About Mathematics in India.... Dr. Sanjay Mishra, LPU1

Dr. Sanjay Mishra, LPU 1

About Mathematics in India....


Contents

What is Mathematics

History of Mathematics in India

Area of Mathematics

Scope of Mathematics


To those who do not know Mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty of nature.... If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.

Richard Feynman. 1918-1988.American physicist.

The Character of Physical Law

Mathematics is the language in which God has written the Universe.


Real Word Problem

Mathematical Model

Solution of Mathematical

Model

Solution of Real Word Problem

Formulate

Solve

Interpret

Test


History of Mathematics in India

VedicClassicalMedieval to Mughal

periodBorn in 1800sBorn in 1900s


Vedic MathematicianBaudhāyana, (fl. c. 800 BCE)

He was an Indian mathematician, who was most likely also a priest. Author of the earliest Sulba Sūtra — appendices to the Vedas giving rules for the construction of altars — called the

Baudhāyana Śulbasûtra, which contained several important mathematical results.

He is older than the other famous mathematician Āpastambha. He belongs to the Yajurveda school.

He is accredited with calculating the value of pi to some degree of precision, and with discovering what is now known as the Pythagorean theorem.


Vedic MathematicianKātyāyana (c. 3rd century C)

He was a Sanskrit grammarian, mathematician and Vedic priest who lived in ancient India.

He composed one of the later Sulba Sutras.

A series of nine texts on the geometry of altar constructions, dealing with rectangles, right-sided triangles, rhombuses, etc.


Vedic MathematicianYajnavalkya

He was a legendary sage of Vedic India.

Author of the Shatapatha Brahmana.

Important contributions to both philosophy, including the

apophatic teaching of 'neti neti', and to astronomy.

Describing the 95-year cycle to synchronize the motions of

the sun and the moon.


Classical MathematicianAryabhata (476–550 CE)

Aryabhata was born (in 476 BC) in Taregna which is a small town in Bihar, India, about 30 km (19 mi) from Patna.

He was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy.

His most famous works are the Āryabhaṭīya (499 CE, when he was 23 years old) and the Arya-siddhanta.


Classical MathematicianAryabhata (476–550 CE)

His major work, Aryabhatiya, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature and has survived to modern times.

The mathematical part of the Aryabhata covers arithmetic, algebra, plane trigonometry, and spherical trigonometry.

It also contains continued fractions, quadratic equations, sums-of-

power series, and a table of sines.Aryabhata set up an Astronomical

Observatory in the Sun Temple 6th century.


Classical MathematicianAryabhata(476–550 CE)

French mathematician Georges Ifrah explains that knowledge of zero was implicit in Aryabhata's place-value system as a place holder for the powers of ten.

He worked on the approximation for pi (π), and may have come to the conclusion that π is irrational.

He provided elegant results for the summation of series of squares and cubes.


Classical MathematicianAryabhata II (c. 920 – c. 1000)

He was an Indian mathematician and astronomer, and the

author of the Maha-Siddhanta. The numeral II is given to him

to distinguish him from the earlier and more influential Āryabhaṭa.

He worked on topics related to mathematical astronomy as like

the longitudes of the planets, lunar and solar eclipses, the

estimation of eclipses, the lunar crescent, the rising and setting of

the planets, association of the planets with each other and with

the stars.


Classical MathematicianAryabhata II (c. 920 – c. 1000)

He worked on geometry, geography and algebra, which were

applied to calculate the longitudes of the planets. In about twenty

verses in the treatise, he gives elaborate rules to solve the

indeterminate equation: by = ax + c.

He played a vital role in it by constructing a sine table, which

was accurate up to five decimal places.


Classical MathematicianBhāskara (c. 600 – c. 680) He was born at Bori, in Parbhani district of Maharashtra state in

India in 7th century.Bhaskara is considered the most important scholar of

Aryabhata's astronomical school.He was apparently the first to write numbers in the Hindu-

Arabic decimal system with a circle for the zero.He gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata's work.


Classical MathematicianBrahmagupta (598–668 CE)

Brahmagupta is believed to have been born in 598 AD in Bhinmal city in the state of Rajasthan of Northwest India.

Brahmagupta's most famous work is his Brahmasphutasiddhanta.

Brahmagupta was the first to use zero as a number. He gave rules to compute with zero.

Brahmagupta used negative numbers and zero for computing. The modern rule that two negative numbers multiplied together equals a positive number first appears in Brahmasputasiddhanta.

Brahmagupta gave the solution of the general linear equation.


Classical MathematicianBrahmagupta (598–668 CE)

Four fundamental operations (addition, subtraction, multiplication and division) were known to many cultures before Brahmagupta. This current system is based on the Hindu Arabic number system and first appeared in Brahmasputa siddhanta.

