In mathematics, one can hardly study any mathematical concept unless it is clearly and rigorously defined. For example, without the definition the fundamental group, it is almost impossible to teach anything serious about it.
But as far as real numbers were concerned, they were taught very early without any clear definition was even given. Definitions of real numbers using say Dedekind cut were never introduced until in say Mathematical Analysis.
Pedagogically speaking, how is it even possible to teach about a mathematical concept (real number) before clearly defining it?