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Topic: Construction of real numbers

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Real number

Dedekind cut

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	Real Numbers
		The real numbers are the central object of study in real analysis.
		He meant that the real numbers form the largest Archimedean field in the sense that every other Archimedean field is a subfield of R. Thus R is "complete" in the sense that nothing further can be added to it without making it no longer an Archimedean field.
		The nonexistence of a subset of the reals with cardinality strictly in between that of the integers and the reals is known as the continuum hypothesis.
	www.metu.edu.tr /~e128327/real.htm (1866 words)

	Real number Summary
		While the numbers used for this purpose are generally decimal fractions representing rational numbers, writing them in decimal terms suggests they are an approximation to a theoretical underlying real number.
		The non-existence of a subset of the reals with cardinality strictly between that of the integers and the reals is known as the continuum hypothesis.
		The reals carry a canonical measure, the Lebesgue measure, which is the Haar measure on their structure as a topological group normalised such that the unit interval [0,1] has measure 1.
	www.bookrags.com /Real_number (4419 words)

	Math Help - Algebra - Number Systems - Technical Tutoring
		We are going to avoid a precise construction of the real numbers, since that is normally a subject of advanced math courses (and does very little to get either good grades or solve practical math problems).
		For one thing, every bounded infinite sequence of numbers has at least one limit point (in ordinary language, when a sequence is defined so that infinitely many sequence numbers fall into a finite section of the real line, there is at least one real number which is the limit of an infinite number of them).
		The square root of -1 is not a real number.
	www.hyper-ad.com /tutoring/math/algebra/numbers.html (1166 words)

	Dedekind cut Summary
		Real analysis, which studies the theoretical foundations of the calculus of real-valued functions, depends upon an accurately defined real number concept.
		Wherever a cut occurs and it is not on a real rational number, an irrational number (which is also a real number) is created by the mathematician.
		A construction similar to Dedekind cuts is used for the construction of surreal numbers.
	www.bookrags.com /Dedekind_cut (1240 words)

	Real numbers 2
		Though Cauchy understood that a real number could be obtained as the limit of rationals, he did not develop his insight into a definition of real numbers or a detailed description of the properties of real numbers.
		His definition of a real number was made in terms of convergent sequences of rational numbers and is explained in [Casopis Pest.
		Similarly Cantor realised that if he wants the line to represent the real numbers then he has to introduce an axiom to recover the connection between the way the real numbers are now being defined and the old concept of measurement.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Real_numbers_2.html (2626 words)

	The Nature of the Real Numbers
		The real numbers in the interval [0,1) are then infinite sequences of digits with the provision that infinite sequences of the form.500000....
		The denumerable cardinality of the generalized algebraic numbers is established by a process similar to that used in establishing the denumerability of the rational numbers.
		A circular definition that a generalized algebraic number is a root of a poynomial with generalized algebraic number coefficients would lead to the argument that π is a generalized algebraic number because it is the solution to the linear equation x - π = 0.
	www.sjsu.edu /faculty/watkins/reals.htm (1445 words)

	Real number - Psychology Wiki
		Real numbers may be rational or irrational; algebraic or transcendental; and positive, negative, or zero.
		The non-existence of a subset of the reals with cardinality strictly between that of the integers and the reals is known as the continuum hypothesis.
		The reals carry a canonical measure, the Lebesgue measure, which is the Haar measure on their structure as a topological group normalised such that the unit interval [0,1] has measure 1.
	psychology.wikia.com /wiki/Real_number (2417 words)

	Construction of the real numbers@Everything2.com
		There are several ways to construct the real numbers in terms of sets; This writeup will present the method that Paul Bernays used.
		This method takes a construction of the rational numbers (go read that now!) and defines a "real number" as a set of rational numbers that meets certain criteria.
		However, construction of the rational numbers showed that we can represent rational numbers as fraction triplets, as equivalence classes of fraction triplets, or as canonical examples from each equivalence class.
	www.everything2.com /index.pl?node_id=1099596 (700 words)

