Talk:E (mathematical constant) and Brigham Young Jr.: Difference between pages
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{{otheruses2|Brigham Young}} |
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English name = Brigham Young, Jr.| |
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{{maths rating|frequentlyviewed=yes|class=GA|importance=Top|field=basics}} |
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birth_name=Brigham Young, Jr.| |
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{{ArticleHistory |
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birth_date={{birth date|1836|12|18}}| |
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birthplace=[[Kirtland, Ohio|Kirtland]], [[Ohio]]| |
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|action1date= 19 June 2007 |
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death_date={{death date and age|1903|04|11|1836|12|18}}| |
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|action1link=Talk:E (mathematical constant)#GA review |
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deathplace=[[Salt Lake City]], [[Utah]]| |
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president_who_called=[[Brigham Young]]| |
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|action1oldid= 137806575 |
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apostledate={{death date and age|1864|02|04|1836|12|18}}| |
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ordination_reason=Brigham Young's discretion<ref>Brigham Young ordained three of his sons in 1864 without adding them to the [[Quorum of the Twelve Apostles (LDS Church)|Quorum of the Twelve Apostles]]. However, Young, Jr. became a member of the Quorum of the Twelve in 1868 after [[Heber C. Kimball]] died and [[George A. Smith]] was removed from the Quorum to join the [[First Presidency (LDS Church)|First Presidency]].</ref>| |
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end_date={{death date and age|1903|04|11|1836|12|18}}| |
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|action2date= 18 July 2007 |
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|action2link=Talk:E (mathematical constant)#GA review Mk.2 |
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reorganization=[[George Albert Smith]] ordained |
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|action3date=August 31, 2007 |
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|action3link=Talk:E (mathematical constant)#GA status reviewed |
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*[[/Archive 1|Archive 1]]: Jan 2001 - Jan 2006 |
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*[[/Archive 2|Archive 2]]: Feb 2006 - May 2007 |
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'''Brigham Young, Jr.''' ([[December 18]], [[1836]]–[[April 11]], [[1903]]) served as [[President of the Quorum of the Twelve Apostles|president]] of the [[Quorum of the Twelve Apostles (LDS Church)|Quorum of the Twelve Apostles]] of [[The Church of Jesus Christ of Latter-day Saints]] (LDS Church) from 1899 until his own death. His tenure was interrupted for one week in 1901 when [[Joseph F. Smith]] was the president of the Quorum. |
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==Early life== |
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== Harlan Brothers' papers == |
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Young was born in [[Kirtland, Ohio|Kirtland]], [[Ohio]], the son of [[Brigham Young]] and [[Mary Ann Angell]]. Young's twin sister Mary died at age seven from the effects of injuries received at age two in a wagon accident.<ref name = Jenson>[[Andrew Jenson|Jenson, Andrew]]. ''Biographical Encyclopedia of the Church of Jesus Christ of Latter-day Saints'' (Salt Lake City, Utah: [[Deseret Book]] and A. Jenson Historical Co., 1901–1936) '''1''':121.</ref> At age twelve Young drove an ox cart across the plains, reaching [[Salt Lake City]] in 1848.<ref name = Jenson/> Young served as a guard and scout in the following years operating in [[Salt Lake Valley]] and the surrounding canyons.<ref name = Jenson/> |
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Young married Catherine Curtis Spencer, a daughter of [[Orson Spencer]] with the exact same name as her mother, on [[15 November]] [[1855]].<ref name = Jenson/> |
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Recently, [[User:Hjb|Hjb]] annotated the following formula with a reference: |
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In Utah Territory, Young became a member of the reconstituted [[Nauvoo Legion]]. He was involved in the rescue of the Willie and Martin Companies of [[Mormon pioneers|Mormon handcart pioneers]]. He also served in the [[Utah War]] with the troops that worked to halt the advance of [[Johnston's Army]].<ref name = Flake>Flake, Lawrence R. "Brigham Young, Jr." in Garr, Arnolds K., [[Donald Q. Cannon]] and [[Richard O. Cowan]], ed., ''Encyclopedia of Latter-day Saint History'' (Salt Lake City, Utah: [[Deseret Book]], 2000) p. 1379–1380.</ref> |
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n 1861, Young was made a member of the Salt Lake [[Stake (Latter Day Saints)|Stake]] [[High council (Latter Day Saints)|high council]].<ref name = Jenson/> |
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:<math>e=\lim_{n \to \infty} \left [ \frac{(n+1)^{n+1}}{n^n}- \frac{n^n}{(n-1)^{n-1}} \right ]</math> |
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==Early years as a general authority== |
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The reference is "[[Harlan J. Brothers|H. J. Brothers]] and J. A. Knox, New closed-form approximations to the Logarithmic Constant e. ''The Mathematical Intelligencer'', Vol. 20, No. 4, 1998; pages 25-29.". Presumably [[User:Hjb|Hjb]] is Harlan J. Brothers himself. |
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Brigham Young ordained Young an apostle at the young age of 27 in 1864. However, he was not placed in the Quorum of the Twelve Apostles until four years later in 1868. Young Jr. also served as a counselor to his father in the [[First Presidency (LDS Church)|First Presidency]] of the church from [[April 8]], [[1873]] until his father's death on [[August 29]], [[1877]]. |
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==Missions to Europe== |
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I would have no objection to the reference, except that I have reviewed the paper, and it does not appear to contain the specified formula. It does contain many equivalent formulas, but (as has been noted on this page and elsewhere several times before) many formulas are equivalent to this one, including the definition itself: |
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From 1862 to 1863 Young served as a [[Mormon missionary|church missionary]] in [[England]], spending most of the time in [[London]].<ref name = Jenson/> During this time he also accompanied [[Joseph F. Smith]] on a trip to [[Paris]], [[France]].<ref>Jenson. ''Biographical Encyclopedia''. '''1''':66.</ref> |
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In 1864, Young returned to Europe, this time with his wife Catherine as his companion. He was an assistant to mission president [[Daniel H. Wells]]. In 1865, when Wells left for Utah, Young succeeded him as president of the European Mission.<ref name = Jenson/> |
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:<math>\lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n</math> |
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As [[Mission president|president]] of the [[Europe]]an [[Mission (LDS Church)|Mission]] of the church in 1866 and 1867, Young preached in [[France]], [[Switzerland]], [[Germany]], [[Denmark]], [[Sweden]], [[Norway]], [[Russia]], the [[United Kingdom]] and [[Ireland]]. Young also oversaw the emigration of British [[Latter-day Saint]]s to Utah Territory.<ref>[[George Q. Cannon|Cannon, George Q.]] and [[Wilford Woodruff]], ''Faith Promoting Series: Gems For the Young Folks'', p. 19</ref> It was from a conversation as Young was about to return to Utah at the end of his time as mission president that [[Charles W. Penrose]] wrote the hymn "Beautiful Zion For Me".<ref>[[Heber J. Grant]], [http://search.ldslibrary.com/article/view/236250 ''Conference Report'', April 1926, p. 147].</ref> |
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In any case, it is not clear to me what value there is in citing the formula with a paper that does not actually mention that formula. |
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From October 1890 until February 1893 Young served for a second time as the president of the European Mission.<ref>[[B. H. Roberts|Roberts, B. H.]] ''[[Comprehensive History of the Church of Jesus Christ of Latter-day Saints]]'' (Provo, Utah: [[Brigham Young University Press]], 1965) p. 89</ref> The mission was headquartered in [[Liverpool]], England and Young directly supervised missionary work in the [[British Isles]] while also serving as a leader over the mission presidents of the various missions on the [[European continent]]. |
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It is also possible that I am missing something; perhaps the formula appears in the paper and I didn't see it. I would welcome corrections. |
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==Colonization and church assignment in America== |
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Any discussion on this? -- [[User:Dominus|Dominus]] 22:44, 11 March 2007 (UTC) |
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In the western [[United States]], Young was involved in the colonization of [[Cache Valley]], southern Utah and the extension of Mormon settlements into [[New Mexico]] and [[Arizona]]. Young was also involved at times with the LDS settlements in the Mexican state of [[Chihuahua (state)|Chihuahua]].<ref name = Flake/> |
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In 1867, Young was involved with the formation of the [[Deseret Sunday School Union]] to provide centralized direction to the [[Sunday school]]s of the church.<ref>Poleman, B. Lloyd. "Sunday School" in Ludlow, Daniel H., ed., ''[[Encyclopedia of Mormonism]]'' (New York: MacMillan, 1992) p. 1425</ref> |
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Hello. Thanks for your attentiveness. However, the formula, dubbed the "Power Ratio Method," appears on page 26, approximation number 4. |
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During 1868, Young acted as his father's agent in finding workers for the Utah portion of the [[First Transcontinental Railroad]].<ref name = Jenson/> |
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[[User:Hjb|Hjb]] 07:35, 12 March 2007 (UTC) |
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From 1869 until 1877, Young presided over the Latter-day Saints in Cache Valley, closely assisted by [[William B. Preston (Mormon)|William B. Preston]] who was serving as the regional presiding bishop.<ref>[[Andrew Jenson|Jenson, Andrew]]. ''Encyclopedic History of the Church of Jesus Christ of Latter-day Saints''. (Salt Lake City, Utah: [[Deseret News Press]], 1941, p. 105.</ref> During this time Young co-owned a feed and livery stable in [[Soda Springs, Idaho]] with Solomon Hale.<ref>Jenson. ''Biographical Dictionary''. Vol. 2, p. 168</ref> |
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:It does indeed. I wonder now how I missed it. Thank you! -- [[User:Dominus|Dominus]] 11:12, 12 March 2007 (UTC) |
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In 1877, Young, [[Erastus Snow]] and [[Wilford Woodruff]] dedicated parts of the [[St. George Temple]] under the general direction of Brigham Young.<ref>Woodruff, Wilford. "Living By the Spirit" in Stuy, H. Bryan, ed., ''Collected Discourses, 1888–1898'' (Glendale, California and Woodland Hills, Utah: B. H. S. Publishing, 1987–1992) Vol. 5.</ref> |
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:Thanks for seeking advice on this question! I would be '''against''' including this formula. e is such a fundamental constant that it must occur in hundreds of similar formulas. including them all would make the article impenetrable and detract from understanding rather than contribute to it. |
