Integer Products

Why

We want sums to follow those of natural numbers.¹

Definition

Consider $\eqc{(a, b)}, \eqc{(b, c)} \in \Z $. The integer product of $\eqc{(a, b)}$ with $\eqc{(b, c)}$ is $\eqc{(ac + bd, ad + bc)}$.²

Notation

We denote the product of $\eqc{(a, b)}$ and $\eqc{(c, d)}$ by $\eqc{(a, b)} \cdot \eqc{(b, c)}$ So if $x, y \in \Z $ then the sum of $x$ and $y$ is $x\cdot y$. As with natural products, we often drop the $\cdot $ and write $xy$ for $x\cdot y$.

1. Future editions will modify this. ↩︎
2. One needs to show that this is well-defined. The account will appear in future editions. ↩︎