Real number that can be computed within arbitrary precision
Not to be confused with constructible number.
π can be computed to arbitrary precision, while almost every real number is not computable.
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 Emile 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.
^Mazur, Stanisław (1963). Grzegorczyk, Andrzej; Rasiowa, Helena (eds.). Computable analysis. Rozprawy Matematyczne. Vol. 33. Institute of Mathematics of the Polish Academy of Sciences. p. 4.
^van der Hoeven (2006).
^Pour-El, Marian Boykan; Richards, Ian (1983). "Noncomputability in analysis and physics: a complete determination of the class of noncomputable linear operators". Advances in Mathematics. 48 (1): 44–74. doi:10.1016/0001-8708(83)90004-X. MR 0697614.
^Rogers, Hartley, Jr. (1959). "The present theory of Turing machine computability". Journal of the Society for Industrial and Applied Mathematics. 7: 114–130. MR 0099923.{{cite journal}}: CS1 maint: multiple names: authors list (link)
^P. Odifreddi, Classical Recursion Theory (1989), p.8. North-Holland, 0-444-87295-7
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PMID 19797653. S2CID 17187000. Manin, Yu. I. (1980). Vychislimoe i nevychislimoe [Computable and Noncomputable] (in Russian). Soviet Radio. pp. 13–15. Archived from...
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