Indian Contribution to Mathematics Dr Indulal G. Department of Mathematics St. Aloysius College, Edathua Alappuzha, Kerala 689573 indulalgopal@yahoo.com October 20,2010 Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Indian Mathematics OR Hindu Mathematics Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Indian Mathematics OR Hindu Mathematics Classical period (400 AD to 1200 AD) Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Indian Mathematics OR Hindu Mathematics Classical period (400 AD to 1200 AD) Ancient and medieval Indian mathematical works - In Sanskrit Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Indian Mathematics OR Hindu Mathematics Classical period (400 AD to 1200 AD) Ancient and medieval Indian mathematical works - In Sanskrit Used by upper Class of Society Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Indian Mathematics OR Hindu Mathematics Classical period (400 AD to 1200 AD) Ancient and medieval Indian mathematical works - In Sanskrit Used by upper Class of Society Not so popular as today. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India VM D 5}6"DN£ 5}6"lDN\ 5}6"FN 5}6"D]NrI N[£ 5}6":I 5}6"DFNFI 5}6"D[J FJlXQ I T[ VM D XgTL£ XgTL£ XgTL£ Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India ³¢ ÉâVHÎÆ£ ÉâVHÎßÆ¢ ÉâVHÞÆí ÉâVHÎáƺcçÄ£ ÉâVHØc ÉâVHÎÞÆÞÏ ÉâVHçÎÕÞÕÖß×cçÄ ³¢ ÖÞLߣ ÖÞLߣ ÖÞLߣ Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India ∞+∞=∞ ∞−∞=∞ Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Karl Friedrich Gauss (1777-1855) Mathematics is the Queen of all Sciences Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India ÏÅÞ Öß¶Þ ÎÏâøÞÃÞ¢ ÈÞ·ÞÃÞ¢ ÎÃçÏÞ ÏÅÞ ÄÅÞ çÕÆÞ¢·ÖÞdØñÞÃÞ¢ ·ÃßÄ¢ ÎâViÈß ØíÅßÄ¢ çÕÆ·ÃßÄ¢_ (1500 BC)Ü·ÞÇÎÙV×ß Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Rich history from 4000 BC Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Rich history from 4000 BC Orally transmitted until 500 BC Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Rich history from 4000 BC Orally transmitted until 500 BC Memorizing the text Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Rich history from 4000 BC Orally transmitted until 500 BC Memorizing the text Both orally and in manuscript form after 500 BC - Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Rich history from 4000 BC Orally transmitted until 500 BC Memorizing the text Both orally and in manuscript form after 500 BC The oldest extant mathematical document Bakhshali Manuscript, discovered in 1881 Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Important contributions by scholars like Aryabhata, Brahmagupta, and Bhaskara II. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Important contributions by scholars like Aryabhata, Brahmagupta, and Bhaskara II. Decimal number system and the Binary number system Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Important contributions by scholars like Aryabhata, Brahmagupta, and Bhaskara II. Decimal number system and the Binary number system Early contributions to the study of the concept of zero , negative numbers, arithmetic, and algebra. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Important contributions by scholars like Aryabhata, Brahmagupta, and Bhaskara II. Decimal number system and the Binary number system Early contributions to the study of the concept of zero , negative numbers, arithmetic, and algebra. Modern definitions of sine and cosine Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Fields of Indian Mathematics Arithmetic Decimal system, Negative numbers - Brahmagupta Zero - Hindu-Arabic numeral system Binary numeral system, the modern positional notation numeral system, Floating point numbers - Kerala school of astronomy and mathematics Number theory, Infinity - Yajur Veda Transfinite numbers, Irrational numbers -Shulba Sutras Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Fields of Indian Mathematics Geometry Square roots - Bakhshali approximation Cube roots - Mahavira Pythagorean triples - Sulba Sutras; Baudhayana and Apastamba Pascal’s triangle - Pingala Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Fields of Indian Mathematics Algebra Quadratic equations - Sulba Sutras, Aryabhata, and Brahmagupta Cubic equations and Quartic equations - Mahavira and Bha-skara II Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Fields of Indian Mathematics General Mathematics and Trigonometry Fibonacci numbers and Earliest forms of Morse code - Pingala Logarithms, indices - Jaina Mathematics Algorithms - Aryabhata and Brahmagupta Trigonometric functions - Surya Siddhanta and Aryabhata Trigonometric series - Madhava and Kerala school Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Prehistory Excavations at Harappa, Mohenjo-daro and other sites of the Indus Valley Civilization Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Prehistory Excavations at Harappa, Mohenjo-daro and other sites of the Indus Valley Civilization Evidence of the use of "practical mathematics". Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Prehistory Excavations at Harappa, Mohenjo-daro and other sites of the Indus Valley Civilization Evidence of the use of "practical mathematics". Bricks whose dimensions were in the proportion 4:2:1, considered favorable for the stability of a brick structure Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Prehistory Excavations at Harappa, Mohenjo-daro and other sites of the Indus Valley Civilization Evidence of the use of "practical mathematics". Bricks whose dimensions were in the proportion 4:2:1, considered favorable for the stability of a brick structure Used a standardized system of weights Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Prehistory Excavations at Harappa, Mohenjo-daro and other sites of the Indus Valley Civilization Evidence of the use of "practical mathematics". Bricks whose dimensions were in the proportion 4:2:1, considered favorable for the stability of a brick structure Used a standardized system of weights Produced weights in regular geometrical shapes. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Prehistory Excavations at Harappa, Mohenjo-daro and other sites of the Indus Valley Civilization Evidence of the use of "practical mathematics". Bricks whose dimensions were in the proportion 4:2:1, considered favorable for the stability of a brick structure Used a standardized system of weights Produced weights in regular geometrical shapes. Standardize measurement of length to a high degree of accuracy Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Samhitas and Brahmanas Evidence for the use of large numbers Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Samhitas and Brahmanas Evidence for the use of large numbers Yajurvedasam.hita- (1200-900 BCE) Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Samhitas and Brahmanas Evidence for the use of large numbers Yajurvedasam.hita- (1200-900 BCE) Numbers as high as 1012 were included in the texts Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Samhitas and Brahmanas Evidence for the use of large numbers Yajurvedasam.hita- (1200-900 BCE) Numbers as high as 1012 were included in the texts Mantra at the end of the annahoma performed during the asvamedha Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India ÖÄ¢ H undre d - 102 ØÙdØ¢ T housa nd - 103 ¥ÏáÄ¢ T e n t housa nds - 104 ÈßÏáÄ¢ One La k h - 105 dÉÏáÄ¢ T e n La k hs - 106 ¥VÌáÆ¢ One Crore - 107 ÈÏÞVÌáÆ¢ T e n Crore s - 108 ØÎádÆ¢ H undre d Crore s - 109 ÎÇc¢ T housa nd Crore s - 1 0 10 ¥L¢ T e n t housa nd Crore s - 1 0 11 ÉøÞVÇ¢ One la k h Crore s - 1 0 12 Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Sulba Su-tras( 700 - 400 BC) List rules for the construction of sacrificial fire altars Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Sulba Su-tras( 700 - 400 BC) List rules for the construction of sacrificial fire altars Sulba Su-tras contain the earliest extant verbal expression of the Pythagorean Theorem Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Sulba Su-tras( 700 - 400 BC) List rules for the construction of sacrificial fire altars Sulba Su-tras contain the earliest extant verbal expression of the Pythagorean Theorem ... The diagonal rope (aksnaya–rajju) of an oblong (rectangle) produces both which the flank (parsvamani) and the horizontal tiryanmani) ropes produce separately ...... Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Sulba Su-tras( 700 - 400 BC) List rules for the construction of sacrificial fire altars Sulba Su-tras contain the earliest extant verbal expression of the Pythagorean Theorem ... The diagonal rope (aksnaya–rajju) of an oblong (rectangle) produces both which the flank (parsvamani) and the horizontal tiryanmani) ropes produce separately ...... Contain lists of Pythagorean triples Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Sulba Su-tras( 700 - 400 BC) List rules for the construction of sacrificial fire altars Sulba Su-tras contain the earliest extant verbal expression of the Pythagorean Theorem ... The diagonal rope (aksnaya–rajju) of an oblong (rectangle) produces both which the flank (parsvamani) and the horizontal tiryanmani) ropes produce separately ...... Contain lists of Pythagorean triples Contain statements about squaring the circle and circling the square. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Baudhayana( 700 BC) Composed the Baudhayana Sulba Sutra Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Baudhayana( 700 BC) Composed the Baudhayana Sulba Sutra Contains examples of simple Pythagorean triples and Pythagorean theorem Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Baudhayana( 700 BC) Composed the Baudhayana Sulba Sutra Contains examples of simple Pythagorean triples and Pythagorean theorem Manava Sulba Sutra by Manava ( 750-650 BC) and the Apastamba Sulba Sutra by Apastamba ( 600 BC), Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Baudhayana( 700 BC) Composed the Baudhayana Sulba Sutra Contains examples of simple Pythagorean triples and Pythagorean theorem Manava Sulba Sutra by Manava ( 750-650 BC) and the Apastamba Sulba Sutra by Apastamba ( 600 BC), √ 2=1+ 1 3 + 1 3·4 − 1 3·4·34 ≈ 1.4142156 · · · Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Baudhayana( 700 BC) Composed the Baudhayana Sulba Sutra Contains examples of simple Pythagorean triples and Pythagorean theorem Manava Sulba Sutra by Manava ( 750-650 BC) and the Apastamba Sulba Sutra by Apastamba ( 600 BC), √ 2=1+ 1 3 + 1 3·4 − 1 3·4·34 ≈ 1.4142156 · · · Main objective of the Sulbasutras was to describe the constructions of altars and the geometric principles involved in them. