There are mathematical definitions of the terms "real number", "rational number", etc., but there is no mathematical definition of the word "number".

> we can approximate it with better and better precision

In one of the three common formal definitions of the real numbers, that's what a real number is: a Cauchy sequence of rational numbers, which approximate that real number with increasing precision. (Well, a real is an equivalence class of such sequences.)

(The other two common definitions are the Dedekind reals and the reals as the unique complete ordered field.)