When someone introduced the symbol ?, they did it to make it easier to describe a REAL number which was not easy to describe with the pre-existing symbols in real analysis. No one claims that the addition of ? into the toolbox of real analysis, or numbers like ?/2, required a migration to an entirely other number system divorced from the reals. They simply did it because it was not easy to describe ? with the other symbols. I have done the same thing with $\widehat\infty-b$; it was not easy to describe that number with the pre-existing symbols. However, people say that I have invented a whole new number system even while the person who invented ? was not accused of that, nor was Stevin when he introduced decimal fractions, nor was anyone who added to the pre-existing body of tools in real analysis. The tool box increased and increased and increased over the years, but then when I threw inf-hat in there, all of the charlatans on this website say, "Nuh-uh! New tool box! Meds! Take ur schizo!"