How to check if the number after decimal point of the result of a division goes to infinity? Particularly, is this number any special? I divide a number by a prime number and I observed the division in the below picture. I suspect that the result, quotient, goes to infinity. But how can I check if so? The big calculator showed the same repetitive pattern goes until 90000 digit only as it was apparently limited by the program.
How to check if the number after decimal point of the result of a division goes to infinity?
Also, as you see there is a repetitive pattern in the quotient, which is 473684210526315789. Is there anything mathematically special about this repetition?
Is there any other similar example division that gives a result going to infinity with a pattern like this?
Thank you.


The decimal expansion of a rational is always eventually periodic. Indeed, assume $r = \frac{a}{b}$ and without loss of generality $b > a$. Long division gives you the decimal expansion of $r$:
\begin{align*}
10a &= r_1 \cdot b + c_1\\
10c_1 &= r_2 \cdot b + c_2\\
10c_2 &= r_3 \cdot b + c_3\\
&\ \ \ \vdots
\end{align*}
where $0 \leq c_i \leq r-1$, i.e. $r = 0.r_1r_2r_3...$
By the pidgeonhole principle, we must have $c_i = c_j$ for some $j > i$. Therefore the expansion just repeats after the $j$-th decimal and the period is exactly the sequence of numbers between the $i$-th and the $j$-th digit. Note that the period can also be $0$, which means that the decimal expansion is finite.
