No.
It is just one element of a set which contains infinitely many elements.
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The set of rational numbers is a subset of the set of real numbers. That means that every rational number is a real number, but not every real number is rational. The square root of 2 is an example of a real number that isn't rational; that is, it can't be expressed as the quotient of two integers.
The set of rational numbers is the union of the set of fractional numbers and the set of whole numbers.
The set of rational numbers is represented by Q.
A square root is not a number system. Square roots of non-negative numbers may be rational or irrational, but they all belong to the set of real numbers. The square roots of negative numbers do not. To include them, the number system needs to be extended to the complex numbers.
Probably the ancient Egyptians who discovered that the diagonal of a unit square was not a rational number. And then discovered other such numbers.