In mathematics, a non-Archimedean ordered field is an ordered field

Ordered field

In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and...

 that does not satisfy the Archimedean property

Archimedean property

In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some ordered or normed groups, fields, and other algebraic structures. Roughly speaking, it is the property of having no infinitely large or...

. Examples are the Levi-Civita field

Levi-Civita field

In mathematics, the Levi-Civita field, named after Tullio Levi-Civita, is a non-Archimedean ordered field, i.e., a system of numbers containing infinite and infinitesimal quantities...

, the hyperreal number

Hyperreal number

The system of hyperreal numbers represents a rigorous method of treating the infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form1 + 1 + \cdots + 1. \, Such a number is...

s, the surreal number

Surreal number

In mathematics, the surreal number system is an arithmetic continuum containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number...

s, the Dehn field, and the field of rational function

Rational function

In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...

s with real coefficients with a suitable order.

Definition

The Archimedean property

Archimedean property

In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some ordered or normed groups, fields, and other algebraic structures. Roughly speaking, it is the property of having no infinitely large or...

 is a property of certain ordered fields such as the rational number

Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

s or the real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s, stating that every two elements are within an integer multiple of each other. If a field contains two positive elements for which this is not true, then must be an infinitesimal, greater than zero but smaller than any integer unit fraction

Unit fraction

A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/n...

. Therefore, the negation of the Archimedean property is equivalent to the existence of infinitesimals.

Applications

Hyperreal field

Hyperreal number

The system of hyperreal numbers represents a rigorous method of treating the infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form1 + 1 + \cdots + 1. \, Such a number is...

s, non-Archimedean ordered fields containing the real numbers as a subfield, may be used to provide a mathematical foundation for non-standard analysis

Non-standard analysis

Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...

.

Max Dehn

Max Dehn

Max Dehn was a German American mathematician and a student of David Hilbert. He is most famous for his work in geometry, topology and geometric group theory...

 used the Dehn field, an example of a non-Archimedean ordered field, to construct non-Euclidean geometries

Non-Euclidean geometry

Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...

 in which the parallel postulate

Parallel postulate

In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry...

 fails to be true but nevertheless triangles have angles summing to .

The field of rational functions over can be used to construct an ordered field which is complete (in the sense of convergence of Cauchy sequences) but is not the real numbers. Sometimes the term complete is used to mean that the least upper bound property

Least-upper-bound property

In mathematics, the least-upper-bound property is a fundamental property of the real numbers and certain other ordered sets. The property states that any non-empty set of real numbers that has an upper bound necessarily has a least upper bound ....

 holds. With this meaning of complete there are no complete non-Archemedian ordered fields. The subtle distinction between these two uses of the word complete is occasionally a source of confusion.