Why are rational numbers closed under addition?

Answer 1

Suppose x and y are rational numbers.That is, x = p/q and y = r/s where p, q, r and s are integers and q, s are non-zero.

Then x + y = ps/qs + qr/qs = (ps + qr)/qs

The set of integers is closed under multiplication so ps, qr and qs are integers;

then, since the set of integers is closed addition, ps + qr is an integer;

and q, s are non-zero so qs is not zero.

So x + y can be represented by a ratio of two integers, ps + qr and qs where the latter is non-zero.