Al-Khwarizmi

Muhammad ibn Musa al-Khwarizmi or . Ibn Khaldun notes in his Prolegomena: "The first to write on this discipline [algebra] was Abu 'Abdallah al-Khuwarizmi. After him, there was Abu Kamil Shuja' b. Aslam. People followed in his steps." In the introduction to his critical commentary on Robert of Chester's Latin translation of al-Khwārizmī's ''Algebra'', L.C. Karpinski notes that Abū Ja'far Muḥammad ibn Mūsā refers to the eldest of the Banū Mūsā brothers. Karpinski notes in his review on (Ruska 1917) that in (Ruska 1918): "Ruska here inadvertently speaks of the author as Abū Ga'far M. b. M., instead of Abū Abdallah M. b. M." Donald Knuth writes it as and quotes it as meaning "literally, 'Father of Abdullah, Mohammed, son of Moses, native of Khwārizm,'" citing previous work by Heinz Zemanek.}} (; ), often referred to as simply al-Khwarizmi, was a polymath who produced vastly influential Arabic-language works in mathematics, astronomy, and geography. Hailing from Khwarazm, he was appointed as the astronomer and head of the House of Wisdom in the city of Baghdad around 820 CE.

His popularizing treatise on algebra, compiled between 813–33 as ''Al-Jabr (The Compendious Book on Calculation by Completion and Balancing)'', presented the first systematic solution of linear and quadratic equations. One of his achievements in algebra was his demonstration of how to solve quadratic equations by completing the square, for which he provided geometric justifications. Because al-Khwarizmi was the first person 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), he has been described as the father or founder of algebra. The English term ''algebra'' comes from the short-hand title of his aforementioned treatise ( , ). His name gave rise to the English terms ''algorism'' and ''algorithm''; the Spanish, Italian, and Portuguese terms ; and the Spanish term and Portuguese term , both meaning "digit".

In the 12th century, Latin-language translations of al-Khwarizmi's textbook on Indian arithmetic (), which codified the various Indian numerals, introduced the decimal-based positional number system to the Western world. Likewise, ''Al-Jabr'', translated into Latin by the English scholar Robert of Chester in 1145, was used until the 16th century as the principal mathematical textbook of European universities.

Al-Khwarizmi revised ''Geography'', the 2nd-century Greek-language treatise by the Roman polymath Claudius Ptolemy, listing the longitudes and latitudes of cities and localities. He further produced a set of astronomical tables and wrote about calendric works, as well as the astrolabe and the sundial. Al-Khwarizmi made important contributions to trigonometry, producing accurate sine and cosine tables and the first table of tangents. Provided by Wikipedia