How write a periodic number as a fraction? What I call as a periodic number is for exemple
$$0.\underbrace{13}_{period}131313...$$ or   $$42.\underbrace{465768}_{period}465768465768.$$
So how can we put theses numbers like a integer fractional, i.e. of the form $\frac{a}{b}$ with $a,b\in\mathbb Z$ ?

Multiplying your number by a suitable power of $10$ we can make some parts the nummber jump to the left of the decimal point leaving identical fractional part. That is $10^mx$ and $x$ have the same fractional part. SO their difference is an integer $a$: That is $a= (10^k-1)x$, this shows $x$ is a rational number.
