I know addition, multiplication and subtraction are definitely defined.
However division does not work on this set because if a = some integer and b = 0 then a/b is not in the rational numbers.
Is this correct?
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Yes, this is correct. Division is a binary operation on $\mathbb{Q} ^*$ however. – Joel Cohen Sep 15 '16 at 02:59
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One can define infinitely many such operations, of course. Is the question about the commonly used operations with widely accepted names? – Andris Birkmanis Sep 15 '16 at 03:05