Proof
(related to Proposition: Multiplication of Integers Is Commutative)
- By the definition of integers, the integers \(x,y \in \mathbb Z\) are identified by pairs \(x:=[a,b]\), \(y:=[c,d]\) of natural numbers \(a,b,c,d\in\mathbb N\).
- We have, therefore, $x\cdot y:=[a,b] \cdot [c,d].$
- $=[ac + bd,~ ad + bc],$ by definition of integer multiplication.
- $=[ca + db,~ da + cb],$ because multiplying natural numbers is commutative.
- $=[ca + db,~ cb + da],$ because adding natural numbers is commutative.
- $=[c,d]\cdot [a,b]$ by definition of integer multiplication.
- $=y\cdot x$ by definition of integers.
∎
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References
Bibliography
- Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013
