Takshashila University, established in the 6th century BCE, is widely regarded as the oldest university in the world. It was a renowned center of learning and scholarship, attracting students from across the globe.

Aryabhata, a renowned mathematician and astronomer of the 5th century CE, is credited with developing the concept of zero and incorporating it into the Indian number system. This revolutionary idea had a profound impact on mathematics.

Sanskrit was the primary language of instruction at ancient Indian mathematical institutions. It was the language of scholars and intellectuals, and many mathematical texts and treatises were written in Sanskrit.

Jantar Mantar, a series of astronomical observatories built by Maharaja Jai Singh II in the 18th century, was renowned for its contributions to astronomy and astrology. These observatories were used to make accurate observations of celestial bodies and calculate astronomical data.

Ancient Indian mathematical institutions covered a wide range of topics, including arithmetic, geometry, algebra, and astronomy. These institutions emphasized the practical applications of mathematics in fields such as architecture, engineering, and surveying.

The Kerala School of Astronomy and Mathematics, active from the 14th to the 16th century, made significant contributions to the development of calculus. Mathematicians like Madhava of Sangamagrama and Nilakantha Somayaji developed concepts such as the Taylor series expansion and the idea of infinitesimals.

Ancient Indian mathematical institutions served multiple purposes. They aimed to train mathematicians and scholars, promote scientific research, and preserve and transmit mathematical knowledge from one generation to another.

The Ujjain School of Astronomy, active from the 5th to the 12th century, made significant contributions to the development of trigonometry. Mathematicians like Aryabhata and Varahamihira developed trigonometric functions and tables, which were essential for astronomical calculations.

Mathematics played a dual role in ancient Indian society. It was used for religious rituals, such as calculating auspicious dates and times, and it was also used for practical applications, such as architecture, engineering, and surveying.

The Pataliputra School of Mathematics, active from the 3rd century BCE to the 3rd century CE, made significant contributions to the development of algebra. Mathematicians like Aryabhata and Brahmagupta developed algebraic methods for solving equations, including quadratic and indeterminate equations.

The primary source of mathematical knowledge in ancient India was mathematical texts and treatises. These texts were written by mathematicians and scholars and contained detailed explanations of mathematical concepts, methods, and applications.

The Varanasi School of Mathematics, active from the 6th to the 12th century, made significant contributions to the development of number theory. Mathematicians like Bhaskara I and Bhaskara II developed methods for solving Diophantine equations and studied properties of numbers.

Ancient Indian mathematical institutions contributed to the global development of mathematics in several ways. They shared their knowledge with other civilizations through trade and cultural exchange, they attracted scholars from other countries who came to study at their institutions, and their mathematical texts were translated into other languages, making them accessible to a wider audience.

Ancient Indian mathematical institutions faced several challenges, including lack of funding, political instability, and natural disasters. These challenges often hindered their ability to conduct research and preserve mathematical knowledge.

Ancient Indian mathematical institutions contributed to the development of modern mathematics in several ways. Their ideas were incorporated into modern mathematical theories, their methods were used to solve modern mathematical problems, and their texts were studied by modern mathematicians.