---
tag: 0QSPNASE50
kind: statement
role: theorem
name: The reals form an ordered field
msc: [12J15]
uses: [0FGENPMPJX, 0337CYB0VD, 010JCV9J4E]
sources:
  - url: https://en.wikipedia.org/wiki/Ordered_field
    locator: Examples of ordered fields
---

## Statement

The [[real numbers|08J2M1JYAG]] $\mathbb{R}$, with their addition, multiplication and [[order|0337CYB0VD]] $\le$, form an [[ordered field|010JCV9J4E]].
That is, $\le$ is a total order with two properties for all real numbers $a$, $b$ and $c$.
If $a \le b$, then $a + c \le b + c$. If $0 \le a$ and $0 \le b$, then $0 \le a b$.

## Remarks

The ordered field $\mathbb{R}$ contains $\mathbb{Q}$ as an ordered subfield.
The order of $\mathbb{R}$ is the only order that makes $\mathbb{R}$ an ordered field. The nonnegative reals are exactly the squares.

## Explanations

### L1 human

All the usual rules for inequalities hold for the real numbers.
For example, from $a \le b$ you get $a + 5 \le b + 5$, and from $a \ge 0$ and $b \ge 0$ you get $a b \ge 0$.