Rational numbers

The rational numbers are those which may be expressed as the ratio between two integers, where the denominator is not equal to zero. They are commonly called `fractions'.

Mathematically we may define them as an ordered pair of integers (a,b), with b not equal to zero. We define the following operations:

(a,b) + (c,d) = (a*d + b*c,b*d)

(a,b) * (c,d) = (a*c,b*d)

To conform to our expectation that 2/4 = 1/2, we consider (a,b) to be equivalent to (c,d) if, and only if, a*d = b*c.

So defined, the set of rational numbers, Q, forms a field. It may be shown that Q is the smallest field which contains the integers.

See also:

• real numbers