In the following, Marvin Minsky defines the numbers to be computed in a manner similar to those defined by Alan Turing in 1936; i.e., as "sequences of digits interpreted as decimal fractions" … See more

A real number a is computable if it can be approximated by some computable function $${\displaystyle f:\mathbb {N} \to \mathbb {Z} }$$ in the following manner: given any positive integer See more

Not computably enumerable
Assigning a Gödel number to each Turing machine definition produces a subset $${\displaystyle S}$$ of the natural numbers corresponding to the computable numbers and identifies a surjection from See more

Turing's original paper defined computable numbers as follows:
A real number is computable if its digit sequence can be produced by some algorithm or Turing machine. The algorithm takes an integer $${\displaystyle n\geq 1}$$ as … See more

The computable numbers include the specific real numbers which appear in practice, including all real algebraic numbers, as well as e, π, and many other transcendental numbers. Though the computable reals exhaust those reals we can calculate or … See more

Computer packages representing real numbers as programs computing approximations have been proposed as early as 1985, under the name "exact arithmetic". Modern examples include the CoRN library (Coq), and the RealLib package (C++). A … See more

• Constructible number
• Definable number
• Semicomputable function
• Transcomputational problem See more