Muhammad ibn Musa al-Khwarizmi (c. 780–850) probably came to Baghdad from the region of Khwarezm — today's western Uzbekistan — and worked in the milieu of the House of Wisdom under Caliph al-Ma'mun. The institution was at once a caliphal library and a translation bureau: Greek, Syriac, Pahlavi, and Sanskrit works were being rendered into Arabic, producing what was probably the largest single-language scientific corpus the world had then assembled. Al-Khwarizmi was not just a transmitter of this body of knowledge; he was the first author to lay down a mathematical method that bears his own stamp.

His Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabala, written around 820, turned equation-solving from an arithmetic accounting task into a symbolic method. "Al-jabr" in Arabic means "restoring, completing" — moving a term from one side of an equation to the other. "Al-muqabala" is comparing and simplifying like terms on the two sides. These two operations are still the exact core of school algebra. The book reduced linear and quadratic equations to six standard forms and gave a general solution for each. The variables were not yet written as x but in words; the method, though, was universal.

His second major contribution was to settle the Indian positional decimal system — the digits 0–9, and zero as a place-holder — into the working language of Islamic mathematics. The book in which he explained it, On Indian Numbers, survives only in a 12th-century Latin translation; the first words of that title were "Algoritmi de numero Indorum" — "Algoritmi on Indian numbers". In later centuries, Europeans began calling any step-by-step procedure an "algorithm", from the author's name. In the same passage of history al-jabr crossed over as "algebra".

The joint transmission of these three things — algebra, algorithm, zero — is a rare convergence in the history of mathematics. Algebra moved thinking from concrete numbers to abstract operations. The idea of an algorithm made the finite sequence of steps that solves a problem an object in its own right (the foundation of computer science). The decimal system with zero made calculation possible for ordinary people for the first time in history — anyone who has tried to multiply 23 by 47 in Roman numerals understands what was given here. The mathematical foundations of the Renaissance were laid in this system, learned by Italian merchants through al-Andalus.