Classical Indian mathematics (c. 400â€”1600)

Main article: Indian mathematics

See also: History of the Hindu-Arabic numeral system

Aryabhata

Aryabhata

The Surya Siddhanta (c. 400) introduced the trigonometric functions of sine, cosine, and inverse sine, and laid down rules to determine the true motions of the luminaries, which conforms to their actual positions in the sky. The cosmological time cycles explained in the text, which was copied from an earlier work, corresponds to an average sidereal year of 365.2563627 days, which is only 1.4 seconds longer than the modern value of 365.25636305 days. This work was translated to Arabic and Latin during the Middle Ages.

Aryabhata in 499 introduced the versine function, produced the first trigonometric tables of sine, developed techniques and algorithms of algebra, infinitesimals, differential equations, and obtained whole number solutions to linear equations by a method equivalent to the modern method, along with accurate astronomical calculations based on a heliocentric system of gravitation. An Arabic translation of his Aryabhatiya was available from the 8th century, followed by a Latin translation in the 13th century. He also computed the value of π to the fourth decimal place as 3.1416. Madhava later in the 14th century computed the value of π to the eleventh decimal place as 3.14159265359.

In the 7th century, Brahmagupta identified the Brahmagupta theorem, Brahmagupta's identity and Brahmagupta's formula, and for the first time, in Brahma-sphuta-siddhanta, he lucidly explained the use of zero as both a placeholder and decimal digit and explained the Hindu-Arabic numeral system. It was from a translation of this Indian text on mathematics (around 770) that Islamic mathematicians were introduced to this numeral system, which they adapted as Arabic numerals. Islamic scholars carried knowledge of this number system to Europe by the 12th century, and it has now displaced all older number systems throughout the world. In the 10th century, Halayudha's commentary on Pingala's work contains a study of the Fibonacci sequence and Pascal's triangle, and describes the formation of a matrix.

In the 12th century, Bhaskara first conceived differential calculus, along with the concepts of the derivative, differential coefficient and differentiation. He also stated Rolle's theorem (a special case of the mean value theorem), studied Pell's equation, and investigated the derivative of the sine function. From the 14th century, Madhava and other Kerala School mathematicians, further developed his ideas. They developed the concepts of mathematical analysis and floating point numbers, and concepts fundamental to the overall development of calculus, including the mean value theorem, term by term integration, the relationship of an area under a curve and its antiderivative or integral, tests of convergence, iterative methods for solutions of non-linear equations, and a number of infinite series, power series, Taylor series and trigonometric series. In the 16th century, Jyeshtadeva consolidated many of the Kerala School's developments and theorems in the Yuktibhasa, the world's first differential calculus text, which also introduced concepts of integral calculus. Mathematical progress in India became stagnant from the late 16th century onwards due to subsequent political turmoil.