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 from $\frac{p}{q}$, what can you say about the prime factors of $q$?
i) $43.123456789$

Answer 1

$43.123456789$ is terminating

So, it would be a rational number

$43.123456789=\frac{43.123456789}{1000000000}$

$=\frac{43.123456789}{{10}^9}$

$=\frac{43.123456789}{(2\times 5{)}^9}$

$=\frac{43.123456789}{2^9\times 5^9}$

Hence $43.123456789$ is now in the form $\frac{p}{q}$

And the prime factors of q are in terms of $2$ and $5$