Archimedes' axiom

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Illustration of the Archimedean axiom: No matter how small the segment A is, if you only lay these segments together sufficiently often, the total length will be greater than for segment B.

The so-called Archimedean axiom is named after the ancient mathematician Archimedes , but it is older and was formulated by Eudoxus of Knidos in his theory of sizes. In modern terms it reads as follows:

For every two quantities there is a natural number with .

Geometrically, the axiom can be interpreted as follows: If you have two segments on a straight line, you can surpass the larger of the two if you only remove the smaller one often enough.

An ordered group or an ordered body , in which the Archimedean axiom applies, is called Archimedean (an) ordered .

For the field of real numbers it is sometimes introduced axiomatically . However, one can prove with the axioms of an ordered body and the supremum axiom ( every upwardly restricted subset of the body has a supremum ) that the real numbers are ordered Archimedes.

Proof from the supremum axiom for an ordered body

Be it

Assertion: For each there is a natural number such that it holds.

Counter-assumption: There is a such that for all natural numbers

From the opposite assumption it follows that for all natural numbers there is an upper bound for . With the supremum axiom it follows from this the existence of a smallest upper bound . But if it applies to all natural numbers , then it also applies, and thus also to all natural numbers . But then there is also an upper bound for . Because of , is not a smallest upper bound, which contradicts the definition of . Thus the counter-assumption must be wrong and the claim is proven.

Consequences from the Archimedean axiom

For every number there is such that and . It follows that for every there exists a unique number with

It is denoted by or . There is also a clearly defined number with

which is denoted by or . This also applies: for everyone there is a with and therefore vice versa . This relationship is useful in analysis, for example to demonstrate the convergence or divergence of sequences .

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

Archimedean ordered groups

An ordered group is a group with a (here written additively) link and an order structure compatible with the group structure .

The two axioms apply to the order structure:

Für alle  gilt , das heißt  ist reflexiv.
Aus  und  folgt  für alle , das heißt  ist transitiv.

In addition there is the axiom of group compatibility :

Aus  folgt  für alle .

An ordered group is ordered according to Archimedes if:

Zu je zwei Elementen  und  der Gruppe mit  existiert eine natürliche Zahl  mit .

Holder's theorem

Every Archimedean ordered group is commutative and isomorphic to an additively ordered subgroup of .

The figure is for a group linkage written with e> 0 and additively

an isomorphism of G into an additive ordered subgroup of , where for and and for and .

The element e can be used as a unit with which each group element can be "measured". This means: For each element of the group there is one such that .

Example : The intervals in music theory form an Archimedean ordered commutative group and can all be measured with the unit octave or cent . See: tone structure .

Classification : Either an Archimedean ordered group G is of the form G = {0} or G = {…, −3a, −2a, −a, 0, a, 2a, 3a, ...} (isomorphic to the additive group of the whole Numbers) or there is no smallest element, which is specified in the following.

There is a with for each element . (If there is no minimal positive , then there is certainly a with for each . If you can choose. If there is a with and if holds for the inequality .)

Non-Archimedean arranged bodies

An example of an arranged field in which the axiom of Archimedes does not apply is the field of hyperreal numbers studied in nonstandard analysis .

A simpler example consists of the rational functions over the rational (or real) number field, which are ordered so that is greater than all numbers (this can be done in a unique way).


In the elements in Book 3 Proposition 16, Euclid gives an explicit example of quantities that do not fulfill Archimedes' axiom, so - called horn - shaped angles that are formed by touching curved curves, in Euclid's example by a circle and its tangent. They only appear at this point in the elements.

Web links

Wikibooks: Math for Non-Freaks: Archimedes' Axiom  - Learning and Teaching Materials

Individual evidence

  1. ↑ Narrated in: Euclid, elements V, definition 4: That they have a relationship to one another is said of quantities which, when multiplied, exceed one another.
  2. Otto Hölder The axioms of quantity and the doctrine of measure , Ber. Verh. Saxon Society of Sciences Leipzig, Math. Phys. Class, Volume 53, 1901, pp. 1-64.
  3. Alexander Gennadjewitsch Kurosch lectures on general algebra. Harri Deutsch, Zurich 1964.
  4. ^ Euclid, Book 3, Proposition 16, in David Joyce
  5. ^ Felix Klein Elementary Mathematics from the Higher Viewpoint , Springer Verlag, Volume 2, pp. 221f.