Computable number

A real number is computable if an algorithm can produce rational approximations within any requested positive error. This includes every rational number and many irrational numbers.

Cauchy sequence: A sequence whose terms become arbitrarily close to one another as the indices grow. Effective Cauchy sequences provide a standard way to represent computable reals.
Real number: An element of the number system extending the rational numbers to include limits and irrational magnitudes. Computable numbers form a precisely defined subset of the real numbers.
Noncomputable number: A real number for which no algorithm produces rational approximations to every requested accuracy. It fails exactly the effective approximation condition that defines computable numbers.
Computable analysis: The study of effective computation on real numbers, functions, and other infinite mathematical objects. It develops a general framework for algorithms operating on computable reals.
Fast-converging sequence: A sequence with a known error bound that shrinks at a specified rate. A rapid error bound lets an algorithm know when an approximation is accurate enough.
Rational approximation: A rational number chosen to stand in for another number within a specified error. Computability is defined by the ability to produce such approximations on demand.
Chaitin's constant: A real number encoding the halting probability of a universal prefix-free program. Its digits cannot be computed by an algorithm, despite its precise mathematical definition.
Numerical analysis: The study of algorithms for approximating mathematical problems with finite computations. Its error estimates help turn computable-real definitions into practical numerical methods.
Turing machine: An abstract model of computation that manipulates symbols according to finite rules. A Turing machine formalizes the algorithm that produces each rational approximation.
Absolute error: The nonnegative distance between an approximate value and the value being approximated. The requested accuracy is a bound on the absolute error of the output.

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Countable setBroader topic: Algorithms form a countable collection, so computable reals are countable.