Why is a repeating decimal a rational number? 
$$\frac{1}{3}=.33\bar{3}$$ 

is a rational number, but the $3$ keeps on repeating indefinitely (infinitely?). How is this a ratio if it shows this continuous pattern instead of being a finite ratio?  
I understand that $\pi$ is irrational because it extends infinitely without repetition, but I am confused about what makes $1/3=.3333\bar{3}$ rational. It is clearly repeating, but when you apply it to a number, the answers are different: $.33$ and $.3333$ are part of the same concept, $1/3$, yet:
$.33$ and $.3333$ are different numbers:  
$.33/2=.165$ and $.3333/2=.16665$, yet they are both part of $1/3$. 
How is $1/3=.33\bar{3}$ rational?

$$\begin{align}
0.3333333333333\ldots &= 0.3 +0.03 +0.003 +0.0003+ \ldots\\
&=\frac{3}{10} + \frac{3}{100} + \frac{3}{1000}+ \frac{3}{10000} +\ldots
\end{align}$$
If you know the sum of a geometric sequence, that is:
$$a+aq+aq^2+aq^3+\ldots = \frac{a}{1-q} \quad\text{ if $|q| < 1$}$$
you can use it to conclude that for $q = \frac{1}{10}$:
$$\frac{3}{10} + \frac{3}{100} + \frac{3}{1000}+ \frac{3}{10000} +\ldots =\frac{\frac{3}{10}}{1-\frac{1}{10}}=\frac{1}{3}
$$
