The set of integers can be constructed as an equivalence relation over the natural numbers using the the binary operation of addition, and a similar process yields the rationals from integers and multiplication. I'm curious what happens if I try to continue this sequence with next logical step: extending the rationals via an equivalence relation equating pairs of exponentials of rational numbers raised to to rational powers.
I know that after suitably defining the relation to exclude indeterminate forms like 0^0, the resulting set will be isomorphic to a proper superset of the rational numbers, since it includes the square root of 2, and a proper subset of the real numbers since numbers like pi can't be represented (it occurs to me I don't actually know how to prove this, though I do know how to prove pi is irrational). So my question is just what is this set of numbers? Does it have a name? And lastly, why isn't it mentioned in less advanced mathematics classes (presumably it lacks sufficiently useful algebraic properties, maybe?)?