Why are integers subset of reals? In most programming languages, integer and real (or float, rational, whatever) types are usually disjoint; 2 is not the same as 2.0 (although most languages do an automatic conversion when necessary). In addition to technical reasons, this separation makes sense -- you use them for quite different purposes.
Why did they choose to say $\mathbb{Z} \subset \mathbb{R}$ in math? In other words, why are 2 and 2.0 considered the same?
When you are working in $\mathbb{R}$, does it make any difference whether some elements, eg. 2.0, also belong to $\mathbb{Z}$ or not?

There are very many reasons, but first of all, if the real numbers did not contain the integers then it would be very very very difficult to do elementary arithmetic.
Computers don't have any way to represent "real numbers", they can represent only integers and rationals.
