Computable number

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, effective numbers, computable reals, or recursive reals. The concept of a computable real number was introduced by Émile Borel in 1912, using the intuitive notion of computability available at the time.

Equivalent definitions can be given using μ-recursive functions, Turing machines, or λ-calculus as the formal representation of algorithms. The set of computable numbers is sometimes denoted by 𝕂. The idea of computable numbers can be further extended to computable complex numbers(or gaussian computable numbers) - whose both real and imaginary components are computable. 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

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 [is] one for which there is a Turing machine which, given n on its initial tape, terminates with the nth digit of that number [encoded on its tape].

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 state 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 a numerical value be attained to within any given accuracy.