25 February, 2022

**ONLINE EVENT**Azadi Ka Amrit Mahotsav is an initiative of the Government of India to celebrate and commemorate 75 years of progressive India and the glorious history of it’s people, culture and achievements.

After a prolonged struggle spread over more than 100 years on the back of several movements spearheaded by individuals and communities across the country, India successfully ousted the foreign rulers from the Indian subcontinent in 1947.

India’s journey from the beginning of time is dotted with consequential events, each of which added to the idea of India in a very unique and inevitable manner. For long, these events have been stories limited to folklore or a part of a historian’s collective.

It is time we walk down the annals of history and celebrate achievements which we didn’t know belong to India; claim our contribution which innately belongs to Bharat and take forward the continuum of India’s contribution towards making the world a better place for humanity to thrive and coexist.

Ministry of Culture is proud to announce Dhara: An ode to Indian Knowledge System, a series of programs powered by lecture demonstrations, celebrating and showcasing India’s contribution and achievements across diverse fields. Our first event under this series is dedicated to ‘India’s Contribution to Mathematics Through Ages’

Mathematics in India has a very rich, long and hallowed history. Starting from the most elementary thing in mathematics namely the representation of numbers, through the way of expressing recursive relations, to arriving at the solutions of indeterminate equations, to the development of sophisticated techniques in handling the infinite and the infititesimals, Indian mathematicians have made remarkable contributions.

Sulbasutras, the oldest extant texts (∼ 800 BCE), explicitly state and make use of the so-called Pythagorean theorem besides giving various interesting approximations to surds. Following this, Pingala’s Chandassastra (∼ 3rd cent. BCE), a text that deals with the prosody, lays foundations for various combinatorial techniques. By the time of Aryabhata (c. 499 CE), the Indian mathematicians were fully conversant with most of the mathematics that we currently teach in our schools, which include the algorithms for extracting square root and cube root based on the decimal place-value system.

Among other things, Aryabhata also presented the differential equation of sine function in its finite-difference form and a method for solving the linear indeterminate equation. Brahmagupta (c. 628), for the first time in the history of mathematics, fully discusses the arithmetic operations with zero. He also introduces the profound ‘bhavana’ law of composition for solving quadratic indeterminate equations. Apart from some of these important landmarks in the evolution of arithmetic, geometry, and algebra, significant contributions have also been made in the development of trigonometry.

The Kerala School of astronomy and mathematics pioneered by Madhava (c. 1340–1420) discovered the infinite series for pi (π)—the so-called Gregory-Leibniz series)—and other trigonometric functions. The series for π/4 being an excruciatingly slowly converging series, Madhava also came up with several brilliant fast convergent approximations to it. This School is also credited with the introduction of non-geocentric planetary models. These two things, namely the introduction of infinite series, and non-geocentric planetary models are in fact, hailed as the hallmarks of the genesis of modern science in Europe a few centuries later. Today, if the modern scholarship is aware of some of these significant achievements of Indians in the development of mathematics, it is primarily because of the dedicated work of few great scholars such as BB Datta, KS Shukla, KV Sarma and so on in the last hundred years.

Much of their painstaking work was largely carried out voluntarily, with hardly any support from the institutions of higher learning. Unfortunately, most Indians are not aware of these remarkable contributions made by Indians to the development of mathematics. The series of talks to be delivered by eminent scholars from all over the world, aims to provide a glimpse of this rich mathematical heritage of India.

**K. Ramasubramanian IIT Bombay**

Inaugural Session

10.00 -10.15 | Welcome + Introducing the Theme of the Conference | ||

10.15 -10.30 | Prof. Rajive Kumar, Member Secretary of AICTE | Address by Chief Guest | |

