Archimedean property

The so-called Archimedean axiom is named after the ancient mathematician Archimedes, but it is older and has been formulated by Eudoxus of Cnidus in size gauge. In modern precise it is as follows:

Geometric can interpret the axiom so: If you have two lines on a straight line, so you can see the larger surpass both, if the smaller removes only often enough.

A ( to ) parent group or a (an) ordered field in which the Archimedean axiom applies is, Archimedean ordered ( to ).

For the field of real numbers, it is sometimes introduced axiomatically. One can, however, with the axioms of a parent body and the Supremumsaxiom (Any upward bounded subset of the body has a supremum ) prove that the real numbers are ordered Archimedean.

Evidence from the Supremumsaxiom for a parent body

It should be

Claim: For each there is a natural number such that applies.

Opposite assumption: There is a such that for all natural numbers

From the opposite assumption it follows that for all natural numbers is an upper bound for. With the Supremumsaxiom it follows the existence of a least upper bound. But is valid for all natural numbers, so shall also, and hence for all natural numbers. But then also an upper bound for. Because, that is not a least upper bound, which is in contradiction to the definition of. Thus, the opposite assumption must be false and the assertion is proved.

Consequences of the Archimedean axiom

For every number, so that and. It follows that there is a uniquely determined for each number with

It is denoted by or. Similarly, there exists a uniquely determined number with

Which is denoted by or. This is also true that for all there exists a with and therefore reversed. Analysis in this connection is useful for detecting, for example, convergence or divergence of sequences.

Furthermore, it follows from the Archimedean axiom, that there are always two real numbers is a rational number with and that the set of natural numbers in the body is not bounded above.

Set of Hölder

Each Archimedean ordered group G is commutative and isomorphic to an additive subgroup of parent.

It is a with e> 0 and additively written group linking the picture

An isomorphism of G in an additive ordered subset of, where and for and and for.

The element e can be used as a unit, with each group element x can be " measured ". This means that for every element x of the group r exists such that.

Example: The intervals in music theory form an Archimedean ordered commutative group and can all be measured by the unit octave or cents. See: clay structure (mathematical description).

Classification, either a Archimedean ordered group G of the form G = {0 } or G = { ..., -3a, -2a ,-a, 0, a, 2a, 3a, ...} ( isomorphic to the additive group of integers ) or there is no smallest element, which is specified below.

For each element a > 0 there exists a b such that 0 < 2b 0 certainly a c with 0 < c a applies to b = a -. c the inequality 0 < 2b = 2a - 2c < 2a - a = a )