The following real numbers have decimal expansions as given below. In each case, state whether they are rational or not. If they are rational and expressed in the form $\dfrac{p}{q}$, where p, q are integers, q ≠ 0 and p, q are co-prime, then what can you say about the prime factors of q ?

$\begin{matrix} \text{(i)} & 37.09158 \\[1.5em] \text{(ii)} & 423.\overline{04567}\\[1.5em] \text{(iii)} & 8.9010010001… \\[1.5em] \text{(iv)} & 2.3476817681… \\[1.5em] \end{matrix}$

(i) 37.07158

This can be written as 37.09158 = $\dfrac{3709158}{100000}$

Since , it is terminating decimal

It is Rational number and the prime factors of its denominator q will be 2 or 5 or both .

(ii) $423.\overline{04567}$

since it has non-terminating recurring decimal,

$423.\overline{04567}$ = 423.0456704567…

It is a rational number which is non-terminating and repeating. Its denominator q will have prime factors other than 2 or 5.

(iii) 8.9010010001…

Since, it is non-terminating non-repeating decimal number

∴ It is not a Rational number.

(iv) 2.3476817681… = $2.34\overline{7681}$

Since, it is a non-terminating repeating decimal number,

∴ It is a Rational number and its denominator q will have prime factors other than 2 or 5.