A computable number are numbers that can accepted by computers if computers had unlimited precision. Basically a number is computable if there exists a computer or a turing machine that can produce approximate up to digits.
Like the natural and rational numbers there is a countably infinite number of computable numbers. This is because there is a countable number of Turing machines or computer programs.
More amazingly numbers such as , and are computable even though they are not natural, integer or rational. This is because these numbers have algorithms that can compute them to great precision. The problem is only the amount of time and storage that can be dedicated to computing these constants
However since computable numbers are only countably infinite this means that there exists numbers that cannot be computed. An example of these are the Chaitin constants which do not have algorithms that can produce approximations.