The decimal expansions of some real numbers are given below. In each case, decide whether they are rational or not. If they are rational, write it is the form 'p/q'. What can you say about the prime factors of 'q'?

Question

The decimal expansions of some real numbers are given below. In each case, decide whether they are rational or not. If they are rational, write it is the form $\frac{p}{q}$. What can you say about the prime factors of $q$?

(A) 0.140140014000140000 ....

(B) $0 . \overline{16}$

Answer 1

(A) We have $0.140140014000140000 \ldots$. a non-terminating and non-repeating decimal expansion. So it is irrational it cannot be written in the form of $\frac{p}{q}$.

(B) We have, $0 . \overline{16}$ a non-terminating but repeating decimal expansion. So it is rational.

Let $x \quad=0 . \overline{16}$

Then, $\quad x=0.1616$ ........(i)

$100 x=16.1616$ ........(ii)

On subtracting equation (i) from (ii), we get

$100 x-x  =16.1616-0.1616$

$99 x  =16 \Rightarrow x=\frac{16}{99}.$

The denominator $(q)$ has factors other than 2 or 5.