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:i do however support inclusion of the ''classic'' formula (1 +1/n)^n and it's fair to ask why. here are few reasons 1) a single formula, rather then burdening the intellect, is likely to have an illuminating impact. this is the right level of detail to strive for in an encyclapedia article. 2) the classic formula is taught in the schools. its presence then will act more as a reminder to readers, rather than representing new information requiring assimilation. 3) the classic formula is a very general result and is largely responsible for e's importance. the suggested formula - while it may be the right tool to use for a special class of problems - hasn't been shown to have the same impact in a broad range of cases. |
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--[[User:Philopedia|Philopedia]] ([[User talk:Philopedia|talk]]) 08:46, 5 May 2008 (UTC) |
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== relation with pi == |
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see: [[Talk:Pi#T-shirt_equation]] |
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— [[User:Xiutwel|Xiutwel]] <small>[[User_talk:Xiutwel|(talk)]]</small> 10:51, 2 April 2007 (UTC) |
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:Approximate mathematical "coincidences" like this one between powers of π and powers of ''e'' are a dime a dozen. Come back to me when you find one where they match to one thousand places. Then we can look for a proof that they are the same. [[User:JRSpriggs|JRSpriggs]] 09:32, 3 April 2007 (UTC) |
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== Known digits == |
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I disgree in cutting out the calculation of known digits. I think it is interesting and relevant (however it does need to be referenced). I would also like to see a similar section in pi. <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Jim77742|Jim77742]] ([[User talk:Jim77742|talk]] • [[Special:Contributions/Jim77742|contribs]]) {{{2|}}}</small><!-- Template:Unsigned --> |
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:I agree it's interesting and relevant. But each item needs a good reference. I tried to find one on von Neumann, and can see why someone marked it dubious. Work on it some before putting it back, or add cn tags to ones you can't find refs for, so someone will know to work on them. I calculated e to 5000 digits with a basic program once in 1970, but had an error due to not leaving enough digits for carries at some point; time to get cracking on the 50 billion or so record... [[User:Dicklyon|Dicklyon]] 05:35, 16 April 2007 (UTC) |
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The known digits table is incomplete, there is no entry for 1,000,000 million digits, which were calculated by Robert AH Prins in 1992 using a PL/I program running on an IBM 3048 at his then employer Willis Corroon in Ipswich - needlessly to say they were not pleased he did this. [[User:Robert AH Prins|Robert AH Prins]] 26 June 2007 <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:193.244.32.140|193.244.32.140]] ([[User talk:193.244.32.140|talk]] • [[Special:Contributions/193.244.32.140|contribs]]) 08:25, 26 June 2007</small><!-- {{signed|193.244.32.140|08:25, 26 June 2007}} --> |
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==revert of edit== |
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Why did you guys rever the edit of the external [http://abrau.durso.googlepages.com/mathematicalconstants link] back to the [http://67.49.215.31/constants.htm link] that is broken, which I put in there like a year back in this [http://en.wikipedia.org/w/index.php?title=E_%28mathematical_constant%29&diff=45531692&oldid=45459782 here] --[[User:BorisFromStockdale|BorisFromStockdale]] 02:07, 7 May 2007 (UTC) |
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:It fails [[WP:RS]], and seems unnecessary as [[WP:EL]], even if we had some external assurance it was correct. It's '''your''' site, isn't it? — [[User:Arthur Rubin|Arthur Rubin]] | [[User_talk:Arthur_Rubin|(talk)]] 02:31, 7 May 2007 (UTC) |
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::Shouldn't we also remove the IP site with 10 million digits for the same reason? I can never get it to load, so I can't even see if it's attributed or reliable, but I expect not if it doesn't even have a domain name. [[User:Dicklyon|Dicklyon]] 03:13, 7 May 2007 (UTC) |
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:::Yes. — [[User:Arthur Rubin|Arthur Rubin]] | [[User_talk:Arthur_Rubin|(talk)]] 03:25, 7 May 2007 (UTC) |
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::::Yes, it is my site, but you guys let it in in 2006 and let it stay there for a year. --[[User:BorisFromStockdale|BorisFromStockdale]] 03:51, 7 May 2007 (UTC) |
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:::::Thanks for pointing that out. We'll try not to make the same mistake twice. [[User:Dicklyon|Dicklyon]] 04:21, 7 May 2007 (UTC) |
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By the way, what exactly does it violate in the [WP:EL]]? --[[User:BorisFromStockdale|BorisFromStockdale]] 03:55, 7 May 2007 (UTC) |
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:[[Wp:el#Advertising_and_conflicts_of_interest]] where it says "You should avoid linking to a website that you own, maintain or represent, even if the guidelines otherwise imply that it should be linked. If the link is to a relevant and informative site that should otherwise be included, please consider mentioning it on the talk page and let neutral and independent Wikipedia editors decide whether to add it. This is in line with the conflict of interest guidelines." So, if you think it should be linked, please explain here, in terms on the criteria at [[Wp:el#What_to_link]]. Please also explain where the digits come from, and why we should see it as authoritative. -- [[User:Dicklyon|Dicklyon]] 04:21, 7 May 2007 (UTC) |
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Basically the digits come form Mathematica 5.0<br /> |
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I used the commands: <br /> |
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abc = N[GoldenRatio, 20000000];<br /> |
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Export["C:\my_files\golden.txt", abc, "table"];<br /> |
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This produces a .txt file. Then zipped it using a regular zip compresstion built into Total Commander 6.56. This reduced the file sizes to about 50% of the original. |
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this way I calculated the 3 constants on the website (e, golden ratio, pi). |
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Well, the website is being served by google through their google page creator, so it should handle the bandwidth. Unfortunately I do not think that google allows the upload of more than 10 MB files, so 20 million digits is the limit for what I can put up in one peace... --[[User:BorisFromStockdale|BorisFromStockdale]] 08:31, 7 May 2007 (UTC) |
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:I see. Wouldn't it be much simpler to just say in the article that anyone who needs lots of digits can get them trivially in a one-liner in Mathematica, and show the command? [[User:Dicklyon|Dicklyon]] 14:24, 7 May 2007 (UTC) |
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::And how many people can afford Mathematica? [[User:Fredrik|Fredrik Johansson]] 16:01, 7 May 2007 (UTC) |
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I was thinking of just aggregating the digits for several important constants on the same website. I think that having it all in one place might be useful for some people. Anyway, the website is still there, I will be upgrading(adding new constants) to it. If you guys want to incude it to this or to any other articles, Great. If not then ... --[[User:BorisFromStockdale|BorisFromStockdale]] 21:58, 8 May 2007 (UTC) |
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==GA review== |
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Overall the article looks to be in pretty good. A few notes: |
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* The Feynman quote needs a more complete citation, but is it really even necessary here? Seems a bit like fluff, but it could (should?) be moved into a footnote for "one of the most important formulas in mathematics:". {{tick|18}} '''Done''' |
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* The maximum value "f(e)=..." seems a bit superfluous, {{tick|18}} '''Done''' and perhaps the two properties related to f(x)=x^(1/(x^n)) and its special case could be combined. |
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* Why is one of the four equivalent definitions listed in the properties section as the most common? Just delete it, and perhaps add a note about the commonness in the definitions section above. |
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* All the continuous fractions and infinite series seem like overkill. Reduce it down to the most important ones. |
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* The "Non-mathematical uses of e" seems like a [[WP:TRIVIA|trivia]] section that should be incorporated or deleted. {{tick|18}} '''Done''' |
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I'll put the nomination on hold until these are resolved one way or the other. --[[User:Flex|Fl<font color="green">e</font>x]] ([[User_talk:Flex|talk]]/[[Special:Contributions/Flex|contribs]]) 03:05, 9 June 2007 (UTC) |
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:Here are [http://books.google.com/books?as_q=jewel&num=10&btnG=Google+Search&as_epq=remarkable+formula&as_oq=&as_eq=&as_libcat=0&as_brr=0&as_vt=&as_auth=&as_pub=&as_drrb=c&as_miny=&as_maxy=&as_isbn= some book sources] for the Feynman quote if anyone wants to add one. [[User:Dicklyon|Dicklyon]] 04:24, 9 June 2007 (UTC) |
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::I found a citation for it on the article for Feynman. —[[User:Disavian|Disavian]] ([[User talk:Disavian|<sup>talk</sup>]]/[[Special:Contributions/Disavian|<sub>contribs</sub>]]) 05:02, 9 June 2007 (UTC) |
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:I don't see a place where a certain definition is listed as the most common. Are you talking about the beginning of the section, where it discusses the "e is its own derivative" property? I think that's just stating the property by using one of the definitions above. —[[User:Disavian|Disavian]] ([[User talk:Disavian|<sup>talk</sup>]]/[[Special:Contributions/Disavian|<sub>contribs</sub>]]) 05:18, 9 June 2007 (UTC) |
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:I don't know which infinite sums are important. Would someone else like to make that decision? —[[User:Disavian|Disavian]] ([[User talk:Disavian|<sup>talk</sup>]]/[[Special:Contributions/Disavian|<sub>contribs</sub>]]) 08:46, 9 June 2007 (UTC) |
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:Would you be willing to restore the "Non-mathematical uses" section? (Perhaps under a different section heading?) It didn't feel like trivia, at least not in the sense that these facts needed to be merged into the History section. I think the reviewer may have been objecting to the organization of the material in this section as a bulletted list, although with a little effort I'm sure it can be tied together in prose. Also, I don't feel that the History section is an appropriate place for Google to be incorporated, as the company had nothing to do with the history of ''e'' aside from paying homage to the number. An unrelated note: At [[WP:WPM]], [[User:Geometry guy]] made a promising suggestion of moving the "Representations of e" out as a subarticle. I think this is an excellent idea. [[User:Silly rabbit|Silly rabbit]] 10:48, 9 June 2007 (UTC) |
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:There is a reference in the "Notes" section S. M. Ruiz 1997 which doesn't attach to anything. [[User:Silly rabbit|Silly rabbit]] 11:01, 9 June 2007 (UTC) |
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Regarding the "Non-mathematical uses" section: it felt like trivia to me because it was a list of unconnected factoids that don't really have anything to do with definitions or uses of ''e'' proper. (It's like if someone named a character in their novel after [[Alex Trebek]] -- that would likely be a relevant fact to incorporate into the article on the novel, but it would not be appropriate on his page. Substitute e for Alex and Google for the novel.) As it stood, this section seemed little different than a "Miscellanea" or "Cultural references" section (cf. [[WP:TRIVIA]] and [[Wikipedia:Handling_trivia#Trivia_and_lists]]), but renamed it seems slightly better. I'd still say it should be deleted, but I'll leave it up to you all. |
