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 $p$, you say about the prime factors of $q$?
(i) $43.123456789$ (ii) $\mathrm{0.120120012000120000.}..$ (iii) $43.\overline{123456789}$

Answer 1

(i) 43.123456789

It has certain number of digits, so they can be represented in form of $\frac{p}{q}$.

Hence they are rational number.

As they have certain number of digits and the number which has certain

number of digits is always terminating number and for terminating number

denominator has prime factor $2$ and $5$ only.

(ii) 0.120120012000120000. . .

Here the prime factor of denominator Q will has a value which is not equal to

$2$ or $5$.

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

(iii) $43.\overline{123456789}$

Here the prime factor of denominator Q will has a value which is apart from

$2$ or $5$, some other factor also.

So, it is an rational number, $0.123456789$ repeating again and again. It is non- terminating.