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In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers or the computable reals or recursive reals. Equivalent definitions can be given using μ-recursive functions, Turing machines, or λ-calculus as the formal representation of algorithms. The computable numbers form a real closed field and can be used in the place of real numbers for many, but not all, mathematical purposes. ==Informal definition using a Turing machine as example== In the following, Marvin Minsky defines the numbers to be computed in a manner similar to those defined by Alan Turing in 1936; i.e., as "sequences of digits interpreted as decimal fractions" between 0 and 1: :"A computable number () one for which there is a Turing machine which, given ''n'' on its initial tape, terminates with the ''nth'' digit of that number (on its tape )." (Minsky 1967:159) The key notions in the definition are (1) that some ''n'' is specified at the start, (2) for any ''n'' the computation only takes a finite number of steps, after which the machine produces the desired output and terminates. An alternate form of (2) – the machine successively prints all n of the digits on its tape, halting after printing the nth – emphasizes Minsky's observation: (3) That by use of a Turing machine, a ''finite'' definition – in the form of the machine's table – is being used to define what is a potentially-''infinite'' string of decimal digits. This is however not the modern definition which only requires the result be accurate to within any given accuracy. The informal definition above is subject to a rounding problem called the table-maker's dilemma whereas the modern definition is not. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「computable number」の詳細全文を読む スポンサード リンク
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