Archimedean field

In mathematics

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

, an Archimedean field is an ordered field

Ordered field

In mathematics, an ordered field is a field together with a total ordering of its elements that agrees in a certain sense with the field operations...

 with the Archimedean property

Archimedean property

In abstract algebra, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some groups, fields, and other algebraic structures....

, named after the ancient Greek mathematician Archimedes

Archimedes

Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity...

 of Syracuse

Syracuse, Italy

Syracuse is a historic city in southern Italy, the capital of the province of Syracuse. The city is famous for its rich Greek history, culture, amphitheatres, architecture and association to Archimedes, playing an important role in ancient times as one of the top powers of the Mediterranean world;...

.

In an ordered field F we can define the absolute value of an element x in F in the usual way by setting |x| = x for nonnegative x and |x| = −x for negative x. Then, an Archimedean field F is one such that for any non-zero x in F there exists a natural number

Natural number

In mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {, , , ...} according to the traditional definition or the set of non-negative integers {, 1, 2, ...} according to...

 n such that
The real number

Real number

In mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...

s form an Archimedean field.

In mathematics

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

, an Archimedean field is an ordered field

Ordered field

In mathematics, an ordered field is a field together with a total ordering of its elements that agrees in a certain sense with the field operations...

 with the Archimedean property

Archimedean property

In abstract algebra, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some groups, fields, and other algebraic structures....

, named after the ancient Greek mathematician Archimedes

Archimedes

Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity...

 of Syracuse

Syracuse, Italy

Syracuse is a historic city in southern Italy, the capital of the province of Syracuse. The city is famous for its rich Greek history, culture, amphitheatres, architecture and association to Archimedes, playing an important role in ancient times as one of the top powers of the Mediterranean world;...

.

In an ordered field F we can define the absolute value of an element x in F in the usual way by setting |x| = x for nonnegative x and |x| = −x for negative x. Then, an Archimedean field F is one such that for any non-zero x in F there exists a natural number

Natural number

In mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {, , , ...} according to the traditional definition or the set of non-negative integers {, 1, 2, ...} according to...

 n such that
The real number

Real number

In mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...

s form an Archimedean field. Moreover, it can be proved that any Archimedean field is isomorphic

Isomorphism

In abstract algebra, an isomorphism is a bijective map f such that both f and its inverse f ⁻¹ are homomorphisms, i.e., structure-preserving mappings....

 (as an ordered field) to a subfield of the real numbers. Any complete Archimedean field is isomorphic

Isomorphism

In abstract algebra, an isomorphism is a bijective map f such that both f and its inverse f ⁻¹ are homomorphisms, i.e., structure-preserving mappings....

 (as an ordered field) to the field of real numbers.

Archimedean fields are important in the axiomatic construction of the real numbers.

Non-archimedean fields

Non-archimedean fields with infinitesimal and infinitely large elements are also possible. The p-adic number

P-adic number

In mathematics, the p-adic number systems were first described by Kurt Hensel in 1897. For each prime number p, the p-adic number system extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...

 fields furnish one family of examples. An example using rational functions is presented at Archimedean property

Archimedean property

In abstract algebra, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some groups, fields, and other algebraic structures....

. Another example is the hyperreal numbers of nonstandard analysis.