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Also, are there naturally occurring instances of ''e'' in biology or other fields besides finance and math proper? If so, perhaps the compound interest section could be expanded into an "Applications" section to reflect that. --[[User:Flex|Fl<font color="green">e</font>x]] ([[User_talk:Flex|talk]]/[[Special:Contributions/Flex|contribs]]) 15:48, 9 June 2007 (UTC) |
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::See [[exponential growth]]. [[User:Pmanderson|Septentrionalis]] <small>[[User talk:Pmanderson|PMAnderson]]</small> 23:41, 9 June 2007 (UTC) |
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:Reply to first paragraph: I think a "Pop culture" section can have some value if it's done properly, and I have provided some rather limited connection between the three facts that were listed as bullets before. I would personally like to see expansion rather than deletion, but in some kind of more encyclopedic direction. Anyway, merging the google references into the "History" section was not the way to go about incorporating this into the article in a harmonious way: in fact, it had the opposite effect (from Euler to Google?!) Nevertheless, if the section isn't headed anywhere, interesting though it may be, perhaps you're right that it should be deleted. |
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:On the second point, yes: it would be nice to find some applications of ''e''. The trouble is that most applications seem to focus on the natural exponential and logarithm. The significance of the numerical constant ''e'' is difficult to disentangle from these ideas. I, too, am eager to see suggestions and edits in this direction, though. [[User:Silly rabbit|Silly rabbit]] 16:01, 9 June 2007 (UTC) |
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I think the article's looking better and better. On the motivation section, I'd suggest that it seems too specific to e's calculus properties and that there are other motivations (e.g., e's occurrence in certain natural and probability problems). This could perhaps be resolved by putting the history section first or expanding the motivation section (and/or the history section?) to include it's other motivators. --[[User:Flex|Fl<font color="green">e</font>x]] ([[User_talk:Flex|talk]]/[[Special:Contributions/Flex|contribs]]) 14:26, 11 June 2007 (UTC) |
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: Thanks for the suggestion. In hindsight, this is the "obvious" thing to do, but I couldn't quite see how to suitably organize the article beforehand. Please let us know if you have any other organizational suggestions! [[User:Silly rabbit|Silly rabbit]] 14:50, 11 June 2007 (UTC) |
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::The article seems to be in a bit of flux right now, thanks in part to my review, I suppose (but cf. also [[User:Pmanderson|Septentrionalis]]'s comment below). If it settles in the next day or three, please leave a note on my talk page, and I'll come back and take another look to finish the GA process. I think it's pretty close to GA, but stability is also a factor. --[[User:Flex|Fl<font color="green">e</font>x]] ([[User_talk:Flex|talk]]/[[Special:Contributions/Flex|contribs]]) 18:48, 11 June 2007 (UTC) |
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::I failed this article for now. Feel free to renominate it when it gets to a good resting place. --[[User:Flex|Fl<font color="green">e</font>x]] ([[User_talk:Flex|talk]]/[[Special:Contributions/Flex|contribs]]) 17:00, 19 June 2007 (UTC) |
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===[[Representations of e]]=== |
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Representations of ''e'' is now live, so the material here should be summarized in prose, with one or two supporting formulas. I don't know enough about the history, level of interest, and applications of these techniques to comment on them, aside from the sophomoric "There are many ways to represent ''e''..." (etc.) Is there an expert among us? [[User:Silly rabbit|Silly rabbit]] 11:13, 9 June 2007 (UTC) |
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:Good split. It really cleaned up this (main) article. —[[User:Disavian|Disavian]] ([[User talk:Disavian|<sup>talk</sup>]]/[[Special:Contributions/Disavian|<sub>contribs</sub>]]) 20:28, 9 June 2007 (UTC) |
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::This is a serious loss to the article; it would be better to withdraw the Bad Articles nomination, and proceed directly to [[Wikipedia:Scientific peer review]] than to disfigure it in this manner. [[User:Pmanderson|Septentrionalis]] <small>[[User talk:Pmanderson|PMAnderson]]</small> 23:45, 9 June 2007 (UTC) |
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== Probability application == |
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Consider a slot machine that pays off one time in a million. If you play the slot machine one million times, you can [[expected value|expect]] to win once. But you have a 1/''e'' probability of winning nothing. |
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Perhaps this is worth mentioning as a natural appearance of ''e'' in a fairly simple problem not obviously related to compound interest. -- [[User:Dominus|Dominus]] 05:36, 10 June 2007 (UTC) |
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: Good one. I had thought about including an application of ''e'' (as opposed to its relationship with exponential growth). I came up with [[derangement]]s, but this is much easier. [[User:Silly rabbit|Silly rabbit]] 10:43, 10 June 2007 (UTC) |
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::Actually, they are much the same problem; the derangement problem is a lottery which one guest may be expected to win by getting his own hat. There is a real difference: no two guests can get the ''same'' hat, but that's a second-order term. [[User:Pmanderson|Septentrionalis]] <small>[[User talk:Pmanderson|PMAnderson]]</small> 17:09, 10 June 2007 (UTC) |
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== first citation is bogus == |
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" The number e is one of the most important numbers in mathematics" is backed up by the citation: It was described by Richard Feynman as "[...] the most remarkable formula in mathematics [...], our jewel." Source: Feynman, Richard [June 1970]. "Chapter 22: Algebra", The Feynman Lectures on Physics: Volume I, p.10. |
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e is a number not a formula. --[[User:C S|C S]][[User talk:C S| (Talk)]] 00:23, 12 June 2007 (UTC) |
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:As I recall, that passage from Feynman is talking about the Euler identity e^{i\theta} = \cos \theta + i \sin \theta. Probably somewhere in there he mentions how all the important constants of mathematics are in that identity when you plug in \theta = \pi. So it may be possible to fix the cite. --[[User:C S|C S]][[User talk:C S| (Talk)]] 00:36, 12 June 2007 (UTC) |
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::I copied the ref from [[Richard Feynman]]. If it's wrong, then it should be fixed there too. —[[User:Disavian|Disavian]] ([[User talk:Disavian|<sup>talk</sup>]]/[[Special:Contributions/Disavian|<sub>contribs</sub>]]) 03:29, 12 June 2007 (UTC) |
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:::I can't find any corresponding comment or ref in the Feynman article. But it's pretty clear that the quote is doesn't fit the way it's used here, so I'll take it out until someone who has the source consults it and figures out a more appropriate use for his jewel comment. He also had another jewel comment about QED, which you can find in GBS. [[User:Dicklyon|Dicklyon]] 04:57, 12 June 2007 (UTC) |
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::::My bad, the article was [[Leonhard Euler]]; it's also used on [[Contributions of Leonhard Euler to mathematics]]. —[[User:Disavian|Disavian]] ([[User talk:Disavian|<sup>talk</sup>]]/[[Special:Contributions/Disavian|<sub>contribs</sub>]]) 16:58, 21 June 2007 (UTC) |
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== Peer review == |
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I'm transcluding the peer review to this page so it will gather more attention from the many editors who frequent this talk page. Per typical Peer Review ettiquite, respond to and/or implement the reviewer's suggestions resonably quickly so that reviewers can identify and comment on new issues. —[[User:Disavian|Disavian]] ([[User talk:Disavian|<sup>talk</sup>]]/[[Special:Contributions/Disavian|<sub>contribs</sub>]]) 16:16, 21 June 2007 (UTC) |
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{{Wikipedia:Scientific peer review/E (mathematical constant)}} |
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==First sentence seems a little weak?== |
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The article begins with: |
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:"The [[mathematical constant]] '''''e''''' is the unique [[real number]] such that the [[tangent line]] to the graph of the [[exponential function]] ''y'' = ''e''<sup>''x''</sup> at ''x'' = 0 is the line ''y'' = 1 + ''x''." |
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While no doubt true, this seems a fairly arbitrarily-chosen property to lead off with, and not one which gives readers (especially the less technical readers) any real feel for what e is all about. Also, the definition is circular because "the" exponential function is defined in terms of e, so a definition of e shouldn't really refer to it. I feel this intro could be improved. Matt 22:34, 6 July 2007 (UTC). |
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:The original first sentence defined ''e'' to be the base of the natural log. This was rejected because it appeared to be circular. (I didn't feel it was, but trying to explain why this so to non-mathematicians seems impossible.) This sentence was suggested because it makes a link with the displayed image. Also, you don't need ''e'' to define an exponential function. For instance, ''y''=2<sup>''x''</sup> doesn't involve ''e''. What makes ''e'' special is that the associated exponential function has unit slope at ''x''=0. [[User:Silly rabbit|Silly rabbit]] 22:45, 6 July 2007 (UTC) |
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::You are talking about ''an'' exponential function; I emphasised ''the'' exponential function, which is the term used in the definition and AFAIK always means exp(x). In fact, my original statement that the exponential function depends on already having defined ''e'' is nonsense, because obviously you can define it via the series. However, I still think the current wording is wrong, or at least misleading, because it implies that there are lots of different forms of ''the'' exponential function (effectively ''a''^''x'' for any constant ''a''), and that ''e'' is the constant we need to use to satisfy the slope condition. I am not sure that this is correct. |
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::: Ah, this is then a miscommunication over the use of the word "the" in the first sentence. Point taken. [[User:Silly rabbit|Silly rabbit]] 00:13, 7 July 2007 (UTC) |
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::::No, it's not about the word "the", it's about a circular definition. Why not just say "The [[mathematical constant]] '''''e''''' is the unique [[real number]] y such that ''y'' = ''e''<sup>1</sup>"?[[User:Johnlv12|John Lawrence]] 18:25, 12 September 2007 (UTC) |
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::::After thinking some more, I don't think it's circular. It would be more clear that it was not circular if it was written as |
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::::"The [[mathematical constant]] '''''e''''' is the unique [[real number]] ''a'' such that the [[tangent line]] to the graph of the [[exponential function]] ''y'' = ''a''<sup>''x''</sup> at ''x'' = 0 is the line ''y'' = 1 + ''x''." |
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::::The way it is currently written, it looks like you need to know how to calculate e^x. Written this way, it is now clear that you need to know how to calculate ''a''^''x'' for an arbitrary ''a'', how to find the tangent line at ''x''=0, then locate the unique value of ''a'' that makes the tangent line what you want. However, if you don't know how to find the tangent line, good luck finding '''''e'''''. I think a more constructive definition such as either |
