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	Field (mathematics) - the free encyclopedia (Site not responding. Last check: 2007-10-15)
		A field is a commutative ring (F, +, *) such that 0 does not equal 1 and all elements of F except 0 have a multiplicative inverse.
		For a given field F, the set F(X) of rational functions in the variable X with coefficients in F is a field; this is defined as the set of quotients of polynomials with coefficients in F.
		An algebraic extension of a field F is the smallest field containing F and a root of an irreducible polynomial p(x) in F[x].
	www.free-web-encyclopedia.com /?t=Field_(mathematics) (1266 words)

	Science Fair Projects - Field (mathematics)
		In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers.
		Fields are important objects of study in algebra since they provide the proper generalization of number domains, such as the sets of rational numbers, real numbers, or complex numbers.
		The concept of a field is of use, for example, in defining vectors and matrices, two structures in linear algebra whose components can be elements of an arbitrary field.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Subfield (1476 words)

	FIELD (MATHEMATICS) FACTS AND INFORMATION (Site not responding. Last check: 2007-10-15)
		Fields are important objects of study in algebra since they provide a useful generalization of many number systems, such as the rational numbers, real numbers, and complex_numbers.
		A structure which satisfies all the properties of a field except for commutativity, is today called a ''division_ring'' or sometimes a ''skew field'', but also ''non-commutative field'' is still widely used.
		An algebraic_number_field is a finite field extension of the rational_numbers Q, that is, a field containing Q which has finite dimension as a vector_space over Q.
	www.enablepay.com /field_(mathematics) (1494 words)

	Number - the free encyclopedia (Site not responding. Last check: 2007-10-15)
		Elements ofalgebraic function fields of finite characteristic behave in many ways like numbers andare often regarded as a kind of number by number theorists.
		Superreal, hyperreal and surreal numbers extend the realnumbers by adding infinitesimal and infinitely large numbers.
		The arithmetical operations of numbers, such as addition, subtraction, multiplicationand division, are generalized in the branch of mathematics called abstract algebra; one obtainsthe groups, rings and fields.
	www.free-web-encyclopedia.com /?t=Number (406 words)

	Superreal Number [Definition] (Site not responding. Last check: 2007-10-15)
		The superreal numbers compose a more inclusive category than hyperreal numberIn mathematics, particularly in non-standard analysis and mathematical logic, hyperreal numbers or nonstandard reals (usually denoted as R) denote an ordered field* which is a proper extension of the ordered field of real numbers R and which satisfies the transfer principle.
		Algebraic numbersIn mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form...
		Surreal numbersIn mathematics, the surreal numbers are a field containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similiar to superreal numbers and hyperreal numbers.
	www.wikimirror.com /Superreal_number (1842 words)

	Complex number - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-15)
		In mathematics, the term "complex" when used as an adjective means that the field of complex numbers is the underlying number field considered, for example complex matrix, complex polynomial and complex Lie algebra.
		To see that these properties characterize C as a topological field, one notes that P ∪ {0} ∪ -P is an ordered Dedekind-complete field and thus can be identified with the real numbers R by a unique field isomorphism.
		In applied fields, the use of complex analysis is often used to compute certain real-valued improper integrals, by means of complex-valued functions.
	www.bucyrus.us /project/wikipedia/index.php/Complex_number (3078 words)

	Online Encyclopedia and Dictionary - Hyperreal number
		It is also stronger than that of being a superreal field in the sense of Dales and Woodin.
		As a real closed field with cardinality the continuum, it is isomorphic as a field to R but is not isomorphic as an ordered field to R.
		The hyperreal fields we obtain in this case are called ultrapowers of R and are identical to the ultrapowers constructed via free ultrafilters in model theory.
	fact-archive.com /encyclopedia/Hyperreal_number (2067 words)

	Number theory Information (Site not responding. Last check: 2007-10-15)
		Elements of function fields of finite characteristic (algebra)characteristic behave in some ways like numbers and are often regarded as a kind of number by number theorists.
		Superreal numberSuperreal, hyperreal numberhyperreal and surreal numbers extend the real numbers by adding infinitesimal and infinitely large numbers.
		The arithmetical operations of numbers, such as addition, subtraction, multiplication and division, are generalized in the branch of mathematics called abstract algebra; one obtains the group (mathematics)groups, ring (algebra)rings and field (mathematics)fields.
	www.echostatic.com /index.php?title=Number_theory (522 words)

