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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.
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Is the set of irrational numbers closed under subtraction?
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.
Are rational numbers closed under subtraction?
Yes. They are closed under addition, subtraction, multiplication. The rational numbers WITHOUT ZERO are closed under division.
Why do whole number require an extension?
The set of whole numbers is not closed under division (by non-zero whole numbers).
Are irrational numbers closed under 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.
Is the set of rational numbers closed under division and Why?
No. Zero is a rational number, but division by zero is not defined.
