#### 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 p /q what can you say about the prime factors of q?

(1) 43.123456789

(2) 0.120120012000120000. . .

(3) `43.bar(123456789)`

#### Solution

(1) 43.123456789

It is a rational number and q will have factors of 2 and 5 only.

(2) 0.120120012000120000. . .

It is an irrational number as it is non-terminating and non-repeating.

(3) `43.bar(123456789)`

It is rational number, as it is non-terminating but repeating. The factors of q will be, apart from 2 and 5, some other factor also.

Is there an error in this question or solution?

Solution The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. Concept: Revisiting Rational Numbers and Their Decimal Expansions.