Question

The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form , what can you say about the prime factor of q?

(i) 43.123456789 (ii) 0.120120012000120000… (iii)

Answer 1

(i) 43.123456789

Since this number has a terminating decimal expansion, it is a rational number of the form and q is of the form

i.e., the prime factors of q will be either 2 or 5 or both.

(ii) 0.120120012000120000 …

The decimal expansion is neither terminating nor recurring. Therefore, the given number is an irrational number.

(iii)

Since the decimal expansion is non-terminating recurring, the given number is a rational number of the form and q is not of the form i.e., the prime factors of q will also have a factor other than 2 or 5.