<Li> Fibonacci numbers: This sequence was first described by Virahanka (c. 700 AD), Gopāla (c. 1135), and Hemachandra (c. 1150), as an outgrowth of the earlier writings on Sanskrit prosody by Pingala (c. 200 BC). </Li> <Li> Zero, symbol: Indians were the first to use the zero as a symbol and in arithmetic operations, although Babylonians used zero to signify the' absent' . In those earlier times a blank space was used to denote zero, later when it created confusion a dot was used to denote zero (could be found in Bakhshali manuscript). In 500 AD circa Aryabhata again gave a new symbol for zero (0). </Li> <Li> Law of signs in multiplication: The earliest use of notation for negative numbers, as subtrahend, is credited by scholars to the Chinese, dating back to the 2nd century BC . Like the Chinese, the Indians used negative numbers as subtrahend, but were the first to establish the "law of signs" with regards to the multiplication of positive and negative numbers, which did not appear in Chinese texts until 1299 . Indian mathematicians were aware of negative numbers by the 7th century, and their role in mathematical problems of debt was understood . Mostly consistent and correct rules for working with negative numbers were formulated, and the diffusion of these rules led the Arab intermediaries to pass it on to Europe., for example (+) × (-) = (-), (-) × (-) = (+) etc . </Li> <Li> Madhava series: The infinite series for π and for the trigonometric sine, cosine, and arctangent is now attributed to Madhava of Sangamagrama (c. 1340--1425) and his Kerala school of astronomy and mathematics . He made use of the series expansion of arctan ⁡ x (\ displaystyle \ arctan x) to obtain an infinite series expression for π . Their rational approximation of the error for the finite sum of their series are of particular interest . They manipulated the error term to derive a faster converging series for π . They used the improved series to derive a rational expression, 104348 / 33215 (\ displaystyle 104348 / 33215) for π correct up to eleven decimal places, i.e. 3.14159265359 (\ displaystyle 3.14159265359). Madhava of Sangamagrama and his successors at the Kerala school of astronomy and mathematics used geometric methods to derive large sum approximations for sine, cosine, and arctangent . They found a number of special cases of series later derived by Brook Taylor series . They also found the second - order Taylor approximations for these functions, and the third - order Taylor approximation for sine . </Li>

Inventions by india in the field of science in last 10 years