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{{Main|Repeating decimal|Cyclic number}}
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{{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.&nbsp;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&nbsp;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>&nbsp;−&nbsp;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&nbsp;.&nbsp;.&nbsp;.&nbsp;."'', 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]]

Latest revision as of 23:02, 31 December 2014

A lagging computer is really annoying plus is quite a headache. Almost each individual that utilizes a computer faces this problem several time or the other. If the computer furthermore suffers within the same issue, you'll find it hard to continue working as normal. In such a situation, the thought, "what must I do to make my PC run quicker?" is recurring plus infuriating. There's a answer, yet!

Document files enable the consumer to input information, images, tables and different elements to enhance the presentation. The just issue with this structure compared to alternative file kinds like .pdf for example is its ability to be easily editable. This means that anyone viewing the file will change it by accident. Also, this file formatting is opened by additional programs but it refuses to guarantee that what you see inside the Microsoft Word application may still become the same whenever we see it utilizing another system. However, it is nonetheless preferred by most computer consumers for its ease of employ and features.

Registry cleaning is important because the registry will receive crowded plus messy whenever it's left unchecked. False entries send the running system searching for files and directories which have long ago been deleted. This takes time and uses precious resources. So, a slowdown inevitably occurs. It is especially noticeable whenever you multitask.

The computer was very quick when we first purchased it. Because the registry was rather clean plus free of mistakes. After time, your computer starts to run slow plus freezes up now plus then. Because there are errors accumulating in it and some info is rewritten or even completely deleted by a incorrect uninstall of programs, wrong operations, malware or additional things. That is the reason why your computer performance decreases gradually and become fairly unstable.

Another choice whenever arresting the 1328 error is to wash out your PC's registry. The registry is important because it is actually where settings plus files employed by Windows for running are stored. As it is frequently used, breakdowns and cases of files getting corrupted are not uncommon. Also as a result of the way it's configured, the "registry" gets saved inside the wrong fashion continually, which makes a program run slow, eventually causing your PC to suffer from a series of errors. The best system 1 could use in cleaning out registries is to use a reliable tuneup utilities system. A registry cleaner can find out plus repair corrupted registry files plus settings allowing one's computer to run usually again.

Active X controls are utilized across the entire spectrum of computer plus web technologies. These controls are referred to as the building blocks of the web plus because the glue that puts it all together. It is a standard which is chosen by all developers to create the web more beneficial and interactive. Without these control guidelines there would basically be no public web.

Google Chrome is my lifeline and to the day happily. My all settings and analysis related bookmarks were saved in Chrome and stupidly I did not synchronize them with the Gmail to shop them online. I could not afford to install new variation and sacrifice all my function settings. There was no method to retrieve the old settings. The just way left for me was to miraculously fix it browser in a method that all information plus settings stored inside it are recovered.

Another important system you'll like to receive is a registry cleaner. The registry is a huge list of everything installed on the computer, and Windows references it whenever it opens a system or utilizes a device attached to your computer. If you delete a program, its registry entry must equally be deleted, however, often it's not. A registry cleaner will do away with these old entries so Windows may look the registry faster. It furthermore deletes or corrects any entries that viruses have corrupted.