	Online Encyclopedia and Dictionary - Surreal number
		In mathematics, the surreal numbers are a field containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similar to superreal numbers and hyperreal numbers.
		By limiting the construction to a Grothendieck universe, a set is obtained, rather than a class, with an honest field with the cardinality of some strongly inaccessible cardinal.
		Finally it can be shown that the generalized operations on the equivalence classes have the desired algebraic properties, i.e., the equivalence classes plus their ordering and the algebraic operations constitute an ordered field, with the caveat that they do not form a set but a proper class, see below.
	fact-archive.com /encyclopedia/Surreal_number (3152 words)

	Read about Surreal number at WorldVillage Encyclopedia. Research Surreal number and learn about Surreal number here! (Site not responding. Last check: 2007-10-15)
		The class of games is more general than the surreals, and has a simpler definition, but lacks some of the nicer properties of surreal numbers.
		field, but the class of games does not.
		Finally, he developed the multiplication operator, and proved that the surreals are actually a field, and that it includes both the reals and ordinals.
	encyclopedia.worldvillage.com /s/b/Surreal_number (2959 words)

	Articles - Real number (Site not responding. Last check: 2007-10-15)
		The set R is a field, i.e., addition, subtraction, multiplication and division are defined and have the usual properties.
		However, an ordered group (and a field is a group under the operations of addition and subtraction) defines a uniform structure, and uniform structures have a notion of completeness (topology); the description in the section Completeness above is a special case.
		Ordered fields extending the reals are the hyperreal numbers and the surreal numbers; both of them contain infinitesimal and infinitely large numbers and thus are not Archimedean.
	www.kamero.net /articles/Real_number (1653 words)

	HYPERREAL NUMBER FACTS AND INFORMATION (Site not responding. Last check: 2007-10-15)
		However, a 2003 paper by Kanovei and Shelah shows that there is a definable, countably saturated (meaning ω-saturated, but not of course countable) elementary extension of the reals, which therefore has a good claim to the title of ''the'' hyperreal numbers.
		The condition of being a hyperreal field is a stronger one than that of being a real_closed_field strictly containing R.
		As a real_closed_field with cardinality the continuum, it is isomorphic as a field to R but is ''not'' isomorphic as an ordered field to R.
	www.southcountryequity.com /hyperreal_number (2025 words)

	Number - Open Encyclopedia (Site not responding. Last check: 2007-10-15)
		Roots of polynomials with rational coefficients lead to algebraic numbers.
		Elements of algebraic function fields of finite characteristic behave in many ways like numbers and are often regarded as a kind of number by number theorists.
		Many languages have the concept of grammatical number, an attribute of certain words and phrases that affects their syntactic usage and meaning.
	open-encyclopedia.com /Number (460 words)

	Rational number
		The rationals are the smallest field with characteristic 0: every other field of characteristic 0 contains a copy of
		the field of roots of rational polynomials, is the algebraic numbers.
		The rational numbers are an important example of a space which is not locally compact.
	www.askfactmaster.com /Rational_number (712 words)

	Hyperreal Number [Definition] (Site not responding. Last check: 2007-10-15)
		Types and properties of sequences A subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements....
		F is a formally real field such that any algebraic extension of F is not formally real.
		The quotient field F of A is a superreal field if F strictly contains the real numbers, so that F is not order isomorphic to, thoug...
	www.wikimirror.com /Hyperreal_number (4432 words)

	Real number - InformationBlast
		The field R is ordered, i.e., there is a total order ≥ such that, for all real numbers x, y and z:
		It's easy to see that no ordered field can be lattice complete, because it can have no largest element (given any element z, z + 1 is larger), so this is not the sense that is meant.
		Occasionally, formal elements +∞ and -∞ are added to the reals to form the extended real number line, a compact space which is not a field but retains many of the properties of the real numbers.
	www.informationblast.com /Real_Numbers.html (1974 words)

	Hypercomplex number - One Language (Site not responding. Last check: 2007-10-15)
		More precisely, they form finite-dimensional algebras over the real numbers.
		But none of these extensions forms a field, essentially because the field of complex numbers is algebraically closed — see fundamental theorem of algebra.
		The quaternions, octonions and sedenion can be generated by the Cayley-Dickson construction.
	www.onelang.com /encyclopedia/index.php/Hypercomplex_number (170 words)