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::::a) the unique real number <math>a</math> such that <math>\lim_{h\to 0}\frac{a^h-1}{h} =1</math> |
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::::or |
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::::b) <math>\lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n</math> |
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::::are better. Although these definitions require an understanding of limits, either seem more accessible than the definition now. In fact, the key concept behind the definition written now lies within suggestion a). Since both definitions are equivalent and I think b) is the easier to understand, I suggest to replace the sentence with |
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::::"The [[mathematical constant]] '''''e''''' is the [[real number]] defined by <math>\lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n</math>. An approximate value of '''''e''''' can be found by calculating this expression for a large value of <math>n</math>, for example <math>\left(1+\frac{1}{1000}\right)^{1000} = 1.001</math> multipled by itself <math>999</math> times <math>= 2.71...</math>"[[User:Johnlv12|John Lawrence]] 16:38, 15 October 2007 (UTC) |
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From 1877 to 1880 Young served as an editor of the ''[[Deseret News]]'' along with [[George Q. Cannon]].<ref>Jenson. ''Encyclopedic History''. p. 187</ref> |
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In 1878, Young and [[Moses Thatcher]] selected the site for the Latter-day Saint settlement in the [[Star Valley]] of [[Wyoming]]. In August 1878, Young dedicated the valley as a place for the gathering of the Latter-day Saints.<ref>''[[Church News]]'', [[1992-08-08]], p. Z5.</ref> |
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In February 1883, Young went on a tour among the [[Navajo]] and [[Hopi]] peoples with many other church leaders, including [[Heber J. Grant]].<ref>Grant, [http://search.ldslibrary.com/article/view/191814 ''Conference Report'', Oct. 1942, p. 25].</ref> |
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::On the second point, your phrasing referring to the slope seems much better to me than stating the equation of the tangent line. The slope is the key thing (linking in to the importance in calculus), and the equation of the tangent line is just a relatively unimportant consequence of that. Matt 00:04, 7 July 2007 (UTC). |
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In 1883, Young convinced the residents of Jonesville, Arizona to rename it Lehi.<ref>Jenson. ''Encyclopedic History''. p. 426</ref> It is today part of [[Mesa, Arizona]]. |
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::: To condense that into a concrete suggestion, then, you would like to see the sentence rephrased in terms of the slope? Ok. But this alone doesn't seem to justify such a lengthy debate. How would ''you'' define ''e'' in the first line? The original version, which I favor, is [http://en.wikipedia.org/w/index.php?title=E_%28mathematical_constant%29&oldid=139366716 here]. [[User:Silly rabbit|Silly rabbit]] 00:13, 7 July 2007 (UTC) |
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==Family== |
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::::: Well, on the first point I wrote one sentence querying the use of the tangent line equation in the definition, you suggested something else, and I wrote two sentences saying I preferred your version. I don't see how that constitutes a "lengthy debate". Wd u rthr i rote lk ths? I would define e as equal to 1/0! + 1/1! + 1/2! + ..., but I didn't want to suggest that as it is seems less accessible than what we have at the moment. There are potential pitfalls with any definition that relies ''solely'' on expressions such as a^x, because to know what that means we first have to define what the "^" operator means for real numbers, and usually that itself involves defining exp and log first (though you could get around it by using the limit of rational approximations to x I guess). I am no mathematician though. Matt 00:53, 7 July 2007 (UTC) |
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Young practiced [[plural marriage]]. His fist wife was Catherine Curtis Spencer. Among their children was Brigham Spencer Young, who would later serve as president of the Northwestern States Mission of the church.<ref>[http://search.ldslibrary.com/article/view/234151 ''Conference Report'', Oct. 1926, p. 2].</ref> |
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In 1857 Young married his second wife, Jane Carrington, a daughter of [[Albert Carrington]].<ref name = Jenson/> |
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::::I'd suggest changing it to be more like that caption, omitting "the exponential function" and the equation for the tangent line. [[User:Dicklyon|Dicklyon]] 00:32, 7 July 2007 (UTC) |
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Young's wife Abigail Stevens was one of his younger wives. Their daughter Klara Young Cheney, born in [[Fruitland, New Mexico]] in 1894, turned 100 years old in 1994.<ref>''[[Church News]]'', [[1994-12-24]].</ref> Abigail and Brigham Jr.'s last daughter, Marian Young, was also born at Fruitland on [[15 January]] [[1899]]. She died on [[22 November]] [[2004]], less than two months short of her 106th birthday. She was the last grandchild of Brigham Young to die.<ref>''[[Church News]]'', [[2004-12-04]], p. Z12.</ref> |
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:::::Again, without referencing the exponential function, how are we supposed to know what something raised to the power of x means when x is not a rational number? Is it OK to just gloss over this? Matt 00:58, 7 July 2007 (UTC). |
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::::::In e^x it doesn't matter that e is irrational, and the x can be dealt with by taking the slope using a limit approached by a sequence of rational values of x that approaches 1, like e^((N+1)/N). So, sure sweep it under the rug is OK by me. [[User:Dicklyon|Dicklyon]] 01:23, 7 July 2007 (UTC) |
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:::::::With my limited experience, I would expect a differentiable function to be defined for all real x (in the relevant range), but technically I don't see why your method shouldn't work. I've changed the intro along the lines suggested to refer to the slope rather than the equation of the line, and remove the reference to "the exponential function". Please feel entirely free to change further as you see fit! Matt 02:05, 7 July 2007 (UTC). |
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::::::::Yes, of course. But if you only know how to do rational powers, you can define the value at other points as limits of those, or you can define the deriviative using limits of differences using only rational args. Many ways to get there... [[User:Dicklyon|Dicklyon]] 03:06, 7 July 2007 (UTC) |
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:::::::::Right. After looking again at the new intro, I've reinstated the link to [[exponential function]], as I think having mentioned f(x) = e^x we really ought to say what that's called. I've tried to do it in such a way as not to give the impression that "the exponential function" is a whole family of functions, which was my quibble with the original text in this respect. Matt 19:47, 7 July 2007 (UTC). |
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==Politics== |
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== GA review Mk.2 == |
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Young served several terms in the [[Utah Territorial Legislature]].<ref name = Jenson/> |
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==President of the Quorum of the Twelve Apostles== |
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The reviewer(whoever he/she will be) may add comments here.--[[User:Cronholm144|Cronholm]]<sup>[[User talk:Cronholm144|144]]</sup> 06:31, 18 July 2007 (UTC) |
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Young was ordained an apostle before [[Joseph F. Smith]] but was not placed in the Quorum of the Twelve Apostles until after Joseph F. Smith. However, It was not until 1900 that a clear decision was made giving Joseph F. Smith seniority in the Quorum of the Twelve Apostles, since he had been a member of the First Presidency since becoming an apostle.<ref>Cowan, Richard O. ''The Church in the 20th Century''. (Salt Lake City, Utah: Bookcraft, 1985) p. 14</ref> Young had served as the [[president of the Quorum of the Twelve Apostles]] since the previous president, [[Franklin D. Richards (Mormon apostle)|Franklin D. Richards]], had died on [[1899-12-09]]. When church president [[Lorenzo Snow]] died on [[1901-10-10]], Joseph F. Smith served as president of the Quorum until he was made church president on [[1901-10-17]]. When Smith became president, Young again assumed the position of president of the Quorum of the Twelve. Young is the only person to have served two non-consecutive terms as president of the Quorum. |
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:Well, it looks like we were passed without any comments. —[[User:Disavian|Disavian]] ([[User talk:Disavian|<sup>talk</sup>]]/[[Special:Contributions/Disavian|<sub>contribs</sub>]]) 20:36, 18 July 2007 (UTC) |
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::That's kind of anticlimactic. :P [[User:Silly rabbit|Silly rabbit]] 20:40, 18 July 2007 (UTC) |
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:Now hold on just one minute. Did you actually think i wouldn't make a statement? ;-) I would like to start off with excellent job everyone. Give yourselves a pat on the back. I noticed that you went into depth and included plenty of clear equations showing how E works. This and the writing prose made it over all a good article. Consequently the article gets a shiny. [[User:Thedagomar|Dagomar]] 21:39, 18 July 2007 (UTC) |
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==Death== |
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== Another graphical example == |
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Young died in [[Salt Lake City, Utah|Salt Lake City]], [[Utah]] at age 66. |
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[[Image:BrighamYoungJrGrave.jpg|right|thumb|200px|Brigham Young, Jr.'s grave marker.]] |
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I have always liked the function <math> x^y = y^x </math> . It is multi-valued, with the trivial solution being x=y. But there is a second curve that looks somewhat like the first-quadrant rectangular hyperbola of y=1/x (See [[:Image:HyperbolaRect01.png]]), except that it asymtotically approaches x=1 and y=1 instead of x=0 and y=0. The straight line and the curved line intersect at (e,e). It is probably not very satisfying to the purist, but graphically it is very compelling. Perhaps we should include this in the article. Objections?--[[User:SallyForth123|SallyForth123]] 07:48, 4 August 2007 (UTC) |
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==References== |
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: I like that function as well—I'm trying to remember the name of the paper I read that discussed that function in some detail, in case you need it as a ref. Perhaps it will come to me. In any case, make a picture and add a paragraph to the article! |
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{{reflist}} |
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: [[User:CRGreathouse|CRGreathouse]]<small> ([[User talk:CRGreathouse|t]] | [[Special:Contributions/CRGreathouse|c]])</small> 01:09, 14 August 2007 (UTC) |
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== See also == |
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== Exponential function definition == |
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* [[Richards-Young family]] |
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{{start box}} |
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I was looking through this page to verify that <math>e^x</math> can also be defined as <math> lim_{x \to \infty} (1+ \frac{x}{n})^n </math>. I couldn't find this information for quite a while until I came upon the exponential function page ( http://en.wikipedia.org/wiki/Exponential_function ). Since the series definition for the exponential function, e^x, is included on this page, it might be relevant to include this second definition as well. At the very least, a "see also" section containing a link to the exponential function on wikipedia would be useful. |
