Q:

Show that a real number in $(0,1]$ is rational if and only if it has a repeating decimal representation. 

A decimal expression is said to be repeating if it ends in a repeating pattern of digits.  For example, the following are repeating decimal expressions:
$$.333..., .1231333..., 123121312131213...$$
Show that a real number in $(0,1]$ is rational if and only if it has a repeating decimal representation. 
Find all decimal representations for the rational numbers $1/5$ and $10/13$
Do I need to prove this.  Can I just say that $1/5 = 0.2$ and $10/13$ is $0.7692307692307...$

A:

Any periodic decimal can be written as a geometric series, where the sum formula is then a rational expression. 
The other way around, any rational number $\frac mn$ can be rewritten as $10^{-k}\cdot \frac pq$ with $gcd(p,q)=1$ and $q$ containing no factors $2$ or $5$. Set $d=\phi(q)$ the value of Eulers totient function, then per Fermat's little theorem, $10^{d}\equiv 1\pmod{q}$, that is, there is some number $q'$ with $qq'=10^{d}-1$.
This now allows to write the fraction $\frac mn=10^{-k}\cdot\frac {pq'}{10^{d}-1}$ as a periodic decimal with period $d$.

