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,^[1] effective numbers,^[2] 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.^{[citation needed]}