The word algebra comes from the title of a book called Al-Jabr, written around the year 820 by a Persian mathematician named Muhammad ibn Musa al-Khwarizmi at the House of Wisdom in Baghdad — and the word algorithm comes from the Latinized version of his name

Two of the most substantially used words in the current global vocabulary of mathematics and computing come from the same 9th-century Persian scholar working in the same Baghdad library across approximately the same twenty-year period between 813 and 833 CE. The first — “algebra” — is derived from the title of the specific book he wrote around the year 820, whose Arabic name (Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wal-muqābala, “The Compendious Book on Calculation by Completion and Balancing”) was subsequently abbreviated in medieval Latin translation to the specific fragment al-jabr, meaning “restoration” or “completion” — the particular arithmetic operation of moving a subtracted term across an equation to render it positive. The second — “algorithm” — is derived from the Latinised form of his own name, which medieval European translators rendered variously as Algoritmi, Algorismi, or Algorismus, in reference to the specific systematic step-by-step arithmetic procedures his separate 9th-century book on Hindu numeral calculation had introduced to European mathematical practice.

The scholar was Abu Ja’far Muhammad ibn Musa al-Khwarizmi (c. 780 – c. 850 CE), and he worked at what the subsequent Islamic and European intellectual historiography has come to describe as the House of Wisdom (Bayt al-Hikma) in Baghdad, the multi-institutional library, translation bureau, and research academy that the Abbasid Caliph al-Ma’mun (reigned 813-833) had substantially expanded from a smaller institution founded by his father Harun al-Rashid in the late 8th century. The House of Wisdom’s specific institutional function, in the specific 9th-century context of the Islamic Golden Age, was to serve as the centralised repository for the systematic Arabic translation of essentially the entire surviving corpus of Greek philosophical, mathematical, medical, and astronomical scholarship — supplemented by the systematic Arabic translation of comparable Sanskrit, Persian, and Syriac scientific texts — for the specific purpose of preserving, extending, and eventually transmitting the accumulated pre-Islamic scientific inheritance of the Mediterranean and South Asian world through the Arabic-speaking Islamic empire that had, in the two centuries between the 632 CE death of the Prophet Muhammad and the 820 CE height of al-Ma’mun’s reign, come to control substantially the entire land area between the Iberian Peninsula and the Indus River.

Al-Khwarizmi’s specific institutional role at the House of Wisdom, according to the MacTutor History of Mathematics Archive at the University of St Andrews’ biographical entry on al-Khwarizmi, was that of a working scholar attached to the translation and research program — jointly with the three Banu Musa brothers (Muhammad, Ahmad, and al-Hasan ibn Musa), who were the substantially most prominent scientific patrons of the al-Ma’mun period — with substantive responsibilities across the specific combined disciplines of algebra, geometry, astronomy, and geography. Around 820, al-Ma’mun appointed him head of the House of Wisdom’s library, which contained by that point the substantially largest collection of scientific manuscripts assembled anywhere in the world since the destruction of the Great Library of Alexandria in late antiquity. In his position at the library, al-Khwarizmi had systematic access to both the surviving Greek mathematical corpus (Euclid, Archimedes, Diophantus, Ptolemy) and the substantially different Hindu mathematical tradition (which the House of Wisdom had begun translating from Sanskrit during al-Ma’mun’s reign), and could therefore synthesise elements of both into what would become a substantively new mathematical framework that neither predecessor tradition had produced.

What the algebra book actually did

The specific mathematical innovation of the Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wal-muqābala, per Britannica’s entry on al-Khwarizmi and his mathematical contributions, was to develop the first systematic treatment of the discipline that would subsequently be called algebra as a distinct branch of mathematics rather than as a specific solution technique embedded within arithmetic or geometry. Al-Khwarizmi identified six canonical forms of linear and quadratic equations, provided general methods for solving each form, offered geometric proofs to validate the solutions, and applied the resulting techniques to substantive practical problems including Islamic inheritance calculations (which required the systematic division of estates among multiple heirs under complex Quranic legal requirements), commercial calculations, and land measurement. The two specific operations that gave the book its title — al-jabr (moving a subtracted term across an equation to render it positive; roughly, “restoration”) and al-muqabala (subtracting equal terms from both sides of an equation; roughly, “balancing”) — were the fundamental algebraic manipulations that the book systematically taught. The specific term al-jabr, translated into medieval Latin in Robert of Chester’s 1145 CE Toledo translation as algebra, subsequently became the standard European name for the entire branch of mathematics that the book had substantially founded.

How his name became algorithm

The specific etymological transmission of al-Khwarizmi’s name into the modern English word “algorithm” proceeded through a substantially more circuitous linguistic path. As detailed by the Davidson Institute of Science Education’s summary of al-Khwarizmi’s specific contributions to modern mathematical vocabulary, al-Khwarizmi’s separate book on Hindu-Arabic numeral arithmetic — a text whose original Arabic version has been lost but whose Latin translation (Algoritmi de numero Indorum, “Al-Khwarizmi on the Hindu Art of Reckoning”) survived — introduced European medieval mathematics to the Indian decimal positional numeral system, including the concept of zero, that had been substantially unknown in Roman-numeral-based Latin arithmetic. Across the subsequent four centuries, European medieval mathematicians used the specific Latin term algorismus (derived from the Latinised form of al-Khwarizmi’s name in the book’s title) to refer to the entire discipline of decimal arithmetic — the substantially new calculation method that European commerce and science were progressively adopting to replace Roman numeral arithmetic across the 12th-15th centuries. The word entered Middle English as augrim or algorism, migrated to Old French as augorisme, and eventually evolved (with a spelling change apparently influenced by the Greek word arithmos, “number,” in the 17th century) into the modern form algorithm. The specific expansion of the term from its original narrow meaning (“Hindu-Arabic decimal arithmetic”) to its current general meaning (“any systematic step-by-step computational procedure”) occurred progressively across the 17th, 18th, and 19th centuries as the concept of formal computational method became increasingly central to mathematics, logic, and eventually computer science.

What one 9th-century scholar produced

The specific historical fact that essentially every computer program written in essentially every programming language on essentially every computing device currently operating anywhere on Earth is executing what modern computer science calls “algorithms” — and that the modern English word “algorithm” derives from the Latinised name of a 9th-century Persian mathematician working at a Baghdad research library approximately 1,200 years ago — is one of the more substantial single continuities in the recorded intellectual history of the human species. Per the Lowell Milken Center’s summary of Muhammad ibn Musa al-Khwarizmi’s contributions to mathematical vocabulary and practice, al-Khwarizmi’s specific 9th-century Baghdad work productively synthesised the Greek geometric-mathematical tradition (which had substantially peaked in the 3rd century BCE), the Hindu decimal-arithmetic tradition (which had developed independently in India across the same 3rd-century BCE to 5th-century CE period), and the specific practical mathematical problems that the Islamic legal, commercial, and astronomical infrastructure of the 9th-century Abbasid Caliphate required to be solved. The specific transmission chain by which al-Khwarizmi’s synthesis subsequently reached medieval Europe — Arabic-to-Latin translations by Robert of Chester in Segovia in 1145 CE and by Gerard of Cremona in Toledo across the second half of the 12th century, followed by the progressive adoption of both his algebra and his Hindu-Arabic arithmetic across European universities between the 13th and 16th centuries — is essentially the specific historical mechanism by which the modern Western mathematical tradition that underlies contemporary global science acquired the two fundamental conceptual tools (algebra and algorithms) that carry the etymological signature of the 9th-century Persian scholar who produced them.