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Topic: Extended real number line

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	Extended real number line - Wikipedia, the free encyclopedia
		The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers).
		The extended real number line is denoted R or [−∞, +∞].
		This limiting behaviour is similar to the limit of a function at a real number, except that there is no "number" to which x is approaching.
	en.wikipedia.org /wiki/Extended_real_number_line (817 words)

	Real number - FreeEncyclopedia (Site not responding. Last check: 2007-10-21)
		The real numbers are, intuitively, numbers that are in one-to-one correspondence with the points on a line -- the number line[?].
		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.
		It's impossible to explicitly specify a non-recursive number; the Russian school[?] of constructivism assumes that all real numbers are recursive.
	openproxy.ath.cx /re/Real_number.html (2236 words)

	Learn more about Real number in the online encyclopedia. (Site not responding. Last check: 2007-10-21)
		Real numbers could be constructed as the topological completion of rational numbers.
		Occasionally, formal elements +∞ and -∞ are added to the reals to form the extended real number line, a compact space which is not a field anymore but retains many of the properties of the real numbers.
		Self-adjoint operatorss on a Hilbert space (for example, self-adjoint square complex matrices) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their eigenvalues are real and they form a real associative algebra.
	www.onlineencyclopedia.org /r/re/real_number.html (2010 words)

	Number line - Wikipedia, the free encyclopedia
		A number line is a one-dimensional graph in which the integers are shown as specially-marked points evenly spaced on a line.
		The number line is most often represented as being horizontal (though it need not be so).
		Customarily, positive numbers lie on the right side of zero, and negative numbers lie on the left side of zero (though it need not be so).
	en.wikipedia.org /wiki/Number_line (133 words)

	Knowledge King - Extended real number line (Site not responding. Last check: 2007-10-21)
		The extended real number line is obtained from the real number line R by adding two elements: +∞ and -∞ (which are not considered to be real numbers).
		The extended real number line is denoted by
		The extended real number line turns into a totally ordered set by defining -∞ ≤ a ≤ +∞ for all a.
	www.knowledgeking.net /encyclopedia/e/ex/extended_real_number_line.html (469 words)

	Unit interval (Site not responding. Last check: 2007-10-21)
		As a topological space, it is homeomorphismhomeomorphic to the extended real number line.
		The unit interval is a dimensionone-dimensional analytical manifold with boundary {0,1}, carrying a standard orientation (manifold)orientation from 0 to 1.
		As a subset of the real numbers, its Lebesgue measure is 1.
	www.infothis.com /find/Unit_interval (505 words)

	Extended real number line -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		is obtained from the (Click link for more info and facts about real number line) real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers).
		The extended real number line is denoted by R or [−∞,+∞].
		The extended real number line turns into a (Click link for more info and facts about totally ordered set) totally ordered set by defining −∞ ≤ a ≤ +∞ for all a.
	www.absoluteastronomy.com /encyclopedia/e/ex/extended_real_number_line.htm (664 words)

	Extended real number line - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-10-21)
		is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers).
		The extended real number line is denoted by R or.
		R is a compact Hausdorff space homeomorphic to the unit interval.
	encyclopedia.worldsearch.com /extended_real_number_line.htm (515 words)

	Lattice - FreeEncyclopedia (Site not responding. Last check: 2007-10-21)
		The supremum is given by the union and the infimum by the intersection of subsets.
		The unit interval [0,1] and the extended real number line, with the familiar total order and the ordinary suprema and infima.
		The infimum is given by the intersection of convex sets and the supremum by the convex hull of the union.
	openproxy.ath.cx /la/Lattice.html (913 words)

	Real number (Site not responding. Last check: 2007-10-21)
		In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the pointpoints on an infinite line—the number line.
		Real numbers may be rational numberrational or irrational numberirrational; algebraic numberalgebraic or transcendental numbertranscendental; and positive numberpositive, negative numbernegative, or zero/.
		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 groupArchimedean.
	www.infothis.com /find/Real_number (2292 words)

	More Real Number Paradoxes — Mathematics — PAIAS (Site not responding. Last check: 2007-10-21)
		The paradox is that this means there can be at most countable real numbers in the unit interval (easily extendable to the entire real number line), as opposed to the standardly accepted non-denumerable real numbers.
		real numbers (with something akin to the topological discontinuities we get when projecting spaces onto lower dimensional subspaces), but the arithmetic convergence requirement takes away that ability.
		The reals (will) have their own granularity, and one of the casualties there will be the Bolzano-Weierstrass theorem with its “cluster points”, since the granularity that derives from the actually non-vanishing remainder ensures that we can always find a non-null punctured neighborhood around any point with no numbers/points in it whatsoever.
	paias.org /Mathematics/Paradoxes/morerealnumberparadoxes.htm (1468 words)

