algorithm

The word algorithm preserves the name of Muhammad ibn Musa al-Khwarizmi (c. 780–c. 850), a Persian scholar who worked at the House of Wisdom in Baghdad under the Abbasid caliphs. Around 825 he wrote a treatise explaining calculation with the Hindu-Arabic numerals, including zero and place value. The Arabic original is lost, but a twelfth-century Latin version opens with the words “Dixit Algorizmi” (“Thus spoke al-Khwarizmi”), and European readers turned his name into a label for the new method of reckoning. His other major work, on solving equations by balancing and restoring terms, gave English the word algebra.

The idea is far older than the word. Babylonian clay tablets record step-by-step procedures for arithmetic and for approximating square roots. In Book VII of the Elements, written about 300 BCE, Euclid described a method for finding the greatest common divisor of two numbers by repeated subtraction, still taught today as Euclid's algorithm. The sieve of Eratosthenes, another Greek technique, picks out prime numbers by crossing off multiples. These methods share the defining feature of an algorithm: a finite list of exact steps that produces a result without requiring fresh insight at each stage.

In 1843 Ada Lovelace published notes on Charles Babbage's proposed Analytical Engine that included a detailed procedure for computing Bernoulli numbers, often described as the first algorithm written for a machine. The concept received a rigorous mathematical definition in the 1930s. Alonzo Church used his lambda calculus, and Alan Turing, in a 1936 paper, imagined a simple abstract machine reading and writing symbols on a tape. Their work showed that some problems, such as deciding in general whether a given program will ever stop running, cannot be solved by any algorithm at all.

With electronic computers, algorithm became a central term of computer science. Donald Knuth's multivolume The Art of Computer Programming, begun in 1968, treated algorithms as objects to be analyzed for correctness and efficiency. Computer scientists compare algorithms by how their running time grows with the size of the input, a measure written in big O notation. Well-known examples include quicksort, devised by Tony Hoare around 1959, Edsger Dijkstra's method for finding shortest paths in a network, and PageRank, the link-ranking system that Larry Page and Sergey Brin developed at Stanford in the late 1990s.

Since the 2010s the word has also taken on a looser public meaning. People speak of “the algorithm” when they mean the automated systems that rank posts, videos, and products on platforms such as YouTube, Instagram, and TikTok. This usage has fed debates about algorithmic bias, in which systems trained on historical data reproduce unfair patterns in hiring, lending, or policing, and about how much of daily life should be shaped by rules that few people can see or question.