Q: Is an irrational number closed under division?

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

No, it is not. Root2 and root 8 are each irrational. Root8 / root2 =2. 2 is not a member of the set.

The whole numbers are not closed under division! The statement is false since, for example, 2/3 is not a whole number.

If a set is closed under an operation. then the answer will be a part of that set. If you add, subtract or multiply any two rational numbers you get another national number. But when it comes to division, it is closed except for one number and that is ZERO. eg 3.56 (rational number) ÷ 0 = no answer. Since no answer is not a rational number, that rational numbers are not closed under the operation of division.

No, whole numbers are not closed under division. It is possible to divide one whole number by another whole number and get a result which is not a whole number, for example, 1/2. One divided by two is a half.

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No; here's a counterexample to show that the set of irrational numbers is NOT closed under subtraction: pi - pi = 0. pi is an irrational number. If you subtract it from itself, you get zero, which is a rational number. Closure would require that the difference(answer) be an irrational number as well, which it isn't. Therefore the set of irrational numbers is NOT closed under subtraction.

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. You can well multiply two irrational numbers and get a result that is not an irrational number.

No, it is not. Root2 and root 8 are each irrational. Root8 / root2 =2. 2 is not a member of the set.

No. 4 root 2 and 2 root 2 are both irrational. Divide the first by the second you get 2. Which is not a member of the set of irrational numbers.

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.

It cannot be closed under the four basic operations (addition, subtraction, multiplication, division) because it is indeed possible to come up with two negative irrational numbers such that their sum/difference/product/quotient is a rational number, indicating that the set is not closed. You will have to think of a different operation.

No, they are not. An irrational number subtracted from itself will give 0, which is rational.

Yes

The whole numbers are not closed under division! The statement is false since, for example, 2/3 is not a whole number.

If a set is closed under an operation. then the answer will be a part of that set. If you add, subtract or multiply any two rational numbers you get another national number. But when it comes to division, it is closed except for one number and that is ZERO. eg 3.56 (rational number) ÷ 0 = no answer. Since no answer is not a rational number, that rational numbers are not closed under the operation of division.

No. Zero is a rational number, but division by zero is not defined.