1. About Mathematics in India....
Dr. Sanjay Mishra, LPU

2. Contents What is Mathematics History of Mathematics in India
Area of Mathematics
Scope of Mathematics
Dr. Sanjay Mishra, LPU

3. Mathematics is the language in which God has written the Universe.
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 American physicist. The Character of Physical Law
Mathematics is the language in which God has written the Universe.
Dr. Sanjay Mishra, LPU

4. Solution of Real Word Problem Solution of Mathematical Model
Formulate
Real Word Problem
Mathematical Model
Test
Solve
Solution of Real Word Problem
Solution of Mathematical Model
Interpret
Dr. Sanjay Mishra, LPU

5. History of Mathematics in India
Vedic
Classical
Medieval to Mughal period
Born in 1800s
Born in 1900s
Dr. Sanjay Mishra, LPU

6. Vedic Mathematician Baudhāyana, (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.
Dr. Sanjay Mishra, LPU

7. Vedic Mathematician Kātyāyana (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.
Dr. Sanjay Mishra, LPU

8. Vedic Mathematician Yajnavalkya
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.
Dr. Sanjay Mishra, LPU

9. Classical Mathematician
Aryabhata (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.
Dr. Sanjay Mishra, LPU

10. Classical Mathematician
Aryabhata (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.
Dr. Sanjay Mishra, LPU

11. Classical Mathematician
Aryabhata(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.
Dr. Sanjay Mishra, LPU

12. Classical Mathematician
Aryabhata 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.
Dr. Sanjay Mishra, LPU

13. Classical Mathematician
Aryabhata 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.
Dr. Sanjay Mishra, LPU

14. Classical Mathematician
Bhāskara (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.
Dr. Sanjay Mishra, LPU

15. Classical Mathematician
Brahmagupta (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.
Dr. Sanjay Mishra, LPU

16. Classical Mathematician
Brahmagupta (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.
Dr. Sanjay Mishra, LPU

17. Classical Mathematician
Bhāskara II ( )
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.
Dr. Sanjay Mishra, LPU

18. Classical Mathematician
Bhāskara II ( )
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.
Dr. Sanjay Mishra, LPU

19. Classical Mathematician
Bhāskara II ( )
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.
Dr. Sanjay Mishra, LPU

20. 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,
Mahendra Suri (14th century)
Shankara Variyar (c. 1530)
Dr. Sanjay Mishra, LPU

21. 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)
Dr. Sanjay Mishra, LPU

22. 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.
Dr. Sanjay Mishra, LPU

23. 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.
Dr. Sanjay Mishra, LPU

24. 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.
Dr. Sanjay Mishra, LPU

25. 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.
Dr. Sanjay Mishra, LPU

26. 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.
Dr. Sanjay Mishra, LPU

27. 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.
Dr. Sanjay Mishra, LPU

28. Born in 1900s Tirukkannapuram Vijayaraghavan (1902 – 1955)
Dattaraya Ramchandra Kaprekar (1905 – 1986)
Sarvadaman Chowla (1907–1995)
Lakkoju Sanjeevaraya Sharma ( )
Subrahmanyan Chandrasekhar (1910–1995)
S. S. Shrikhande (born 1917)
Harish-Chandra (1920–1983)
Calyampudi Radhakrishna Rao (born 1920)
Mathukumalli V. Subbarao (1921–2006)
Dr. Sanjay Mishra, LPU

29. Born in 1900s P. 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)
Dr. Sanjay Mishra, LPU

30. 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)
Dr. Sanjay Mishra, LPU

31. Born in 1900s Navin 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
Dr. Sanjay Mishra, LPU

32. 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.
Dr. Sanjay Mishra, LPU

33. Area of Mathematics A traditional division of Mathematics
Pure Mathematics
Studied for its intrinsic interest
Applied Mathematics
Which can be directly applied to real world problems
Dr. Sanjay Mishra, LPU

34. 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.
Dr. Sanjay Mishra, LPU

35. Area of Mathematics The top level subjects under the MSC are:
General / foundations
00: General (Includes topics such as recreational mathematics, philosophy of mathematics and Mathematical modeling.)
01: History and biography
03: Mathematical logic and foundations, including model theory, computability theory, set theory, proof theory, and algebraic logic.
Dr. Sanjay Mishra, LPU

36. Area of Mathematics Discrete mathematics / algebra 05: Combinatorics
The top level subjects under the MSC are:
Discrete mathematics / algebra
05: Combinatorics
06: Order theory
08: General algebraic systems
11: Number theory
12: Field theory and polynomials
13: Commutative rings and algebras
Dr. Sanjay Mishra, LPU

37. Area of Mathematics Discrete mathematics / algebra
The top level subjects under the MSC are:
Discrete mathematics / algebra
14: Algebraic geometry
15: Linear and multilinear algebra; matrix theory
16: Associative rings and associative algebras
17: Non-associative rings and non-associative algebras
18: Category theory; homological algebra
19: K-theory
20: Group theory and generalizations
22: Topological groups, Lie groups, and analysis upon them
Dr. Sanjay Mishra, LPU

38. Area of Mathematics The top level subjects under the MSC are: Analysis
26: Real functions, including derivatives and integrals
28: Measure and integration
30: Complex functions, including approximation theory in the complex domain
31: Potential theory
32: Several complex variables and analytic spaces
33: Special functions
34: Ordinary differential equations
35: Partial differential equations
37: Dynamical systems and Ergodic theory
39: Difference equations and functional equations
Dr. Sanjay Mishra, LPU

39. Area of Mathematics The top level subjects under the MSC are: Analysis
40: Sequences, series, summability
41: Approximations and expansions
42: Harmonic analysis, including Fourier analysis, Fourier transforms, trigonometric approximation, trigonometric interpolation, and orthogonal functions
43: Abstract harmonic analysis
44: Integral transforms, operational calculus
45: Integral equations
46: Functional analysis, including infinite-dimensional holomorphy, integral transforms in distribution spaces
47: Operator theory
49: Calculus of variations and optimal control; optimization (including geometric integration theory)
Dr. Sanjay Mishra, LPU

40. Area of Mathematics The top level subjects under the MSC are:
Geometry and topology
51: Geometry
52: Convex geometry and discrete geometry
53: Differential geometry
54: General topology
55: Algebraic topology
57: Manifolds
58: Global analysis, analysis on manifolds (including infinite-dimensional holomorphy)
Dr. Sanjay Mishra, LPU

41. Area of Mathematics The top level subjects under the MSC are:
Applied mathematics / other
60 Probability theory and stochastic processes
62 Statistics
65 Numerical analysis
68 Computer science
70 Mechanics (including particle mechanics)
74 Mechanics of deformable solids
76 Fluid mechanics
78 Optics, electromagnetic theory
80 Classical thermodynamics, heat transfer
81 Quantum theory
Dr. Sanjay Mishra, LPU

42. Area of Mathematics The top level subjects under the MSC are:
Applied mathematics / other
82 Statistical mechanics, structure of matter
83 Relativity and gravitational theory, including relativistic mechanics
85 Astronomy and astrophysics
86 Geophysics
90 Operations research, mathematical programming
91 Game theory, economics, social and behavioral sciences
92 Biology and other natural sciences
93 Systems theory; control, including optimal control
94 Information and communication, circuits
97 Mathematics education
Dr. Sanjay Mishra, LPU