	Real number (Site not responding. Last check: 2007-10-21)
		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.city-search.org /re/real-number.html (2464 words)

	Metric space (Site not responding. Last check: 2007-10-21)
		R (where R is the set of real numbers).
		Some authors use the extended real number line and allow the distance function d to attain the value ∞.
		A metric space M is called bounded if there exists some number r > 0 such that d(x,y) ≤ r for all x and y in M (not to be confused with "finite", which refers to the number of elements, not to how far the set extends; finiteness implies boundedness, but not conversely).
	usapedia.com /m/metric-space.html (1196 words)

	Yellowtail - Animated, Real-Time Pattern Playback
		No randomness was employed in the procedural animation of the "curlies." Instead, their animated behavior is strictly determined by the shape and speed of the mark when it was drawn.
		Nevertheless, because each line repeats according to its own natural period, the complex phase relationships of the different marks produce the effect of an ever-changing yet coherent animated texture.
		Since then, a number of researchers and companies have developed spectrogram-based drawing systems for the analysis and resynthesis of sound.
	www.flong.com /yellowtail/excerpt (2369 words)

	Real-Time Rendering Resources (Site not responding. Last check: 2007-10-21)
		Here is a comparison generated using his program of lines improperly drawn with gamma=1.0 (note the severe roping and Moíre patterns) and properly drawn with gamma=2.2.
		Optimizer includes a number of model decimators and is free for download; also see the white paper on it.
		A white paper and a number of related demos of Matrox's hardware displacement mapping feature, used in their Parhelia chipset, are available.
	www.realtimerendering.com (11594 words)

	Infimum (Site not responding. Last check: 2007-10-21)
		In analysis the infimum or greatest lower bound of a set S of real numbers is denoted by inf(S) and is defined to be the biggest real number that is smaller than or equal to every number in S.
		If S is empty, we define inf(S) = ∞ (see extended real number line).
		An important property of the real numbers is that every set of real numbers has an infimum (any bounded nonempty subset of the real numbers has an infimum in the non-extended real numbers).
	www.theezine.net /i/infimum.html (378 words)

	Supremum (Site not responding. Last check: 2007-10-21)
		An important property of the real numbers is that every set of real numbers has a supremum.
		This is sometimes called the supremum axiom and expresses the completeness of the real numbers.
		He was coming with the speed of a whirlwind, and heading interest that can scarcely be imagined.
	www.termsdefined.net /su/supremum.html (467 words)

	What are the Real Numbers? (Site not responding. Last check: 2007-10-21)
		The so-called 'Real Number System' is an attempt to extend the ideas of number and arithmetic (which originate in counting) to describe more general measurements such as length, weight and temperature.
		On the other hand, if that was the bad news, the good news is that the operations of arithmetic (defined for whole numbers in terms of combinations of sets) can be extended in a natural way to fractions and also to those "limits of fractions" that we need to account for all possible measurements.
		One way to define the Real Number System is to "construct" it by identifying real numbers with sequences of approximating rationals and showing that the arithmetic operations do extend consistently.
	www.langara.bc.ca /mathstats/resource/onWeb/precalculus/reals/intro.htm (353 words)

	Daily Nightly: Another night on location - Nightly News with Brian Williams - MSNBC.com
		Cars are lined up down the street waiting for ice, water and MREs and they tell me its been like that everyday from open to close.
		The number of automatic weapons (with muzzles exposed) is still striking, and the roadblocks are so random and plentiful that we are recommending that people leave hours in advance of their arrival times...
		The line will be staffed by FBI personnel and the information collected will be analyzed, investigated, and shared with state, local, and federal law enforcement personnel, the bureau says.
	www.msnbc.msn.com /id/9314188 (7271 words)

	Bit intrinsic to extract sign of a real number - possible?
		Swizzling floating point numbers through the integer register file is kind of an old-fashioned trick that normally leads to slower code on current processors.
		Perhaps BTEST could be extended to testing the sign bit of a real variable.
		Even without the extra line of code, bit testing is unlikely provide any worthwhile performance gain as the real*real seems to be costing next to no CPU time on my system.
	www.codecomments.com /message260390.html (2527 words)

