Question

Find the rational numbers whose decimal expansion are : (i) $0.142857$ (ii) $0.\overline{142857}$. Are these two same?

Answer 1

(i) Let $r=0.142857$ and $r$ can be rewritten as follows:

$r=0.142857=\frac{142857}{1000000}.....\left(1\right)$

(ii) Let $r=0.\overline{142857}$ then ${10}^{6}r$ will be:

${10}^{6}r={10}^6\times 0.\overline{142857}=142857.\overline{142857}$

Now we find ${10}^{6}r-r$ as follows:

${10}^{6}r-r=142857.\overline{142857}-0.\overline{142857}=142857.0\Rightarrow r({10}^6-1)=142857\Rightarrow 999999r=142857\Rightarrow r=\frac{142857}{999999}\Rightarrow r=\frac{1}{7}.....\left(2\right)$

From equations $\left(1\right)$ and $\left(2\right)$ we get that the two rational numbers are not the same.