You have it backwards. Integers are a subset of real numbers.
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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.
Yes.
A subset is a smaller set that is part of a larger set. For example, the set of animals contains the subset of reptiles, the subset of mammals, and various others. Or in mathematics, the set of real numbers contains the subset of positive integers, the subset of negative integers, the subset of rational numbers, etc.
The natural numbers (ℕ) are a subset of the integers (ℤ) which are a subset of the rational numbers (ℚ) which are a subset of the real numbers (ℝ): ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ → ℕ ⊂ ℝ and ℤ ⊂ ℝ as well as ℚ ⊂ ℝ
Irrational Numbers, Rational Numbers, Integers, Whole numbers, Natural numbers