alKaraji

Al-Karaji, also spelled al-Karaji or al-Karaji of Khwarezm, was a Persian mathematician active during the Islamic Golden Age, roughly in the late 10th to early 11th centuries. He is regarded as an important figure in the development of algebra, contributing to its transition from a problem-solving practice to a systematic manipulation of algebraic expressions.

His work is associated with extending algebra to include polynomials and powers, and with introducing methods that treat algebra as a discipline of manipulating symbols and identities rather than solely solving equations. He is often credited with introducing the use of mathematical induction to prove algebraic identities, including forms related to the binomial theorem for natural exponents. His approach emphasized deriving and employing polynomial expansions to reduce problems to simpler components, influencing later scholars in the Islamic mathematical tradition.

Al-Karaji wrote several algebraic and arithmetic treatises, though precise details and surviving titles are scattered in historical sources. What remains of scholarship on his work tends to emphasize his methodological contributions—namely, rigorous manipulation of expressions, systematic handling of powers, and the early use of inductive reasoning in proving algebraic facts. His writings helped lay groundwork that would be developed further by subsequent mathematicians in the medieval Islamic world and later in Europe during the Renaissance.

Overall, al-Karaji is regarded as an important link in the historical progression of algebra, notable for advancing symbolic manipulation and introducing inductive proofs that would shape later mathematical practice.