Brahmasphuṭasiddhanta is the very first book that mentions zero as a number, hence Brahmagupta is considered as the man who found zero.

He gave rules of using zero with negative and positive

numbers.Brahmagupta's most famous result in geometry is his formula

for cyclic quadrilaterals. Given the lengths of the sides of any cyclic quadrilateral, Brahmagupta gave an approximate and an exact formula for the area.


Classical MathematicianBhāskara II (1114-1185) He was born near Vijjadavida (Bijāpur in modern Karnataka).

His main work Siddhānta Shiromani which is divided into four parts called Lilāvati, Bijaganita, Grahaganita and Golādhyāya.

He is particularly known in the discovery of the principles of

differential calculus and its application to astronomical problems

and computations.

Preliminary concept of mathematical analysis.

Preliminary concept of infinitesimal calculus, along with

notable contributions towards integral calculus.


Classical MathematicianBhāskara II (1114-1185)

Stated Rolle's theorem, a special case of one of the most

important theorems in analysis, the mean value theorem. Traces of

the general mean value theorem are also found in his works.

Bhaskara's arithmetic text Lilavati covers the topics of

definitions, arithmetical terms, interest computation, arithmetical

and geometrical progressions, plane geometry, solid geometry, the

shadow of the gnomon, methods to solve indeterminate equations,

and combinations.


Classical MathematicianBhāskara II (1114-1185)

His Bijaganita ("Algebra") was a work in twelve chapters. It

was the first text to recognize that a positive number has two

square roots (a positive and negative square root).

Bhaskara derived a cyclic, chakravala method for solving

indeterminate quadratic equations of the form ax² + bx + c = y.

Bhaskara's method for finding the solutions of the problem

Nx² + 1 = y² (the so-called "Pell's equation") is of considerable

importance.

He also discovered spherical trigonometry, along with other

interesting trigonometrical results.


Medieval to Mughal Period (13th to 18th century)

Narayana Pandit

Madhava of Sangamagrama

Parameshvara (1360–1455), discovered drk-ganita, a

mode of astronomy based on observations, Madhava's

Kerala school

Nilakantha Somayaji,1444-1545

Mahendra Suri (14th century)

Shankara Variyar (c. 1530)


Medieval to Mughal Period (13th to 18th century)

Raghunatha Siromani, (1475–1550), Logician,

Navadvipa school

Jyeshtadeva , 1500–1610, Author of Yuktibhāṣā,

Madhava's Kerala school

Achyuta Pisharati, 1550–1621,

Astronomer/mathematician, Madhava's Kerala school

Munishvara (17th century)

Kamalakara (1657)

Jagannatha Samrat (1730)


Born in 1800s

Ramchundra (1821 – 1880)

He was British India's first major mathematician.

His book, Treatise on Problems of Maxima and Minima, was promoted by the prominent mathematician Augustus De Morgan.


Born in 1800s

Ganesh Prasad (1876 – 1935)He was an Indian mathematician who specialized in the theory

of potentials, theory of functions of a real variable, Fourier series and the theory of surfaces.

He was trained at the Universities of Cambridge and Göttingen and on return to India he helped develop the culture of mathematical research in India.

The mathematical community of India considers Ganesh Prasad as the Father of Mathematical Research in India.

He was also an educator taking special interest in the advancement of primary education in the rural areas of India.


Born in 1800s

Srīnivāsa Rāmānujan (22 December 1887 – 26 April 1920)He was an Indian mathematician and

autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series and continued fractions.

He was Independently compiled nearly 3900 results during his short lifetime.


Born in 1800s

Srīnivāsa Rāmānujan (22 December 1887 – 26 April 1920)He stated results that were both

original and highly unconventional, such as the Ramanujan prime and the Ramanujan theta function, and these have inspired a vast amount of further research.

He was youngest Fellow of the Royal Society, Londan and Trinity College, Cambridge.


Born in 1900s

Harish-Chandra (11 October 1923 – 16 October 1983)He was an Indian mathematician, who

did fundamental work in representation theory, especially Harmonic analysis on semisimple Lie groups.

He was a member of the National Academy of Sciences of the U.S. and a Fellow of the Royal Society.


Born in 1900s

Harish-Chandra (11 October 1923 – 16 October 1983)He was the recipient of the Cole

Prize of the American Mathematical Society, in 1954.

The Indian National Science Academy honoured him with the Srinivasa Ramanujan Medal in 1974.

The Indian Government named the Harish-Chandra Research Institute, an institute dedicated to Theoretical Physics and Mathematics, after him.