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{{s-rel}} |
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[[User:Jamned|Jamned]] 04:04, 16 August 2007 (UTC) |
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{{succession box |
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| title= [[President of the Quorum of the Twelve Apostles]] |
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== missing limit formula == |
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| years= [[December 9]], [[1899]]–[[October 10]], [[1901]] |
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| before=[[Franklin D. Richards (Mormon apostle)|Franklin D. Richards]] |
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From the limit formula |
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| after= [[Joseph F. Smith]] |
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:<math>e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n</math> |
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}} |
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follows the useful fact that |
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{{succession box |
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:<math>e = \lim_{n\to\infty} \left( 1 - \frac{1}{n} \right)^{-n}</math> |
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| title= [[President of the Quorum of the Twelve Apostles]] |
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I miss this formula in the article. |
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| years= [[October 17]], [[1901]]–[[April 11]], [[1903]] |
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| before=[[Joseph F. Smith]] |
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The two formulas may be written together as. |
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| after= [[Francis M. Lyman]] |
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:<math>e = \lim_{|n|\to\infty} \left( 1 + \frac{1}{n} \right)^{n}</math> |
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}} |
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[[User:Bo Jacoby|Bo Jacoby]] 07:03, 17 August 2007 (UTC). |
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{{succession box | |
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title= [[Quorum of the Twelve Apostles (LDS Church)|Quorum of the Twelve Apostles]] |
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== Definition is circular? == |
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| years= 1868–[[April 8]], [[1873]]<br />[[August 29]], [[1877]]–[[April 11]], [[1903]] |
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| before=[[Joseph F. Smith]] |
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The definition given of "e" in the opener is circular as it involves e^x. With that we don't need to bother with a derivative, we can just say what's the value of e^x at 1 since anything to 1 is itself? But that's of course still just as circular. Although "e^x" can be defined without first defining "e", it just looks bad since we see the symbol "e" in both places. [[User:Mike4ty4|mike4ty4]] 18:27, 17 August 2007 (UTC) |
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| after= [[Albert Carrington]] |
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}} |
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:Various people have already attempted to make a good non-circular one-line definition. To address your concern, the definition could be rephrased to something like: The number ''e'' is the unique value of the constant ''a'' for which the function ''f''(''x'') = ''a''<sup>''x''</sup> has slope 1 at ''x''=0. I believe this is already implicit in the definition, but if you prefer, it can also be spelled out more explicitly. (An earlier version defined ''e'' in terms of the natural logarithm. This was also not circular, but other editors seemed to feel that it was.) [[User:Silly rabbit|Silly rabbit]] 20:29, 17 August 2007 (UTC) |
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{{end box}} |
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A similar thing is done with the definition of [[Natural_logarithm|natural logarithm]] when the author states: <blockquote>In simple terms, the natural logarithm of a number x is the power to which e would have to be raised to equal x</blockquote><p>Here, one function is defined by its inverse.</p><p>[[User:Zgozvrm|Zgozvrm]] 22:04, 24 August 2007 (UTC)</p> |
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:The definition is not circular unless the circle is closed, e.g. if '''ln''' were defined in terms of '''exp''' and '''exp''' in terms of '''ln'''. It is OK to define '''ln''' as the inverse function of '''exp''' provided that '''exp''' is defined in a way that does not depend on '''ln''', e.g. define '''exp''' as the limit of a power series. [[User:JRSpriggs|JRSpriggs]] 01:59, 25 August 2007 (UTC) |
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==GA status reviewed== |
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As part of [[Wikipedia:WikiProject Good articles/Project quality task force/Sweeps|GA Sweeps]], I have reviewed the status of this article. I believe this article is accurate due to my background in calculus, but requires more references. Therefore, I decided to keep this article as GA. [[User:OhanaUnited|<span style="font-weight:bold"><font color="#0000FF">OhanaUnited</font></span>]][[User talk:OhanaUnited|<span style="font-weight:bold"><font color="green"><sup>Talk page</sup></font></span>]] 02:26, 1 September 2007 (UTC) |
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==infinite series== |
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number e is the sum of the infinite series e=1/0+1/1+1/2-------- |
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in the summation 1/0 is it not equal to infinity. am i missing something[[User:70.131.68.114|70.131.68.114]] 21:44, 2 October 2007 (UTC)g.g.subramanian |
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:That's <math>e=\sum_{n=0}^\infty \frac{1}{n!}</math>, where the "!" represents the [[factorial]] function. — [[User:Arthur Rubin|Arthur Rubin]] | [[User_talk:Arthur_Rubin|(talk)]] 22:04, 2 October 2007 (UTC) |
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== First Sentence == |
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"In mathematics, e denotes one of the most important irrational numbers in mathematics,[1] along with the additive identity 0, the multiplicative identity 1, the imaginary unit i, and π, the ratio of the circumference of any circle to its diameter. e can be defined in a number of ways, several of which are shown below." |
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To me, this sentence seems to imply that 0, 1 and i are irrational numbers, which they are not. Or am I missing something? [[Special:Contributions/222.153.7.37|222.153.7.37]] ([[User talk:222.153.7.37|talk]]) 15:41, 7 December 2007 (UTC) |
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==Suggestion from history== |
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More clarity may be obtained by using #5 in the alternative characterizations. |
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After all, when [[Gregoire de Saint-Vincent]] set about making a quadrature of the hyperbola xy = 1, he discovered the facts we now associate with the [[Natural logarithm]]. Though we teach calculus with derivatives first, then integrals, in this case the integral and natural logarithm are the definitive concept. The number e just happens to be the projection point on the asymptote where the quadrature reaches one.[[User:Rgdboer|Rgdboer]] ([[User talk:Rgdboer|talk]]) 22:21, 13 December 2007 (UTC) |
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==The Intro is inadequate== |
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The current version of the Intro is weak and inadequate for all the reasons mentioned above, and more. I emphatically dissent from any claimed "consensus" and submit that the Intro should read as follows: |
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:"'''e''' denotes one of the most important [[number]]s in all of [[mathematics]],<ref>{{cite book | title = An Introduction to the History of Mathematics | author = Howard Whitley Eves | year = 1969 | publisher = Holt, Rinehart & Winston | url = http://books.google.com/books?id=LIsuAAAAIAAJ&q=%22important+numbers+in+mathematics%22&dq=%22important+numbers+in+mathematics%22&pgis=1 }}</ref> The only numbers of comparable significance are the [[additive identity]] [[0 (number)|0]], the [[multiplicative identity]] [[1 (number)|1]], the [[imaginary unit]] ''i'', and [[pi|π]], the [[ratio]] of the [[circumference]] of any [[circle]] to its [[diameter]]. ''e'' can be defined in a number of ways, several of which are shown below. |
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:"''e'' is one of the two most important [[irrational number]]s in mathematics (π being the other). Hence ''e'' cannot be stated as a finite or repeating decimal. The first 20 digits in the [[decimal|decimal representation]] of ''e'' are: |
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:2.71828 18284 59045 23536... |
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:"''e'' is also [[transcendental number|transcendental]]. |
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:"''e'' occurs frequently in [[calculus]], [[differential equation]]s, [[mathematical analysis|analysis]], the theory of [[complex number]]s, [[probability]], [[statistics]], [[physics]], [[chemistry]], [[engineering]], and [[economics]]. ''e'' appears in the [[exponential function]] ''e''<sup>x</sup>, one of the most common [[function]]s in all of mathematics. This function is potentially useful whenever change over time, or growth and decay, are treated mathematically. |
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:"''e'' is occasionally called '''Euler's number''' to honor the [[Switzerland|Swiss]] [[mathematician]] [[Leonhard Euler]] who first proposed the notation ''e'' and who discovered many results that involve it. ''e'' has also been called or '''Napier's constant''' in honor of the [[Scotland|Scottish]] mathematician [[John Napier]] who introduced [[logarithm]]s. (''e'' is not to be confused with γ – the [[Euler–Mascheroni constant]], sometimes called ''Euler's constant''.)" |
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This entry also does not convey much of the richness and beauty of Eli Maior's remarkable book. Getting this entry right is very important, because while most of us finish secondary school with some idea of the centrality of π, that is not the case for ''e''. Only in grad school did I see the light re ''e''.[[Special:Contributions/132.181.160.42|132.181.160.42]] ([[User talk:132.181.160.42|talk]]) 00:46, 14 December 2007 (UTC) |
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== right align column == |
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on the column of known digits at certain times, I believe we should right align it to allow easier comparisons of order of magnitude <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Michael miceli|Michael miceli]] ([[User talk:Michael miceli|talk]] • [[Special:Contributions/Michael miceli|contribs]]) 04:42, 2 January 2008 (UTC)</small><!-- Template:Unsigned --> <!--Autosigned by SineBot--> |
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== Rational approximation of e == |
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Maybe I'm not looking hard enough, but is there a rational approximation of e, such as 22/7 for pi? --[[User:Steerpike|Steerpike]] ([[User talk:Steerpike|talk]]) 16:36, 20 January 2008 (UTC) |
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:The continued fraction doesn't have early big terms like the one for pi has, so you don't get any amazing accurate convergents. But they're all tabulated: http://www.research.att.com/~njas/sequences/A007676 and http://www.research.att.com/~njas/sequences/A007677, which has: |
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:Numerators: 2, 3, 8, 11, 19, 87, 106, 193; Denominators: 1, 1, 3, 4, 7, 32, 39, 71; which gives 2/1, 3/1, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, the last two of which are 2.71794872 and 2.71830986. The next one needs much better integers to get another half digit of accuracy: 1264/465 = 2.71827957. [[User:Dicklyon|Dicklyon]] ([[User talk:Dicklyon|talk]]) 18:38, 20 January 2008 (UTC) |