	[No title]
		group (and a field is a group under the operations of addition and subtraction) defines a uniform structure, and uniform structures have a notion of
		Archimedean field, and indeed one often hears the phrase "complete Archimedean field" instead of "complete ordered field".
		extended real number line, a compact space which is not a field but retains many of the properties of the real numbers.
	en-cyclopedia.com /wiki/Real_number (1750 words)

	Hyperreal number - Iridis Encyclopedia (Site not responding. Last check: 2007-10-15)
		Being a hyperreal field is a stronger condition than being a real closed field strictly containing
		\Bbb{R}, and also stronger than being a superreal field in the sense of Dales and Woodin.
		The map st is locally constant, which entails that its derivative is identically zero and that it is continuous with respect to the order topology on the finite hyperreals.
	www.iridis.com /Hyperreal_numbers (1712 words)

	Category:Field theory - Wikipedia, the free encyclopedia
		Over $95,000 has been donated since the drive began on 19 August.
		Field theory is a branch of mathematics which studies the properties of fields.
		This page was last modified 03:06, 14 October 2004.
	en.wikipedia.org /wiki/Category:Field_theory (60 words)

	Rational number - Linix Encyclopedia (Site not responding. Last check: 2007-10-15)
		In mathematics, the term "rational something" means that the underlying field considered is the field $\mathbb{Q} of rational numbers .$
		The set $\mathbb{Q}, together with the addition and multiplication operations shown above, forms a$ field, the quotient field of the integers $\mathbb{Z} .$
		The rationals are the smallest field with characteristic 0: every other field of characteristic 0 contains a copy of $\mathbb{Q} .$
	web.linix.ca /pedia/index.php/Rational_number (940 words)

	Superreal number - Encyclopedia Glossary Meaning Explanation Superreal number (Site not responding. Last check: 2007-10-15)
		Superreal number - Encyclopedia Glossary Meaning Explanation Superreal number.
		\Bbb{R}, though they may be isomorphic as fields.
		The orginal Superreal number article can be editet
	www.encyclopedia-glossary.com /en/Superreal-number.html (229 words)

	Knowledgebase : Complex_number : Mpageni.com : wikipedia.org reflection (Site not responding. Last check: 2007-10-15)
		The set of all complex numbers is usually denoted by C, or in flboard bold by
		So defined, the complex numbers form a field, the complex number field, denoted by C.
		The treatment of resistors, capacitors, and inductors can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance.
	mpageni.com /fun/knowledge/index.php?title=Complex_number (2800 words)

	Constructible pharmacy number (Site not responding. Last check: 2007-10-15)
		Thus, the pharmacy set K of all constructible complex numbers forms a field, a subfield of the pharmacy field of pharmacy algebraic numbers.
		pharmacy These facts can be used to pharmacy characterize the field of constructible numbers, because, in essence, the equations defining lines and circles are no worse than quadratic.
		One should note pharmacy that it is true, (but not pharmacy obvious to show) that the converse is false — this is not a sufficient condition for constructibility.
	www.onlinepharmacy-now.com /site45/Eu/Euclidean_number.html (985 words)

	birding facts Birding Resources by the Fat Birder
		I love the superreal images where fine detail even in backgrounds give images a surreal feel and are as accurate in their portraiture as a fieldguide illustration.
		James Coe is best known as author and illustrator of the Golden field guide, Eastern Birds.© He has contributed illustrations to numerous other field guides, including the Macmillan guide Birds of North America: Western Region and Birds of New Guinea, and to Frank Gill`s widely-used college textbook Ornithology.
		View how sketches, preparatory drawings and a gallery of wildlife paintings are evolved from initial field encounters, detailed information about creative working methods with wildlife watching tips, painting stages from initial brush stroke to completion, oils in progress.
	www.fatbirder.com /links/images_and_sound/art_and_artists.html (3756 words)

	Hyperreal number (Site not responding. Last check: 2007-10-15)
		, and also stronger than being a superreal field in the sense of Dales and Woodin.
		The hyperreal fields we obtain in this case are called ultrapowers of
		As a real closed field with cardinality the continuum, it is isomorphic as a field to
	www.worldhistory.com /wiki/H/Hyperreal-number.htm (1745 words)

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