I know that there are infinitely many number of real numbers between (1,2), (2,3) and so on and I know that there's no meaning to compare the number of real numbers between 2 ranges of real numbers but I've not been able to find fault with the below argument without bringing the concept of infinity. The argument goes like this:
Since each real number between (4,9) can be mapped to a unique real number between (2,3), after taking the square root, can't we say that the number of real numbers between (4,9) (whatever that number may be) is equal to the number of real numbers between (2,3)?
If we extend this argument, does this not mean that the number of real numbers between (4,5) is smaller than that between (2,3)!
You must define the terms and expressions that you use first. If, when you say that two sets $A$ and $B$ of real numbers have the same number of elements what you mean is that there is a bijection between them, then, yes, $(4,9)$ and $(2,3)$ have the same number of elements. Actualy, any two open (or closed, for that matter) intervals have the same number of elements. Take $(2,3)$ and $(4,5)$, for instance, and consider the bijection $f\colon(2,3)\longrightarrow(4,5)$ defined by $f(x)=x+2$.
Analogously, if you want to say that one set of real numbers is smaller than another one, you must explain first what is it that you mean by that.