The map is not the territory, and a way of notating a number is not the number itself.

Decimal expansions are convenient for most purposes, but they have this property that sometimes there are two different ways of notating the same number. This is unfortunate, but unless you can come up with a better notation (for Dedekind cuts over the rationals, or some other way of constructing the real numbers), we have to live with it.

(Recurring decimals are not "approximation". They're notation for specific individual real numbers)