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Panini( 500 -400 BC) Provided more than 4,000 rules that describe the Sanskrit of his day completely Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Panini( 500 -400 BC) Provided more than 4,000 rules that describe the Sanskrit of his day completely Intellectual achievements of all time. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Panini( 500 -400 BC) Provided more than 4,000 rules that describe the Sanskrit of his day completely Intellectual achievements of all time. Boolean logic of the null operator, and of context free grammars Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Panini( 500 -400 BC) Provided more than 4,000 rules that describe the Sanskrit of his day completely Intellectual achievements of all time. Boolean logic of the null operator, and of context free grammars A precursor of the Backus-Naur form (used in the description modern programming languages). Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Panini( 500 -400 BC) Provided more than 4,000 rules that describe the Sanskrit of his day completely Intellectual achievements of all time. Boolean logic of the null operator, and of context free grammars A precursor of the Backus-Naur form (used in the description modern programming languages). Used a finite number of rules. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Panini( 500 -400 BC) Provided more than 4,000 rules that describe the Sanskrit of his day completely Intellectual achievements of all time. Boolean logic of the null operator, and of context free grammars A precursor of the Backus-Naur form (used in the description modern programming languages). Used a finite number of rules. Anticipated the logical framework of modern computers. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Jaina Mathematics (400 BCE - 200 CE) Jaina texts on mathematical topics were composed after the 6th century BCE Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Jaina Mathematics (400 BCE - 200 CE) Jaina texts on mathematical topics were composed after the 6th century BCE Freed Indian mathematics from its religious and ritualistic constraints Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Jaina Mathematics (400 BCE - 200 CE) Jaina texts on mathematical topics were composed after the 6th century BCE Freed Indian mathematics from its religious and ritualistic constraints Classified numbers into three classes: enumerable, innumerable and infinite Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Jaina Mathematics (400 BCE - 200 CE) Jaina texts on mathematical topics were composed after the 6th century BCE Freed Indian mathematics from its religious and ritualistic constraints Classified numbers into three classes: enumerable, innumerable and infinite Devised notations for simple powers Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Jaina Mathematics (400 BCE - 200 CE) Jaina texts on mathematical topics were composed after the 6th century BCE Freed Indian mathematics from its religious and ritualistic constraints Classified numbers into three classes: enumerable, innumerable and infinite Devised notations for simple powers Use the word shunya to refer to zero Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India ØâøcdɼíÈÞÉÄß 6 0 0 BC èÕÖÞÜß·ÃßÄ¢ 3 0 0 BC ØñÞȹí·ØâdÄ¢ 3 0 0 -2 0 0 BC ©ÎÞØbÞÄß 1 5 0 BC ¥çÈÞçÏÞ·ÆbÞøØâdÄ¢ 2 0 0 BC - 1 0 0 CE ÖÄí¶w޷΢ 200 C ÍdÆÌÞÙÞÕßØ¢ÙßÄ M a t he m a t ic ia ns ÍdÆÌÞÙá 2 9 8 BC ÏÞÄßdÕß×cÞºÞøc 1 7 6 BC Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Pingala ( 300-200 BCE) A musical theorist who authored the Chandas Shastra Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Pingala ( 300-200 BCE) A musical theorist who authored the Chandas Shastra Developed advanced mathematical concepts for describing prosody Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Pingala ( 300-200 BCE) A musical theorist who authored the Chandas Shastra Developed advanced mathematical concepts for describing prosody Presented the first known description of a Binary numeral system. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Pingala ( 300-200 BCE) A musical theorist who authored the Chandas Shastra Developed advanced mathematical concepts for describing prosody Presented the first known description of a Binary numeral system. Pascal triangle and Binomial coefficients Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Pingala ( 300-200 BCE) A musical theorist who authored the Chandas Shastra Developed advanced mathematical concepts for describing prosody Presented the first known description of a Binary numeral system. Pascal triangle and Binomial coefficients Contains the basic ideas of Fibonacci numbers Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Katyayana ( 3rd century BCE) Last of the Vedic mathematicians Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Katyayana ( 3rd century BCE) Last of the Vedic mathematicians Wrote the Katyayana Sulba Sutra Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Katyayana ( 3rd century BCE) Last of the Vedic mathematicians Wrote the Katyayana Sulba Sutra geometry, the general Pythagorean theorem and a computation of the square root of 2 correct to five decimal places.. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Oral tradition Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Oral tradition Memorization Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Oral tradition Memorization Vedic Period Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Oral tradition Memorization Vedic Period The Written Tradition: Prose Commentary Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Classical Period (400 CE - 1200CE) Aryabhata, Varahamihira, Brahmagupta, Bhaskara I, Mahavira, and Bhaskara II Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Classical Period (400 CE - 1200CE) Aryabhata, Varahamihira, Brahmagupta, Bhaskara I, Mahavira, and Bhaskara II Works included both astronomical and mathematical contributions Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Surya Siddhanta - 400 CE Authorship unknown Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Surya Siddhanta - 400 CE Authorship unknown First use of Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Surya Siddhanta - 400 CE Authorship unknown First use of Sine (Jya). Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Surya Siddhanta - 400 CE Authorship unknown First use of Sine (Jya). Cosine (Kojya). Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Surya Siddhanta - 400 CE Authorship unknown First use of Sine (Jya). Cosine (Kojya). Inverse sine (Otkram jya). Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Surya Siddhanta - 400 CE Authorship unknown First use of Sine (Jya). Cosine (Kojya). Inverse sine (Otkram jya). Tangent. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Surya Siddhanta - 400 CE Authorship unknown First use of Sine (Jya). Cosine (Kojya). Inverse sine (Otkram jya). Tangent. Secant. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Surya Siddhanta - 400 CE Authorship unknown First use of Sine (Jya). Cosine (Kojya). Inverse sine (Otkram jya). Tangent. Secant. Computed the average length of the year as 365.2563795 days Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Aryabhata I (476-550) Author of Aryabhatiyam Quadratic equations Trigonometry The value of π correct to 4 decimal places. Arya Siddhantam Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Aryabhata I (476-550) Introduced the trigonometric functions. Gave methods of calculating their approximate numerical values. Earliest tables of sine and cosine values, in 3.75 intervals from 0 to 90, to 4 decimal places of accuracy. Trigonometric formula sin(n + 1)x − sin nx = sin nx − sin(n − 1)x − Dr.Indulal G.̇ S.A. College, Edathua 1 225 Mathematics from India sin nx. Aryabhata I (476-550) Introduced the trigonometric functions. Gave methods of calculating their approximate numerical values. Earliest tables of sine and cosine values, in 3.75 intervals from 0 to 90, to 4 decimal places of accuracy. Trigonometric formula sin(n + 1)x − sin nx = sin nx − sin(n − 1)x − Dr.Indulal G.̇ S.A. College, Edathua 1 225 Mathematics from India sin nx. Aryabhata I (476-550) Continued fractions. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Aryabhata I (476-550) Continued fractions. Solutions of simultaneous quadratic equations. Whole number solutions of linear equations by a method equivalent to the modern method. General solution of the indeterminate linear equation . Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Aryabhata I (476-550) Continued fractions. Solutions of simultaneous quadratic equations. Whole number solutions of linear equations by a method equivalent to the modern method. General solution of the indeterminate linear equation . Proposed for the first time, a heliocentric solar system with the planets spinning on their axes and following an elliptical orbit around the Sun. ( Galelio by 1540’s) Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Aryabhata I (476-550) Continued fractions. Solutions of simultaneous quadratic equations. Whole number solutions of linear equations by a method equivalent to the modern method. General solution of the indeterminate linear equation . Proposed for the first time, a heliocentric solar system with the planets spinning on their axes and following an elliptical orbit around the Sun. ( Galelio by 1540’s) Accurate calculations for astronomical constants Solar eclipse. Lunar eclipse. The formula for the sum of the cubes, which was an important step in the development of integral calculus. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Varahamihira (505-587) Authored Pancha Siddhanta Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Varahamihira (505-587) Authored Pancha Siddhanta ØâøcØßiÞL¢ çøÞεØßiÞL¢ ÉìÜàÖØßiÞL¢ ÕØß×íÀØßiÞL¢ èÉÄÞÎÙØßiÞL¢ Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Varahamihira (505-587) Sine and Cosine tables to 4 decimal places of accuracy Gave the formulae: sin2 (x) + cos2 (x) = 1 π sin(x) = cos −x 2 1 − cos(2x) = sin2 (x) 2 Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Seventh and Eighth Centuries(600 - 700) Arithmetic and Algebra, began to emerge Later called "pathiganitham" and "beejaganitham" Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Brahmagupta, (604-) Authored "Brahma Sphuta Siddhanta" (628 CE) Chapter 12, contained 66 Sanskrit verses Basic operations and Practical mathematics Famous theorem on the diagonals of a cyclic quadrilateral Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Brahmagupta, (604-) Theorem If a cyclic quadrilateral has diagonals that are perpendicular to each other, then the perpendicular line drawn from the point of intersection of the diagonals to any side of the quadrilateral always bisects the opposite side. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Brahmagupta, (604-) Theorem The area A of a cyclic quadrilateral with sides of lengths a, b, c, d respectively, is given by p A = (s − a)(s − b)(s − c)(s − d ) Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Brahmagupta, (604-) He gave the first explicit solution of the quadratic equation Was able to make progress in finding (integral) solutions of Pell’s equation x 2 − Ny 2 = 1 Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Bhaskara I (c. 600-680) Authored Mahabhaskariya, Aryabhatiya-bhashya and Laghu-bhaskariya. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Bhaskara I (c. 600-680) Authored Mahabhaskariya, Aryabhatiya-bhashya and Laghu-bhaskariya. 1 2 3 Solutions of indeterminate equations. A rational approximation of the sine function. A formula for calculating the sine of an acute angle without the use of a table, correct to 2 decimal places. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India 800 - 1100 CE Virasena (9th century) authored Dhavala ¥Viç»Æ¢ - Log 2 dÄßµç»Æ¢ - Log 3 ºÄáVç»Æ¢ - Log 4 Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Mahavira Acharya ( 800 - 870) Authored Ganit Saara Sangraham Zero, Squares, Cubes, square roots, cube roots, and the series extending beyond these Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Mahavira Acharya ( 800 - 870) Authored Ganit Saara Sangraham Zero, Squares, Cubes, square roots, cube roots, and the series extending beyond these Plane geometry, Solid geometry, Problems relating to the casting of shadows, the area of an ellipse and quadrilateral inside a circle. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Mahavira Acharya ( 800 - 870) Authored Ganit Saara Sangraham Zero, Squares, Cubes, square roots, cube roots, and the series extending beyond these Plane geometry, Solid geometry, Problems relating to the casting of shadows, the area of an ellipse and quadrilateral inside a circle. The square root of a negative number do not exist ( An observation which "European Community was not aware") Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Mahavira Acharya ( 800 - 870) Sum of a series whose terms are squares of an arithmetical progression ( Gauss - 20th Century) Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Mahavira Acharya ( 800 - 870) Sum of a series whose terms are squares of an arithmetical progression ( Gauss - 20th Century) Gave empirical rules for area and perimeter of an ellipse. Solved cubic equations. ( Girolamo Cardano (1501-1576)) Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Mahavira Acharya ( 800 - 870) Sum of a series whose terms are squares of an arithmetical progression ( Gauss - 20th Century) Gave empirical rules for area and perimeter of an ellipse. Solved cubic equations. ( Girolamo Cardano (1501-1576)) Solved quartic equations. ( Del Fero - 1590) Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Mahavira Acharya ( 800 - 870) Sum of a series whose terms are squares of an arithmetical progression ( Gauss - 20th Century) Gave empirical rules for area and perimeter of an ellipse. Solved cubic equations. ( Girolamo Cardano (1501-1576)) Solved quartic equations. ( Del Fero - 1590) Solved some quintic equations and higher-order polynomials. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Mahavira Acharya ( 800 - 870) Sum of a series whose terms are squares of an arithmetical progression ( Gauss - 20th Century) Gave empirical rules for area and perimeter of an ellipse. Solved cubic equations. ( Girolamo Cardano (1501-1576)) Solved quartic equations. ( Del Fero - 1590) Solved some quintic equations and higher-order polynomials. Solved indeterminate quadratic equations, cubic equations, higher order equations. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Shridhara (c. 870 - 930) Authored Nav Shatika, Tri Shatika and Pati Ganita Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Shridhara (c. 870 - 930) Authored Nav Shatika, Tri Shatika and Pati Ganita Rule for finding the volume of a sphere. Formula for solving quadratic equations. Methods of summation of different arithmetic and geometric series Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Aryabhata II (920 - 1000) Authored Maha-Siddhanta Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Aryabhata II (920 - 1000) Authored Maha-Siddhanta Numerical mathematics (Ank Ganit) Algebra. Solutions of indeterminate equations Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Shripati Mishra (1019 -1066) Authored Siddhanta Shekhara and Ganita Tilakam Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Shripati Mishra (1019 -1066) Authored Siddhanta Shekhara and Ganita Tilakam Permutations and combinations and general solution of the simultaneous indeterminate linear equation. Nemichandra Siddhanta Chakravati (c. 1100) Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Bhaskara II (1114 - 1185) Authored ØßiÞLÖßçøÞÎÃß ÜàÜÞÕÄß Ì༷ÃßÄ¢ ç·Þ{ÞÇcÞÏ¢ ·ãÙ·ÃßÄ¢ µøõáÄâÙÜ¢ Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Bhaskara II (1114 - 1185) Authored ØßiÞLÖßçøÞÎÃß ÜàÜÞÕÄß Ì༷ÃßÄ¢ ç·Þ{ÞÇcÞÏ¢ ·ãÙ·ÃßÄ¢ µøõáÄâÙÜ¢ Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Bhaskara II (1114 - 1185) - Arthmetic Interest computation Arithmetical and geometrical progressions Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Bhaskara II (1114 - 1185) - Arthmetic Interest computation Arithmetical and geometrical progressions Plane geometry Solid geometry Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Bhaskara II (1114 - 1185) - Arthmetic Interest computation Arithmetical and geometrical progressions Plane geometry Solid geometry The shadow of the moon Solutions of combinations Gave a proof for division by zero being infinity. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Bhaskara II (1114 - 1185) - Calculus Conceived of differential calculus. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Bhaskara II (1114 - 1185) - Calculus Conceived of differential calculus. Discovered the derivative. Discovered the differential coefficient. Developed differentiation. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Bhaskara II (1114 - 1185) - Calculus Conceived of differential calculus. Discovered the derivative. Discovered the differential coefficient. Developed differentiation. Stated Rolle’s theorem, a special case of the mean value theorem . Derived the differential of the sine function. - ( Newton Leibniz 17th Century) Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Bhaskara II (1114 - 1185) - Calculus Conceived of differential calculus. Discovered the derivative. Discovered the differential coefficient. Developed differentiation. Stated Rolle’s theorem, a special case of the mean value theorem . Derived the differential of the sine function. - ( Newton Leibniz 17th Century) Computed π correct to 5 decimal places. Calculated the length of the Earth’s revolution around the Sun to 9 decimal places. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Kerala Mathematics (1300-1600) Madhava of Sangamagrama Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Kerala Mathematics (1300-1600) Madhava of Sangamagrama Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri , Chithrabhanu and Achyuta Panikkar Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Kerala Mathematics (1300-1600) Madhava of Sangamagrama Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri , Chithrabhanu and Achyuta Panikkar Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Madhava School of Mathematics( 1340-1630) Ø¢·Îd·ÞÎ ÎÞÇÕ Î 1340 ‐ 1425 Õ ÕGçÛøß ß ÉøçÎÎÖbøX X 13 360 ‐ 1460 0 Æ ÆÞçÎÞÆ ÆøX §{ÏÄí § 1410 ‐ 151 10 Èàܵ µÃíÀ çØÞÎÎÏÞ¼ß çç¼c×íÀ ÀçÆÕX X 1443 ‐ 1560 1500 ‐ 1621 º ºßdÄÍÞ ÞÈá 1550‐‐ ¥ ¥ºcáÄM Mß×Þø ø¿ß 1550 ‐1621 ÖCø ÕÞøccV 1500 15600 Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Kerala School of Mathematics( 1340-1630) Vararuci (4 th century CE) Candravakyas Haridatta (7 th century CE) Parahita system Govindasvami (c. 800 – c. 860) Mahabhaskariya Sankaranarayana (9 th century CE) Observatory at Kodungallur Govinda Bhattathiri (1237–1295 CE) Dasadhyayi Sangamagrama Madhava (fl.1340 – 1425) Venvaroha Paramesvara (fl.1360 – 1460) Drigganita Damodara (fl.1410 – 1510) Nilakantha Somayaji (fl.1443 – 1560) Tantrasangraha Jyesthadeva (fl.1500 – 1610) Yuktibhasa Sankara Acyuta Variar Pisharati (fl.1500 – 1560) (fl.1550 – 1621) Kriya-kramakari Citrabhanu (fl.1550) Puthumana Somayaji Karanapadhati (1733) Sankara Varman (1774 - 1839) Sadratnamala Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Madhava of Sangamagrama (1340-1425) Discovered the infinite series for circular and trigonometric functions Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Madhava of Sangamagrama (1340-1425) Discovered the infinite series for circular and trigonometric functions Discovered the infinite series expansion for tan−1 x - ( Gregory Series -1671) Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Madhava of Sangamagrama (1340-1425) Discovered the infinite series for circular and trigonometric functions Discovered the infinite series expansion for tan−1 x - ( Gregory Series -1671) The series π 4 =1− 1 3 + 1 5 − Dr.Indulal G.