	Jeff’s Space and Time » Blog Archive » Almost Real (Site not responding. Last check: )
		However, unlike the Dedekind construction, the construction proceeds directly from the integers to the real numbers bypassing the intermediate construction of the rational numbers.
		While the gradual acceptance of numbers to measure geometrical quantities was a useful development in thinking about geometrical problems, this approach did not have a rigorous basis until Dedekind’s construction of the field of real numbers, and until then there was no adequate substitute for Eudoxus’s theory of proportion.
		Although it is tempting to attribute to Euclid a “concept of real numbers in the back of his mind,” he dealt “only with those magnitudes constructible by ruler and compass, while Dedekind made the amazing mental leap of considering the set of all Dedekind cuts, which for Euclid would have been inconceivable” (463).
	jefflindstrom.com /blog/2006/03/27/almost-real (6301 words)

	Real number - encyclopedia article - Citizendium
		A real number may be either rational or irrational; either algebraic or transcendental; and either positive, negative, or zero.
		The first says that real numbers comprise a field, with addition and multiplication as well as division by nonzero numbers, which can be totally ordered in a way compatible with addition and multiplication.
		While the numbers used for this purpose are generally decimal fractions representing rational numbers, writing them in decimal terms suggests they are an approximation to a theoretical underlying real number.
	en.citizendium.org /wiki/Real_numbers (2608 words)

	Real number - Wikipedia, the free encyclopedia
		In mathematics, the real numbers may be described informally as a number that can be given by an infinite decimal representation, such as 2.4871773339….
		The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and the square root of 2, and can be represented as points on an infinitely long number line.
		Real numbers are so called to distinguish them from imaginary and complex numbers; early mathematicians used "imaginary" where we would now use "complex".
	en.wikipedia.org /wiki/Real_numbers (2595 words)

	Construction of real numbers - Wikipedia, the free encyclopedia
		A practical and concrete representative for an equivalence class representing a real number is provided by the representation to base b -- in practice, b is usually 2 (binary), 8 (octal), 10 (decimal) or 16 (hexadecimal).
		The real numbers form the largest subfield that is Archimedean (meaning that no real number is infinitely large).
		Real numbers are defined as the equivalence classes of this relation.
	en.wikipedia.org /wiki/Construction_of_real_numbers (1528 words)

	Springer Online Reference Works
		A number is a pair of sets of numbers with the property that no member of its left set is larger or equal to a member of its right set.
		A number is less or equal to another number in case no member of the left set of the first number is less or equal to the second number and no member of the right set of the second number is less or equal to the first.
		The Dedekind construction was invented to close the gaps in between the rational numbers.
	eom.springer.de /S/s091430.htm (1055 words)

	real number - Anarchopedia (Site not responding. Last check: )
		Informally, the "real numbers" are the rational numbers with all the holes plugged.
		If we view the rational numbers as being contained in the set of cuts, via the correspondence $q \leftrightarrow \Gamma(q)$ , then the set of cuts must therefore be a proper superset of the rational numbers.
		It is possible to develop a version of analysis that doesn't depend on the axiom of choice, based on what are called the constructible reals.
	meta.anarchopedia.org /Real_number (1187 words)

	Dedekind's Real Numbers
		The successive expansions of the number classes are motivated on the one hand by the wish to extend the operations valid in one system to the inverse operations or more generalized operations and on the other hand by the objective to have a number system which is closed under these new operations.
		The rational numbers were introduced in a creative act by the Greeks, based on the analogy between the rational numbers and the ratio of magnitudes.
		Also the real numbers were in use long before their characterization in terms of rational numbers.
	www.colorado.edu /StudentGroups/PhilosophyClub/reals.htm (1860 words)

