Nope.
Quick example: e (2.71828) is irrational. Therefore 2*e is irrational making both of them elements of the set of irrational numbers. However, dividing the two, e/(2*e), gives you 1/2, which is a rational number.
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Irrational numbers are not closed under any of the fundamental operations. You can always find cases where you add two irrational numbers (for example), and get a rational result. On the other hand, the set of real numbers (which includes both rational and irrational numbers) is closed under addition, subtraction, and multiplication - and if you exclude the zero, under division.
no it is not
None.
No, the natural numbers are not closed under division. For example, 2 and 3 are natural numbers, but 2/3 is not.
Yes. The set of real numbers is closed under addition, subtraction, multiplication. The set of real numbers without zero is closed under division.