Foundations of Mathematics

Mathematical and Formal Logic

What is an algorithm?

The word “algorithm” is a distortion of Muhammad ibn Musa al-Khuwarizmi’s name (783-c. 850; also seen as al-Khowarizmi and al-Khwarizmi), the Persian mathematician who wrote about algebraic methods (for more about al-Khuwarizmi, see “History of Mathematics”). In general, an algorithm is a specific set of instructions that, if followed correctly, will lead to a recognizable end result. Simply put, a recipe is an example of an algorithm. For example, if there are two different recipes for making apple pie—one calling for cutting fresh apples for the filling, the other calling for apples from a can—the end results will be the same: an apple pie.

In mathematics, most algorithms include a finite sequence of steps that repeat, or require decisions using logic and comparisons until the final result is found (often called a computation). The best example is the long-division algorithm, in which the remainders of partial divisions are carried to the next digit or digits. For example, in the division of 1,347 by 8, a remainder of 5 in the division of 13 by 8 is placed in front of the 4, and 8 is then divided into “54,” and so on. More advanced use of algorithms are found in a type of logic called metamathematics (see below).



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