̇ S.A. College, Edathua 1 7 + . . . ( Leibniz - 1670) Mathematics from India Madhava of Sangamagrama (1340-1425) Discovered the infinite series for circular and trigonometric functions Discovered the infinite series expansion for tan−1 x - ( Gregory Series -1671) The series π 4 =1− x3 3! 2 1 − x2! sin x = x − x5 5! 4 + x4! + 1 3 + 1 5 x7 7! 6 − x6! − − 1 7 + . . . ( Leibniz - 1670) + · · · and + · · · ( Newton Power series - 1680) cos x = - (DeMoivre 1707 - 1738) - ( Euler - 1748) Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Vattasseri Parameswaran (1360-1460) Authored Goladeepika Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Vattasseri Parameswaran (1360-1460) Authored Goladeepika Much contributed to astronomy, recorded many eclipses and predicted many with high degree of accuracy. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Vattasseri Parameswaran (1360-1460) Authored Goladeepika Much contributed to astronomy, recorded many eclipses and predicted many with high degree of accuracy. An early version of the mean value theorem in Lilavathy Bhashya Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Vattasseri Parameswaran (1360-1460) Authored Goladeepika Much contributed to astronomy, recorded many eclipses and predicted many with high degree of accuracy. An early version of the mean value theorem in Lilavathy Bhashya First mathematician to give the radius of a circle inscribed in a cyclic quadrilateral ( L’Huilier (1782)) Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Neelakantha Somayaji (1444-1544) Neelakanta - Tantrasangraha Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Neelakantha Somayaji (1444-1544) Neelakanta - Tantrasangraha Computed the path of planets ( Johannes Kepler 1609 ) Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Neelakantha Somayaji (1444-1544) Neelakanta - Tantrasangraha Computed the path of planets ( Johannes Kepler 1609 ) Revised the heliocentric model Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India ç·Þ{ØÞø¢ ‐ Astronomical Procedures ØßiÞLÆVMâ ‐ Astronomical constants ºdw»ÞÏÞ·ÃßÄ¢ ‐ Eclipses ¦øcÍ¿àÏÍÞ×c¢ d·ÙÈßVHÏ¢ ‐ Solar, Lunar Eclipses Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Citrabhanu (c. 1530) Gave integer solutions to 21 types of systems of two simultaneous Diophantine equations in two unknowns Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Citrabhanu (c. 1530) Gave integer solutions to 21 types of systems of two simultaneous Diophantine equations in two unknowns x +y =a x −y =b xy = c x2 + y2 = d x2 − y2 = e x3 + y3 = f x3 − y3 = g Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Citrabhanu (c. 1530) Gave integer solutions to 21 types of systems of two simultaneous Diophantine equations in two unknowns x +y =a x −y =b xy = c x2 + y2 = d x2 − y2 = e x3 + y3 = f x3 − y3 = g Deduced several astronomical phenomenon Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Jyeshtadeva(1500-1610) Authored Yukthibhasha Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Jyeshtadeva(1500-1610) Authored Yukthibhasha Computed the infinite series for pi and sin,their numerical values Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Jyeshtadeva(1500-1610) Authored Yukthibhasha Computed the infinite series for pi and sin,their numerical values Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Thrikkandiyoor Achyuta Pisharati (1550-1621) Discovered the technique of ’the reduction of the ecliptic’. Authored Sphuta-nirnaya , Raasi-gola-sphuta-neeti Lunar and solar eclipses Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Srinivasa Iyengar Ramanujan (1887-1920) No formal training in pure mathematics Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Srinivasa Iyengar Ramanujan (1887-1920) No formal training in pure mathematics Made substantial contributions to Mathematical analysis, Number theory, Infinite series and Continued fractions Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Srinivasa Iyengar Ramanujan (1887-1920) No formal training in pure mathematics Made substantial contributions to Mathematical analysis, Number theory, Infinite series and Continued fractions Compiled nearly 3900 results Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Srinivasa Iyengar Ramanujan (1887-1920) No formal training in pure mathematics Made substantial contributions to Mathematical analysis, Number theory, Infinite series and Continued fractions Compiled nearly 3900 results √ ∞ 1 2 2 X (4k)! (1103 + 26390k) = π 9801 (k!)4 3964k k=0 Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Srinivasa Iyengar Ramanujan (1887-1920) No formal training in pure mathematics Made substantial contributions to Mathematical analysis, Number theory, Infinite series and Continued fractions Compiled nearly 3900 results √ ∞ 1 2 2 X (4k)! (1103 + 26390k) = π 9801 (k!)4 3964k k=0 " ∞ X cos (nθ) 1+2 cosh (nπ) n=1 #−2 " ∞ X cosh (nθ) + 1+2 cosh (nπ) Dr.Indulal G.