What “computable” means for a number
A real number is called computable if there is a finite, precise procedure that can produce approximations to it as accurately as requested. The procedure need not print every decimal digit one by one: it might instead return enough digits, or a fraction close enough to the number, to meet a specified error limit.
This distinction matters because a calculation that takes a million years is still, in principle, a calculation. An uncomputable number has no single algorithm that can reliably provide approximations to arbitrary precision. More computing power can speed up a recipe; it cannot supply a recipe that does not exist.
Why most real numbers have no recipe
Every computer program is a finite string of symbols. There are infinitely many such strings, but they can be listed in a sequence: first the short programs, then longer ones, in some systematic order. So there are at most countably many programs—and therefore at most countably many numbers that programs can compute.
The real numbers, by contrast, cannot be completely listed. A classic diagonal argument shows why: imagine a list of all infinite decimal expansions, then construct a new decimal whose first digit differs from the first number’s first digit, whose second differs from the second number’s second digit, and so on. The new number cannot be anywhere on the list. Thus the real numbers are uncountable, and the computable ones are only a tiny subset in this precise sense.
A limit on what computation can promise
The argument does not tell us how to identify a particular uncomputable number, or even guarantee that its digits look unusual. A number can be perfectly well-defined yet lack an algorithm for generating its decimal digits. Some constructions connect such numbers to questions no program can decide in general, such as whether another program will eventually stop running.
The practical consequence is a boundary on computation, not merely a shortage of time or hardware: some mathematically specified answers cannot be produced by any general-purpose recipe at all.
