Integer Sums

Why

We want sums to follow those of natural numbers.¹

Definition

Consider $\eqc{(a, b)}, \eqc{(c, d)} \in \Z $. We define the integer sum of $\eqc{(a, b)}$ with $\eqc{(c, d)}$ as $\eqc{(a + c, b + d)}$.²

Notation

We denote the sum of $\eqc{(a, b)}$ and $\eqc{(c, d)}$ by $\eqc{(a, b)} + \eqc{(b, c)}$ So if $x, y \in \Z $ then the sum of $x$ and $y$ is $x + y$.

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