I think I understand what you mean, so let me dial back "anything involving the real numbers". If you are representing or thinking of "sqrt(2)" as "the positiv

I think I understand what you mean, so let me dial back "anything involving the real numbers". If you are representing or thinking of "sqrt(2)" as "the positive solution to x^2 = 2", then you preserve algebraic properties. But you generally (correct me if I'm wrong) don't get to know whether it's bigger or smaller than something else of the form "the _choose_uniquely_ solution to _some_equation_" unless you rely on an argument where you invoke approximations.