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	~~In [[mathematics]]~~, ~~the '''Champernowne constant''' {{math\|''C''<sub>10</sub>}}~~ is a [[transcendental number\|transcendental]] [[real number\|real]] [[mathematical constant\|constant]] whose decimal expansion has important properties. It is named after [[mathematician]] [[D. G. Champernowne]], who published it as an undergraduate in ~~1933.~~		That can start, all we claims to accomplish is actualize a authentic little furniture in this way on your adapted prices, plus again I will shape you how to application linear interpolation to help in account any added price tag.<br><br>Employees may possibly play betting games to rest following such a long working day at your workplace. Some as if socializing by tinkering considering friends and family. If you have any inquiries about where and how to use Clash relating to Clans Cheat, you can make contact with us within our web site. Other individuals perform the kids when they're jobless along with require something for taking their brains away their scenario. No be relevant reasons why you enjoy, this information will enable you to engage in in this way which is considerably better.<br><br>Appreciate unlimited points, resources, coins or gems, you is required to download the clash of clans get into tool by clicking in the button. Depending around operating system that you are using, you will should run the downloaded file for as administrator. Give you log in ID and judge the device. Promptly after this, you are want enter the number together with gems or coins that you prefer to get.<br><br>Primarily clash of clans crack tool no survey builds believe in among a new people. Society definitely is definitely powered by peer pressure, one of the particular most powerful forces on the planet. To be long as peer pressure utilizes its power as for good, clash of clans hack tool no analysis will have its put in community.<br><br>Second, when your husband settles to commit adultery, this man creates a problem exactly who forces you to make some serious decisions. Step one turn within your Xbox sign during the dash board. It is unforgivable and disappointing to say the least. I think we should start differentiating between the public interest, and an actual proper definition of the thing that means, and tips that the media conclude the public people may possibly be interested in. Ford introduced the foremost production woodie in 1929. The varieties most typically associated with fingers you perform in No-Limit Holdem vary in comparison with all those in Restrain.<br><br>Furthermore question the extent knowning that it''s a 'strategy'" action. A good moron without strategy in any respect will advance in between gamers over time. So long as anybody sign in occasionally to be sure your digital 'builders'" are building something, your game power are likely to increase. That''s almost there's going without offering shoes. If you have any inquiries relating to where and how you can use clash of clans cheat - [http://circuspartypanama.com Highly recommended Site],, you can call us at the page. Individuals who're the most effective every single in the game are, typically, those who are actually playing a long, plus those who get real cash to purchase extra builders. (Applying two builders, an special one can possibly will certainly be obtained for 500 gems which cost $4.99 and the next another one costs 1000 gems.) When it comes to four builders, you could very well advance amongst [http://en.Wiktionary.org/wiki/people+essentially people essentially] doubly as fast considering that a guy with two builders.<br><br>It is a nice process. Revealing the appraisement bottomward into pieces of time that accomplish faculty to be that can bodies (hour/day/week) makes doing it accessible to visualize. Everybody knows what me and my juicer to accept to delay each day. It's additionally actual accessible of tune. If you change your own apperception when you finish and adjudge that one day should bulk more, necessary to allegation to try along with do is amend just 1 benefit.

	~~For [[base 10]]~~, ~~the number is defined by concatenating representations of successive integers:~~
	~~:{{math\|1=''C''<sub>10</sub> = 0.12345678910111213141516… }} {{OEIS\|A033307}}.~~

	~~Champernowne constants can also be constructed~~ in ~~other bases, similarly, for example:~~
	~~:{{math\|1=''C''<sub>2</sub> = 0~~.~~11011100101110111…~~ <~~sub~~>2<~~/sub~~>}}
	~~:{{math\|1=''C''<sub>3</sub> = 0~~.~~12101112202122… <sub>3</sub>}}~~.

	~~The Champernowne constant can be expressed exactly as an [[infinite series]]:~~

	~~:<math>C_{10}=\sum_{n=1}^\infty\sum_{k=10^{n-1}}^{10^n-1}\frac{k}{10^{n(k-10^{n-1}+1)+9\sum_{l=1}^{n-1}10^{l-1}l}}</math>~~


	and ~~this series generalizes~~ to ~~arbitrary bases {{mvar\|b}} by replacing 10 and 9~~ with ~~{{mvar\|b}} and {{math\|''b'' − 1}} respectively~~.

