Computables

Computables, also known as computable numbers or recursive numbers, are real numbers for which there exists an algorithm that can compute their decimal digits to any desired precision. In essence, a computable number is one that can be described and manipulated by a computer program. This means that for any given computable number, one can write a set of instructions that, when followed, will produce the digits of that number one by one, as far as needed.

The concept of computability is deeply rooted in the foundations of computer science and mathematical logic. It is closely related to the idea of effective calculability, where a computation is considered effective if it can be performed by a human following a finite set of rules. Alan Turing's work on Turing machines provided a formal definition of computability, and a number is considered computable if its decimal expansion can be generated by a Turing machine.

Not all real numbers are computable. For example, the digits of transcendental numbers like pi and e are computable because algorithms exist to generate them. However, numbers that are defined by non-computable processes or that contain "random" or "uncomputable" information are not computable. The set of computable numbers is countably infinite, meaning that despite being infinite, they can be put into a one-to-one correspondence with the natural numbers. This property has significant implications in various fields of mathematics and theoretical computer science.