No, an integer is a whole number. For example, 2 is an integer but 2.25 is not. -48 is an integer but -48.5 is not.
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Starting at the top, we have the real numbers. The rational numbers is a subset of the reals. So are the irrational numbers. Now some rationals are integers so that is a subset of the rationals. Then a subset of the integers is the whole numbers. The natural numbers is a subset of those.
Irrational Numbers, Rational Numbers, Integers, Whole numbers, Natural numbers
You have it backwards. Integers are a subset of real numbers.
There are infiitelt many subsets of irrational numbers. One possible subset is the set of all positive irrational numbers.
A subset of integers.