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No. A number cannot be closed under addition: only a set can be closed. The set of rational numbers is closed under addition.
Quite simply, they are closed under addition. No "when".
Yes they are closed under multiplication, addition, and subtraction.
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.
It is not closed under taking square (or other even) roots.