en>Wtmitchell: Reverted edits by 2001:E68:440C:1D99:7101:F36:9E79:7E6E (talk) (HG)

2014-12-31T22:02:55Z

Reverted edits by 2001:E68:440C:1D99:7101:F36:9E79:7E6E (talk) (HG)

en>SPACKlick: Reverted 1 edit by 117.214.145.122 (talk) to last revision by ClueBot NG. (TW)

2014-01-23T13:20:23Z

Reverted 1 edit by 117.214.145.122 (talk) to last revision by ClueBot NG. (TW)

← Older revision		Revision as of 15:20, 23 January 2014
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	~~All person who wrote all~~ of the ~~article~~ is ~~called Leland but it~~'~~s not all most masucline name~~ . ~~To go~~ to ~~karaoke~~ is the ~~thing he loves most~~ of ~~all~~. ~~He works~~ as a ~~cashier~~. ~~His wife~~ and ~~him live inside~~ of ~~Massachusetts~~ and ~~he has everything that he conditions there~~. He'~~s not godd at design but may also want to check our website~~: http://~~circuspartypanama~~.com~~<br><br>Here is my blog post~~ [http://~~circuspartypanama~~.com ~~triche clash of clans~~]		{{Main\|Repeating decimal\|Cyclic number}}
			{{Redirect\|1/7 (number)\|the number 7\|7 (number)}}
			{{Infobox number
			\| number = 142857
			}}
			'''142857''' is the six [[repeating decimal\|repeating digits]] of '''1/7''', <math> 0.\overline{142857} </math>, and is the best-known [[cyclic number]] in base 10.<ref>[http://www.daviddarling.info/encyclopedia/C/cyclic_number.html "Cyclic number"], ''The Internet Encyclopedia of Science''</ref><ref>Michael W. Ecker, [http://links.jstor.org/sici?sici=0049-4925(198303)14%3A2%3C105%3ATALOCN%3E2.0.CO%3B2-Y "The Alluring Lore of Cyclic Numbers"], ''The Two-Year College Mathematics Journal'', Vol.14, No.2 (March 1983), pp. 105–109</ref><ref>[http://planetmath.org/encyclopedia/CyclicNumber.html Cyclic number], PlanetMath</ref><ref>{{cite web \|url=http://findarticles.com/p/articles/mi_hb4870/is_200508/ai_n17913296 \|title=Go figure (cyclic numbers) \|author=Hogan, Kathryn \|publisher=''Australian Doctor'' \|date=August 2005 \|archiveurl=http://web.archive.org/web/20071224191059/http://findarticles.com/p/articles/mi_hb4870/is_200508/ai_n17913296 \|archivedate=24 December 2007}}</ref> If it is [[multiplication\|multiplied]] by 2, 3, 4, 5, or 6, the answer will be a [[cyclic permutation]] of itself, and will correspond to the repeating digits of '''2/7''', '''3/7''', '''4/7''', '''5/7''', or '''6/7''' respectively.

			== Calculations ==
			: 1 × 142,857 = 142,857
			: 2 × 142,857 = 285,714
			: 3 × 142,857 = 428,571
			: 4 × 142,857 = 571,428
			: 5 × 142,857 = 714,285
			: 6 × 142,857 = 857,142
			: 7 × 142,857 = 999,999

			If you multiply by an integer greater than 7, there is a simple process to get to a cyclic permutation of 142857. By adding the rightmost six digits (ones through hundred thousands) to the remaining digits and repeating this process until you have only the six digits left, it will result in a cyclic permutation of 142857

			: 142857 × 8 = 1142856
			: 1 + 142856 = 142857

			: 142857 × 815 = 116428455
			: 116 + 428455 = 428571

			: 142857<sup>2</sup> = 142857 × 142857 = 20408122449
			: 20408 + 122449 = 142857

			Multiplying by a multiple of 7 will result in 999999 through this process

			: 142857 × 7<sup>4</sup> = 342999657
			: 342 + 999657 = 999999

			If you square the last three digits and subtract the square of the first three digits, you also get back a cyclic permutation of the number.

