Aryabhata, a renowned mathematician and astronomer of the 5th century CE, is widely regarded as the pioneer who introduced the concept of zero and its use in the decimal system.

In ancient India, mathematics was primarily driven by practical applications, such as astronomy, architecture, engineering, and commerce. It was seen as a tool to solve real-world problems and make accurate predictions.

The decimal system, with its concept of place value and the use of zero, was a crucial development that greatly simplified mathematical calculations. It paved the way for the development of calculus and other advanced mathematical concepts.

The Nyaya school of philosophy, founded by Gautama, placed a strong emphasis on logic and reasoning. It considered mathematics as a fundamental tool for understanding the structure and principles of the universe.

Brahmagupta, a renowned mathematician of the 6th century CE, made significant contributions to number theory. He discovered perfect numbers and developed methods for solving indeterminate equations.

In ancient India, mathematics was seen as a reflection of the inherent order and harmony of the universe. It was believed that by studying mathematics, one could gain insight into the fundamental principles that govern the cosmos.

The concept of sine, introduced by Indian mathematicians, played a crucial role in the development of trigonometry. It enabled the calculation of angles and distances in triangles, which had practical applications in astronomy, surveying, and navigation.

The concept of Brahman in Indian philosophy refers to the ultimate reality, the underlying unity and interconnectedness of all things in the universe. It is often seen as the source and essence of all existence.

Aryabhata, in addition to his contributions to mathematics, also made significant advances in astronomy. He proposed a heliocentric model of the solar system, where the Earth and other planets revolve around the Sun.

The Vedanta school of philosophy, particularly the Advaita Vedanta tradition, emphasized the importance of personal experience and direct realization of the ultimate truth, often referred to as Brahman or Atman.

The quadratic formula, developed by Indian mathematicians, provided a general method for solving quadratic equations. This formula had a profound impact on the development of algebra and enabled the solution of a wide range of mathematical problems.

The concept of Karma in Indian philosophy refers to the law of cause and effect, where one's actions in this life determine their fate in the next. It emphasizes the cyclical nature of existence and the interconnectedness of all beings.

Bhaskara II, a renowned mathematician of the 12th century CE, made significant contributions to combinatorics. He developed Pascal's triangle and explored various combinatorial problems, including permutations and combinations.

The concept of Vairagya in Indian philosophy refers to the renunciation of worldly desires and attachments. It is seen as a path to spiritual liberation and the realization of the true nature of reality.

The mean value theorem, discovered by Indian mathematicians, is a fundamental result in calculus. It provides a connection between the average value of a function over an interval and the value of the function at a specific point within that interval.