Construction of $\mathbb{Z}$ from $\mathbb{N}$ via equivalence relation For $(a,b)$ and $(c,d)$ in $\Bbb{N}\times \Bbb{N}$, we define a relation 
$$(a,b) \sim (c,d) \Leftrightarrow a+d=b+c$$
and this is an equivalence relation on $\Bbb{N}\times \Bbb{N}$.
We define $\mathbb{Z}= \Bbb{N}\times \Bbb{N} / {\sim}$. 
This is the outline of the construction on $\mathbb{Z}$. It's ok!
But in this construction, one can define $\underline{-3=[(0,3)], -2=[(0,2)]}$ and so on..
I can't understand the meaning of these underlined terms.
Can some one explain a bit more?  
 A: The pair $(a,b)$ is related to the pair $(c,d)$, and one states that by saying $(a,b)\sim(c,d),$ precisely if $a-b = c-d.$ But it's not being stated that way because we don't yet have subtraction: because we haven't yet defined negative numbers we cannot find such things as $5-8.$ Hence one says instead $a+d=b+c.$
The notation $[(0,3)]$ denotes the $\sim$-class to which the pair $(0,3)$ belongs. Notice that
$$
(0,3) \sim(1,4)\sim(2,5)\sim(3,6) \sim(4,7)\sim \cdots
$$
We do not state here that this is also equivalent to $(-1,2)$ because at this point we do not yet have negative integers.
Then we identify the negative number $-3$ with this class of ordered pairs of positive integers -- just the ones whose subtraction would yield $-3.$
The point of all this is to show that if we know only about nonnegative integers, then the introduction of negative integers after that will not introduce any logical inconsistencies because this whole process shows how to reduce statements about arithmetic of negative integers to statements about arithmetic of nonnegative integers.