	Power Line: March 2005 Archives
		Earlier today, we noted that Michelle Malkin has identified a number of newspapers that ran the Washington Post's story on the memo, but in a version that (unlike the one that appeared in the Post itself) explicitly attributed the document to the party's leadership.
		The bottom line: both ABC News and the Washington Post are now disavowing any claim that the alleged "talking points memo" was authored by a Republican, let alone that it was some kind of official Republican strategy memo.
		The powers-that-be at the college extended the voting deadline this morning from April 11 to April 22 -- or so I was advised in a telephone conversation with a college representative this morning.
	powerlineblog.com /archives/2005_03.php (12731 words)

	real number - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "real number" is defined.
		Real Number : The Computational Beauty of Nature [home, info]
		Phrases that include real number: nonstandard real number, axiomatic real number, extended real number line, extended real number system, real number or real number, more...
	www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=real+number (216 words)

	Humusoft: Extended Real Time Toolbox (Product Description)
		The Extended Real Time Toolbox brings the power of MATLAB and Simulink to the real world.
		The Extended Real Time Toolbox simplifies this as much as possible: the only thing you need to do is to set the parameters in the driver's GUI, and if you use your board with factory-default settings you don't need to change anything at all.
		The serial input and output commands are almost identical to their file counterparts and the serial line is treated almost identically to a file.
	www.humusoft.com /rt (741 words)

	C'mere, AquaFresh! \| MetaFilter
		And although my 'real' name is not immediately recognizable as a brand, I did once own a cassette recorder with my name on it.
		Nearly two hundred years ago an English couple on a extended honeymoon tour of Europe named their two daughters after the cities in which they were born - Parthenope and Florence.
		I really can't believe these names are so widespread that 3 mefiters would know of independent cases, so I hereby declare those names Urban Legends.
	www.metafilter.com /mefi/29566 (7600 words)

	AlterSlash ~ the unofficial SlashDot digest (Site not responding. Last check: 2007-10-21)
		But the bitch of it is that the chronic pain they suffered rewired their brains to feel chronic pain and a lot of them still have quite a bit of pain after their amputations, even though the affected limb is gone.
		The obvious one being extended range (up to 15 miles), which means that establishing a few towers would pretty much make the entire city connected.
		I really hope that someone digs out the old installer and makes it work with 2.0 so we can get back to advancing linux software instead of stepping backwards by getting rid of the installer.
	alterslash.org (12054 words)

	Amazon.com: DVD: The Lord of the Rings - The Fellowship of the Ring (Platinum Series Special Extended Edition ... (Site not responding. Last check: 2007-10-21)
		Extremely useful are the chapter menus that indicate which scenes are new or extended.
		But what really makes this film work is that the hobbits seem hobbit size and after the first time you see Frodo and Gandalf together you never give it a second thought.
		Tolkien's world is clearly male dominated, the Lady Galadriel and Eowyn being relatively minor figures in the tale despite their respective powers, and I can appreciate the idea that Arwen should be more than the elf babe who shows up and marries the King at the end.
	www.amazon.com /exec/obidos/tg/detail/-/B000069F5H?v=glance (3261 words)

	[No title]
		Embedded RealPlayer Extended Functionality Guide at http://service.real.com/help/library/index.html explains how to use JavaScript or VBScript to extend the functionality of the Netscape Plug-in or ActiveX control.
		The JVM is required only when extending plug-in functionality with JavaScript.
		When no clip is playing, only the specified background color shows in the window.
	service.real.com /help/library/guides/production/htmfiles/control.htm (2464 words)

	Scoop - New Zealand News
		The general vibe of the afternoon was cheery with Mr Peters flashing a magnificent smile as he went up for a handshake with the Governor-General.
		And when she achieved Hurricane status a few short hours late she became the 12th hurricane of 2005, also equaling the previous record for a single season.
		All I caught was the scent of tequila, and that was coming from me. There really isn't much happening down there at the moment...
	www.scoop.co.nz (2031 words)

	Newsline 1991-1997
		Fiber optic line cut creates phone service outage.
		FCC assigns rulemaking number to ARRL request for National Band Planning.
		Ham radio seems to be growing again, with decline of morse code class licensees dropping in numbers.
	www.tapr.org /newsline.html (7865 words)

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