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## Proposition: Addition of Integers

Any integer $$x\in\mathbb Z$$ is by the corresponding proposition an equivalence class
$x:=[a,b],$
where $$a$$ and $$b\neq 0$$ denote natural numbers representing the equivalence class $$x$$. Given two integers $$x:=[a,b]$$, $$y:=[c,d]$$, $$a,b,c,d\in \mathbb N$$, the addition of integers is defined by using the addition of the natural numbers $$a+c$$ and $$b+d$$:

$\begin{array}{rcl} x+y&:=&[a+c,b+d], \end{array}$

where $$[a+c,b+d]$$ is also an integer, called the sum of the integers $$x$$ and $$y$$. The sum exists and is well-defined, i.e. it does not depend on the specific representatives $$[a,b]$$ and $$[c,d]$$ of $$x$$ and $$y$$.

| | | | | created: 2014-09-21 14:25:51 | modified: 2017-08-17 23:09:45 | by: bookofproofs | references: [696]

## 7.Proposition: Existence of Inverse Integers With Respect to Addition

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### Bibliography (further reading)

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