In
abstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, the
superreal numbers are a class of extensions of the real numbers, introduced by H. Garth Dales and
W. Hugh WoodinWilliam Hugh Woodin is an American mathematician and set theorist at University of California, Berkeley. He has made many notable contributions to the theory of inner models and determinacy. A type of large cardinal, the Woodin cardinal, bears his name.-Biography:Born in Tucson, Arizona, Woodin...
as a generalization of the
hyperreal numberThe 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 and primarily of interest in
non-standard analysisNon-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:...
,
model theoryIn mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
, and the study of
Banach algebraIn mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space...
s. The
fieldIn abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
of superreals is itself a subfield of the
surreal numberIn 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.
Dales and Woodin's superreals are distinct from the super-real numbers of
David O. TallDavid Orme Tall is a mathematics education theorist at the University of Warwick. One of his most influential works is the joint paper with Vinner Concept image and concept definition.... The "concept image" is a notion in cognitive theory. It consists of all the cognitive structure in the...
, which are lexicographically ordered fractions of
formal power seriesIn mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...
over the reals.
Formal Definition
Suppose X is a
Tychonoff spaceIn topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces.These conditions are examples of separation axioms....
, also called a T
3.5 space, and C(X) is the algebra of continuous real-valued functions on X. Suppose P is a
prime idealIn algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
in C(X). Then the factor algebra A = C(X)/P is by definition an integral domain which is a real algebra and which can be seen to be
totally orderedIn set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
. The
field of fractionsIn abstract algebra, the field of fractions or field of quotients of an integral domain is the smallest field in which it can be embedded. The elements of the field of fractions of the integral domain R have the form a/b with a and b in R and b ≠ 0...
F of A is a
superreal field if F strictly contains the real numbers
, so that F is not order isomorphic to
.
If the prime ideal P is a maximal ideal, then F is a field of
hyperreal numberThe 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.