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


In addition there is the axiom of group compatibility :

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


An ordered group is ordered according to Archimedes if:

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


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.