Linear congruential generator - Revision history

en>Niceguyedc: WPCleaner v1.34 - Repaired 1 link to disambiguation page - (You can help) - C11

2014-12-28T04:13:58Z

WPCleaner v1.34 - Repaired 1 link to disambiguation page - (You can help) - C11

← Older revision		Revision as of 06:13, 28 December 2014
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en>Maxal: binary mod, not congruence

2014-02-06T19:07:40Z

binary mod, not congruence

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en>OsmanRF34: /* External links */ spam rm

2014-01-31T13:03:53Z

External links: spam rm

← Older revision		Revision as of 15:03, 31 January 2014
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			[[Image:Dedekind cut sqrt 2.svg\| thumb\| right\| 350px\| Dedekind used his cut to construct the [[irrational number\|irrational]], [[real number]]s.]]

			In [[mathematics]], a '''Dedekind cut''', named after [[Richard Dedekind]], is a [[partition of a set\|partition]] of the [[rational number]]s into two non-empty parts ''A'' and ''B'', such that all elements of ''A'' are less than all elements of ''B'', and ''A'' contains no [[greatest element]].

			If ''B'' has a smallest element among the rationals, the ''cut'' corresponds to that rational. Otherwise, that cut defines a unique irrational number which, loosely speaking, fills the "gap" between ''A'' and ''B''. In other words, ''A'' contains every rational number ''less than'' the cut, and ''B'' contains every rational number ''greater than'' the cut. An irrational cut is equated to an irrational number which is in neither set. Every real number, rational or not, is equated to one and only one cut of rationals.

			{{quote\|Whenever, then, we have to do with a cut produced by no rational number, we create a new ''irrational'' number, which we regard as completely defined by this cut ... . From now on, therefore, to every definite cut there corresponds a definite rational or irrational number ....\|Richard Dedekind, ''Continuity and Irrational Numbers'', Section IV}}

			More generally, a Dedekind cut is a partition of a [[totally ordered set]] into two non-empty parts (''A'' and ''B''), such that ''A'' is closed downwards (meaning that for all ''a'' in ''A'', ''x'' ≤ ''a'' implies that ''x'' is in ''A'' as well) and ''B'' is closed upwards, and ''A'' contains no greatest element. See also [[completeness (order theory)]].

			In particular, as discussed below, Dedekind cuts among the real numbers may be considered defined as cuts among the rationals. It turns out that every cut of reals is identical to the cut produced by a specific real number (which can be identified as the smallest element of the ''B'' set). In other words, the [[number line]] where every [[real number]] is defined as a Dedekind cut of rationals is a [[Complete metric space\|complete]] [[linear continuum\|continuum]] without any further gaps.

			Dedekind used the German word ''Schnitt'' (cut) in a visual sense rooted in Euclidean geometry. When two straight lines cross, one is said to ''cut'' the other. Dedekind's construction of the number line ensures that two crossing lines always have one point in common because each of them defines a Dedekind cut on the other.

			==Representations==
			It is more symmetrical to use the (''A'',''B'') notation for Dedekind cuts, but each of ''A'' and ''B'' does determine the other. It can be a simplification, in terms of notation if nothing more, to concentrate on one 'half' — say, the lower one — and call any downward closed set ''A'' without greatest element a "Dedekind cut".

			If the ordered set ''S'' is complete, then, for every Dedekind cut (''A'', ''B'') of ''S'', the set ''B'' must have a minimal element ''b'',
			hence we must have that ''A'' is the [[interval (mathematics)\|interval]] ( −∞, ''b''<nowiki>)</nowiki>, and ''B'' the interval <nowiki>[</nowiki>''b'', +∞<nowiki>)</nowiki>.
			In this case, we say that ''b'' ''is represented by'' the cut (''A'',''B'').

			The important purpose of the Dedekind cut is to work with number sets that are ''not'' complete. The cut itself can represent a number not in the original collection of numbers (most often [[rational number]]s). The cut can represent a number ''b'', even though the numbers contained in the two sets ''A'' and ''B'' do not actually include the number ''b'' that their cut represents.

