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	Hyperreal number - Wikipedia, the free encyclopedia
		In 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.
		In the case of the hyperreals, a long historical delay in their development was caused by uncertainty among mathematicians as to exactly which properties could be retained, and which would have to be given up.
		The self-consistent development of the hyperreals turned out to be possible if every true first-order logic statement that uses basic arithmetic (the natural numbers, plus, times, comparison) and quantifies only over the real numbers was assumed to be true in a reinterpreted form if we presume that it quantifies over hyperreal numbers.
	en.wikipedia.org /wiki/Hyperreal_number (3027 words)

	Number article - Number quantity whole numbers natural numbers counting negative integers - What-Means.com (Site not responding. Last check: 2007-11-07)
		The most familiar numbers are the whole numbers {0, 1, 2,...} denoted by W and the natural numbers {1, 2, 3,...} used for counting and denoted by N.
		Newer developments are the hyperreal numbers and the surreal numbers, which 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 groups, rings and fields.
	www.what-means.com /encyclopedia/Number (447 words)

	Kids.net.au - Encyclopedia Hyperreal number - (Site not responding. Last check: 2007-11-07)
		The hyperreal numbers or nonstandard reals (usually denoted as R) are an extension of the real numbers* R that adds infinitely large as well as infinitesimal numbers to R.
		The study of these numbers, their functions and properties is called nonstandard analysis which some find more intuitive than standard real analysis.
		The hyperreals are defined in such a way that every first-order logic statement that uses basic arithmetic (the natural numbers, plus, times, comparison) and quantifies only over the real numbers is also true if we presume that they quantify over hyperreal numbers.
	www.kids.net.au /encyclopedia-wiki/hy/Hyperreal_number (937 words)

	Surreal number - Wikipedia, the free encyclopedia
		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.
		This number is equivalent with the ordinal number with the same name.
		The surreal numbers were originally motivated by studies of the game Go, and there are numerous connections between popular games and the surreals.
	en.wikipedia.org /wiki/Surreal_number (3179 words)

	Encyclopedia: Hyperreal number
		In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line.
		Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...
		In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third).
	www.nationmaster.com /encyclopedia/Hyperreal-number (4734 words)

	hyperreal number
		Any of a colossal set of numbers, also known as nonstandard reals, that includes not only all the real numbers but also certain classes of infinitely large (see infinity) and infinitesimal numbers as well.
		Hyperreals emerged in the 1960s from the work of Abraham Robinson who showed how infinitely large and infinitesimal numbers can be rigorously defined and developed in what is called nonstandard analysis.
		But in the hyperreal system, it turns out that that each real number is surrounded by a cloud of hyperreals that are infinitely close to it; the cloud around zero consists of the infinitesimals themselves.
	www.daviddarling.info /encyclopedia/H/hyperreal_number.html (249 words)

	Encyclopedia: Hyperreal numbers
		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 hyperreals are to be defined in such a way that every true first-order logic statement that uses basic arithmetic (the natural numbers, plus, times, comparison) and quantifies only over the real numbers is also true in a reinterpreted form if we presume that it quantifies over hyperreal numbers.
		A hyperreal number x is called finite (or limited by some authors) if there exists a natural number n such that -n < x < n; otherwise, x is called infinite.
	www.nationmaster.com /encyclopedia/Hyperreal-numbers (1757 words)

	Real number - FreeEncyclopedia (Site not responding. Last check: 2007-11-07)
		Real numbers may be expressed by decimal fractions, such as 324.823211247...; it is recursive[?] if the digits can be specified by a recursive algorithm.
		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[?].
		Negative numbers began to be generally accepted in the 1600s and were invented by Muslim mathematicians.
	openproxy.ath.cx /re/Real_number.html (2236 words)

	real number
		Real numbers stand in one-to-one correspondence with the points on a continuous line, known as the real number line, that stretches from zero to infinity in both directions.
		The name "real number" is a retronym, coined by René Descartes in response to the concept of imaginary numbers.
		Number systems that are even more general that the real numbers include the complex numbers and, of much more recent discovery, hyperreal numbers and surreal numbers.
	www.daviddarling.info /encyclopedia/R/real_number.html (182 words)

	ipedia.com: Hyperreal number Article (Site not responding. Last check: 2007-11-07)
		In mathematical logic, the hyperreal numbers or nonstandard reals (usually denoted as R) is an ordered field which is a proper extension of the ordered field of real numbers* R.
		The study of these numbers, their functions and properties is called nonstandard analysis; some find it more intuitive than standard real analysis.
		The hyperreals are defined in such a way that every true first-order logic statement that uses basic arithmetic (the natural numbers, plus, times, comparison) and quantifies only over the real numbers is also true if we presume that it quantifies over hyperreal numbers.
	www.ipedia.com /hyperreal_number.html (1030 words)