	Real numbers 3
		Some of the intuitive difficulties that began to be felt revolved around the fact that the real numbers were not countable, that is, they could not be put in 1-1 correspondence with the natural numbers.
		Cantor proved that the real numbers were not countable in 1874.
		In The Construction of Decimals Normal in the Scale of Ten published in the Journal of the London Mathematical Society in 1933, Champernowne proved that his number was normal in base 10.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Real_numbers_3.html (2182 words)

	More on Real Numbers
		Mathematicians use the symbol R (or alternatively, \Bbb{R}, the letter "R" in flboard bold) to represent the set of all real numbers.
		In fact, the cardinality of the reals is 2ω (see cardinality of the continuum), i.e., the cardinality of the set of subsets of the natural numbers.
		The real numbers can be generalized and extended in several different directions.
	www.artilifes.com /real-numbers.htm (2134 words)

	Logical Constructions (Stanford Encyclopedia of Philosophy)
		With the numbers defined, for example, two as the class of all two membered sets, or pairs, the properties of numbers could be derived by logical means alone.
		In any case, constructions do not appear as the referents of logically proper names, and so by that account are not part of the fundamental “furniture” of the world.
		More generally, however, the use of set theoretic constructions became widespread among philosophers, and continues in the construction of set theoretic models, both in the sense of logic where they model formal theories, and as objects of interest in their own right.
	plato.stanford.edu /entries/logical-construction (2366 words)

	Realty Times: Numbers Adding Up In Real Estate
		In basic terms, the numbers say home values are soaring and property demand is likely to remain strong for years to come.
		The other number that should interest everyone comes from Fannie Mae: In ads broadcast in the Washington area, Fannie Mae says that "The population is growing, and with it the American Dream of homeownership.
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	realtytimes.com /printrtpages/20030916_numbers.htm (583 words)

	Real Numbers
		Consider the set of rational numbers {1, 1.4, 1.41, 1.414, 1.4142,...} converging to the square root of 2.
		The importance for us is that this property is one of the most basic properties of the real numbers, and it distinguishes the real from the rational numbers (which do not have this property).
		In order to prove this theorem we need to know what exactly the real numbers are, and we have indeed given two possible constructions at the beginning of this section..
	pirate.shu.edu /projects/reals/infinity/reals.html (813 words)

	Our Playground: The Real Numbers and Their Development - Wikiversity
		Occurences like this were actually kept very secret, and even up to the nineteenth century, irrational numbers, numbers that could not be expressed as the ratio of two whole amounts, were viewed with suspicion as in some way not being "real" numbers.
		As a matter of fact, it may be disturbing to realize that there are, in a sense that will be made clearer if you continue on to study real analysis, "more" irrational numbers than rational numbers on our number line.
		These abstract numbers that completely fill in all possible lengths on the number line, and including their negatives, are what we call the real numbers.
	en.wikiversity.org /wiki/Our_Playground:_The_Real_Numbers_and_Their_Development (3077 words)

	Construction of the rational numbers@Everything2.com
		However, set theorists like to construct everything in terms of sets, and the rational numbers are no exception.
		This construction follows the method of Paul Bernays, an intermediate step in his construction of the real numbers.
		This is perfectly acceptable for a construction of the real numbers.
	everything2.com /index.pl?node_id=1097746 (818 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 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.
		We constructed a field of numbers deliberately in such a manner that it would look almost like the real numbers, except for a few special properties that we wanted to be different.
	mathforum.org /dr.math/faq/analysis_hyperreals.html (9036 words)

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	Surreal Numbers
		Cantor built the infinite ordinal numbers and Dedekind constructed the real numbers from the rationals using "cuts".
		This shall be the first rule: Every number corresponds to two sets of previously created numbers, such that no member of the left set is greater than or equal to any member of the right set.
		Many of the numbers I have used have used the empty set as either L or R. The formula for these numbers is no different, and the calculations are no more difficult.
	www.usna.edu /MathDept/wdj/surreal_numbers.html (2262 words)

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