Integer

Definition

$\mathbb{Z}$ The zero, a positive Natural Number, or a negative integer

The $\mathbb{Z}$ form the smallest Ring containing natural numbers

Construction

$(a,b)\sim (c,d)\iff a+d=b+c$ Define an Equivalence Relation $\sim$ on the set of ordered pairs $(a,b)\in \mathbb{N}\times \mathbb{N}$ of natural numbers,

The set of Integers, denoted by $\mathbb{Z}$, is defined as the set of equivalence classes under the relation $\sim$ An integer is represented by the Equivalence Class $[(a,b)]$ of ordered pairs under the $\sim$

We denote the integer as $\forall (a,b)\in \mathbb{N}\times \mathbb{N},[(a,b)]:=\{\begin{array}{ll}a-b& a>b\\ 0& a=b\\ -(b-a)& a<b\end{array}$

Operations

$[(a,b)]+[(c,d)]:=[(a+c,b+d)]$ $[(a,b)]-[(c,d)]:=[(a+d,b+c)]$ $[(a,b)]\cdot [(c,d)]:=[(ac+bd,ad+bc)]$ $-[(a,b)]:=[(b,a)]$