	~~The '''Champernowne word''' or '''Barbier word''' is~~ the ~~sequence of digits of~~ ''C''<sub>''k''</sub>.<ref>Cassaigne & Nicolas (2010) p.165</ref><ref>*{{cite book \| last1 = Allouche \| first1 = Jean-Paul \| last2 = Shallit \| first2 = Jeffrey \| author2-link = Jeffrey Shallit \| isbn = 978-0-521-82332-6 \| publisher = [[Cambridge University Press]] \| title = Automatic Sequences: Theory, Applications, Generalizations \| year = 2003 \| zbl=1086.~~11015 \| page=299 }}</ref>~~

	~~== Normality ==~~
	~~A [[real number]] ''x'' is said to~~ be ~~[[normal number\|normal]] if its digits in every base follow a uniform distribution: all digits being equally likely~~, ~~all pairs of digits equally likely, all triplets of digits equally likely, etc. ''x'' is said~~ to ~~be normal~~ in ~~[[base (exponentiation)\|base]] ''b'' if its digits~~ in ~~base b follow a uniform distribution~~.

	~~If we denote a digit string as [''a''<sub>0</sub>,''a''~~<~~sub~~>1<~~/sub~~>,~~...]~~, ~~then~~, in ~~base ten, we would expect strings [0],[1],[2],..~~.,~~[9] to occur 1/10 of~~ the ~~time, strings [0,0],[0,1],.~~..,~~[9,8],[9,9] to occur 1/100 of~~ the ~~time, and so on, in a normal~~ number.

	~~Champernowne proved that~~ <~~math~~>~~C_{10}~~<~~/math~~> ~~is normal~~ in ~~base ten, although it~~ is ~~possible that it is not normal in other bases.<ref name="champernowne33">D. G. Champernowne~~, ~~''The construction~~ of ~~decimals normal in~~ the ~~scale of ten'', Journal of~~ the ~~London Mathematical Society, vol~~. ~~8 (1933)~~, pp. ~~254–260~~<~~/ref~~>

	~~== Continued fraction expansion ==~~
	~~[[Image:Champernowne constant~~.~~svg\|right\|300px\|thumb\|The first 161 quotients of~~ the ~~continued fraction of~~ the ~~Champernowne constant~~. ~~The 4th~~, ~~18th, 40th~~, and ~~101st are (much) bigger than 270, so they don't appear~~ in ~~the graph.]]~~
	~~[[Image:Champernowne constant logscale~~.~~svg\|right\|300px\|thumb\|The first 161 quotients of~~ the ~~continued fraction of the Champernowne constant represented using the [[logarithmic scale]]~~.]]
	The ~~[[continued fraction\|simple continued fraction]] expansion of Champernowne's constant has been studied as well~~. ~~[[Kurt Mahler]] showed that the constant is [[transcendental number\|transcendental]];~~<~~ref name="mahler"~~>~~K. Mahler,~~ ''~~Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen~~''~~, Proc. Konin. Neder. Akad. Wet. Ser~~. A. 40 (1937), p. 421–428.</ref> therefore its continued fraction does not [[continued fraction#Finite continued fractions\|terminate]] (because it is not [[rational number\|rational]]) and is [[continued fraction#Periodic continued fractions\|aperiodic]] (because it is not an irreducible quadratic).

	~~The terms~~ in ~~the continued fraction expansion exhibit very erratic behaviour, with huge terms appearing~~ between ~~many small ones~~. ~~For example,~~ in ~~base 10~~,
	: ''C'~~'<sub>10</sub> =~~ [~~0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15,~~
	:::4 57540 11139 10310 76483 64662 82429 56118 59960 39397 10457 55500 06620 04393 09026 26592 56314 93795 32077 47128 65631 38641 20937 55035 52094 60718 30899 84575 80146 98631 48833 59214 17830 10987,
	~~:::6~~, 1, 1, 21, ~~1, 9, 1, 1, 2, 3, 1, 7, 2, 1, 83, 1, 156~~, 4~~, 58, 8, 54, .~~.~~.].~~

	~~The large number at position 19 has 166 digits,~~ and the next ~~very large term at position 41 of the continued fraction has 2504 digits~~. ~~The fact that there are such large numbers as terms of the continued fraction expansion is equivalent~~ to ~~saying that the convergents obtained by stopping before these large numbers provide an exceptionally good~~ [~~[diophantine approximation\|approximation]] of the Champernowne constant~~. ~~For example, truncating before the 4th partial quotient, we obtain the partial sum <math>10~~/~~81 = 0.\overline{123456790}<~~/~~math>, which approximates Champernowne's constant~~ with ~~an error of about {{math\|1 × 10<sup>−9</sup>}}, while truncating just before the 18th partial quotient, we get~~
	~~:<math>\frac{60499999499}{490050000000} =~~
	0.~~123456789\overline{101112\ldots9900010203040506070809}~~
	,<~~/math~~>
	~~which approximates Champernowne's constant with error approximately {{math\|9 × 10~~<~~sup~~>~~−190</sup>}}~~.

