Write in the form of a fraction (if possible) the number:

$$
x=0,5123412341234123412341234123412341234 \ldots
$$

Can you generalize this method to all real numbers with a periodic decimal expansion? And conversely
$$
\begin{aligned}
& 10 x=5.123412341234123412341234123412341234 \ldots \text { and } \\
& 100000 x=51234.12341234123412341234123412341234 \ldots
\end{aligned}
$$

So $100000 x-10 x=99990 x=51229$ and finally:

$$
x=\frac{51229}{99990}
$$

which is irreducible. This method generalizes easily and shows that any number with a periodic decimal expansion can be written as a fraction. Conversely, in the decimal expansion calculation, there are only a finite number of remainders, and thus the decimal expansion of a fraction is necessarily periodic.