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Encyclopedia > Madhava of Sangamagrama
Madhavan
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 േകരളം 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 േകരളം 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 (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. ...

Contents

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 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
Explanation of the sine rule in Yuktibhasa

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:യുക്തിഭാഷ ; meaning — 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 (1768-1830), 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 θ 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 etymologically-related 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:യുക്തിഭാഷ ; meaning — 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 (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. ...

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 – 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 περιφέρεια, 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 περιφέρεια, 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

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 (1114-1185), also known as Bhaskara II and Bhaskara Achārya (Bhaskara the teacher), was an Indian mathematician-astronomer. ...


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

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. 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 േകരളം in the native language Malayalam) is a state on the Western Coast of south-western India. ... South India is a linguistic-cultural 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) (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. ... Melpathur Narayana Bhattathiri (1559-1632), third student of Achyuta Pisharati, was a member of Madhava of Sangamagramas school of Astronomy and Mathematics. ... 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. ... 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 (1559-1632), 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 Polish-Lithuanian Commonwealth after Zygmunt III Waza death November 16 - Battle of Lützen... 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 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 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:യുക്തിഭാഷ ; meaning — 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 േകരളം 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. ...

See also


  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
  More results at FactBites »

 
 

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