integration.

• Term by term integration.

• The solution of transcendental equations by iteration

• Approximation of transcendental numbers by continued fractions.

• Tests of convergence of infinite series.

Madhava laid the foundations for the development of calculus, which was then further developed by his successors at the Kerala School. Sir Isaac Newton, PRS, (4 January [O.S. 25 December 1642] 1643 – 31 March [O.S. 20 March] 1727) was an English physicist, mathematician, astronomer, alchemist, inventor and natural philosopher who is generally regarded as one of the most influential scientists in history. ... Lower-case pi The mathematical constant π is a real number which may be defined as the ratio of a circles circumference (Greek περιφέρεια, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. ... Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ... Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ... Note: A theta probe is a device for measuring soil moisture. ... In classical geometry, a radius of a circle or sphere is any line segment with one endpoint on the circle (i. ... For the geometric term, see diameter. ... The circumference is the distance around a closed curve. ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ... Integration may be any of the following: In the most general sense, integration may be any bringing together of things: the integration of two or more economies, cultures, religions (usually called syncretism), etc. ... A transcendental function is a function which does not satisfy a polynomial equation whose coefficients are themselves polynomials. ... Iteration is the repetition of a process, typically within a computer program. ... In mathematics, a transcendental number is any real number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ... The integral test for convergence is a method used to test infinite series of nonnegative terms for convergence. ... Integral and differential calculus is a central branch of mathematics, developed from algebra and geometry. ... Madhava founded the Kerala school, which included as its prominent members Parameswara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ...

He also extended some results found in earlier works, including those of Bhaskara. Bhāskara (1114-1185), also called Bhāskara II and BhāskarāAchārya (Bhaskara the teacher) was an Indian mathematician. ...

Kerala School of Astronomy and Mathematics

Main article: Kerala School

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 (1559-1632). 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 non-linear 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. Madhava founded the Kerala school, which included as its prominent members Parameswara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ... A map of Southern India, its rivers, regions and water bodies. ... Parameshvara (परमेश्वर) (1360-1425) was a major mathematician of the Kerala school. ... Neelakanta Somayaji (1444-1545) was a member of the Kerala school of astronomy and mathematics and was a student of Damodara. ... Jyestadeva (1500-1610), was an astronomer of the Kerala school founded by Madhava of Sangamagrama and a student of Damodara. ... Achyuta Pisharati (1550–1621) was a renowned Sanskrit grammarian, astrologer and mathematician of his time. ... This 14th-century 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. ... // Overview Events Romulus Augustus, Last Western Roman Emperor 410: Rome sacked by Visigoths 452: Pope Leo I allegedly meets personally with Attila the Hun and convinces him not to sack Rome 439: Vandals conquer Carthage At some point after 440, the Anglo-Saxons settle in Britain. ... Events 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 Polish-Lithuanian Commonwealth after Zygmunt III Waza death November 16 - Battle of Lützen... Integral and differential calculus is a central branch of mathematics, developed from algebra and geometry. ... 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). ... Integration may be any of the following: In the most general sense, integration may be any bringing together of things: the integration of two or more economies, cultures, religions (usually called syncretism), etc. ... 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 non-linear control systems non-linear video editing See also: Aleksandr Mikhailovich Lyapunov Dynamical system External links http://www. ... In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ...

Jyeshtadeva consolidated the Kerala School's discoveries in the Yuktibhasa, the world's first calculus text.

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. Ayurveda (आयुर्वेद Sanskrit: ayu—life; veda—knowledge of) or ayurvedic medicine is a comprehensive system of medicine, first described by Charaka around the beginning of the Common Era, and based on a holistic approach rooted in earlier Vedic culture. ... Kerala (IPA: ; Malayalam: േകരളം — ) is a state on the tropical Malabar 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). ...

See also

• Indian mathematics
• List of Indian mathematicians
• Kerala School
• Mathematical analysis
• Calculus
• History of calculus
• Taylor series
• Parameswara
• Neelakanta Somayaji
• Jyeshtadeva
• Achyuta Pisharati
• Melpathur Narayana Bhattathiri

The chronology of Indian mathematics spans from the Indus Valley civilization (3300-1500 BC) and Vedic civilization (1500-500 BC) to modern India. ... The chronology of Indian mathematics spans from the Indus valley civilization and the Vedas to Modern times. ... Madhava founded the Kerala school, which included as its prominent members Parameswara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ... Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ... Integral and differential calculus is a central branch of mathematics, developed from algebra and geometry. ... Though the origins of integral calculus are generally regarded as going no farther back than to the ancient Greeks, there is evidence that the ancient Egyptians may have harbored such knowledge amongst themselves as well (see Moscow and Rhind Mathematical Papyri). ... As the degree of the Taylor series rises, it approaches the correct function. ... Parameswara (1344-1424) was a Malay Srivijayan prince who escaped the Majapahit capture of Palembang, on Sumatra in the 1390s. ... Neelakanta Somayaji (1444-1545) was a member of the Kerala school of astronomy and mathematics and was a student of Damodara. ... Jyestadeva (1500-1610), was an astronomer of the Kerala school founded by Madhava of Sangamagrama and a student of Damodara. ... Achyuta Pisharati (1550–1621) was a renowned Sanskrit grammarian, astrologer and mathematician of his time. ...

External links

• Madhava's mathematical contributions by J J O'Connor and E F Robertson
• Neither Newton nor Leibnitz - The Pre-History of Calculus and Celestial Mechanics in Medieval Kerala
• Madhava's Contributions by Ian Pearce