10.30 -11.05 | Manjul Bhargava |
Keynote Address | |

11.05 -11.10 | Vote of Thanks |

11.10 -11.40 | Tea Break |

Session 1

Ancient Period

11.40 -12.10 | Parthasarathi Mukhopadhyay | Zero & Decimal Place Value System |

12.10 -12.40 | Jean Michel Delire | Geometry in Sulbasutras |

12.40 -13.10 | MD Srinivas | Combinatorics in Piṅgala’s Chandas-sastra |

13.10 -14.00 | Lunch Break |

Session 2

Classical Period

14.00 -14.30 | Amartya Kumar Dutta | Some Landmarks in Indian Algebra |

14.30 -15.00 | Clemency Montelle | Jyotpatti : Trigonometry in India |

15.00 -15.30 | Avinash Sathaye | Indeterminate Equations in Indian Algebra |

15.30 -16.00 | Tea Break |

Session 3

Contributions of Kerala School

16.00 -16.30 | K Ramasubramanian | Madhava’s Infinite Series for π (C) |

16.30 -17.00 | MS Sriram | Calculus of Trigonometric Functions |

Session 4

Valedictory Session

17.00 -17.30 | SG Dani | Pioneering Historians of Indian Mathematics |

17.30 -18.00 | K Ramasubramanian | Summary & Vote of Thanks |

- Manjul Bhargava Beyond Zero: A survey of some of India’s fundamental contributions to Mathematics
- Parthasarathi Mukhopadhyay Zero & Decimal Place Value System
- Jean Michel Delire Geometry in Sulbasutras
- M. D. Srinivas Combinatorics in Pingala’s Chandas-sastra
- Amartya Kumar Dutta Indian Algebra
- Clemency Montelle
*Jyotpatti :*Trigonometry in India - Avinash Sathaye Indeterminate Equations in Indian Algebra
- K. Ramasubramanian Madhava’s Infinite Series for π
## K. Ramasubramanian

### Madhava’s Infinite Series for π

Mathematics in India has a very rich, long and hallowed history. Starting from the most elementary thing in mathematics namely the representation of numbers, through the way of expressing recursive relations, to arriving at the solutions of indeterminate equations, to the development of sophisticated techniques in handling the infinite and the infititesimals, Indian mathematicians have made remarkable contributions.

Sulbasutras, the oldest extant texts (∼ 800 BCE), explicitly state and make use of the so-called Pythagorean theorem besides giving various interesting approximations to surds. Following this, Pingala’s Chandassastra (∼ 3rd cent. BCE), a text that deals with the prosody, lays foundations for various combinatorial techniques. By the time of Aryabhata (c. 499 CE), the Indian mathematicians were fully conversant with most of the mathematics that we currently teach in our schools, which include the algorithms for extracting square root and cube root based on the decimal place-value system. Among other things, Aryabhata also presented the differential equation of sine function in its finite-difference form and a method for solving the linear indeterminate equation. Brahmagupta (c. 628), for the first time in the history of mathematics, fully discusses the arithmetic operations with zero. He also introduces the profound ‘bhavana’ law of composition for solving quadratic indeterminate equations.

Apart from some of these important landmarks in the evolution of arithmetic, geometry, and algebra, significant contributions have also been made in the development of trigonome- try. The Kerala School of astronomy and mathematics pioneered by Madhava (c. 1340–1420) discovered the infinite series for pi (π)—the so-called Gregory-Leibniz series)—and other trigonometric functions. The series for π/4 being an excruciatingly slowly converging series, Madhava also came up with several brilliant fast convergent approximations to it. This School is also credited with the introduction of non-geocentric planetary models. These two things, namely the introduction of infinite series, and non-geocentric planetary models are in fact, hailed as the hallmarks of the genesis of modern science in Europe a few centuries later.

Today, if the modern scholarship is aware of some of these significant achievements of Indians in the development of mathematics, it is primarily because of the dedicated work of few great scholars such as BB Datta, KS Shukla, KV Sarma and so on in the last hundred years. Much of their painstaking work was largely carried out voluntarily, with hardly any support from the institutions of higher learning. Unfortunately, most Indians are not aware of these remarkable contributions made by Indians to the development of mathematics. The series of talks to be delivered by eminent scholars from all over the world, aims to provide a glimpse of this rich mathematical heritage of India.

- MS Sriram Calculus of Trigonometric Functions
- S. G. Dani Pioneering Historians of IndianMathematics