Why we are classifing number system?

Answer 1

The simplest subset have the simplest properties. The range of properties increases as the compexity of the subset increases.

Take the property "closure". For a binary operation denoted by #, a set is said to be closed if for any elements x and y from the set, x#y is also a member of the set.

Then, counting numbers are closed under addition or multiplication but not subtraction. You need the whole set of integers for that. But that is not closed under division. So you extend the subset to all rationals. So far so good.

Now you have closure under the 4 basic operations, including squaring. But what about the reverse operation? Taking square roots? No. So add in the irrationals which gives you the set of real numbers.

But the reals are not closed under square roots of negative numbers. So extend to the set of complex numbers. Hope you get the drift.