i think the sets you are talking about are whole numbers, integers, rational numbers, irrational numbers, real numbers.
-1.567 is not an integer because integers don't have decimal parts.
-1.567 is a rational number because it is terminating.
-1.567 is not an irrational number because it is terminating.
-1.567 is a real number because it does not contain any imaginary part.
I hope it helps! :)