The rationals form an ordered field

Builds on The rationals form a field Ordered field

Statement

Define an order on the rational numbers 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.

Remarks

The order on $\mathbb{Q}$ extends the order on the integers. 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 $\mathbb{Q}$ .

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Integers object
Ordered field definition
Rational numbers object
Ring of integers statement
The rationals form a field statement
The rationals form an ordered field statement

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Tag	`09RFKMBQ63`
URI	`m2://09RFKMBQ63`
Kind	statement, theorem
Writing	W0 stub
MSC	12J15
Status	This copy of the site has no build data. Run `m2 status 09RFKMBQ63`.
Uses	The rationals form a field, Ordered field

The rationals form an ordered field

Statement

Remarks

Reading path

Where this leads

Further reading

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