This is a good point, but an even more basic issue is that the question "what is a number" is a matter of definition. There isn't a "correct" definition of numbers; only one that we've accepted as standard. The accepted definition of a "real number" is actually quite complicated [1], and it's certainly not easy to convey why this complexity is necessary. Other definitions are also possible [2], but nonstandard.
The simplest definition is: a finite decimal ak ... a1.b1 ... bh is defined to be a fraction and an infinite decimal is defined to be a limit. You'd still have to define what a limit is, but that is somewhat more intuitive.