Wait, what? So you reject the integers as well as the imaginary numbers, so “-25” necessarily means the unary minus operator applied to the natural number 25, yielding a symbolic representation that has no sensible interpretation on its own as a number?

Okayyyy, let’s keep going with this then. If you have operators p and q, and their inverses b and d, defined on the natural numbers, under what conditions do the operators commute with their inverses? That is, if b(p(x)) = p(b(x)) = x, and d(q(x)) = q(d(x)) = x, for what operators is b(q(p(x)) = q(x) and all other variants like that.

Or, in the language of function composition, where 1 is the identity function and . is the function composition operator, given b.p = p.b = 1 and d.q = q.d = 1, when is b.q.p = q, p.q.b = q, d.p.q = p, and q.p.d = p?

In particular, you are claiming the this is true when b = square root, q = unary minus, and p = square, yes?