			For example if ''A'' and ''B'' only contain [[rational numbers]], they can still be cut at √2 by putting every negative rational number in ''A'', along with every non-negative number whose square is less than 2; similarly ''B'' would contain every positive rational number whose square is greater than or equal to 2. Even though there is no rational value for √2, if the rational numbers are partitioned into ''A'' and ''B'' this way, the partition itself represents an [[irrational number]].

			==Ordering of cuts==
			Regard one Dedekind cut (''A'', ''B'') as ''less than'' another Dedekind cut (''C'', ''D'') if ''A'' is a proper subset of ''C''. Equivalently, if ''D'' is a proper subset of ''B'', the cut (''A'', ''B'') is again ''less than'' (''C'', ''D''). In this way, set inclusion can be used to represent the ordering of numbers, and all other relations (''greater than'', ''less than or equal to'', ''equal to'', and so on) can be similarly created from set relations.

			The set of all Dedekind cuts is itself a linearly ordered set (of sets). Moreover, the set of Dedekind cuts has the [[supremum\|least-upper-bound]] property, i.e., every nonempty subset of it that has any upper bound has a ''least'' upper bound. Thus, constructing the set of Dedekind cuts serves the purpose of embedding the original ordered set ''S'', which might not have had the least-upper-bound property, within a (usually larger) linearly ordered set that does have this useful property.

			==Construction of the real numbers==
			{{See also\|Construction of the real numbers#Construction by Dedekind cuts}}
			A typical Dedekind cut of the [[rational number]]s is given by

			:<math>A = \{ a\in\mathbb{Q} : a^2 < 2 \lor a < 0 \},</math>
			:<math>B = \{ b\in\mathbb{Q} : b^2 > 2 \land b > 0 \}.</math>

			This cut represents the [[irrational number]] √2 in Dedekind's construction. To establish this truly, one must show that this really is a cut and that it is the square root of two. However, neither claim is immediate. Showing that it is a cut requires showing that for any positive rational <math>x\,</math> with <math>x \times x < 2\,</math>, there is a rational <math>y\,</math> with <math>x<y\,</math> and <math>y \times y <2\,.</math> The choice <math>y=\frac{2x+2}{x+2}\,</math> works. Then we have a cut and it has a square no larger than 2, but to show equality requires showing that if <math>r\,</math> is any rational number less than 2, then there is positive <math>x\,</math> in <math>A</math> with <math>r<x \times x\,</math>.

			Note that the equality ''b''<sup>2</sup> = 2 cannot hold since [[Square root of 2#Proofs of irrationality\|√2 is not rational]].

			==Generalizations==
			A construction similar to Dedekind cuts is used for the construction of [[surreal number]]s.

			===Partially ordered sets===
			{{Main\|Dedekind–MacNeille completion}}
			More generally, if ''S'' is a [[partially ordered set]], a ''completion'' of ''S'' means a [[complete lattice]] ''L'' with an order-embedding of ''S'' into ''L''. The notion of ''complete lattice'' generalizes the least-upper-bound property of the reals.

			One completion of ''S'' is the set of its ''downwardly closed'' subsets, ordered by [[subset\|inclusion]]. A related completion that preserves all existing sups and infs of ''S'' is obtained by the following construction: For each subset ''A'' of ''S'', let ''A''<sup>u</sup> denote the set of upper bounds of ''A'', and let ''A''<sup>l</sup> denote the set of lower bounds of ''A''. (These operators form a [[Galois connection]].) Then the [[Dedekind–MacNeille completion]] of ''S'' consists of all subsets ''A'' for which (''A''<sup>u</sup>)<sup>l</sup> = ''A''; it is ordered by inclusion. The Dedekind-MacNeille completion is the smallest complete lattice with ''S'' embedded in it.

			==References==
			*Dedekind, Richard, ''Essays on the Theory of Numbers'', "Continuity and Irrational Numbers," Dover: New York, ISBN 0-486-21010-3. Also [http://www.gutenberg.org/etext/21016 available] at Project Gutenberg.

			==External links==
			* {{springer\|title=Dedekind cut\|id=p/d030530}}

			[[Category:Order theory]]
			[[Category:Rational numbers]]

68.118.158.98: /* Period length */

2012-08-14T19:03:38Z

Period length

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