Al-Khwarizmi's popularizing treatise on algebra (The Compendious Book on Calculation by Completion and Balancing, c. 813–833 CE[8]:171) presented the first systematic solution of linear and quadratic equations. One of his principal achievements in algebra was his demonstration of how to solve quadratic equations by completing the square, for which he provided geometric justifications.[7]:14 Because he was the first to treat algebra as an independent discipline and introduced the methods of "reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation),[9] he has been described as the father[10][11][12] or founder[13][14] of algebra. The term algebra itself comes from the title of his book (specifically the word al-jabrmeaning "completion" or "rejoining").[15]His name gave rise to the terms algorism and algorithm.[16] His name is also the origin of (Spanish) guarismo[17]and of (Portuguese) algarismo, both meaning digit.
In the 12th century, Latin translations of his textbook on arithmetic (Algorithmo de Numero Indorum) which codified the various Indian numerals, introduced the decimal positional number system to the Western world.[18] The Compendious Book on Calculation by Completion and Balancing, translated into Latin by Robert of Chester in 1145, was used until the sixteenth century as the principal mathematical text-book of European universities.[19][20][21][22]
In addition to his best-known works, he revised Ptolemy's Geography, listing the longitudes and latitudes of various cities and localities.[23]:9 He further produced a set of astronomical tables and wrote about calendaric works, as well as the astrolabe and the sundial.[24]:669 He also made important contributions to trigonometry, producing accurate sine and cosine tables, and the first table of tangents.
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