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::Thank you! --[[User:Steerpike|Steerpike]] ([[User talk:Steerpike|talk]]) 20:03, 20 January 2008 (UTC) |
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: Also see [[Talk:E (mathematical constant)/Archive_2#Inaccurate_value.3F]] -- [[User:Dominus|Dominus]] ([[User talk:Dominus|talk]]) 14:37, 5 March 2008 (UTC) |
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From the cited continued fraction in the main article |
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:<math> e = [[ 1 , \textbf{0} , 1 , 1, \textbf{2}, 1, 1, \textbf{4}, 1 , 1 , \textbf{6}, 1, \ldots]] \, </math> |
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e clearly has a pattern that pi lacks. After the 1st term we have 1/1; the 4th, 3/1; the 7th, 19/7; and the 10th, 193/71. To get the next important rational approximation to e, note the following patterns: |
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1 + 3*6 = 19 and 1 + 1*6 = 7. Also, 3 + 19*10 = 193 and 1 + 7*10 = 71. |
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As [[User:Dicklyon|Dicklyon]] points out, the next fraction is 1264/465; this is followed by (193+1264)/(71+465) = 1457/536, followed by the great approximation (1264+1457)/(465+536) = 2721/1001. Guess what? |
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19 + 193*14 = 2721 and 7 + 71*14 = 1001. Clearly the next important one is |
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193 + 2721*18 = 49171 and 71 + 1001*18 = 18089. This continued fraction gets them all: |
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:<math>e = 1+\cfrac{2}{1+\cfrac{1}{6+\cfrac{1}{10+\cfrac{1}{14+\cfrac{1}{\ddots\,}}}}}</math> |
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[[User:Glenn L|Glenn L]] ([[User talk:Glenn L|talk]]) 09:42, 5 March 2008 (UTC) |
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== First sentence (again) == |
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I hesitate to rehash old discussions, but the first sentence does strike me as unideal. Admittedly, there is a chicken-or-egg thing between which is the more primitive definition, that of ''e'' or that of raising to a power. But I point out that for ''general'' (complex) ''a'', ''a<sup>x</sup>'' is ''defined'' to be ''e<sup>ln(a) x</sup>'', so there may be some ambiguity here. I always thought ''e'' was defined as the root of |
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:<math> \int_{1}^{x} \frac{\mathrm{d} z}{z} = 1 \;\!</math> |
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and that the properties of the exponential function (including its derivative) followed from that. |
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It's been a while though, so I admit perhaps my memory isn't quite up to the task here. Thoughts? [[User:Baccyak4H|Baccyak4H]] ([[User talk:Baccyak4H|Yak!]]) 19:19, 21 March 2008 (UTC) |
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: This is old ground. Many moons ago, ''e'' was defined (in the first sentence) as the base of the natural logarithm. To my mind, this is the cleanest approach to the constant, and the most compelling motivation. However, the inevitable chorus arose, claiming that the definition is "circular" because, it was claimed, the natural logarithm requires the exponential function to define it. Well, of course it isn't true, but one can never satisfy the naysayers. Ultimately, the present suboptimal version of the first sentence was settled on. You're welcome to try your hand at improving it, though. [[User:Silly rabbit|<font color="#c00000">silly rabbit</font>]] ([[User talk:Silly rabbit|<span style="color:#FF823D;font-family:Monotype Corsiva;cursor:help"><font color="#c00000">talk</font></span>]]) 19:38, 21 March 2008 (UTC) |
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:: Hmm, I guess I would agree, as my so called definition coincides with it being the base of the natural logarithm (and avoiding circularity to boot). But I am not in a boat rocking mood. If I become so, I'll chime in here first. Thanks for the feedback. [[User:Baccyak4H|Baccyak4H]] ([[User talk:Baccyak4H|Yak!]]) 02:02, 22 March 2008 (UTC) |
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:::There are many equivalent definitions. But for the base of the natural log be to one of them, you have to have an independent definition of the natural log. You can say it's the log base such that the derivative of ln(x) at 1 is 1, that would be equivalent to the current definition we have, but not as direct. Or you can say that ln(x) is the antiderivative of 1/x, which leads to that integral definition above. Pick one. I like the one we have now because I could see how to illustrate it. [[User:Dicklyon|Dicklyon]] ([[User talk:Dicklyon|talk]]) 02:04, 22 March 2008 (UTC) |
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::::Yes, I always have taken the natural log to be defined to be the antiderivative as above, which prompted my query here. However, your point about illustration is ''very'' well taken, so I will not make the perfect the enemy of the good. [[User:Baccyak4H|Baccyak4H]] ([[User talk:Baccyak4H|Yak!]]) 02:42, 22 March 2008 (UTC) |
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== Italics == |
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Is there any reason why ''e'' is set in italics? See for example the [http://www.iupac.org/standing/idcns/italic-roman_dec99.pdf IUPAC guidance]. |
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—DIV ([[Special:Contributions/128.250.80.15|128.250.80.15]] ([[User talk:128.250.80.15|talk]]) 04:15, 7 April 2008 (UTC)) |
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:Yes, I'd say that's why. Are you reading it differently than I am? [[User:Dicklyon|Dicklyon]] ([[User talk:Dicklyon|talk]]) 04:27, 7 April 2008 (UTC) |
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::Paragraph 7 says "The symbols π, e (base of natural logarithms), i (square root of minus one), etc. are always roman". It really bothers me to see stuff such as <math>\displaystyle e^{ \frac{eV}{k_{\scriptscriptstyle \text{B}}T}}</math> where the two ''e''s refer to different things. --<span style="font-family: monospace; font-weight: 600; color: #00F; background-color: #FFF"> [[User:Army1987|Army]][[1987]][[User talk:Army1987| (t ]][[Special:Contributions/Army1987|— c)]]</span> 01:17, 11 October 2008 (UTC) |
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:::Yep, looks like he was reading it differently that I was; I missed that detail. Should fix... [[User:Dicklyon|Dicklyon]] ([[User talk:Dicklyon|talk]]) 03:53, 11 October 2008 (UTC) |
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== Proposed change to introductory sentence == |
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I propose to change the intro, which currently reads thus: |
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<blockquote> |
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The [[mathematical constant]] '''''e''''' is the unique [[real number]] such that the value of the [[derivative]] (slope of the [[tangent]] line) of the function ''f''(''x'') = ''e<sup>x</sup>'' at the point ''x'' = 0 is exactly 1. |
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</blockquote> |
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to this: |
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<blockquote> |
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The [[mathematical constant]] '''''e''''' is the unique [[real number]] such that the function ''e<sup>x</sup>'' has the same value as its [[derivative|slope]], for all values of ''x''.[http://www.vias.org/calculus/08_exp-log_functions_03_01.html] |
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</blockquote> |
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I think this better conveys a memorable, comprehensible, geometric interpretation of ''e''. Also, as a followon to that sentence, one might add this: |
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<blockquote> |
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More generally, the only functions which are equal to their own [[derivative]]s are of the form ''Ce<sup>x</sup>'', where ''C'' is a constant.[http://www.vias.org/calculus/08_exp-log_functions_06_01.html] |
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</blockquote> |
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This is a really important general statement, and the previous sentence leads nicely into it.[[User:JCLately|JCLately]] ([[User talk:JCLately|talk]]) 03:45, 16 April 2008 (UTC) |
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::Your proposed new definition can be shown, with enough work, to follow from the existing simpler definition. I believe that a definition should be as simple as possible (but no simpler). [[User:Dicklyon|Dicklyon]] ([[User talk:Dicklyon|talk]]) 05:44, 16 April 2008 (UTC) |
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:::The existing "definition" can be shown (with less work, I believe) to be a consequence of my proposed first sentence. So if the two statements are mathematically equivalent, on what basis is one simpler than the other? I didn't present my lead sentence as a definition, although it could be so considered, and that is specifically mentioned in the reference that I provided. No reference has been offered that the previous formulation is widely accepted as the definition of ''e'', but it isn't my intention to get into a semantic argument as to what constitutes the definition of ''e'', either historically or according to some formal criterion. |
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:::As it currently reads, the lead sentence, while correct and reasonably concise, is not particularly useful as a general statement, it is not especially memorable, and it fails to convey the very special, non-arbitrary character of ''e''. If a person were to read only one sentence of this article, which one better encapsulates the essence of ''e''? Whether or not my formulation is "simpler", whether or not one regards it as definitive, I think it says far more about what is ''e'', and what is its significance. |
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:::The caption on the accompanying graphic is virtually identical to the previous intro sentence, so what's the point of saying the same thing twice? We can keep the graphic and its caption, but I think my proposed change to the intro sentence would be an improvement. [[User:JCLately|JCLately]] ([[User talk:JCLately|talk]]) 01:52, 17 April 2008 (UTC) |
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::::The purpose of the figure and its caption is to illustrate the defining characteristic described in the lead sentence. It seems to me that the slope of the tangent line at one point is much simpler and more concise than the general derivative everywhere. [[User:Dicklyon|Dicklyon]] ([[User talk:Dicklyon|talk]]) 02:56, 17 April 2008 (UTC) |
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::::: I agree with Dicklyon. This definition is a more primitive one, requiring only the slope of a tangent line, and should therefore be favored over the definition that requires the notion of the derivative as a function. Futhermore, there is a picture to go along with the current defintion, while it is unclear if a compelling picture can be drawn to illustrate the definition JC has in mind. Finally, let us not forget that this article is on the mathematical constant ''e'', rather than the [[natural exponential function]]. [[User:Silly rabbit|<font color="#c00000">silly rabbit</font>]] ([[User talk:Silly rabbit|<span style="color:#FF823D;font-family:Monotype Corsiva;cursor:help"><font color="#c00000">talk</font></span>]]) 03:04, 17 April 2008 (UTC) |
