---
tag: 0F6FM9Y4Y6
kind: definition
name: Order on the integers
msc: [06F25]
defines: 07J9ZJFECP
uses: [0C4JS9RF5M, 0MWKJ44V5A]
sources:
  - url: https://en.wikipedia.org/wiki/Integer
    locator: Order-theoretic properties
---

## Definition

Let $m$ and $n$ be [[integers|0C4JS9RF5M]].
The notation $m \le n$ means that $n - m$ is a [[natural number|0MWKJ44V5A]], 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|0Y9H52BT03]].
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.