How to represent calculable real numbers?
A number is obtained from natural numbers using addition, subtraction, multiplication, or division if and only if it is rational. You can store rational numbers as pairs $(p,q)$ of integers, which represent the number $p/q$ (so $q \neq 0$). You can further assume that $p,q$ are mutually prime.

Libraries implementing rational numbers using arbitrary-precision integers are available, for example GMP.