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?

Let $$a=0.333333333333333...$$
Then multiply by $10$
$$10a=3.333333333333333...$$
Substract $a$ from both sides
$$10a-a=3.3333333....- 0.3333333...$$
So we get
$$9a=3$$
And therefore
$$a=1/3$$
and it’s rational
