The 9th-Century Baghdad Mathematician Whose Name Became the Word 'Algorithm' and Whose Book Title Became 'Algebra'

Muḥammad ibn Mūsā al-Khwārizmī (c. 780–c. 850) was a mathematician and astronomer at the court of the Abbasid caliph al-Maʾmūn in Baghdad. His name suggests an origin in the Central Asian region of Khwarazm, south of the Aral Sea — modern Khiva, Uzbekistan — though he is documented only at Baghdad. He worked at the Bayt al-Ḥikma — the House of Wisdom — al-Maʾmūn’s state-sponsored translation and research institution that, between approximately 815 and 835, produced the systematic Arabic translation of the surviving Greek, Persian, and Indian scientific corpus.

Al-Khwarizmi’s surviving books are few but consequential.

The algebra book

His al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala (“The Compendious Book on Calculation by Completion and Balancing”), written between approximately 820 and 830, presents the first systematic Arabic-language treatment of what is now called elementary algebra. The book gives general methods for solving linear and quadratic equations in one unknown — the methods of al-jabr (restoration: moving a subtracted quantity to the other side of the equation by adding it to both sides) and al-muqābala (balancing: cancelling equal positive quantities on both sides).

The book uses no symbols. Everything is written out in prose. An equation like x² + 10x = 39 appears as “a square and ten roots are equal to thirty-nine units.” Despite the verbal form, the procedures are general — al-Khwarizmi gives complete methods for the six canonical forms of linear and quadratic equation, with geometric demonstrations of why each method works.

The Latin translation of the title’s word al-jabr produced the European word algebra. The Latin translation of the book itself — by Robert of Chester at Toledo in 1145 — was the textbook that introduced Europe to the discipline. Algebra enters the English language in the 1540s.

The arithmetic book and the word ‘algorithm’

Al-Khwarizmi’s separate book Kitāb al-jamʿ wa-l-tafrīq bi-ḥisāb al-Hind (“The Book of Addition and Subtraction According to the Hindu Calculation”), written around the same period, introduced the Hindu decimal positional numeral system to the Arabic-speaking scientific world. The system uses ten digits (including zero), positional place-value, and a small number of step-by-step computational procedures for the four arithmetic operations.

The Arabic original is lost. The 12th-century Latin translation — beginning Dixit Algoritmi, “Algoritmi said,” with Algoritmi being the Latinisation of al-Khwarizmi — circulated in European universities as the standard introduction to decimal arithmetic.

The 12th-century Latin form of his name was generalised in late-medieval Europe to mean any step-by-step computational procedure. The word algorithm in modern English derives directly from the Latin algoritmi, which is in turn the Latin transliteration of al-Khwarizmi’s geographic name.

The 21st-century word for the foundational concept of computer science is an Arabic-medieval place name.

What else he did

Al-Khwarizmi compiled at al-Maʾmūn’s request the Zīj al-Sindhind, a set of astronomical tables built on Indian and Greek precedents and recalibrated for Baghdad’s latitude. The Zīj remained the standard Islamic astronomical reference for two centuries.

He led al-Maʾmūn’s geographical project that produced the Kitāb ṣūrat al-arḍ (“Book of the Description of the Earth”) — a corrected and updated Arabic version of Ptolemy’s geographical coordinates, with approximately 2,400 cities relocated on the basis of new astronomical observations. The Mediterranean’s length, which Ptolemy had given as 62 degrees, was corrected to approximately 50 degrees — almost exactly the modern value. The correction did not reach European cartography until the 17th century.

He produced the trigonometric sine and cosine tables (using Indian sources) that became the foundation of Islamic-world spherical astronomy.

He died at Baghdad around 850. The location of his grave is unknown. The English words algorithm and algebra are his memorial.