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## 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. $$[1,0],[4,1],\ldots$$ $$\begin{array}{llllll}\{(3,0),&(4,1),&(5,2),&\ldots,&(3+h,h),&~h\in\mathbb N\}\end{array}$$ $$\vdots$$ $$\vdots$$ $$\vdots$$

1 Please note that integers are in fact sets.

2 The concept of negative integers will be introduced in the discussion of order relation for integers.