Al-Karaji

Persian mathematician who freed algebra from geometry by arithmetizing algebraic operations.

Ask the Islamic Golden Age assistant 1 min read · Updated September 9, 2026

Who or what

Abu Bakr Muhammad ibn al-Hasan al-Karaji (c. 953–1029) was a Persian mathematician who worked in Baghdad.

His key works, "Al-Fakhri" and "Al-Badi'," advanced algebra beyond Al-Khwarizmi by treating it as an arithmetic discipline.

Al-Karaji broke from the geometric tradition of algebra, showing that algebraic operations could be performed purely arithmetically.

He discovered the binomial theorem and Pascal's triangle centuries before Pascal.

He explicitly stated the binomial expansion $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$ and extended it to higher powers.

He developed algorithms for finding sums of powers of integers, $\sum_{k=1}^n k$ , $\sum_{k=1}^n k^2$ , and $\sum_{k=1}^n k^3$ .

His inductive reasoning for the binomial coefficients anticipated the modern combinatorial approach.

He introduced the concept of "unknown quantity" ( $shay'$ , "thing") that later became the $x$ in Renaissance algebra.

His arithmetic approach made algebra more systematic and computational, influencing later European algebraists.

His work on sums of powers foreshadowed integral calculus.

Assembled from the ReLU.chat curated knowledge base. These explanations are concise on purpose; check the sources for anything important.