Monthly Archives: June 2014

Rational Numbers and Their Decimal Expansions

We are going to explore when their decimal expansions of rational numbers are terminating and when they are non-terminating, repeating.

Let us consider the following terminating decimal expressions of some rational numbers:
(i) 0.375 (ii) 1.04 (iii) 0.0875 (iv) 12.5 (v) 0.00025

Rational Numbers and Their Decimal Expansions

We see that all terminating decimals taken by us can be expressed as rational numbers whose denominators are powers of 10.

Do you see a pattern in the denominators? It appears that when the decimal expression is expressed in its simplest rational form then pand qare coprime and the denominator (i.e., q) has only powers of 2, or powers of 5, or both. This is because the powers of 10 can only have powers of 2 and 5 as factors.