As others have said, whether a number is rational or not does not depend on the base that it is written in. There are other quite different ways to write numbers, e.g. continued fractions, but these would not affect the rationality either.
However, the base might affect whether or not the representation terminates. In the familiar base 10, the decimal places for p/q will terminate if and only if the prime factorization of q consists of 2s and 5s only. In base 2, the expression will only terminate if q is a power of 2. So, any number whose base 2 representation terminates will terminate in base 10 as well but there are many that terminate in base 10 that do not terminate in base 2 such as 1/10 and 1/100.
So, that computers often use internal formats based on 2 can cause confusion and errors, particularly with financial calculations. Add up 0.01 (e.g. 1 cent) 100 times and then ask whether it is equal to 1.00 (1 dollar). Using float or double, you will probably get the answer no. The designers of Cobol thought of this and it has formats which get it right.