**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]

## 1.**Proof**: *(related to "Addition of Integers")*

## 2.**Proposition**: Addition of Integers Is Associative

## 3.**Proposition**: Addition of Integers Is Commutative

## 4.**Proposition**: Addition of Integers Is Cancellative

## 5.**Proposition**: Existence of Integer Zero (Neutral Element of Addition of Integers)

## 6.**Proposition**: Uniqueness of Integer Zero

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

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

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