			: 857<sup>2</sup> = 734449
			: 142<sup>2</sup> = 20164
			: 734449 − 20164 = 714285

			It is the repeating part in the [[decimal expansion]] of the [[rational number]] 1/7 = 0.{{overline\|142857}}. Thus, multiples of 1/7 are simply repeated copies of the corresponding multiples of 142857:
			: 1 ÷ 7 = 0.{{overline\|142857}}
			: 2 ÷ 7 = 0.{{overline\|285714}}
			: 3 ÷ 7 = 0.{{overline\|428571}}
			: 4 ÷ 7 = 0.{{overline\|571428}}
			: 5 ÷ 7 = 0.{{overline\|714285}}
			: 6 ÷ 7 = 0.{{overline\|857142}}
			: 7 ÷ 7 = [[0.999...\|0.{{overline\|999999}}]] = 1
			: 8 ÷ 7 = 1.{{overline\|142857}}
			: 9 ÷ 7 = 1.{{overline\|285714}}
			: …

			In base 10, 142,857 is a [[Harshad number]] and a [[Kaprekar number]].

			== 1/7 as an infinite sum ==

			There is an interesting pattern of doubling, shifting and addition that gives 1/7.

			<math>\begin{align}1/7\ & = 0.142857142857142857\ldots \\[6pt]

			& = 0.14 + 0.0028 + 0.000056 + 0.00000112 + 0.0000000224 + 0.000000000448 + 0.00000000000896 + \cdots \\[6pt]

			& = \frac{14}{100} + \frac{28}{100^2} + \frac{56}{100^3} + \frac{112}{100^4} + \frac{224}{100^5} + \cdots + \frac{7\times2^N}{100^N} + \cdots \\[6pt]

			& = \left( \frac{7}{50} + \frac{7}{50^2} + \frac{7}{50^3} + \frac{7}{50^4} + \frac{7}{50^5} + \cdots + \frac{7}{50^N} + \cdots \right) \\[6pt]

			& = \sum_{k=1}^\infty \frac{7}{50^k} \end{align}</math>

			Each term is double the prior term shifted two places to the right.

			== Other bases ==
			In some other bases, six-digit numbers with similar properties exist, given by (base<sup>6</sup> − 1)/7. E.g. in [[base 12]] it is 186A35 and [[base 24]] 3A6LDH.

			== Connection to the enneagram ==
			The 142857 number sequence is used in the [[Fourth Way enneagram\|enneagram]] figure, a symbol of the [[Fourth Way\|Gurdjieff Work]] used to explain and visualize the dynamics of the interaction between the two great laws of the Universe (according to [[G. I. Gurdjieff]]), the Law of Three and the Law of Seven. The movement of the numbers of 142857 divided by 1/7, 2/7. etc., and the subsequent movement of the enneagram, are portrayed in Gurdjieff's [[Gurdjieff movements\|sacred dances]] known as the movements.

			==References==
			{{reflist}}
			{{refbegin}}
			* Leslie, John. ''"The Philosophy of Arithmetic: Exhibiting a Progressive View of the Theory and Practice of . . . ."'', Longman, Hurst, Rees, Orme, and Brown, 1820, ISBN 1-4020-1546-1
			* Wells, D. ''[[The Penguin Dictionary of Curious and Interesting Numbers]]'' Revised Edition. London: Penguin Group. (1997): 171–175
			{{refend}}

			==External links==
			*[http://www.142857.com 142857's own site]
			*[http://www.jeorgethedodo.com/huh_142857.asp 142857 Calculator]

			[[Category:Integers\|199e05 142857]]
			[[Category:Rational numbers]]

en>Ahmer Jamil Khan: /* See also */ Added more essencial links

2012-08-27T15:16:27Z

Circular motion - Revision history

en>Wtmitchell: Reverted edits by 2001:E68:440C:1D99:7101:F36:9E79:7E6E (talk) (HG)

en>SPACKlick: Reverted 1 edit by 117.214.145.122 (talk) to last revision by ClueBot NG. (TW)

en>Ahmer Jamil Khan: /* See also */ Added more essencial links