Mathematically we may define them as an ordered pair of integers (a, b), with b not equal to zero. We can define addition and multiplication upon these pairs with the following rules:
To conform to our expectation that 2/4 = 1/2, we define an equivalence relation ~ upon these pairs with the following rule:
So defined, i.e., by the quotient set defined by ~, the set of rational numbers, denoted by Q, forms a field. It may be shown that Q is the smallest field which contains the integers.
-- integer -- irrational number -- real number --