Al-Khwarizmi gave us both 'algebra' and 'algorithm'

Muhammad ibn Musa al-Khwarizmi (c. 780–850), a scholar at Baghdad’s House of Wisdom under Caliph al-Ma’mun, is the source of two everyday words: algebra, from the title of his treatise Hisab al-jabr w’al-muqabala, and algorithm, from the Latin form of his own name.

How did a book title become “algebra”?

The title — roughly “calculation by completion and balancing” — names his two key operations. Al-jabr, completion, removed a negative term by adding it to both sides: his own example turns x² = 40x − 4x² into 5x² = 40x. Al-muqabala, balancing, cancelled like terms. With these he reduced every linear or quadratic equation to one of six standard forms and solved each.

He did it all in words, with no symbols. His best-known problem reads “a square and 10 roots are equal to 39 units”, and he proved the answer geometrically by literally completing a square: four rectangles around a square of side x, four small corner squares, an outer square of area 64 and side 8, which gives a root of 3.

Is he the “father of algebra”? Historians disagree. Diophantus of Alexandria tackled equations centuries earlier, but the historian Solomon Gandz argued al-Khwarizmi had the better claim because he was the first to teach algebra in an elementary form and for its own sake, while Diophantus was mainly concerned with number theory. Others, such as John Crossley, rate his originality lower.

How did his name become “algorithm”?

Through a book that no longer exists in Arabic. His treatise on the Hindu-Arabic numerals — probably the first to use zero as a place holder in positional notation — is lost in the original, even its title. It survives in a 12th-century Latin translation, Algoritmi de numero Indorum, which puts his name in the title. As the text spread through Europe, the name was garbled into algorism, meaning arithmetic with the new numerals, and later into algorithm.

His name points to Khwarizm, south of the Aral Sea, though an epithet recorded by al-Tabari ties him to a district near Baghdad — perhaps only his ancestors came from Central Asia.