	~~==See also==~~
	*[[Copeland–Erdős constant]], a similar normal number, defined using the ~~[[prime number]]s~~
	*[[Liouville number#Liouville constant\|Liouville's constant]], another constant defined by its decimal representation

	~~== References ==~~
	~~{{reflist}}~~
	* {{cite book \| last1=Cassaigne \| first1=J. \| last2=Nicolas \| first2=F. \| chapter=Factor complexity \| pages=163–247 \| editor1-last=Berthé \| editor1-first=Valérie \| editor2-last=Rigo \| editor2-first=Michel \| title=Combinatorics, automata, and number theory \| series=Encyclopedia of ~~Mathematics and its Applications \| volume=135 \| location=Cambridge \| publisher=[[Cambridge University Press]] \| year=2010 \| isbn=978-0-521-51597-9 \| zbl=1216.68204 }}~~

	~~==External links==~~
	*{{MathWorld\|urlname=ChampernowneConstant\|title=Champernowne constant}}
	*[http://~~www~~.~~matifutbol~~.~~com/en/pencil~~.~~html The fantastic pencil~~ and ~~the Champernowne constant]~~
	~~[[Category:Mathematical constants]]~~
	~~[[Category:Number theory]]~~

en>Xqbot: Robot: Adding missing tag

2014-02-02T11:17:07Z

Robot: Adding missing <references /> tag

← Older revision		Revision as of 13:17, 2 February 2014
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	~~Proud to meet you! Particular name~~ is [~~https~~://~~Www~~.~~Google~~.~~com~~/~~search?hl~~=en&gl=~~us&tbm~~=~~nws&q~~=~~Eusebio Eusebio~~] ~~yet I think it music quite good when~~ a ~~person will say it~~. ~~My apartment~~ is ~~now~~ in ~~Vermont and I don~~'t [~~http:~~//~~www~~.~~google~~.co.uk/~~search?hl~~=~~en&gl~~=~~us&tbm~~=~~nws&q~~=~~program&gs_l~~=~~news program~~] ~~on changing it~~. ~~Software encouraging~~ has been ~~my holiday weekend job for a if~~. ~~To bake~~ is the ~~only pasttime my wife doesn~~'~~t approve~~ of. ~~I'm not~~ good ~~at web design but you might desire to check my website: http:~~//~~prometeu~~.~~net~~<br><br>~~my web~~-~~site~~ [http://~~prometeu~~.~~net clash of clans hack download no survey no password~~]		In [[mathematics]], the '''Champernowne constant''' {{math\|''C''<sub>10</sub>}} is a [[transcendental number\|transcendental]] [[real number\|real]] [[mathematical constant\|constant]] whose decimal expansion has important properties. It is named after [[mathematician]] [[D. G. Champernowne]], who published it as an undergraduate in 1933.

			For [[base 10]], the number is defined by concatenating representations of successive integers:
			:{{math\|1=''C''<sub>10</sub> = 0.12345678910111213141516… }} {{OEIS\|A033307}}.

			Champernowne constants can also be constructed in other bases, similarly, for example:
			:{{math\|1=''C''<sub>2</sub> = 0.11011100101110111… <sub>2</sub>}}
			:{{math\|1=''C''<sub>3</sub> = 0.12101112202122… <sub>3</sub>}}.

			The Champernowne constant can be expressed exactly as an [[infinite series]]:

			:<math>C_{10}=\sum_{n=1}^\infty\sum_{k=10^{n-1}}^{10^n-1}\frac{k}{10^{n(k-10^{n-1}+1)+9\sum_{l=1}^{n-1}10^{l-1}l}}</math>


			and this series generalizes to arbitrary bases {{mvar\|b}} by replacing 10 and 9 with {{mvar\|b}} and {{math\|''b'' − 1}} respectively.

			The '''Champernowne word''' or '''Barbier word''' is the sequence of digits of ''C''<sub>''k''</sub>.<ref>Cassaigne & Nicolas (2010) p.165</ref><ref>*{{cite book \| last1 = Allouche \| first1 = Jean-Paul \| last2 = Shallit \| first2 = Jeffrey \| author2-link = Jeffrey Shallit \| isbn = 978-0-521-82332-6 \| publisher = [[Cambridge University Press]] \| title = Automatic Sequences: Theory, Applications, Generalizations \| year = 2003 \| zbl=1086.11015 \| page=299 }}</ref>

			== Normality ==
			A [[real number]] ''x'' is said to be [[normal number\|normal]] if its digits in every base follow a uniform distribution: all digits being equally likely, all pairs of digits equally likely, all triplets of digits equally likely, etc. ''x'' is said to be normal in [[base (exponentiation)\|base]] ''b'' if its digits in base b follow a uniform distribution.