Born in 1900s

Tirukkannapuram Vijayaraghavan (1902 – 1955)Dattaraya Ramchandra Kaprekar (1905 – 1986)Sarvadaman Chowla (1907–1995)Lakkoju Sanjeevaraya Sharma (1907-1998)Subrahmanyan Chandrasekhar (1910–1995)S. S. Shrikhande (born 1917)Harish-Chandra (1920–1983)Calyampudi Radhakrishna Rao (born 1920)Mathukumalli V. Subbarao (1921–2006)


Born in 1900sP. K. Srinivasan (1924-2005)Shreeram Shankar Abhyankar (born 1930)M. S. Narasimhan (born 1932)C. S. Seshadri (born 1932)K. S. S. Nambooripad (born 1935)Vinod Johri (born 1935)S. Ramanan (born 1937)C. P. Ramanujam (1938–1974)V. N. Bhat (1938–2009)


Born in 1900s

S. R. Srinivasa Varadhan (born 1940)

M. S. Raghunathan (born 1941)

Biswatosh Sengupta (born 1944)

Gopal Prasad (born 1945)

Vijay Kumar Patodi (1945–1976)

S. G. Dani (born 1947)

Raman Parimala (born 1948)


Born in 1900sNavin M. Singhi (born 1949)Narendra Karmarkar (born 1957)Manindra Agrawal (born 1966)Madhu Sudan (born 1966)Chandrashekhar Khare (born 1968)Manjul Bhargava (Indian origin American) (born 1974)Akshay Venkatesh (Indian origin Australian) (born 1981)Kannan Soundararajan (born 1982[citation needed])Sucharit Sarkar (born 1983)L. Mahadevan


Area of Mathematics

Mathematics has become a vastly diverse subject over history, and there is a corresponding need to categorize the different areas of mathematics.


Area of Mathematics

A traditional division of Mathematics

Pure MathematicsStudied for its intrinsic

interest

Applied MathematicsWhich can be directly applied to real world

problems


Area of Mathematics

The Mathematics Subject Classification (MSC) is an alphanumerical classification scheme collaboratively produced by staff of and based on the coverage of the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH.

The MSC divides mathematics into over 90 areas, with further subdivisions within each area.


Area of Mathematics

The top level subjects under the MSC are:

General / foundations00: General (Includes topics such as recreational mathematics, philosophy of mathematics and Mathematical modeling.)01: History and biography03: Mathematical logic and foundations, including model theory, computability theory, set theory, proof theory, and algebraic logic.


Area of MathematicsThe top level subjects under the MSC are:

Discrete mathematics / algebra05: Combinatorics06: Order theory08: General algebraic systems11: Number theory12: Field theory and polynomials13: Commutative rings and algebras



Discrete mathematics / algebra14: Algebraic geometry15: Linear and multilinear algebra; matrix theory16: Associative rings and associative algebras17: Non-associative rings and non-associative algebras18: Category theory; homological algebra19: K-theory20: Group theory and generalizations22: Topological groups, Lie groups, and analysis upon them



Analysis26: Real functions, including derivatives and integrals28: Measure and integration30: Complex functions, including approximation theory in the complex domain31: Potential theory32: Several complex variables and analytic spaces33: Special functions34: Ordinary differential equations35: Partial differential equations37: Dynamical systems and Ergodic theory39: Difference equations and functional equations



Analysis40: Sequences, series, summability41: Approximations and expansions42: Harmonic analysis, including Fourier analysis, Fourier transforms, trigonometric approximation, trigonometric interpolation, and orthogonal functions43: Abstract harmonic analysis44: Integral transforms, operational calculus45: Integral equations46: Functional analysis, including infinite-dimensional holomorphy, integral transforms in distribution spaces47: Operator theory49: Calculus of variations and optimal control; optimization (including geometric integration theory)



Geometry and topology51: Geometry52: Convex geometry and discrete geometry53: Differential geometry54: General topology55: Algebraic topology57: Manifolds58: Global analysis, analysis on manifolds (including infinite-dimensional holomorphy)



Applied mathematics / other60 Probability theory and stochastic processes62 Statistics65 Numerical analysis68 Computer science70 Mechanics (including particle mechanics)74 Mechanics of deformable solids76 Fluid mechanics78 Optics, electromagnetic theory80 Classical thermodynamics, heat transfer81 Quantum theory



Applied mathematics / other82 Statistical mechanics, structure of matter83 Relativity and gravitational theory, including relativistic mechanics85 Astronomy and astrophysics86 Geophysics90 Operations research, mathematical programming91 Game theory, economics, social and behavioral sciences92 Biology and other natural sciences93 Systems theory; control, including optimal control94 Information and communication, circuits97 Mathematics education

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About Mathematics in India.... Dr. Sanjay Mishra, LPU1