The rational roots of integers will cause interesting interactions, because when you multiply them you get back integers. In other words, the fact that sqrt(7)

The rational roots of integers will cause interesting interactions, because when you multiply them you get back integers. In other words, the fact that sqrt(7) * sqrt(7) is an integer means that sqrt(7) interacts with the integers when you adjoin it. This is not the case for a transcendental number like pi. More formally: This type of interaction is true for an irrational number if (and only if?) it is the root of a polynomial with integer coefficients. If you adjoin the root of a polynomial P(n) to the integers, the properties of the resulting set of numbers (e.g. factorization being unique) can be related back to properties of P(n). The nature of the connection has to do with the connection between a set of numbers and the set of polynomials of with coefficients from that set, and equivalence relations on that set of polynomials that are induced by P(n).