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No. We go with the proof of a counter-example. pi is a well known irrational number. So is 1/pi. Then pi x (1/pi) = 1, a rational number. If you're not convinced that 1/pi is irrational as well, assume that 1/pi is rational, so that 1/pi = p/q, where p and q are integers and q is not 0 (implicitly, p is also not 0). Then pi = q/p, a contradiction to the fact that pi is not a rational number.

Q: Is the set of irrational numbers closed under multiplication?

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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. The set of rational numbers is closed under addition (and multiplication).

None.

no it is not

Yes.natural numbers are closed under multiplication.It means when the operation is done with natural numbers in multiplication the sum of two numbers is always the natural 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. The set of rational numbers is closed under addition (and multiplication).

No. To say a set is closed under multiplication means that if you multiply any two numbers in the set, the result is always a member of the set. If, say, the 2 numbers are radical 2 and radical 2 we have (1.4142...)(1.4142...) which by definition equals 2. The result is not an irrational number, so the set is not closed.

No. It is not even closed. sqrt(3)*sqrt(3) = 3 - which is rational.

No.

Rational numbers are closed under addition, subtraction, multiplication. They are not closed under division, since you can't divide by zero. However, rational numbers excluding the zero are closed under division.

None.

no it is not

Yes.natural numbers are closed under multiplication.It means when the operation is done with natural numbers in multiplication the sum of two numbers is always the natural number.

No. You can well multiply two irrational numbers and get a result that is not an irrational number.

Yes.

Yes.