Application

Suppose that we know how to work with elements of R on computer. Can we work with elements of Q(R)?

Clearly, a fraction t/n can be represented by the pair (t,n), and the given formulas work for defining product and addition in terms of the operations for R.

Equality amongst fractions also requires a computation: t/n = t'/n' is verified by determining whether tn' = t'n holds.

In the case R = Z there is a unique representative (t,n) for each class with the properties

• gcd(t,n) = 1;
• n > 0.

It is obtained from an arbitrary representative by dividing both numerator and denominator by their common gcd, and also by -1 if necessary to obtain a positive denominator.

Similar observations hold for R = Q[X].