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:::Whatever may have been the purpose of the figure and its caption, the issue I'm raising pertains to the [[Wikipedia:Lead section | Wikipedia guideline on article introductions]]. To wit, "It should establish context, summarize the most important points, explain why the subject is interesting or notable..." and "should be written in a clear, accessible style so as to invite a reading of the full article". I do not see how the present lead sentence is either simpler or more concise than my alternative, but more importantly, the present formulation seems considerably less useful and illuminating. One is required to comprehend the concept of slope (aka derivative) in either case: what is the benefit of a statement about the slope at a particular point, rather than an infinitely more useful statement about the slope at ''any'' point on the same curve? I appreciate your having provided a reference to support this rather unusual definition of ''e'', but I think you'd find many more references to the the way that I put it. |
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:::In further reply to [[User: silly rabbit | silly rabbit]], it would be simple enough to illustrate the more general theorem/definition with a graph showing a family of curves of the function ''f''(''x'') = ''a<sup>x</sup>'' for various choices of ''a'', noting that only for the special value ''a = e'' does the slope equal the value at every point on the curve, including at ''x = 0''. As to your observation that this article is about ''e'', not the exponential function, I don't see how this distinguishes between the two alternatives at hand, since both refer to the same function. Furthermore, it would be pretty hard to do justice to the concept of ''e'' without prominent mention of the exponential function, as they are intimately and inextricably related! [[User:JCLately|JCLately]] ([[User talk:JCLately|talk]]) 21:34, 17 April 2008 (UTC) |
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:::: My own preference is to define ''e'' as the base of the natural logarithm, as indicated above in the talk page. In fact, this was in my own favored version of the lead many moons ago. However, people who are unfamiliar with the many approaches to the subject of [[mathematical analysis]] seemed to feel that this definition was circular. Eventually the present one was settled upon. As you have agreed, a more complicated graph is needed to illustrate the definition of ''e'' you are offering here. Furthermore, it is arguably less geometrical, since it invokes a certain ''global'' property of the function, namely its derivative at every point, rather than the local (in fact, infinitesimal) property of the slope of its tangent line. [[User:Silly rabbit|<font color="#c00000">silly rabbit</font>]] ([[User talk:Silly rabbit|<span style="color:#FF823D;font-family:Monotype Corsiva;cursor:help"><font color="#c00000">talk</font></span>]]) 22:23, 17 April 2008 (UTC) |
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:::::The definition that I found surprisingly often in searching books was pretty much opposite of that. First they define the natural log (as the integral of 1/x, perhaps?). Then they define its inverse function exp(), and then the define e as exp(1) and then show that exp(x) is equal to e^x. |
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:::::I still think that the slope of a tangent line at a point is a much "simpler and more concise" concept than the derivative of a function, and don't get why JCLately says it's not. The former is understandable to persons who don't know a bit of calculus; the latter is not. Neither is the definition of ln(x) via integration. I think it pays for wikipedia to have a definition that can be easily understood by "Algebra II" students. [[User:Dicklyon|Dicklyon]] ([[User talk:Dicklyon|talk]]) 22:44, 17 April 2008 (UTC) |
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::::::: Sometimes it's hard to determine who is replying to whom. I basically agree with everything you just said, for the record. [[User:Silly rabbit|<font color="#c00000">silly rabbit</font>]] ([[User talk:Silly rabbit|<span style="color:#FF823D;font-family:Monotype Corsiva;cursor:help"><font color="#c00000">talk</font></span>]]) 23:22, 17 April 2008 (UTC) |
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:::::OK, I see a point of confusion that is the result of my having used a piped wikilink to [[Derivative | slope]] in my proposed new first sentence. I meant to use the word slope in the sense that is synonymous to derivative, but perhaps it would be clearer to express it as the [[Derivative | slope of the tangent line]]. My intention in the first sentence is to convey the geometrical meaning in such a way that it should be clear to someone ''without'' any understanding of calculus. I don't buy the argument that this more "global" definition is any less geometric than the "local" one, because it tells you something about the shape of the curve ''e<sup>x</sup>'', whereas something that describes only the characteristics of a single point on the curve tells you nothing about its shape. So one could reasonably argue that the broader definition is ''more'' geometric, and it does not require the comprehension of anything more complex than the slope of a tangent line, exactly the same concept used in the present lead sentence. But again, I think this line of argument misses the point that the lead sentence should, above all, express something notable and significant, which offers an immediate insight into why ''e'' is a very special number. As the article says: "The number ''e'' is one of the most important numbers in mathematics". The current article lead doesn't really give the reader a clue as to why this should be, and it is not a particularly useful statement, whether you call it a definition or a theorem. On the other hand, what I am proposing is a statement that directly paraphrases an extremely useful general statement about ''e'' in terms that can be visualized geometrically without an understanding of calculus. |
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:::::I'm not sure how you got the idea that I agreed that a more complicated graph is needed to illustrate the definition of ''e'' that I am offering. In fact, the same diagram could be used, simply by adjusting the caption to note that besides the specific point ''(0, 1)'' at which a tangent is drawn, it is also true that slope = value at every other point on ''e<sup>x</sup>'', and not on any of the other curves. [[User:JCLately|JCLately]] ([[User talk:JCLately|talk]]) 02:19, 18 April 2008 (UTC) |
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::: Since e is transcendental there will naturally be a challenge presenting it to the novice. Appeal to calculus is probably inappropriate when alternatives are available. My preference is the use of area and a nod to the pioneers St-Vincent and de Sarasa. As mentioned above in "A suggestion from history", the constant arose when one wondered when an area determined by the hyperbola xy = 1 becomes equal to one. Now at "Alternative characterizations" in the article there is the Image: hyperbola E.svg which illustrates alternative #5. A different image at [[hyperbolic angle]] could also be used to illustrate for the novice how e is the number where the hyperbolic angle is one. With this approach based on area there would be no immediate need to refer to logarithms (the base of natural log) or to tangents to a curve.[[User:Rgdboer|Rgdboer]] ([[User talk:Rgdboer|talk]]) 23:21, 17 April 2008 (UTC) |
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There was '''not''' consensus for the change JCLately has made to the lead sentence. I'm not going to revert, because I don't feel very strongly one way or the other, but I do prefer the old version over this one. [[User:Silly rabbit|<font color="#c00000">silly rabbit</font>]] ([[User talk:Silly rabbit|<span style="color:#FF823D;font-family:Monotype Corsiva;cursor:help"><font color="#c00000">talk</font></span>]]) 05:52, 24 April 2008 (UTC) |
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== e in computer culture == |
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The article has the text below under the heading above: |
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''In contemporary internet culture, individuals and organizations frequently pay homage to the number e.'' |
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i am tempted to change this. the examples which follow are certainly not relevant internet culture, which nowadays is dominated by facebook and youtube videos of hanna montana. these examples are, instead, excellent examples of nerd culture. |
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please speak up if you have an opinion about my change proposal. |
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--[[User:Philopedia|Philopedia]] ([[User talk:Philopedia|talk]]) 08:32, 5 May 2008 (UTC) |
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== zero fright ? == |
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A graph without origin is like icing without a cake. |
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The graph is missing a zero numeral in each axis. |
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[[User:Jclerman|Jclerman]] ([[User talk:Jclerman|talk]]) 06:02, 19 June 2008 (UTC) |
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== The compound-interest problem == |
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I find it interesting that this example is given using dollars. Which nation's currency would that have been at the time Bernoulli was working on the problem? [[User:Huw Powell|Huw Powell]] ([[User talk:Huw Powell|talk]]) 20:30, 6 September 2008 (UTC) |
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{{LDSApostles}} |
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:Since Jacob Bernouilli was [[Switzerland|Swiss]], it probably would have been [[Swiss franc]]s. [[User:Silly rabbit|<font color="#c00000">siℓℓy rabbit</font>]] ([[User talk:Silly rabbit|<span style="color:#FF823D;font-family:Monotype Corsiva;cursor:help"><font color="#c00000">talk</font></span>]]) 20:35, 6 September 2008 (UTC) |
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{{LDScouncil50}} |
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{{LDSfirstpresidency}} |
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{{LDSpresq12}} |
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{{DEFAULTSORT:Young, Brigham, Jr.}} |
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::Correction: It would have most likely been [[Basel thaler]]s, since the Franc was yet to be introduced. [[User:Silly rabbit|<font color="#c00000">siℓℓy rabbit</font>]] ([[User talk:Silly rabbit|<span style="color:#FF823D;font-family:Monotype Corsiva;cursor:help"><font color="#c00000">talk</font></span>]]) 20:41, 6 September 2008 (UTC) |
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[[Category:1836 births]] |
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[[Category:1903 deaths]] |
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[[Category:American Latter Day Saints]] |
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[[Category:American Mormon missionaries]] |
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[[Category:Members of the Utah Territorial Legislature]] |
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[[Category:Mormon missionaries in Denmark]] |
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[[Category:Mormon missionaries in France]] |
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[[Category:Mormon missionaries in Germany]] |
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[[Category:Mormon missionaries in Ireland]] |
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[[Category:Mormon missionaries in Norway]] |
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[[Category:Mormon missionaries in Russia]] |
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[[Category:Mormon missionaries in Sweden]] |
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[[Category:Mormon missionaries in Switzerland]] |
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[[Category:Mormon missionaries in the United Kingdom]] |
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[[Category:Mormon pioneers]] |
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[[Category:Young family]] |
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Revision as of 03:53, 11 October 2008
| Brigham Young Jr. | |
|---|---|
| File:Byjr.gif | |
| Personal details | |
| Born | Brigham Young, Jr. December 18, 1836 |
| Died | April 11, 1903 (aged 66) |
Brigham Young, Jr. (December 18, 1836–April 11, 1903) served as president of the Quorum of the Twelve Apostles of The Church of Jesus Christ of Latter-day Saints (LDS Church) from 1899 until his own death. His tenure was interrupted for one week in 1901 when Joseph F. Smith was the president of the Quorum.
