Question

Check whether the number 0.318564318564318564….. is rational or irrational.

Answer 1

$\mathrm{0.318564318564318564...}=0.\overline{318564}$.
It is non – terminating and recurring, hence it is a rational number.

Also, we can check by the following method :
$x=0.318564318564318564...\left(i\right)$
Multiplying both sides by 1000000, we get
$1000000x=\mathrm{318564.318564318564...}\left(ii\right)$

Subtracting (i) from (ii), we get
$99999x=318564$
$\Rightarrow x=\frac{318564}{99999}$

We can see that $x$ can be represented in the form $\frac{p}{q}$ where $p$ and $q$ are both integers and $q\ne 0$.
Hence, $x=$ 0.318564318564318564... is a rational number.