Is the dfference of two positive rational number always positive plz help explain.?

Answer 1

Yes.Suppose a and b are two positive rational numbers. Then a can be expressed in the form p/q where p and q are positive integers, and b can be expressed in the form r/s where r and s are positive integers.

Then b - a = r/s - p/q = (qr - ps)/qs.

Now, since p, q, r and s are integers, then

by the closure of the set of integers under multiplications, qr, ps and qs are integers;

q and s are positive => qs is positive, and

by the closure of the set of integers under addition (and subtraction), qr - ps is an integer.

That is, b - a = (qr - ps)/qs is a ratio of two integers, where the denominator of the ratio is positive.