Integer

Set

 definiendum $\mathbb Z \equiv \mathbb N\times\mathbb N\ /\ \{\langle \langle a,b\rangle,\langle m,n\rangle\rangle\ |\ a+n = b+m )\}$

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

Discussion

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

• $0 \equiv [\langle0,0\rangle] = [\langle1,1\rangle] = \dots = [\langle k,k\rangle]$
• $1 \equiv [\langle1,0\rangle] = [\langle2,1\rangle] = \dots = [\langle k+1,k\rangle]$
• $-1 \equiv [\langle0,1\rangle] = [\langle1,2\rangle] = \dots = [\langle k,k+1\rangle]$
• $2 \equiv [\langle2,0\rangle] = [\langle3,1\rangle] = \dots = [\langle k+2,k\rangle]$
• $-2 \equiv [\langle0,2\rangle] = [\langle1,3\rangle] = \dots = [\langle k,k+2\rangle]$
• $3 \equiv [\langle0,3\rangle] = \dots$

where $k$ is any natural number.

Theorems

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

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

Reference

Wikipedia: Integer