	[No title]
		They may in theory be expressed by decimal fractions that have an infinite sequence of digits to the right of the decimal point; these are often (mis-)represented in the same form as 324.823211247...
		Writing them as decimal fractions (which are rational numbers that could be written as ratios, with an explicit denominator) is not only more compact, but to some extent expresses the sense of an underlying real number.
		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 (Site not responding. Last check: 2007-11-07)
		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.
		Put another way, every finite nonstandard real number is "very close" to a unique real number, in the sense that if x is a finite nonstandard real, then there exists one and only one real number st(x) such that x – st(x) is infinitesimal.
		Robert Goldblatt, Lectures on the hyperreals : an introduction to nonstandard analysis, Springer, 1998.
	www.worldhistory.com /wiki/H/Hyperreal-number.htm (2146 words)

	Math Forum: Ask Dr. Math: A Mathematical Essay
		When comparing two hyperreal numbers, a and b, we can form three disjoint sets: the agreement set (set of indices of corresponding equal terms of the sequence) the "a-greater set" (set of indices of corresponding terms greater in a than b), and the "b-greater set" (whatever is left over).
		The set of all hyperreal numbers is denoted by R. The fact that there are infinitely many sequences we can use to denote the same hyperreal* should not bother us, just as 1/2 and 17/34 can be used to denote the same rational number.
		The set R of real numbers is a subset of *R, and a member r of R is the equivalence class identified by the constant sequence r.
	mathforum.org /dr.math/faq/analysis_hyperreals.html (9050 words)

	Learn more about Hyperreal number in the online encyclopedia. (Site not responding. Last check: 2007-11-07)
		Learn more about Hyperreal number in the online encyclopedia.
		Nonetheless these concepts were from the beginning seen as suspect, notably by Bishop Berkeley, and when in the 1800s calculus was put on a firm footing through the development of the epsilon-delta definition of a limit by Augustin Louis Cauchy, Karl Weierstrass and others, they were largely abandoned.
		A non-standard real number x is called finite if there exists a natural number n such that – n
	www.onlineencyclopedia.org /h/hy/hyperreal_number.html (945 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		A real number r is said to be constructible or Euclidean if a line segment of
		Sometimes the negatives of constructible numbers are also called constructible.
		All rational numbers are constructible, and all constructible numbers are algebraic numbers.
	en-cyclopedia.com /wiki/Constructible_number (72 words)

	Real number : Real
		For example, the set of rationals with square less than 2 has a rational upper bound (e.g., 1.5) but no rational least upper bound, because the square root of 2 isn't rational.
		It isn't possible to characterize the reals with first-order logic alone: the Lowenheim-Skolem theorem[?] implies that there exists a countable dense subset of the real numbers satisfying exactly the same sentences in first order logic as the real numbers themselves.
		Occasionally, formal elements +∞ and -∞ are added to the reals to form the extended real number line, a compact space which isn't a field anymore but retains many of the properties of the real numbers.
	www.wordlookup.net /re/real.html (2476 words)

	A Passion for Mathematics
		A Passion for Mathematics is an educational, entertaining trip through the curiosities of the math world, blending an eclectic mix of history, biography, philosophy, number theory, geometry, probability, huge numbers, and mind-bending problems into a delightfully compelling collection that is sure to please math buffs, students, and experienced mathematicians alike.
		Numbers help us glimpse a greater universe normally shielded from our small brains that have not evolved to comprehend fully the mathematical fabric of the universe.
		I hope you'll shiver too as you glimpse numbers ranging from integers, fractions, and radicals to stranger beasts like the transcendental numbers, transfinite numbers, hyperreal numbers, surreal numbers, the "nimbers," the quaternions, biquaternions, sedenions, and octonions.
	sprott.physics.wisc.edu /pickover/passion-math.html (1286 words)

