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 or not. If they are rational, and of them $\frac{p}{q}$, what can you say about the prime factors of $q$?

i) $43.123456789$

ii) $\mathrm{0.120120012000120000....}$

iii) $43.\overline{123456789}$

Answer 1

i) $43.123456789$ is terminating or rational

$43.123456789=\frac{43123456789}{1000000000}=\frac{43123456789}{{10}^9}=\frac{43123456789}{(2\times 5{)}^9}=\frac{43123456789}{2^9\times 5^9}$

This is in the form of $\frac{p}{q}$, and the prime factors of $q$ are in terms of $2$ and $5$.

ii) $\mathrm{0.120120012000120000....}$ is non-terminating and non-repeating, it is irrational. Hence cannot be expressed in the form of $\frac{p}{q}$

iii) $43.\overline{123456789}$ is non-terminating but repeating. So, it would be rational.

Let $x=43.\overline{123456789}$ $\dots \left(1\right)$

$1000000000x=43123456789,\overline{123456789}$ $\dots \left(2\right)$

$\left(2\right)-\left(1\right)\Rightarrow 999999999x=43123456746\Rightarrow x=\frac{43123456746}{999999999}=\frac{43123456746}{3^4\times {37}^1\times {333667}^1}$

In a non-termination repeating expansion of $\frac{p}{q}$, $q$ will have factors $3,37,333667$.