# Archimedes' axiom

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 .${\ displaystyle y> x> 0}$ ${\ displaystyle n \ in \ mathbb {N}}$ ${\ displaystyle nx> y}$

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. ${\ displaystyle \ mathbb {R}}$

## Proof from the supremum axiom for an ordered body

Be it ${\ displaystyle x> 0.}$

Assertion: For each there is a natural number such that it holds. ${\ displaystyle y> x}$${\ displaystyle n}$${\ displaystyle nx> y}$

Counter-assumption: There is a such that for all natural numbers${\ displaystyle y> x}$${\ displaystyle nx \ leq y}$${\ displaystyle n.}$

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. ${\ displaystyle y}$${\ displaystyle n}$${\ displaystyle nx}$${\ displaystyle y_ {0}}$${\ displaystyle nx \ leq y_ {0}}$${\ displaystyle n}$${\ displaystyle \ left (n + 1 \ right) x \ leq y_ {0}}$${\ displaystyle nx \ leq y_ {0} -x}$${\ displaystyle n}$${\ displaystyle y_ {0} -x}$${\ displaystyle nx}$${\ displaystyle y_ {0} -x ${\ displaystyle y_ {0}}$${\ displaystyle y_ {0}}$

## Consequences from the Archimedean axiom

For every number there is such that and . It follows that for every there exists a unique number with ${\ displaystyle x \ in \ mathbb {R}}$${\ displaystyle n_ {1}, n_ {2} \ in \ mathbb {N}}$${\ displaystyle n_ {1}> x}$${\ displaystyle -n_ {2} ${\ displaystyle x \ in \ mathbb {R}}$${\ displaystyle n \ in \ mathbb {Z}}$

${\ displaystyle n \ leq x

It is denoted by or . There is also a clearly defined number with ${\ displaystyle n}$${\ displaystyle \ lfloor x \ rfloor}$${\ displaystyle \ operatorname {floor} (x)}$${\ displaystyle m \ in \ mathbb {Z}}$

${\ displaystyle m-1

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 . ${\ displaystyle \ lceil x \ rceil}$${\ displaystyle \ operatorname {ceil} (x)}$${\ displaystyle \ varepsilon> 0}$${\ displaystyle n \ in \ mathbb {N}}$${\ displaystyle n> 1 / \ varepsilon}$${\ displaystyle 1 / n <\ varepsilon}$

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. ${\ displaystyle a, b \ in \ mathbb {R}, a ${\ displaystyle q \ in \ mathbb {Q}}$${\ displaystyle a ${\ displaystyle \ mathbb {R}}$

## Archimedean ordered groups

An ordered group is a group with a (here written additively) link and an order structure compatible with the group structure . ${\ displaystyle +}$${\ displaystyle \ leq}$

The two axioms apply to the order structure:

Für alle ${\displaystyle x\in G}$ gilt ${\displaystyle x\leq x}$, das heißt ${\displaystyle \leq }$ ist reflexiv.
Aus ${\displaystyle x\leq y}$ und ${\displaystyle y\leq z}$ folgt ${\displaystyle x\leq z}$ für alle ${\displaystyle x,y,z\in G}$, das heißt ${\displaystyle \leq }$ ist transitiv.


In addition there is the axiom of group compatibility :

Aus ${\displaystyle x\leq y}$ folgt ${\displaystyle x+z\leq y+z}$ für alle ${\displaystyle x,y,z\in G}$.


An ordered group is ordered according to Archimedes if:

Zu je zwei Elementen ${\displaystyle x}$ und ${\displaystyle y}$ der Gruppe mit ${\displaystyle y>x>0}$ existiert eine natürliche Zahl ${\displaystyle n\in \mathbb {N} }$ mit ${\displaystyle nx>y}$.


Holder's theorem

Every Archimedean ordered group is commutative and isomorphic to an additively ordered subgroup of . ${\ displaystyle G}$${\ displaystyle \ mathbb {R}}$

The figure is for a group linkage written with e> 0 and additively ${\ displaystyle e \ in G}$

${\ displaystyle x \ mapsto r = \ sup \, \ left \ {{\ frac {z} {n}} \ mid z \ in \ mathbb {Z}, n \ in \ mathbb {N}, z \ cdot e

an isomorphism of G into an additive ordered subgroup of , where for and and for and . ${\ displaystyle \ mathbb {R}}$${\ displaystyle n \ cdot x = \ underbrace {x + x + \ dotsb + x} _ {n {\ text {times}}}}$${\ displaystyle x \ in G}$${\ displaystyle n \ in N}$${\ displaystyle z \ cdot e = -z \ cdot (-e)}$${\ displaystyle z \ in Z}$${\ displaystyle z <0}$

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 . ${\ displaystyle x}$${\ displaystyle x}$${\ displaystyle r}$${\ displaystyle x = r \ cdot e (r \ in \ mathbb {R})}$

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 .) ${\ displaystyle a> 0}$${\ displaystyle b}$${\ displaystyle 0 <2b ${\ displaystyle a}$${\ displaystyle a> 0}$${\ displaystyle c}$${\ displaystyle 0 ${\ displaystyle 2c ${\ displaystyle b = c}$${\ displaystyle 2c = a}$${\ displaystyle b}$${\ displaystyle 0 <2b <2c = a}$${\ displaystyle 2c> a}$${\ displaystyle b = ac}$${\ displaystyle 0 <2b = 2a-2c <2a-a = a}$

## 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). ${\ displaystyle R (x)}$${\ displaystyle x}$

## Historical

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.