Al-Samaw'al ibn Yahya al-Maghribi

Al-Samaw'al ibn Yahya al-Maghribi was born around 1130 CE in Baghdad to a Jewish family of Moroccan origin — his father was a Hebrew poet and scholar. He showed extraordinary mathematical aptitude from childhood, studying under Abu Bakr ibn al-Husayn al-Khawarizmi and other Baghdad mathematicians. By his own account he had mastered most of the mathematics available to him by the age of thirteen, and wrote his most significant work at eighteen.

His masterwork, al-Bahir fi al-Jabr (The Brilliant in Algebra), written around 1148 CE, represents a landmark in the history of mathematics. In it, al-Samaw'al introduced and systematically developed the concept of operating with polynomials of high degree, treating algebraic expressions with negative exponents (negative powers of an unknown) and working with what amounts to the concept of a polynomial ring. He carried out polynomial long division and understood the zero polynomial as a distinct mathematical object.

Al-Samaw'al developed a method equivalent to mathematical induction to prove results about sums of powers, centuries before this technique was formally articulated in European mathematics. He also worked with arithmetic progressions and made significant contributions to combinatorial reasoning, including work on binomial coefficients that anticipates Pascal's triangle.

A central figure in his legacy is his intellectual courage: he engaged critically with earlier algebraists, explicitly identifying errors in al-Karaji's work while building upon it. He later converted to Islam, an event he described in his autobiographical work Decisive Refutation of the Jews and Christians. He traveled widely and died in Maragha, Iran, around 1180 CE. His work profoundly influenced later Islamic mathematics and, through transmission, European algebra.