Charles Babbage sometimes referred as "father of computing". Machines for calculating fixed numerical tasks such as the abacus have existed since antiquity, aiding in computations such as multiplication and division. Further, algorithms for performing computations have existed since antiquity, even before the development of sophisticated computing equipment.

Algorithm example[ edit ] An animation of the quicksort algorithm sorting an array of randomized values. The red bars mark the pivot element; at the start of the animation, the element farthest to the right-hand side is chosen as the pivot.

One of the simplest algorithms is to find the largest number in a list of numbers of random order.

Finding the solution requires looking at every number in the list. From this follows a simple algorithm, which can be stated in a high-level description of English prose, as: If there are no numbers in the set then there is no highest number.

Assume the first number in the set is the largest number in the set. For each remaining number in the set: When there are no numbers left in the set to iterate over, consider the current largest number to be the largest number of the set. Written in prose but much closer to the high-level language of a computer program, the following is the more formal coding of the algorithm in pseudocode or pidgin code: A list of numbers L.

The largest number in the list L. Euclid's algorithm The example-diagram of Euclid's algorithm from T. Heathwith more detail added.

Euclid does not go beyond a third measuring and gives no numerical examples.

Nicomachus gives the example of 49 and He defines "A number [to be] a multitude composed of units": To "measure" is to place a shorter measuring length s successively q times along longer length l until the remaining portion r is less than the shorter length s.

Euclid's original proof adds a third requirement: Euclid stipulated this so that he could construct a reductio ad absurdum proof that the two numbers' common measure is in fact the greatest.

So, to be precise, the following is really Nicomachus' algorithm. A graphical expression of Euclid's algorithm to find the greatest common divisor for and A location is symbolized by upper case letter se. The varying quantity number in a location is written in lower case letter s and usually associated with the location's name.

An inelegant program for Euclid's algorithm[ edit ] "Inelegant" is a translation of Knuth's version of the algorithm with a subtraction-based remainder-loop replacing his use of division or a "modulus" instruction.

Derived from Knuth Depending on the two numbers "Inelegant" may compute the g. The following algorithm is framed as Knuth's four-step version of Euclid's and Nicomachus', but, rather than using division to find the remainder, it uses successive subtractions of the shorter length s from the remaining length r until r is less than s.

The high-level description, shown in boldface, is adapted from Knuth Until the remaining length r in R is less than the shorter length s in S, repeatedly subtract the measuring number s in S from the remaining length r in R. The nut of Euclid's algorithm. Use remainder r to measure what was previously smaller number s; L serves as a temporary location.

S contains the greatest common divisor ]: An elegant program for Euclid's algorithm[ edit ] The following version of Euclid's algorithm requires only six core instructions to do what thirteen are required to do by "Inelegant"; worse, "Inelegant" requires more types of instructions.

The flowchart of "Elegant" can be found at the top of this article. Testing the Euclid algorithms[ edit ] Does an algorithm do what its author wants it to do?

A few test cases usually suffice to confirm core functionality. One source [55] uses and Knuth suggested Another interesting case is the two relatively prime numbers and But exceptional cases must be identified and tested.

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