Integers - Knowlpedia

The integers are the quotient set

\mathbb Z=(\mathbb N\times\mathbb N)/{\sim}, \qquad (a,b)\sim(c,d)\ \Longleftrightarrow\ a+d=b+c,

where $\mathbb N$ denotes the natural numbers. The class $[a,b]$ represents the formal difference $a-b$ .

Arithmetic and notation

Addition and multiplication are defined by

[a,b]+[c,d]=[a+c,b+d],\qquad [a,b][c,d]=[ac+bd,ad+bc].

The zero and unit are $[0,0]$ and $[1,0]$ , and $-[a,b]=[b,a]$ . These operations are independent of the representatives. Identifying $n\in\mathbb N$ with $[n,0]$ gives the usual notation $\mathbb Z=\{\ldots,-2,-1,0,1,2,\ldots\}$ .

Remarks

The integers extend the natural numbers by including additive inverses, and they sit inside the rational numbers via the identification $n = n/1$ . With the usual $\le$ , they form a totally ordered set.

Examples

$-3$ , $0$ , and $14$ are integers.
The equation $x+3=0$ has the integer solution $x=-3$ .