Al-Karaji

(; c. 953 – c. 1029) was a 10th-century Persian mathematician and engineer who flourished at Baghdad. He was born in Karaj, in particular for the beginnings of freeing algebra from geometry. Among historians, his most widely studied work is his algebra book al-fakhri fi al-jabr wa al-muqabala, which survives from the medieval era in at least four copies.

He expounded the basic principles of hydrology and this book reveals his profound knowledge of this science and has been described as the oldest extant text in this field.

He systematically studied the algebra of exponents, and was the first to define the rules for monomials like x, x2, x3 and their reciprocals in the cases of multiplication and division. However, since for example the product of a square and a cube would be expressed, in words rather than in numbers, as a square-cube, the numerical property of adding exponents was not clear.

His work on algebra and polynomials gave the rules for arithmetic operations for adding, subtracting and multiplying polynomials; though he was restricted to dividing polynomials by monomials.

F. Woepcke was the first historian to realise the importance of al-Karaji's work and.

Al-Karaji gave an early formulation of the binomial coefficients and the explicit description of Pascal's triangle. He is also presumed to have discovered the binomial theorem. In a now lost work known only from subsequent quotation by al-Samaw'al, Al-Karaji introduced the idea of argument by mathematical induction. As Katz says