Confusion regarding the definition of addition modulo $n$ on the group $Z_n$. Consider the group $Z_n$$= \{[0],[1],[2],...,[n-1]\}$. Where the elements of $Z_n$ are equivalence classes, and $Z_n$ is a partition of the integers. So, to me I understand it as: If $[r]$ belongs to $Z_n$, then $[r]$ contains all the integers that have a reminder -when divided by n- of $r\pmod n$. Notice the reminder is not just r, it's $r\pmod n$, to guarantee that the notation $ [r] $ still make sense when    $r>n$. Then the definition of addition modulo n is: $[a]+[b] = [a+b]$. Which means that if you add the set with the elements of reminder $a\pmod n$ with the set of the elements with the reminder $b\pmod n$ you get the set of the elements with reminder of $(a+b)\pmod n$, which make sense.
But the problem is that I see books define $[r]$ to mean: the set of numbers with reminder $r$. And then they define $[a]+[b]=[a+b]$. But the problem is $a+b$ could be bigger than $n$, so under this definition of $[r]$, $[a+b]$ means the set of numbers with reminder $a+b$, which doesn't make sense when $a+b>n$. So which is the correct definition of the operation addition modulo n in the context of abstract algebra ?
 A: Well, both are correct. If $a,b$ are integers, the addition mod $n$ is defined as
$$[a]+[b] = [a+b]$$
where $[a+b]$ is the congruence or residue class mod $n$. So if $a+b$ is not a remainder mod $n$, a number from $0$ to $n-1$, then divide $a+b$ by $n$ with remainder:
$a+b =qn+r$, where $0\leq r<n$, and so $$[a+b]=[r].$$
There are two basic facts to use: 
(1) Each integer $a$ lies in the same residue class as its remainder $r$ mod $n$, i.e., if $a=qn+r$ with $0\leq r<n$, then $[a]=[r]$.
(2) For distinct remainders $r,r'$ mod $n$, the residue classes are distinct, i.e., $[r]\ne[r']$.
A: $r$, is simply one of infinitely many representatives of the set of numbers with remainder $r$ on division by $n$. So $a+b$ is just another representative of the set of numbers that have the same remainder as it on division by $n$. We typically use the lowest value, to symbolize the set, but it need not be. We can write any number with the same remainder, as  along division that doesn't completely take every $n$ away at the end. 
