**Proposition**: Multiplication of Integers

According the definition of integers, we can identify integers \(x,y \in \mathbb Z\) with equivalence classes \(x:=[a,b]\), \(y:=[c,d]\) for some natural numbers \(a,b,c,d\in\mathbb N\).

Based on the addition of natural numbers and the multiplication of natural numbers, we define a new multiplication operation “\( \cdot \)” for all integers by setting

\[\begin{array}{ccl}

x\cdot y:=[a,b] \cdot [c,d] &:=& [a\cdot c + b\cdot d,~ a\cdot d + c\cdot b]=[ac + bd,~ ad + bc],

\end{array}

\]

where \([ac + bd,~ ad + bc]\) is also an integer, called the **product** of the integers \(x\) and \(y\). The product 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 14:52:36 | modified: 2019-04-17 06:09:15 | by: *bookofproofs* | references: [696]

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

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

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

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

## 5.**Proposition**: Existence of Integer One (Neutral Element of Multiplication of Integers)

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

## 7.**Proposition**: Multiplying Negative and Positive Integers

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

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