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,[1]effective numbers[2] or the computable reals[3] or recursive reals.[4] The concept of a computable real number was introduced by Émile Borel in 1912, using the intuitive notion of computability available at the time.[5]
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.
^Rogers, Hartley, Jr. (1959). "The present theory of Turing machine computability". Journal of the Society for Industrial and Applied Mathematics. 7: 114–130. MR0099923.{{cite journal}}: CS1 maint: multiple names: authors list (link)
^P. Odifreddi, Classical Recursion Theory (1989), p.8. North-Holland, 0-444-87295-7