Algorithm

• 1. Introduction

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  An algorithm is a finite sequence of mathematically rigorous instructions, used to solve a class of specific problems or to perform a computation. That definition sounds clinical, but it hides a strange lineage. The very word traces back to a single Persian scholar who wrote two books around 825 AD. His name, passed through Latin translators, became the label for an idea that now decides what videos you watch and what stocks get traded. How did a 9th-century method for arithmetic become the backbone of modern computing? And why are the newest machines starting to abandon the rigor that defined the concept in the first place? The answers run from a Sumerian clay tablet to a reinforcement learning system that rewrote a piece of standard software.

• 2. Naming The Idea

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  Muḥammad ibn Mūsā al-Khwārizmī wrote kitāb al-ḥisāb al-hindī, the "Book of Indian computation", around 825 AD. In the early 12th century, Latin translations of his work appeared, carrying the Hindu-Arabic numeral system into Europe. One such text, attributed to Adelard of Bath, was the Liber Algoritmi de numero Indorum. Its opening words were Dixit Algoritmi, meaning "Thus spoke Al-Khwarizmi". The word algorism entered English meaning the use of place-value notation in calculation. It appears in the Ancrene Wisse from around 1225. When Geoffrey Chaucer wrote The Canterbury Tales in the late 14th century, he described augrym stones, pebbles used for place-value sums. In the 15th century the Greek word arithmos, meaning "number", reshaped the Latin into algorithmus. By 1596, Thomas Hood used the form algorithm in English. The name had drifted far from the man, but his contribution went deeper than spelling, as the next chapter shows.

• 3. Steps Before Machines

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  A Sumerian clay tablet found in Shuruppak near Baghdad, dated to around 2500 BC, describes the earliest known division algorithm. Step-by-step procedures for mathematical problems reach back into antiquity across many cultures. During the Hammurabi dynasty around 1800 BC, Babylonian clay tablets set down algorithms for computing formulas and even for predicting astronomical events. Egyptian arithmetic appears in the Rhind Mathematical Papyrus, dated to around 1550 BC. Hellenistic mathematics gave two famous examples. The Sieve of Eratosthenes was described in the Introduction to Arithmetic by Nicomachus. The Euclidean algorithm was first set out in Euclid's Elements, around 300 BC. Al-Khwārizmī, in the 9th century, did something different. In The Compendious Book on Calculation by Completion and Balancing, he moved past single numerical answers to general procedures for algebraic reduction and balancing. He turned mathematics into a mechanical process of well-defined rules. That same century, Al-Kindi wrote A Manuscript On Deciphering Cryptographic Messages, giving the first description of codebreaking by frequency analysis.

• 4. From Clocks To Relays

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  The verge escapement mechanism, producing the tick of mechanical clocks, was a key European invention of the Middle Ages. Accurate automatic machines led to mechanical automata in the 13th century, and eventually to the difference and analytical engines of Charles Babbage and Ada Lovelace in the mid-19th century. Lovelace designed the first algorithm intended for a computer, written for Babbage's analytical engine. That engine is described as the first real Turing-complete computer. Though the full second device was built only decades after her death, Lovelace has been called "history's first programmer". The Jacquard loom, a precursor to punch cards, fed into this lineage alongside telephone switching machines. Ticker tape arrived around the 1870s, punch cards around 1890, and the teleprinter around 1910 with its Baudot code on tape. Telephone-switching networks of electromechanical relays were invented in 1835. These led George Stibitz, in 1937, to build an experimental digital adder. Working at Bell Laboratories, he had found mechanical calculators with gears "burdensome", and tinkered with a better design at home.

• 5. Pinning Down Calculability

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  In 1928, the modern concept of the algorithm began to take formal shape through attempts to solve David Hilbert's Entscheidungsproblem, the decision problem. Researchers tried to define what "effective calculability" or an "effective method" actually meant. The Gödel-Herbrand-Kleene recursive functions arrived across 1930-1934 and 1935. Alonzo Church introduced his lambda calculus in 1936. Emil Post published his Formulation 1 the same year. Alan Turing described his Turing machines across 1936-37 and 1939. These competing frameworks converged on a shared notion of mechanical computation. A Turing machine program can be expressed in several ways: as a sequence of machine tables, as flowcharts, or as rudimentary machine code called "sets of quadruples". Such descriptions sit at three accepted levels. A high-level description ignores how the machine is built. An implementation description explains how the head moves and stores data. A formal description gives the exact state table and list of transitions.

• 6. Counting The Cost

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  Knowing the time, storage, or other cost an algorithm needs is often vital. An algorithm that sums a list of n numbers need only remember two values: the running total and its current position in the list. A binary search outperforms a sequential search when looking up entries in sorted lists. The analysis and study of algorithms forms a discipline of computer science, often pursued abstractly without reference to any specific programming language. Pseudocode suits this work because it is simple and general. Yet most algorithms run on real hardware, and their efficiency gets tested with real code. Empirical testing can uncover unexpected interactions that affect performance, and benchmarks can compare an algorithm before and after optimization. Such tests cannot fully replace formal analysis and are difficult to run fairly. The stakes can be large. A recent innovation in the FFT algorithms used for image processing can cut processing time by up to 1,000 times for medical imaging. The best case of an algorithm is the input that demands the least time and resources; the worst case demands the most.

• 7. Families Of Methods

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  Brute force solves a problem by systematically trying every possible option until the optimal one is found, an approach used to crack passwords or find a shortest path. Divide-and-conquer instead reduces a problem to smaller copies of itself until each is easy, as merge sorting does by splitting and remerging a list. A simpler variant, prune and search, solves only one smaller instance with no merge step; binary search is an example. Recursion invokes itself until a termination condition is met, and the Tower of Hanoi is the classic puzzle solved this way. Every recursive version has an equivalent iterative one, and the reverse holds too. Algorithms also split by hardware. Serial algorithms run one instruction at a time, while parallel algorithms use multiple processors and distributed algorithms span machines on a network. Some problems, called inherently serial, have no parallel version. For optimization, the simplex algorithm tackles linear programming, and dynamic programming caches overlapping subproblems, as the Floyd-Warshall algorithm does for shortest paths. Greedy algorithms such as Kruskal and Prim build minimal spanning trees.

• 8. Machines That Write Algorithms

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  In 2023, Google DeepMind introduced AlphaDev, a reinforcement learning system based on AlphaZero. Reported in a paper in Nature, AlphaDev discovered small sorting algorithms that beat previously known human benchmarks. Those routines were integrated into the LLVM standard C++ sorting library. For decades, the assumed direction of progress ran the other way. Symbolic integration illustrates the old path. In 1961, James Slagle's program SAINT used heuristics to solve 52 of 54 freshman calculus exercises from an MIT textbook, roughly 96 percent. In 1967, Larry Moses's SIN refined those heuristics to 100 percent success, though it stayed heuristic. Then in 1969, Robert Risch introduced the Risch Algorithm with formal guarantees. The traditional trajectory ran from heuristics toward a definitive, guaranteed algorithm. The rise of transformer-based AI has inverted that sequence, with classical algorithms now giving way to heuristics again. In 2025, DeepMind introduced AlphaEvolve, an evolutionary coding agent powered by large language models. It proposes code changes, tests them with automated evaluators, and improves promising candidates over many iterations.