# 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 $$y$$.

Proofs: 1

Definitions: 1 2 3 4 5
Proofs: 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Propositions: 21 22 23 24 25 26 27 28 29 30 31 32

Github: ### References

#### Bibliography

1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013