Some would say that there is no intersection.
However, if the set of Irrational Numbers is considered as a group then closure requires rationals to be a proper subset of the irrationals.
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9.5 is rational. Rational numbers are numbers that can be written as a fraction. Irrational numbers cannot be expressed as a fraction.
-6.3 is rational. Rational numbers are numbers that can be written as a fraction. Irrational numbers cannot be expressed as a fraction.
21.192 is rational. Rational numbers are numbers that can be written as a fraction. Irrational numbers cannot be expressed as a fraction.
If there are no numbers after the 9 it is rational
1.777...(repeating) is rational. Rational numbers are numbers that can be written as a fraction. Irrational numbers cannot be expressed as a fraction.