Medieval Muslims made invaluable contributions to the study of mathematics, and their key role is clear from the many terms derived from Arabic. Perhaps the most famous mathematician was Muhammad ibn Musa al-Khwarizmi (ca. 800-ca. 847), author of several treatises of earth-shattering importance. His book On the Calculation with Hindu Numerals, written about 825, was principally responsible for the diffusion of the Indian system of numeration (Arabic numerals) in the Islamic lands and the West.
Traditional systems had used different letters of the alphabet to represent numbers or cumbersome Roman numerals, and the new system was far superior, for it allowed people to multiply and divide easily and check their work. The merchant Leonardo Fibonacci of Pisa, who had learned about Arabic numerals in Tunis, wrote a treatise rejecting the abacus in favor of the Arab method of reckoning, and as a result, the system of Hindu-Arabic numeration caught on quickly in Central Italy. By the fourteenth century, Italian merchants and bankers had abandoned the abacus and were doing their calculations using pen and paper, in much the same way we do today.
In addition to his treatise on numerals, al-Khwarizmi also wrote a revolutionary book on resolving quadratic equations. These were given either as geometric demonstrations or as numerical proofs in an entirely new mode of expression. The book was soon translated into Latin, and the word in its title, al-jabr, or transposition, gave the entire process its name in European languages, algebra, understood today as the generalization of arithmetic in which symbols, usually letters of the alphabet such as A, B, and C, represent numbers. Al-Khwarizmi had used the Arabic word for "thing" (shay) to refer to the quantity sought, the unknown. When al-Khwarizmi's work was translated in Spain, the Arabic word shay was transcribed as xay, since the letter x was pronounced as sh in Spain. In time this word was abbreviated as x, the universal algebraic symbol for the unknown.
Robert of Chester's translation of al-Khwarzmi's treatise on algebra opens with the words dixit Algorithmi, "Algorithmi says." In time, the mathematician's epithet of his Central Asian origin, al-Khwarizmi, came in the West to denote first the new process of reckoning with Hindu-Arabic numerals, algorithmus, and then the entire step-by-step process of solving mathematical problems, algorithm.