Although the set of integers caters to a larger audience, it is inadequate. This inadequacy has led to the formulation of Rational numbers. Rational numbers are of the form p/q, where p and q are integers and q ≠ 0. The numbers like 2/3,-5/4 are examples of rational numbers. The set of rational numbers are denoted by Q and generally expressed as:
In the set of rational numbers if you consider any of the element say -2/5, we observe that the quotient is - 0.4. Similarly if you consider 7/8, the quotient is 0.875. In both these cases the decimal part is terminating. By terminating, we understand that the division process is coming to an end. Now, in the same set, consider the element 6/7. For this number the quotient is 0.857142857142...... In this case we observe that the decimal part is (i) not terminating and (ii) repeating.
But on occasions we find decimals which neither terminate nor repeat. For instance, consider a number like 65/67. The quotient is of the form 0.970149253..... In this quotient we neither find the decimal terminating nor repeating. Numbers whose decimals are non-terminating and non-repeating are included in a set of numbers called irrational numbers.