My son asked me, why division of integers sometimes produces periodic and sometimes decimal real numbers.
What has come so far to my mind, is that while we use a decimal system, then every non-periodic real value can be represented as something divided by 10, and 10 is a product of 2 and 5.
Maybe there is a known rule/theorem/proof, something like "if ... applies to a/b, then it can't be a periodic value"?
HINT.-WLOG we consider the integer part equal to $0$. You have only three possibilities: $$\begin{cases}1)\space 0.a_1a_2.....a_n\\2)\space 0.\overline{a_1a_2.....a_n}\\3)\space 0.a_1a_2.....a_n\overline{b_1b_2.....b_m}\end{cases}$$ where the overline means periodicity. We make the proof just with concrete examples (you can generalize to any example follow the same procedure) $$\begin{cases}1)\space 0.37=\dfrac{37}{100}=\dfrac{37}{2^2\cdot5^2}\\2)\space0.\overline{37}=\dfrac{37}{100}\left(1+\dfrac{1}{100}+\dfrac{1}{100^2}+\cdots\right)\\3)\space0.5\overline{37}=0.5+0.\overline{37}\end{cases} $$ You can operate in $2)$ finding $\dfrac{37}{99}$ then verify that fractions having as denominator $9$,$99$,$999$,.....,are periodic also. Similarly with $3)$.
Thus the comment above.