$l$-adic expansion of 1/2

Question

I try to understand how to find a $l$-adic expansion of rational numbers. So I tried with $\frac{1}{2}$. Unfortunately I failed.

What I know\think:

We have two cases. What if $l$ is odd? What if $l$ is even?

I have to find:$$\frac{1}{2} = \sum_{i=\infty}^{\infty} z_i\cdot l^{-i},$$ $z_i\in\{ 1,...,l-1\}$ and $l \ge 2$.

Answer 1

If $l$ is even, $$\frac{1}{2}=0.(\frac{l}{2})00\cdots,$$ because $\frac{l}{2}\cdot l^{-1}=\frac{1}{2}.$