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## 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]

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