Algorithms
(after al-Khuwarizmi, the first scholar who had generalized their application)
All About Algorithms with Marilyn Burns
An algorithm is a specified sequence of steps that lead to a particular goal.
The Math Gene: How Mathematical Thinking Evolved and Why Numbers are Like Gossip. Basic Books. p.10.
Here we define algorithm in a customary way, as a finite, step-by-step procedure for accomplishing a task that we wish to complete.
Usiskin, Z. (1998). Paper-and-Pencil Algorithms in a Calculator-and-Computer Age. In the Teaching and Learning of Algorithms in School Mathematics: 1998 NCTM Yearbook (Morrow & Kenney, Eds.). Reston, VA: NCTM. p. 7.
An algorithm is a step-by-step procedure designed to achieve a certain objective in a finite time, often with several steps that repeat or “loop” as many times as necessary. The most familiar algorithms are the elementary school procedures for adding, subtracting, multiplying, and dividing, but there are many other algorithms in mathematics.
From Algorithms in Everyday Mathematics
An algorithm is a precise, systematic method for solving a class of problems.
Maurer, S. B. (1998). What is an Algorithm? What is an Answer? In the Teaching and Learning of Algorithms in School Mathematics: 1998 NCTM Yearbook (Morrow & Kenney, Eds.). Reston, VA: NCTM p. 21.
An algorithm is a computational recipe for the systematic execution of a procedure designed to solve a specific problem that maintains the following:
1. Input data along with a finite set of instructions;
2. A computing agent reacts to the input and instructions and carries out the steps;
3. Intermediate results are stored and used;
4. The computation is carried out in a discrete, stepwise fashion; and
5. The computing agent interprets the set of instructions in such a way that computation is carried out deterministically, without resort to random methods.
Mingus, T. Y. & Grassl, R. M.. (1998). Algorithmic and Recursive Thinking: Current Beliefs and Their Implications for the Future. In the Teaching and Learning of Algorithms in School Mathematics: 1998 NCTM Yearbook (Morrow & Kenney, Eds.). Reston, VA: NCTM p. 33.
ALGORITHM (a systematic technique for solving a problem) was used by Gottfried Wilhelm Leibniz (1646-1716):
Ex cognito hoc velut Algorithmo, ut ita dicam, calculi hujus, quem voco differentialem, omnes aliae aequationes differentiales inveniri possunt per calculum communem, maximae que & minimae, item que tangentes haberi, ita ut opus non sit tolli fractas aut irrationales, aut alia vincula, quod tamen faciendum fuit secundum Methodos hactenus editas. (From this rule, known as an algorithm, so to speak, of this calculus, which I call differential, all other differential equations may be found by means of a general calculus, and maxima and minima, as well as tangents [may be] obtained, so that there may be no need of removing fractions, nor irrationals, nor other aggregates, which nevertheless formerly had to be done in accordance with the methods published up to the present.)
The citation above is from "Nova Methodvs pro maximis et minimis, itemque tangentibus, quae nec fractas, nec irrationales quantitates moratur, & singulare pro illis calculi genus, per G.G.L." (A new method for maxima and minima, as well as tangents, which is not obstructed by fractional or irrational quantities), Leibniz' first published account of the calculus [Acta Eruditorum, vol. 3, pp. 467-473, October 1684], page 469. Both terms "Algorithmo" and "differentialem" are italicized in the original. The English translation is from Evelyn Walker's translation of extracts from Leibniz' memoir found on p. 623 of Smith's Source Book in Mathematics (1929), vol. 2.
Apparently the earliest English translation was carried out by Joseph Raphson in The Theory of Fluxions, Shewing in a compendious manner The first Rise of, and various Improvements made in that Incomparable Method, London, 1715: "Now from this being known as the Algorithm, as I may say of this Calculus, which I call differential, ..." (p.23). The word was then taken up by Euler, for instance in his article 'De usu novi algorithmi in problemate Pelliano solvendo', and its use was then firmly established. [Julio González Cabillón, David Fowler]
The word algorithm is derived from the much older word algorism, and "influenced by the Greek word arithmos (number)," according to the OED2. However, according to Jan Hogendijk of the Utrecht University it was not so influenced. Algorism (meaning "the Hindu-Arabic system of numeration or calculations using it") is derived from the Arabic al-Khowarazmi, the native of Khwarazm (Khiva), surname of the Arab mathematician and astronomer Abu Ja'far Mohammed Ben Musa (c. 780 - c.850).
According to the Theseus Logic, Inc., website, "The term algorithm was not, apparently, a commonly used mathematical term in America or Europe before Markov, a Russian, introduced it. None of the other investigators, Herbrand and Godel, Post, Turing or Church used the term. The term however caught on very quickly in the computing community."
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