Why is it important to have categories of numbers in math?

Answer 1

It is important to know whether the category to which the answer belongs.

For example 45/30 = 1.5

However, if the question was how many 30 seater buses do you need for 45 people, the answer cannot be 1.5. You cannot have half a bus! So the answer needs to belong to the set of whole numbers.

Different mathematical operations result in answers which result in numbers from a more complicated category.

The starting point, presumably, is the set of counting numbers = {1, 2, 3, ... }

The introduction of subtraction extends the set Z.

Next, you introduce division (by non-zero numbers) and you have the set of rational numbers, Q.

But then there are number such as the the square root of 2 and pi which do not belong to Q so there is the class of Irrational Numbers (including transcendental numbers) and together with Q, these make up the set of real number, R.

But R is not enough to deal with, for example, the square root of negative numbers. So the categories need to be extended to C, the set of complex number.

There are further categories: quaternions, for example.