	Research Reports on Transfinite Graphs and Networks (Site not responding. Last check: 2007-11-07)
		The procedure is similar to an ultrapower construction of an internal set from a sequence of subsets of the real line, but now the individual entities are the tips of the branches instead of real numbers.
		After incidences and adjacencies between nonstandard vertices and edges are defined, several formulas regarding numbers of vertices and edges, and nonstandard versions of Eulerian graphs, Hamiltonian graphs, and a coloring theorem are established for these nonstandard graphs.
		Once the transfinite graphs of higher ranks are established, theorems concerning the existence of hyperreal operating points and the satisfaction of Kirchhoff's laws in nonstandard networks of higher ranks can be proven just as they are for nonstandard networks of the first rank.
	www.ece.sunysb.edu /~zeman/researchtgn.html (2624 words)

	Amazon.com: Books: On Numbers and Games (Site not responding. Last check: 2007-11-07)
		His system is so powerful that it includes the "hyperreal" numbers (infinitesimals and such) that emerge (by a very different route, of course) from Abraham Robinson's nonstandard analysis as a trivial special case.
		I am not talking about facts such as whether a specific large number is prime, but about the fundamental definition of what a number is. The appearance of the surreal numbers is one of those mathematical equivalents of a whack on the side of the head.
		Suddenly, numbers are defined as the strengths of positions in certain games, something that is at first strange, but it turns out that the class of objects defined this way includes the real and ordinal numbers.
	www.amazon.com /exec/obidos/tg/detail/-/1568811276?v=glance (2193 words)

	mjgeddes \| Betterhumans > Member (Site not responding. Last check: 2007-11-07)
		Now to understand the notion of p-adic numbers, one first needs to grasp the more ordinary math notions of ‘real-numbers’ and ‘rational numbers’.
		Real numbers are points on an infinite number line. They are continuous – for any two points on an infinite number line you can found another point in between them – you can go on dividing up the line indefinitely.
		Q (the rational numbers) is a sub-set of R (the real numbers). What about p-adic numbers? Well, Q (the rational numbers) is also a sub-set of the p-adics.
	www.betterhumans.com /Members/mjgeddes/Default.aspx (445 words)

	The Assayer: Elementary Calculus: An Approach Using Infinitesimals (Site not responding. Last check: 2007-11-07)
		The real numbers are generalized to make a numer system called the hyperreal numbers, which include infinitesimally small numbers as well as infinitely large ones.
		I suspect that they'd have an easier time with many of the concepts like implicit differentiation, which seems so awkward in the traditional approach, but they might be scared a little by the initial development of the hyperreal number system.
		After that, I began to see the hyperreal numbers as simply another tool for calculating things.
	www.theassayer.org /cgi-bin/asbook.cgi?book=770 (884 words)

	Non-standard analysis - Metaweb (Site not responding. Last check: 2007-11-07)
		Non-standard analysis is that branch of mathematics that is concerned with analysis using the non-Archimedean ordered field of hyperreal numbers.
		It can be seen as the use of model theory to study analysis.
		If we start from the rationals, rather than the real numbers, and divide the ring of non-standard finite rational numbers by the ideal of the infinitesimal rational numbers, we get a field (because it is a maximal ideal) -- the field of real numbers.
	www.metaweb.com /wiki/wiki.phtml?title=Non-standard_analysis (456 words)

	2.6.1 Hyperreal Numbers (Site not responding. Last check: 2007-11-07)
		Hyperreals extend the reals by adding both infinite numbers, such as
		does not, however, contain a third number 0 distinct from +0 and -0.
		The hyperreals are an extension of the reals; they are constructed so that all statements which are provable over the reals are provable over the hyperreals, using a classical proof system.
	www.dgp.toronto.edu /people/mooncake/thesis/node26.html (154 words)

	Introduction to Non-Wimpy Number Systems (Site not responding. Last check: 2007-11-07)
		Learn how to construct and play with two extensions of the real number system that contain infinite and infinitesimal numbers.
		Introduces Abraham Robinson's Hyperreals and illustrates their use in a formulation of probability theory that allows for events with infinitesimal probability.
		We will construct the surreal numbers from the sign sequence point of view, then describe the relationship between this and their game-theoretic origins.
	www.princeton.edu /~adame/nonwimpy.html (131 words)

	Hyperreal number - Metaweb (Site not responding. Last check: 2007-11-07)
		I'm sure these all come into play in cracking codes - a page for the hyperreal number
		The study of these numbers, their functions and properties is called non-standard analysis which some find more intuitive than standard real analysis.
		Jordi Gutierrez Hermoso: Nonstandard Analysis and the Hyperreals — A gentle introduction.
	www.metaweb.com /wiki/wiki.phtml?title=Hyperreal_number (1098 words)

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