Def/Integer

  =Construction=

Set theoretically, every defines::integer can be viewed as an equivalence class of subtraction problems involving natural numbers. If $a,b \in \NN$, we write "$a - b$" (in quotation marks) as a formal subtraction problem, abbreviating the ordered pair $(a,b)$.

We say that two formal subtraction problems "$a-b$" and "$c-d$" are equivalent, if $a+d = b+c$. This defines an equivalence relation $\sim$ on the set $\NN \times \NN$ of formal subtraction problems. We simply write $a-b$ for the equivalence class of the formal subtraction problem "$a-b$".

The set $\ZZ$ is defined as the set $(\NN \times \NN) / \sim$ of equivalence classes of subtraction problems. Furthermore, if $n \in \NN$, we write $-n$ for the equivalence class of the subtraction problem $0-n$.

It can be proven that every element $z \in \ZZ$ is equal to $n$ or $-n$ (or both, if $n = 0$), for a unique natural number $n$.

=Arithmetic operations=

=Order=

=Absolute value=

=Metadata=