Integer Order

Why

We want to order the integers.

Definition

Consider $\eqc{(a, b)}, \eqc{(b, c)} \in \Z $. If $a + d < b + c$, then we say that $\eqc{(a, b)}$ is less than $\eqc{(b, c)}$.¹ If $\eqc{(a, b)}$ is less than $\eqc{(b, c)}$ or equal, then we say that $\eqc{(a, b)}$ is less than or equal to $\eqc{(b, c)}$.

Notation

If $x, y \in \Z $ and $x$ is less than $y$, then we write $x < y$. If $x$ is less than or equal to $y$, we write $x \leq y$.

Positive and negative integers

We call an integer $z$ positive if $z > 0$ and we call $z$ negative if $z < 0$.² We call an integer $z$ nonnegative if $z > 0$ or $z = 0$ and nonpositive if $z < 0$ or $z = 0$.

Notation

We denote the set $\Set{z \in \Z }{z \geq 0_{Z}}$ by $\Z _{++}$.

1. One needs to show that this is well-defined. The account will appear in future editions. ↩︎
2. Some authors use the term positive for the case when $z > 0$ or $z = 0$. We use the term nonnegative in this case. ↩︎