with $a,b,n,m\in \mathbb N$.

For $a \ge b$, we denote $\langle a,b\rangle$ by $a-b$. The structure of the non-negative integers is then that of the natural numbers.

For $a < b$, we have $(b-a)>0$ and we denote $\langle a,b\rangle$ by $-(b-a)$.

So if $[\langle a,b\rangle]$ is the equivalence class of $\langle a,b\rangle$ with respect to the equivalence relation $\{\langle \langle a,b\rangle,\langle m,n\rangle\rangle\ |\ a+n = b+m )\}$, we have

• $ 0 \equiv [(0,0)] = [(1,1)] = \dots = [(k,k)] $

• $ 1 \equiv [(1,0)] = [(2,1)] = \dots = [(k+1,k)] $

• $ -1 \equiv [(0,1)] = [(1,2)] = \dots = [(k,k+1)] $

• $ 2 \equiv [(2,0)] = [(3,1)] = \dots = [(k+2,k)] $

• $ -2 \equiv [(0,2)] = [(1,3)] = \dots = [(k,k+2)] $

• $ 3 \equiv [(0,3)] = \dots $

where $k$ is any natural number.

$-[\langle a,b\rangle]$ is the additive inverse of $[\langle a,b\rangle]$ and can be computed as

$-[\langle a,b\rangle]=[\langle b,a\rangle]$

Wikipedia: Integer

Quotient set

Rational number

Arithmetic structure of natural numbers