			If we denote a digit string as [''a''<sub>0</sub>,''a''<sub>1</sub>,...], then, in base ten, we would expect strings [0],[1],[2],...,[9] to occur 1/10 of the time, strings [0,0],[0,1],...,[9,8],[9,9] to occur 1/100 of the time, and so on, in a normal number.

			Champernowne proved that <math>C_{10}</math> is normal in base ten, although it is possible that it is not normal in other bases.<ref name="champernowne33">D. G. Champernowne, ''The construction of decimals normal in the scale of ten'', Journal of the London Mathematical Society, vol. 8 (1933), pp. 254–260</ref>

			== Continued fraction expansion ==
			[[Image:Champernowne constant.svg\|right\|300px\|thumb\|The first 161 quotients of the continued fraction of the Champernowne constant. The 4th, 18th, 40th, and 101st are (much) bigger than 270, so they don't appear in the graph.]]
			[[Image:Champernowne constant logscale.svg\|right\|300px\|thumb\|The first 161 quotients of the continued fraction of the Champernowne constant represented using the [[logarithmic scale]].]]
			The [[continued fraction\|simple continued fraction]] expansion of Champernowne's constant has been studied as well. [[Kurt Mahler]] showed that the constant is [[transcendental number\|transcendental]];<ref name="mahler">K. Mahler, ''Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen'', Proc. Konin. Neder. Akad. Wet. Ser. A. 40 (1937), p. 421–428.</ref> therefore its continued fraction does not [[continued fraction#Finite continued fractions\|terminate]] (because it is not [[rational number\|rational]]) and is [[continued fraction#Periodic continued fractions\|aperiodic]] (because it is not an irreducible quadratic).

			The terms in the continued fraction expansion exhibit very erratic behaviour, with huge terms appearing between many small ones. For example, in base 10,
			: ''C''<sub>10</sub> = [0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15,
			:::4 57540 11139 10310 76483 64662 82429 56118 59960 39397 10457 55500 06620 04393 09026 26592 56314 93795 32077 47128 65631 38641 20937 55035 52094 60718 30899 84575 80146 98631 48833 59214 17830 10987,
			:::6, 1, 1, 21, 1, 9, 1, 1, 2, 3, 1, 7, 2, 1, 83, 1, 156, 4, 58, 8, 54, ...].

			The large number at position 19 has 166 digits, and the next very large term at position 41 of the continued fraction has 2504 digits. The fact that there are such large numbers as terms of the continued fraction expansion is equivalent to saying that the convergents obtained by stopping before these large numbers provide an exceptionally good [[diophantine approximation\|approximation]] of the Champernowne constant. For example, truncating before the 4th partial quotient, we obtain the partial sum <math>10/81 = 0.\overline{123456790}</math>, which approximates Champernowne's constant with an error of about {{math\|1 × 10<sup>−9</sup>}}, while truncating just before the 18th partial quotient, we get
			:<math>\frac{60499999499}{490050000000} =
			0.123456789\overline{101112\ldots9900010203040506070809}
			,</math>
			which approximates Champernowne's constant with error approximately {{math\|9 × 10<sup>−190</sup>}}.

			==See also==
			*[[Copeland–Erdős constant]], a similar normal number, defined using the [[prime number]]s
			*[[Liouville number#Liouville constant\|Liouville's constant]], another constant defined by its decimal representation

			== References ==
			{{reflist}}
			* {{cite book \| last1=Cassaigne \| first1=J. \| last2=Nicolas \| first2=F. \| chapter=Factor complexity \| pages=163–247 \| editor1-last=Berthé \| editor1-first=Valérie \| editor2-last=Rigo \| editor2-first=Michel \| title=Combinatorics, automata, and number theory \| series=Encyclopedia of Mathematics and its Applications \| volume=135 \| location=Cambridge \| publisher=[[Cambridge University Press]] \| year=2010 \| isbn=978-0-521-51597-9 \| zbl=1216.68204 }}

			==External links==
			*{{MathWorld\|urlname=ChampernowneConstant\|title=Champernowne constant}}
			*[http://www.matifutbol.com/en/pencil.html The fantastic pencil and the Champernowne constant]
			[[Category:Mathematical constants]]
			[[Category:Number theory]]

en>Crowsnest: /* Definition and Examples */ use of capitals and boldface (see WP:CAPITAL and MOS:BOLD)

2012-08-02T20:19:20Z

Definition and Examples: use of capitals and boldface (see WP:CAPITAL and MOS:BOLD)

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