Algebraic Mathematical System

Timeline

1. c. 820–850Baghdad · IraqIn the Compendious Book on Calculation by Completion and Balancing (al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala), al-Khwarizmi systematically proposed the operations of al-jabr (restoration) and al-muqābala (balancing), classifying linear and quadratic equations by their coefficient relations and giving algorithmic solution procedures.
2. c. 1001–1010Baghdad · IraqAround 1001–1010 CE, al-Karaji, in works such as al-Fakhrī, promoted algebra's shift from dependence on geometry to independent computation, introducing polynomial operations and ideas of mathematical induction, marking the mature stage of the Islamic algebraic tradition.
3. c. 1050–1200Samarkand · UzbekistanThe Islamic algebraic system continued to evolve through the schools of Mesopotamia, Persia, and Central Asia, and entered Europe through Latin translations, becoming the basic framework for arithmetic and equation teaching in medieval universities.