Classifying a Real Number

Classifying a real number means testing it against the nested sets in order — natural, whole, integer, rational, irrational — and listing every set it belongs to.

Take √25, 0, −6, 4/2, and π. √25 equals 5: natural, whole, integer, rational. 0: whole, integer, rational — not natural. −6: integer and rational. 4/2 reduces to 2, so it classifies as its reduced value: natural, whole, integer, rational. π: irrational, and nothing else on the list. The sets nest upward — each natural number is whole, each whole number is an integer, each integer is rational — so a number's smallest set fixes all the rest. Run the tests in that order every time, and stop adding labels the moment a test fails: −6 fails the whole-number test, so natural and whole never enter its list.