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) 0.120120012000120000...

(iii) 43.123456789

(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 2^m×5ⁿ

That is, the prime factor of q will be 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) 43.123456789

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 2^m×5ⁿ that is, the prime factors of q will also have a factor other than 2 or 5.