Editing Computable number (section)

{{short description|Real number that can be computed within arbitrary precision}}
{{distinguish|constructible number}}
[[File:10,000 digits of pi - poster.svg|thumb|[[Pi|π]] can be computed to arbitrary precision, while [[almost every]] real number is not computable.]]
In [[mathematics]], '''computable numbers''' are the [[real number]]s that can be computed to within any desired precision by a finite, terminating [[algorithm]]. They are also known as the '''recursive numbers''',<ref>{{cite book
 | last = Mazur | first = Stanisław | author-link = Stanisław Mazur
 | editor1-last = Grzegorczyk | editor1-first = Andrzej | editor1-link = Andrzej Grzegorczyk
 | editor2-last = Rasiowa | editor2-first = Helena | editor2-link = Helena Rasiowa
 | page = 4
 | publisher = [[Institute of Mathematics of the Polish Academy of Sciences]]
 | series = Rozprawy Matematyczne
 | title = Computable analysis
 | url = https://eudml.org/doc/268535
 | volume = 33
 | year = 1963}}</ref> '''effective numbers''',{{sfnp|van der Hoeven|2006}} '''computable reals''',<ref>{{cite journal
 | last1 = Pour-El | first1 = Marian Boykan | author1-link = Marian Pour-El
 | last2 = Richards | first2 = Ian
 | doi = 10.1016/0001-8708(83)90004-X | doi-access=free
 | issue = 1
 | journal = [[Advances in Mathematics]]
 | mr = 697614
 | pages = 44–74
 | title = Noncomputability in analysis and physics: a complete determination of the class of noncomputable linear operators
 | volume = 48
 | year = 1983}}</ref> or '''recursive reals'''.<ref>{{cite journal
 | last = Rogers | first = Hartley, Jr.
 | journal = Journal of the Society for Industrial and Applied Mathematics
 | mr = 99923
 | pages = 114–130
 | title = The present theory of Turing machine computability
 | volume = 7
 | year = 1959| doi = 10.1137/0107009
 }}</ref> The concept of a computable real number was introduced by [[Émile Borel]] in 1912, using the intuitive notion of computability available at the time.<ref>P. Odifreddi, ''Classical Recursion Theory'' (1989), p.8. North-Holland, 0-444-87295-7</ref>

Equivalent definitions can be given using [[μ-recursive function]]s, [[Turing machine]]s, 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.{{Citation needed|date=October 2024}}