If a rational number has a finite decimal representation, then it has a finite representation in base $b$ for any $b>1?$ Is it true that if a  rational number has a finite decimal representation, then it has a finite representation in base $b$ for any $b>1?$
I would like to know if there is a book where this subject is fully detailed.
EDIT: Improve the question!
 A: Theorem. A rational number $\frac{a}{b}$, $\gcd(a,b)=1$, has finite representation in base $k$ if and only if there exists $n\gt 0$ such that $b|k^n$.
Proof. Suppose $b|k^n$. Then $k^n = bq$, so we can write 
$$\frac{a}{b} = \frac{aq}{bq} = \frac{aq}{k^n}$$
which has finite representation.
Conversely, if $\frac{a}{b}$ has finite representation, then we can write it as $\frac{r}{k^n}$ for some $n\gt 0$, hence $br=ak^n$; since $\gcd(a,b)=1$, then $b|k^n$. $\Box$
Corollary. A rational number $\frac{a}{b}$, $\gcd(a,b)=1$, has finite representation in base $k$ if and only if the only primes that divide $b$ also divide $k$.
If we interpret your question as: "if $\frac{a}{b}$ has finite decimal representation, then it has finite representation in base $k$ for any $k\gt 1$", then the answer is "no". $\frac{1}{2}$ does not have finite representation in base $3$. If we interpret it as "if $\frac{a}{b}$ has finite decimal representation, then it has finite representation in base $k$ for some $k$," then the answer is "yes" (though it trivially does in base $10^n$ for any $n$).
A: An $\rm x\in\mathbb{Q}$ has a finite digital representation in base $\rm b$ if and only if $\rm x= n/b^k$ for some integers $\rm n,k$, with $\rm k\ge 0$. Since $\rm x= nm^k /(mb)^k$, $\rm x$ is also has finite representation in base $\rm mb$ for any $\rm m\in\mathbb{N}$.

Wait, do you mean some other base, or any other base? These are distinct questions.
A: $\frac{1}{13}$ in base $10 : 0.0\overline{769230}$
$\frac{1}{13}$ in base $13 : 0.1$
A: Every number with a finite expansion in base $a$ has a finite expansion in base $b$ if and only if there exists a positive integer $n$ such that $a\mid b^n$.
The expansion in base $b$ of every decimal is finite if and only if $10\mid b$.
A: Hardy and Wright has a good analysis of the theory of "decimals" in different bases.
