Real numbers/Ordering axioms/Archimedes/Intervals/Introduction/2/Section
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A field is called an ordered field, if there is a relation (larger than) between the elements of , fulfilling the following properties ( means or ).
- For two elements , we have either or or .
- From and , one may deduce (for any ).
- implies (for any ).
- From and , one may deduce (for any ).
and are, with their natural orderings, ordered fields.
In an ordered field, the following properties hold.
- .
- holds if and only if holds.
- holds if and only if holds.
- holds if and only if holds.
- and imply .
- and imply .
- and imply .
- and imply .
- and imply .
- and imply .
Proof
See exercise.
In an ordered field, the following properties holds.
- From one can deduce .
- From one can deduce .
- For we have if and only if .
- From one can deduce .
- For positive elements the relation is equivalent with .
Let be an ordered field. is called Archimedean, if the following Archimedean axiom holds, i.e. if for every there exists a natural number such that
- For with there exists such that .
- For there exists a natural number such that .
- For two real numbers
there exists a rational number (with
,
) such that
(1). We consider . Because of the Archimedean axiom there exists some natural number with . Since is positive, due to fact also holds. For (2) and (3) see exercise.
For real numbers , , we call
- the closed interval.
- the open interval.
- the half-open interval (closed on the right).
- the half-open interval (closed on the left).