Why is the difference of two rational numbers are rational numbers?

Answer 1

Suppose A and B are two rational numbers.

So A = p/q where p and q are integers and q > 0

and B = r/s where r and s are integers and s > 0.

Then A - B = p/q - r/s = ps/qs - qr/qs = (ps - qr)/qs

Now,

p,q,r,s are integers so ps and qr are integers and so x = ps-qr is an integer

and

y = qs is an integer which is > 0

Thus A-B can be written as a ratio of two integers, x/y where y>0. Therefore, A-B is rational.