**3.** 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 factors of q?

(i) 43.123456789 (ii) 0.1201120012000120000.... (iii)

**Solution:**

(i) 43.123456789

It is rational because decimal expansion is terminating. Therefore, it can be expressed in form where factors of q are of the form where n and m are non-negative integers.

(ii) 0.1201120012000120000...

It is irrational because decimal expansion is neither terminating nor non-terminating repeating.

(iii)

It is rational because decimal expansion is non-terminating repeating. Therefore, it can be expressed in form where factors of q **are not** of the form where n and m are non-negative integers.