---
tags: mth350
---

# Proof that addition in $\mathbb{Z}_n$ is associative

Let $[a],[b],[c] \in \mathbb{Z}_n$. We want to prove that
$$[a] + ([b] + [c]) = ([a] + [b]) +[c]$$

Consider the left hand side, $[a] + ([b] + [c])$. Remember that the definition of addition in $\mathbb{Z}_n$ says that given $[x], [y] \in \mathbb{Z}_n$, $[x] + [y]$ is defined as $[x+y]$ where the "$+$" sign in the latter expression is ordinary addition in $\mathbb{Z}$.

We have:

$$
\begin{align*}
[a] + ([b] + [c]) &= [a] + [b+c] \hskip{.2in} \text{(Definition of addition in $\mathbb{Z}_n$}) \\
&= [a+(b+c)] \hskip{.2in} \text{(Definition of addition in $\mathbb{Z}_n$}) \\
&= [(a+b) + c] \hskip{.2in} \text{(Associativity axiom for regular addition in $\mathbb{Z}$}) \\
&= [a+b] + [c] \hskip{.2in} \text{(Definition of addition in $\mathbb{Z}_n$}) \\
&= ([a] + [b]) + [c] \hskip{.2in} \text{(Definition of addition in $\mathbb{Z}_n$})
\end{align*}
$$

That concludes the proof. $\square$