Early life
Young was born in Kirtland, Ohio, the son of Brigham Young and Mary Ann Angell. Young's twin sister Mary died at age seven from the effects of injuries received at age two in a wagon accident.[2] At age twelve Young drove an ox cart across the plains, reaching Salt Lake City in 1848.[2] Young served as a guard and scout in the following years operating in Salt Lake Valley and the surrounding canyons.[2]
Young married Catherine Curtis Spencer, a daughter of Orson Spencer with the exact same name as her mother, on 15 November 1855.[2]
In Utah Territory, Young became a member of the reconstituted Nauvoo Legion. He was involved in the rescue of the Willie and Martin Companies of Mormon handcart pioneers. He also served in the Utah War with the troops that worked to halt the advance of Johnston's Army.[3]
n 1861, Young was made a member of the Salt Lake Stake high council.[2]
Early years as a general authority
Brigham Young ordained Young an apostle at the young age of 27 in 1864. However, he was not placed in the Quorum of the Twelve Apostles until four years later in 1868. Young Jr. also served as a counselor to his father in the First Presidency of the church from April 8, 1873 until his father's death on August 29, 1877.
Missions to Europe
From 1862 to 1863 Young served as a church missionary in England, spending most of the time in London.[2] During this time he also accompanied Joseph F. Smith on a trip to Paris, France.[4]
In 1864, Young returned to Europe, this time with his wife Catherine as his companion. He was an assistant to mission president Daniel H. Wells. In 1865, when Wells left for Utah, Young succeeded him as president of the European Mission.[2]
As president of the European Mission of the church in 1866 and 1867, Young preached in France, Switzerland, Germany, Denmark, Sweden, Norway, Russia, the United Kingdom and Ireland. Young also oversaw the emigration of British Latter-day Saints to Utah Territory.[5] It was from a conversation as Young was about to return to Utah at the end of his time as mission president that Charles W. Penrose wrote the hymn "Beautiful Zion For Me".[6]
From October 1890 until February 1893 Young served for a second time as the president of the European Mission.[7] The mission was headquartered in Liverpool, England and Young directly supervised missionary work in the British Isles while also serving as a leader over the mission presidents of the various missions on the European continent.
Colonization and church assignment in America
In the western United States, Young was involved in the colonization of Cache Valley, southern Utah and the extension of Mormon settlements into New Mexico and Arizona. Young was also involved at times with the LDS settlements in the Mexican state of Chihuahua.[3]
In 1867, Young was involved with the formation of the Deseret Sunday School Union to provide centralized direction to the Sunday schools of the church.[8]
During 1868, Young acted as his father's agent in finding workers for the Utah portion of the First Transcontinental Railroad.[2]
From 1869 until 1877, Young presided over the Latter-day Saints in Cache Valley, closely assisted by William B. Preston who was serving as the regional presiding bishop.[9] During this time Young co-owned a feed and livery stable in Soda Springs, Idaho with Solomon Hale.[10]
In 1877, Young, Erastus Snow and Wilford Woodruff dedicated parts of the St. George Temple under the general direction of Brigham Young.[11]
From 1877 to 1880 Young served as an editor of the Deseret News along with George Q. Cannon.[12]
In 1878, Young and Moses Thatcher selected the site for the Latter-day Saint settlement in the Star Valley of Wyoming. In August 1878, Young dedicated the valley as a place for the gathering of the Latter-day Saints.[13]
In February 1883, Young went on a tour among the Navajo and Hopi peoples with many other church leaders, including Heber J. Grant.[14]
In 1883, Young convinced the residents of Jonesville, Arizona to rename it Lehi.[15] It is today part of Mesa, Arizona.
Family
Young practiced plural marriage. His fist wife was Catherine Curtis Spencer. Among their children was Brigham Spencer Young, who would later serve as president of the Northwestern States Mission of the church.[16]
In 1857 Young married his second wife, Jane Carrington, a daughter of Albert Carrington.[2]
Young's wife Abigail Stevens was one of his younger wives. Their daughter Klara Young Cheney, born in Fruitland, New Mexico in 1894, turned 100 years old in 1994.[17] Abigail and Brigham Jr.'s last daughter, Marian Young, was also born at Fruitland on 15 January 1899. She died on 22 November 2004, less than two months short of her 106th birthday. She was the last grandchild of Brigham Young to die.[18]
Politics
Young served several terms in the Utah Territorial Legislature.[2]
President of the Quorum of the Twelve Apostles
Young was ordained an apostle before Joseph F. Smith but was not placed in the Quorum of the Twelve Apostles until after Joseph F. Smith. However, It was not until 1900 that a clear decision was made giving Joseph F. Smith seniority in the Quorum of the Twelve Apostles, since he had been a member of the First Presidency since becoming an apostle.[19] Young had served as the president of the Quorum of the Twelve Apostles since the previous president, Franklin D. Richards, had died on 1899-12-09. When church president Lorenzo Snow died on 1901-10-10, Joseph F. Smith served as president of the Quorum until he was made church president on 1901-10-17. When Smith became president, Young again assumed the position of president of the Quorum of the Twelve. Young is the only person to have served two non-consecutive terms as president of the Quorum.
Death
Young died in Salt Lake City, Utah at age 66.
References
- ^ Brigham Young ordained three of his sons in 1864 without adding them to the Quorum of the Twelve Apostles. However, Young, Jr. became a member of the Quorum of the Twelve in 1868 after Heber C. Kimball died and George A. Smith was removed from the Quorum to join the First Presidency.
- ^ a b c d e f g h i j Jenson, Andrew. Biographical Encyclopedia of the Church of Jesus Christ of Latter-day Saints (Salt Lake City, Utah: Deseret Book and A. Jenson Historical Co., 1901–1936) 1:121.
- ^ a b Flake, Lawrence R. "Brigham Young, Jr." in Garr, Arnolds K., Donald Q. Cannon and Richard O. Cowan, ed., Encyclopedia of Latter-day Saint History (Salt Lake City, Utah: Deseret Book, 2000) p. 1379–1380.
- ^ Jenson. Biographical Encyclopedia. 1:66.
- ^ Cannon, George Q. and Wilford Woodruff, Faith Promoting Series: Gems For the Young Folks, p. 19
- ^ Heber J. Grant, Conference Report, April 1926, p. 147.
- ^ Roberts, B. H. Comprehensive History of the Church of Jesus Christ of Latter-day Saints (Provo, Utah: Brigham Young University Press, 1965) p. 89
- ^ Poleman, B. Lloyd. "Sunday School" in Ludlow, Daniel H., ed., Encyclopedia of Mormonism (New York: MacMillan, 1992) p. 1425
- ^ Jenson, Andrew. Encyclopedic History of the Church of Jesus Christ of Latter-day Saints. (Salt Lake City, Utah: Deseret News Press, 1941, p. 105.
- ^ Jenson. Biographical Dictionary. Vol. 2, p. 168
- ^ Woodruff, Wilford. "Living By the Spirit" in Stuy, H. Bryan, ed., Collected Discourses, 1888–1898 (Glendale, California and Woodland Hills, Utah: B. H. S. Publishing, 1987–1992) Vol. 5.
- ^ Jenson. Encyclopedic History. p. 187
- ^ Church News, 1992-08-08, p. Z5.
- ^ Grant, Conference Report, Oct. 1942, p. 25.
- ^ Jenson. Encyclopedic History. p. 426
- ^ Conference Report, Oct. 1926, p. 2.
- ^ Church News, 1994-12-24.
- ^ Church News, 2004-12-04, p. Z12.
- ^ Cowan, Richard O. The Church in the 20th Century. (Salt Lake City, Utah: Bookcraft, 1985) p. 14
See also
Categories:
- 1836 births
- 1903 deaths
- American Latter Day Saints
- American Mormon missionaries
- Members of the Utah Territorial Legislature
- Mormon missionaries in Denmark
- Mormon missionaries in France
- Mormon missionaries in Germany
- Mormon missionaries in Ireland
- Mormon missionaries in Norway
- Mormon missionaries in Russia
- Mormon missionaries in Sweden
- Mormon missionaries in Switzerland
- Mormon missionaries in the United Kingdom
- Mormon pioneers
- Young family