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"Algorithm": A Word from the 9th Century

Abstract

The word “algorithm” derives from the name of Muhammad ibn Musa al-Khwarizmi, a 9th-century Persian mathematician who worked at the House of Wisdom in Baghdad. When his book on arithmetic was translated into Latin in the 12th century, “al-Khwarizmi” became “Algoritmi” in Latin transliteration, and “Algoritmi dixit” (al-Khwarizmi says) became the title-like phrase introducing his methods. “Algorithm” (a procedure for solving a problem by a defined sequence of steps) carries the name of a man who died around 850 CE into every computer science course, textbook, and technical discussion in the world.

Al-Khwarizmi and the House of Wisdom

Muhammad ibn Musa al-Khwarizmi (c. 780–850 CE) worked at the Bayt al-Ḥikma (House of Wisdom) in Baghdad during the Islamic Golden Age under the Abbasid caliphs al-Maʾmūn and al-Mutawakkil. The House of Wisdom was the premier intellectual institution of the medieval world, a library, translation center, and research institution where scholars from across the Islamic world translated and extended Greek, Indian, and Persian scientific and mathematical works.

Al-Khwarizmi wrote two works that transformed European mathematics:

  1. Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala (The Compendious Book on Calculation by Completion and Balancing, c. 820 CE): A systematic treatment of linear and quadratic equations. The word “al-jabr” in the title gave European mathematicians the word “algebra”.

  2. Kitāb al-ḥisāb al-hindī (Book on Hindu Arithmetic, c. 825 CE): An explanation of the Indian positional decimal numeral system, the system we now call Hindu-Arabic numerals, using digits 0–9 with positional value. This book, when translated into Latin in the 12th century, introduced the decimal place-value system to Europe.

The Latin Translation and “Algorismus”

Al-Khwarizmi’s arithmetic book was translated into Latin in the 12th century, possibly by Adelard of Bath. The translator began the text with a phrase in the convention of the era: Dixit Algoritmi (“Al-Khwarizmi says”) using a Latin transliteration of the author’s name. “Algoritmi” (a genitive form, “of al-Khwarizmi”) became attached to the procedures described in the book.

European scholars began using “algorismus” (later “algorism”) to mean the system of arithmetic using Hindu-Arabic numerals with place value, as opposed to calculation using Roman numerals or an abacus. “Algorism” meant specifically: calculation by the new positional number system that al-Khwarizmi had described.

The word’s meaning drifted over centuries: from al-Khwarizmi’s specific numeral system, to any systematic calculation procedure, to the modern computer science definition: a finite sequence of unambiguous instructions for solving a class of problems.

The Mathematical Concept Before the Word

The concept of an algorithm (a defined procedure for solving a problem, guaranteed to terminate in a correct answer) is older than the word. The Algorithm as a Concept traces its history through Euclid’s algorithm for finding the greatest common divisor (c. 300 BCE), Hindu astronomical calculation procedures, and medieval arithmetic. Al-Khwarizmi’s contribution was not the invention of the concept but the systematic codification and transmission of specific algorithmic methods to European mathematics.

The formal definition of what an algorithm is (the question of what is and is not computable) was not settled until Alan Turing’s 1936 paper on computable numbers and Alonzo Church’s lambda calculus. The word “algorithm” is ancient; the theory of algorithms is 20th-century mathematics.

Algebra from the Same Author

The title of al-Khwarizmi’s algebra book (al-jabr wa-l-muqābala) contained two technical terms. Al-jabr meant “completion” or “restoration”, the operation of moving a subtracted quantity to the other side of an equation (e.g., converting x - 2 = 5 to x = 7 by “restoring” the 2). This word, when the text was translated into Latin and Spanish, became “algebra” in European languages.

Al-Khwarizmi’s name thus gave two fundamental mathematical terms to modern science: “algorithm” from his name itself, and “algebra” from the title of one of his books. The history of mathematics and the history of computing share a linguistic debt to a mathematician who died twelve centuries ago.

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