---
tag: 09RFKMBQ63
kind: statement
role: theorem
name: The rationals form an ordered field
msc: [12J15]
uses: [0N2WPCC00T, 010JCV9J4E]
sources:
  - url: https://en.wikipedia.org/wiki/Ordered_field
    locator: Examples of ordered fields
  - url: https://en.wikipedia.org/wiki/Rational_number
    locator: Ordering
---

## Statement

Define an order on the [[rational numbers|0NCJ5WXN6Z]] as follows. For integers $a, c$ and positive integers $b, d$, let $\frac{a}{b} \le \frac{c}{d}$ when $a d \le b c$ in $\mathbb{Z}$.
This relation is well defined, and with it $\mathbb{Q}$ is an [[ordered field|010JCV9J4E]].

## Remarks

The order on $\mathbb{Q}$ extends the [[order on the integers|0F6FM9Y4Y6]].
This is the only order on $\mathbb{Q}$ that makes it an ordered field.
The ordered field $\mathbb{Q}$ is Archimedean, but it is not complete: the set $\{q \in \mathbb{Q} : q^2 < 2\}$ has no least upper bound in $\mathbb{Q}$.

## Explanations

### L1 human

To compare two fractions, write them with positive denominators and cross-multiply.
For example, $2/3 < 3/4$ because $2 \cdot 4 = 8 < 9 = 3 \cdot 3$.