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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 \(a,b,c,d\in\mathbb N\).

\[\begin{array}{ccl}
x\cdot y:=[a,b] \cdot [c,d] &:=& [a\cdot c + b\cdot d,~ a\cdot d + c\cdot b]=[ac + bd,~ ad + bc],
\end{array}
\]

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 \(y\).