A number which can be expressed in the form p/q, q ≠ 0 where p and q are integers, is called a rational number. All integers and the numbers like etc., are rational numbers. Rational numbers can also be represented as terminating decimals like or by non-terminating but repeating decimals like
The non-terminating and non-repeating decimals always represent irrational numbers.
Let us now investigate the form of the integer q in the rational number p/q, which decides whether the decimal representation of p/q is terminating or non-terminating and recurring. This investigation starts with the division lemma named after famous Greek Mathematician Euclid who discovered this important result.