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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 \(\mathbb Z\).


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.

| | | | | created: 2014-09-09 22:11:57 | modified: 2018-05-13 23:38:18 | by: bookofproofs | references: [696]

1.Proof: (related to "Definition of Integers")

2.Proposition: Addition of Integers

3.Definition: Subtraction of Integers

4.Proposition: Multiplication of Integers

5.Proposition: Distributivity Law For Integers

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Bibliography (further reading)

[696] Kramer Jürg, von Pippich, Anna-Maria: “Von den natürlichen Zahlen zu den Quaternionen”, Springer-Spektrum, 2013