Definition of Integers

Proposition: Definition of Integers

Let \((a,b),(c,d)\) be ordered pairs of natural numbers. We consider them equivalent, if there exist a natural number \(h\) such that one ordered pair can be obtained from the other ordered pair by adding \(h\) to both natural numbers of that pair, formally

\[(a,b)\sim (c,d)\quad\Longleftrightarrow\quad\begin{cases}(a+h,b+h)=(c,d)& or\\(a,b)=(c+h,d+h).\end{cases}\]

The relation “\(\sim\)” defined above is an equivalence relation, i.e. for a given ordered pair \((a,b)\in\mathbb N\times\mathbb N\), we can consider a whole set of ordered pairs \((c,d)\in\mathbb N\times\mathbb N\) equivalent to \((a,b)\):

\[x:=\{(c,d)\in\mathbb N\times\mathbb N:\quad( c, d )\sim ( a, b )\}.\]

The set \(x\) is called an integer¹. We say that the ordered pair \((a,b)\in\mathbb N\times\mathbb N\) is representing the integer \(x\). The set of all integers is denoted by \(\mathbb Z\).

Notation

In order to make a difference in notation, we write \([a,b]\), instead of \((a,b)\), if we mean the integer represented by the ordered pair \((a,b)\) rather than the concrete ordered pair \((a,b)\). A more common (e.g. taught in the elementary school) notation is the notation of integers retrieved from the difference \(a-b\), however, the concept of building a difference is not introduced yet (in fact, we have not introduced the concept of negative integers yet²). For the time being, we give a comparison of the different notations to make more clear:

Common integer notation	Alternative integer notations	Set of ordered pairs of natural numbers, each notation stands for
\(\vdots\)	\(\vdots\)	\(\vdots\)
\(-3\)	e.g. \([0,3],[1,4],\ldots\)	\(\begin{array}{llllll}\{(0,3),&(1,4),&(2,5),&\ldots,&(h,3+h),&~h\in\mathbb N\}\end{array}\)
\(-2\)	e.g. \([0,2],[1,3],\ldots\)	\(\begin{array}{llllll}\{(0,2),&(1,3),&(2,4),&\ldots,&(h,2+h),&~h\in\mathbb N\}\end{array}\)
\(-1\)	e.g. \([0,1],[1,2],\ldots\)	\(\begin{array}{llllll}\{(0,1),&(1,2),&(2,3),&\ldots,&(h,1+h),&~h\in\mathbb N\}\end{array}\)
\(0\)	e.g. \([0,0],[1,1],\ldots\)	\(\begin{array}{llllll}\{(0,0),&(1,1),&(2,2),&\ldots,&(h,h),&~h\in\mathbb N\}\end{array}\)
\(1\)	e.g. \([1,0],[2,1],\ldots\)	\(\begin{array}{llllll}\{(1,0),&(2,1),&(3,2),&\ldots,&(1+h,h),&~h\in\mathbb N\}\end{array}\)
\(2\)	e.g. \([1,0],[3,1],\ldots\)	\(\begin{array}{llllll}\{(2,0),&(3,1),&(4,2),&\ldots,&(2+h,h),&~h\in\mathbb N\}\end{array}\)
\(3\)	e.g. \([1,0],[4,1],\ldots\)	\(\begin{array}{llllll}\{(3,0),&(4,1),&(5,2),&\ldots,&(3+h,h),&~h\in\mathbb N\}\end{array}\)
\(\vdots\)	\(\vdots\)	\(\vdots\)