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It follows from the closure of integers under addition and multiplication.

Any rational number can be expressed as a ratio of two integers, where the second is not zero. So two rational numbers may be expressed as p/q and r/s.

A common multiple of their denominators is qs. So the numbers could also have been expressed as ps/qs and qr/qs.

Both these have the same denominator so their sum is (ps + qr)/qs.

Now, because the set of integers is closed under multiplication, ps, qr and qs are integers and because the set of integers is closed under addition, ps + qr is an integer.


Thus the sum has been expressed as a ratio of two integers, ps + qr, and qs and so it is a rational number.


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Q: Why is the multiplication of rational numbers always result in a rational number?
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