Rational numbers are closed under multiplication, because if you multiply any rational number you will get a pattern. Rational numbers also have a pattern or terminatge, which is good to keep in mind.
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Yes. They are closed under addition, subtraction, multiplication. The rational numbers WITHOUT ZERO are closed under division.
Yes. In general, the set of rational numbers is closed under addition, subtraction, and multiplication; and the set of rational numbers without zero is closed under division.
Yes.
No. Zero is a rational number, but division by zero is not defined.
I believe it is because 0 does not have an inverse element.