No; there are infinitely many rational numbers.
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The sum of two rational numbers is rational.From there, it follows that the sum of a finite set of rational numbers is also rational.
Finite, no.
3.14 has a finite number of digits. All numbers with a finite number of digits are rational. Pi has an infinite number of digits, AND the digits don't repeat in a regular pattern. (Numbers with repeating decimals are rational as well.)
Yes - the set of integers is a subset of the set of rational numbers.
A finite set is a set that has numbers you can count. Its not like infinite with no end it has an end.