Ancient Indians Mathematics & Number System

• Finite Difference Interpolation:                                                                                                 The Indian mathematician Brahmagupta presented what is possibly the first instance of finite difference interpolation around 665 CE.
• Algebraic abbreviations:                                                                                                      The mathematician Brahmagupta had begun using abbreviations for unknowns by the 7th century. He employed abbreviations for multiple unknowns occurring in one complex problem.^] Brahmagupta also used abbreviations for square roots and cube roots.
• Basu's theorem:                                                                                                                    The Basu's theorem, a result of Debabrata Basu (1955) states that any complete sufficient statistic is independent of any ancillary statistic.

Number System	Numbers
	0	1	2	3	4	5	6	7	8	9
Gurmukhi	o	੧	੨	੩	੪	੫	੬	੭	੮	੯
Oriya	୦	୧	୨	୩	୪	୫	୬	୭	୮	୯
Bengali	০	১	২	৩	৪	৫	৬	৭	৮	৯
Devanagari	०	१	२	३	४	५	६	७	८	९
Gujarati	૦	૧	૨	૩	૪	૫	૬	૭	૮	૯
Tibetan	༠	༡	༢	༣	༤	༥	༦	༧	༨	༩
Brahmi
Telugu	౦	౧	౨	౩	౪	౫	౬	౭	౮	౯
Kannada	೦	೧	೨	೩	೪	೫	೬	೭	೮	೯
Malayalam	൦	൧	൨	൩	൪	൫	൬	൭	൮	൯
Tamil	೦	௧	௨	௩	௪	௫	௬	௭	௮	௯
Burmese	၀	၁	၂	၃	၄	၅	၆	၇	၈	၉
Khmer	០	១	២	៣	៤	៥	៦	៧	៨	៩
Thai	๐	๑	๒	๓	๔	๕	๖	๗	๘	๙
Lao	໐	໑	໒	໓	໔	໕	໖	໗	໘	໙
Balinese	᭐	᭑	᭒	᭓	᭔	᭕	᭖	᭗	᭘	᭙
Javanese	꧐	꧑	꧒	꧓	꧔	꧕	꧖	꧗	꧘	꧙

The half-chord version of the sine function was developed by the Indian mathematician Aryabhatta.

Brahmagupta's theorem (598–668) states that AF = FD.

• Brahmagupta–Fibonacci identity, Brahmagupta formula, Brahmagupta matrix, and Brahmagupta theorem:                                                                                             Discovered by the Indian mathematician, Brahmagupta (598–668 CE)
• Chakravala method:                                                                                                             The Chakravala method, a cyclic algorithm to solve indeterminate quadratic equations is commonly attributed to Bhāskara II, (c. 1114 – 1185 CE) although some attribute it to Jayadeva (c. 950~1000 CE). Jayadeva pointed out that Brahmagupta’s approach to solving equations of this type would yield infinitely large number of solutions, to which he then described a general method of solving such equations. Jayadeva's method was later refined by Bhāskara II in his Bijaganita treatise to be known as the Chakravala method, chakra (derived from cakraṃ चक्रं) meaning 'wheel' in Sanskrit, relevant to the cyclic nature of the algorithm. With reference to the Chakravala method, E. O. Selenuis held that no European performances at the time of Bhāskara, nor much later, came up to its marvellous height of mathematical complexity.
• Hindu number system:                                                                                                           With decimal place-value and a symbol for zero, this system was the ancestor of the widely used Arabic numeral system. It was developed in the Indian subcontinent between the 1st and 6th centuries CE.
• 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).
• 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).
• 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.
• 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 to obtain an infinite series expression for π.^[128] 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,^[130] for π correct up to eleven decimal places, i.e. . Madhava of Sangamagrama and his successors at the Kerala school of astronomy and mathematics used geometric methods to derive large sum approximations for sine, cosin, and arttangent. 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.
• Pascal's triangle:                                                                                                                 Described in the 6th century CE by Varahamihira and in the 10th century by Halayudha, commenting on an obscure reference by Pingala (the author of an earlier work on prosody) to the "Meru-prastaara", or the "Staircase of Mount Meru", in relation to binomial coefficients. (It was also independently discovered in the 10th or 11th century in Persia and China.)
• Pell's equation, integral solution for:                                                                            About a thousand years before Pell's time, Indian scholar Brahmagupta (598–668 CE) was able to find integral solutions to vargaprakṛiti (Pell's equation): where N is a nonsquare integer, in his Brâhma-sphuṭa-siddhânta treatise.
• Ramanujan theta function, Ramanujan prime, Ramanujan summation, Ramanujan graph and Ramanujan's sum:                                                                   Discovered by the Indian mathematician Srinivasa Ramanujan in the early 20th century.
• Shrikhande graph:                                                                                                             Graph invented by the Indian mathematician S.S. Shrikhande in 1959.
• Sign convention:                                                                                                                     Symbols, signs and mathematical notation were employed in an early form in India by the 6th century when the mathematician-astronomer Aryabhata recommended the use of letters to represent unknown quantities. By the 7th century Brahmagupta had already begun using abbreviations for unknowns, even for multiple unknowns occurring in one complex problem. Brahmagupta also managed to use abbreviations for square roots and cube roots. By the 7th century fractions were written in a manner similar to the modern times, except for the bar separating the numerator and the denominator. A dot symbol for negative numbers was also employed. The Bakhshali Manuscript displays a cross, much like the modern '+' sign, except that it symbolized subtraction when written just after the number affected. The '=' sign for equality did not exist. Indian mathematics was transmitted to the Islamic world where this notation was seldom accepted initially and the scribes continued to write mathematics in full and without symbols.
• Trigonometric functions (adapted from Greek):                                                               The trigonometric functions sine and versine originated in Indian astronomy, adapted from the full-chord Greek versions (to the modern half-chord versions). They were described in detail by Aryabhata in the late 5th century, but were likely developed earlier in the Siddhantas, astronomical treatises of the 3rd or 4th century. Later, the 6th-century astronomer Varahamihira discovered a few basic trigonometric formulas and identities, such as sin^2(x) + cos^2(x) = 1.