̇ S.A. College, Edathua n=1 Mathematics from India #−2 = 3 4 2 Γ π 4 Indian Mathematicians 1 1850 - Sankaravarman 2 1887 - 1920 Sreenivasa Ramanujan 3 1888-1978 Bhoopathy Mohan Sen 4 1893 - 1972 Mahalanobis P C 5 1894 -1974 Sathyendra Natha Bose 6 1905 - 1988 Kapreckar D R 7 1906 - 1964 Roy S N 8 1910 -1987 Raj Chandra Bose 9 1917 - 1980 Kesava Menon 10 1922- Sakunthaladevi 11 1923 - 2002 Dr.Indulal G.̇ S.A. College, Edathua Kapoor J N Mathematics from India Conclusion Vateswaracharya (AD 880) discovered the Newton Gauss Backward interpolation (AD1670) Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Conclusion Vateswaracharya (AD 880) discovered the Newton Gauss Backward interpolation (AD1670) Nilakanta Somayaji ( AD 1444) discovered the Newton Infinite GP Convergent Series (AD1670) Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Conclusion Vateswaracharya (AD 880) discovered the Newton Gauss Backward interpolation (AD1670) Nilakanta Somayaji ( AD 1444) discovered the Newton Infinite GP Convergent Series (AD1670) NIlakanta Somayaji ( AD 1444) discovered Taylor Series (AD 1685) Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Conclusion Vateswaracharya (AD 880) discovered the Newton Gauss Backward interpolation (AD1670) Nilakanta Somayaji ( AD 1444) discovered the Newton Infinite GP Convergent Series (AD1670) NIlakanta Somayaji ( AD 1444) discovered Taylor Series (AD 1685) Govinda swamy ( AD 850) discovered the Newton interpolation (AD 1670) Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Conclusion Vateswaracharya (AD 880) discovered the Newton Gauss Backward interpolation (AD1670) Nilakanta Somayaji ( AD 1444) discovered the Newton Infinite GP Convergent Series (AD1670) NIlakanta Somayaji ( AD 1444) discovered Taylor Series (AD 1685) Govinda swamy ( AD 850) discovered the Newton interpolation (AD 1670) Puthumana Somayaji ( AD 1431) discovered the Newton’s power series (AD 1660) Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Conclusion Vateswaracharya (AD 880) discovered the Newton Gauss Backward interpolation (AD1670) Nilakanta Somayaji ( AD 1444) discovered the Newton Infinite GP Convergent Series (AD1670) NIlakanta Somayaji ( AD 1444) discovered Taylor Series (AD 1685) Govinda swamy ( AD 850) discovered the Newton interpolation (AD 1670) Puthumana Somayaji ( AD 1431) discovered the Newton’s power series (AD 1660) Parameswaracharya (AD1360) discovered the Lhuier power series ( AD 1782) Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Conclusion Vateswaracharya (AD 880) discovered the Newton Gauss Backward interpolation (AD1670) Nilakanta Somayaji ( AD 1444) discovered the Newton Infinite GP Convergent Series (AD1670) NIlakanta Somayaji ( AD 1444) discovered Taylor Series (AD 1685) Govinda swamy ( AD 850) discovered the Newton interpolation (AD 1670) Puthumana Somayaji ( AD 1431) discovered the Newton’s power series (AD 1660) Parameswaracharya (AD1360) discovered the Lhuier power series ( AD 1782) Madhavacharya (AD 1350) discovered the De Moiver’s approximation(AD 1650) Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Conclusion Vateswaracharya (AD 880) discovered the Newton Gauss Backward interpolation (AD1670) Nilakanta Somayaji ( AD 1444) discovered the Newton Infinite GP Convergent Series (AD1670) NIlakanta Somayaji ( AD 1444) discovered Taylor Series (AD 1685) Govinda swamy ( AD 850) discovered the Newton interpolation (AD 1670) Puthumana Somayaji ( AD 1431) discovered the Newton’s power series (AD 1660) Parameswaracharya (AD1360) discovered the Lhuier power series ( AD 1782) Madhavacharya (AD 1350) discovered the De Moiver’s approximation(AD 1650) ......... and many more ........................ Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Conclusion Indian contributions to mathematics have not been given due acknowledgement in modern history Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Conclusion Indian contributions to mathematics have not been given due acknowledgement in modern history Many discoveries and inventions by Indian mathematicians were known to their Western counterparts Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India Conclusion Indian contributions to mathematics have not been given due acknowledgement in modern history Many discoveries and inventions by Indian mathematicians were known to their Western counterparts This mass plagiarism has gone unrecognized due to Eurocentrism. Dr.Indulal G.̇ S.A. College, Edathua Mathematics from India