Why is the sum or product of two rational numbers rational?

Answer 1

Suppose p/q and r/s are rational numbers where p, q, r and s are integers and q, s are non-zero.

Then p/q + r/s = ps/qs + qr/qs = (ps + qr)/qs.

Since p, q, r, s are integers, then ps and qr are integers, and therefore (ps + qr) is an integer.

q and s are non-zero integers and so qs is a non-zero integer.

Consequently, (ps + qr)/qs is a ratio of two integers in which the denominator is non-zero. That is, the sum is rational.

Also p/q * r/s = pr/qs.

Since p, q, r, s are integers, then pr and qs are integers.

q and s are non-zero integers so qs is a non-zero integer.

Consequently, pr/qs is a ratio of two integers in which the denominator is non-zero. That is, the sum is rational.