There is well defined name for "useful reals": Algebraic numbers. Of course the well-definedness necessitates some limit on how the symbolic description looks like (ie. algebraic numbers are roots of polynomials with rational coefficients) because every real number can be described by some arbitrarily complex symbolic notation.
Edit: I vaguely remember that there used to be some name for the intersection of algebraic and real numbers, but I neither can remember it nor can find it on wikipedia.
> every real number can be described by some arbitrarily complex symbolic notation
This seems like it would have to be false, because otherwise the reals would be countable (iterate through every possible 1-character string, then every possible 2 character string, then 3 chars, etc and in a finite (but potentially very very large) amount of time you would come across the description of any real number that can be described).
Edit: I vaguely remember that there used to be some name for the intersection of algebraic and real numbers, but I neither can remember it nor can find it on wikipedia.