Characterization of $\mathbb{Q}$ I have a question. 
As you know, $\mathbb{Q}$ is the set of finite decimals and circulating decimals.
I want to prove this.
Here rational numbers $\mathbb{Q}$ is the set of all fractions. 
The set of finite decimals and circulating decimals is a subset of $\mathbb{Q}$. 
This part is clear.But I do not know how we prove the remaining part. 
Thank you in advance. 
 A: Let $\frac{m}n$ be a rational number, i.e., $m,n\in\Bbb Z$ and $n\ne 0$. There’s no harm in assuming that $0<m<n$: once we handle the rational numbers between $0$ and $1$, the rest are easy.
Imagine doing a long division of $m$ by $n$ to expand $\frac{m}n$ as a decimal. At each stage you get a quotient, the next digit of the decimal expansion, and a remainder. The only possible remainders are the numbers $0,1,2,\dots,n-1$. Either you get a remainder of $0$ at some point, in which case the decimal terminates, or you eventually repeat a remainder. Say that you get the same remainder after finding the $\ell$-th digit as you got after finding the $k$-th digit; then you’re doing the same division to get the $(\ell+1)$-st digit that you did to get the $(k+1)$-st digit, and so on. That is, the block of $\ell-k$ digits starting with the $\ell$-th digit exactly repeats the block starting with the $k$-th digit, and so on.
I’ve left this very informal; it would be a good exercise to try to make formalize it a bit.
Added: Here is an example of what I’m talking about. We wish to expand $\frac27$ as a decimal:
                 0.2857142...  
                -------------
               7)2.0000000...  
                 0  
                 ---  
                 2 0  <-- Remainder of 2  
                 1 4  
                 ----  
                   60  
                   56  
                   ---  
                    40  
                    35  
                    ---  
                     50  
                     49  
                     ---  
                      10  
                       7  
                      ---  
                       30  
                       28  
                       ---  
                        2 <-- Remainder of 2

At this point you’re going to get the same sequence of remainders ($2,6,4,5,1,3$) and the same sequence of digits in the quotient ($2,8,5,7,1,4$) all over again. And then again. Dividing the same thing, in this case $7$, into the same quantity must yield the same result every time.
A: This is another way of looking at the long division method with successive remainders.
The essence of long division is to find an expression $10^sm=k_sn+r_s$ for successive integers $s$, and integers $k_s$ and $0\leq r_s\leq n-1$. If we ever find a $t>s$ with $r_t=r_s$ we can subtract the expressions to eliminate the remainder term, obtaining $$10^s(10^{t-s}-1)m=(k_t-k_s)n$$ or$$\frac m n=\frac{k_t-k_s}{10^s(10^{t-s}-1)}$$
Then use that the right-hand side is a multiple of $\frac 1 {10^{t-s}-1}$, which has a particularly simple form as a recurring decimal to show that $\frac mn$ is also ultimately recurring.
