Order on the integers

Definition

Let $m$ and $n$ be integers. The notation $m \le n$ means that $n - m$ is a natural number, that is, when $n - m \in \{0, 1, 2, \dots\}$. The notation $m < n$ means that $m \le n$ and $m \neq n$.

Remarks

This relation is a total order on $\mathbb{Z}$, and on $\mathbb{N}$ it agrees with the order on the natural numbers. If $m \le n$, then $m + k \le n + k$ for each integer $k$. If $m \le n$ and $0 \le k$, then $m k \le n k$. Thus $\mathbb{Z}$ is an ordered ring. Unlike $\mathbb{N}$, it has no least element.

Explanations

L1 human

On the number line, $m \le n$ means that $m$ is at the same place as $n$ or to the left of it. For example, $-5 \le -2$ because $-2 - (-5) = 3$ is a natural number.