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
