Muhammad ibn Musa al-Khwarizmi

Few details survive about al-Khwarizmi's life (c. 780-850), but his output during the Islamic Golden Age reshaped mathematics. Between 813 and 833, he compiled Al-Jabr, the first systematic treatment of linear and quadratic equations, demonstrating solutions by completing the square with geometric proof. By treating algebra as its own field and formalizing "reduction" and "balancing" - moving terms across the equals sign, canceling like quantities - he earned the title founder of algebra; the English word comes from his book's shorthand title, al-jabr, "completion." His textbook on Indian arithmetic, translated into Latin in the 12th century as Algorithmo de Numero Indorum, introduced the decimal positional system to the West. Translated again in 1145, Al-Jabr served as the principal math text in European universities until the 1500s. He also revised Ptolemy's Geography, produced astronomical tables, and built accurate sine and cosine tables. His latinized name, Algoritmi, became "algorithm."