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As an effective methodan algorithm can be expressed within a finite amount of space and time, [3] and in a well-defined formal language [4] for calculating a function. The transition from one state to the next is not necessarily deterministic ; some algorithms, known as randomized algorithmsincorporate random input.

The concept of algorithm has existed since antiquity. Arithmetic algorithms, such as a division algorithmwas used by ancient Babylonian mathematicians c.

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Later formalizations were framed as attempts to define " effective calculability " [15] or "effective method". The word 'algorithm' has its roots in Latinizing the nisba, indicating his geographic origin, of the name of Persian mathematician Muhammad ibn Musa al-Khwarizmi to algorismus. Aboutal-Khwarizmi wrote an Arabic language treatise on the Hindu—Arabic numeral system click here, which was translated into Latin during the 12th century. In English, it was first used in about and then by Chaucer in English adopted the French term, but it wasn't until the late 19th century that "algorithm" took on the meaning that it has in modern English.

Another early use of the word is fromin a manual titled Carmen de Algorismo composed by Alexandre de Villedieu. It begins with:. Algorism is the art by which at present we use those Indian figures, which number two times five.

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The poem is a few hundred lines long and summarizes the art of calculating with the new styled Indian dice Tali Indorumor Hindu numerals. An informal definition could Thf "a set of rules that precisely defines a sequence of operations", [27] [ need quotation to and Terrorism Israel ] which would include all computer programs including programs that do not perform numeric calculationsand for example any prescribed bureaucratic procedure [28] or cook-book recipe. In general, a program is only an algorithm if it stops eventually [30] - even though infinite loops may sometimes prove desirable. A prototypical example of an algorithm is the Euclidean algorithmwhich is used to determine the maximum common divisor of two integers; an example there are others is described by Literacy Is A Common Problem Around The flowchart above and as an example in a later section.

But humans can do something equally useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the n th member of the setfor arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which Ariund could be followed by a computing machineor by a human who is capable of carrying out only very elementary operations on symbols. An "enumerably infinite set" is one whose elements can be put into one-to-one correspondence with the integers.

Thus Boolos and Jeffrey are saying that an algorithm implies instructions for a process that "creates" output integers from an arbitrary "input" integer or integers that, in theory, can be arbitrarily large. The concept of algorithm is also used to define the notion of decidability —a notion that is central for explaining how formal systems come into being starting Probblem a small set of axioms and rules. In logicthe time that an algorithm requires to complete cannot be measured, as it is not apparently related to the customary physical dimension. From such uncertainties, that characterize ongoing work, stems the unavailability of a definition of algorithm that suits both concrete in some sense and abstract usage of the term.

Algorithms are essential to the way computers process data.

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Many computer programs contain algorithms that detail the specific instructions a computer should perform—in a specific order—to carry out a specified task, such as calculating employees' paychecks or printing students' report cards. Thus, an algorithm can be considered to be any sequence of operations that can be simulated by a Turing-complete system. Authors who assert this thesis include MinskySavage and Gurevich :.

Minsky: "But we will also maintain, with Turing … that any procedure which could "naturally" be called effective, can, in fact, be realized by a simple machine. Although this may seem extreme, the arguments … in its favor are hard to refute".]