https://www.explainxkcd.com/wiki/api.php?action=feedcontributions&user=108.162.215.89&feedformat=atomexplain xkcd - User contributions [en]2021-10-27T00:45:02ZUser contributionsMediaWiki 1.30.0https://www.explainxkcd.com/wiki/index.php?title=1504:_Opportunity&diff=1323271504: Opportunity2016-12-10T02:40:25Z<p>108.162.215.89: /* Explanation */ added reference to what-if xkcd</p>
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<div>{{comic<br />
| number = 1504<br />
| date = March 27, 2015<br />
| title = Opportunity<br />
| image = opportunity.png<br />
| titletext = We all remember those famous first words spoken by an astronaut on the surface of Mars: "That's one small step fo- HOLY SHIT LOOK OUT IT'S GOT SOME KIND OF DRILL! Get back to the ... [unintelligible] ... [signal lost]"<br />
}}<br />
<br />
==Explanation==<br />
This comic is talking about the robotic science platform {{w|Opportunity (rover)|''Opportunity''}}. On January 25, 2004, the Opportunity rover landed on the surface of {{w|Mars}} for the purpose of gathering data about the surface of Mars. Opportunity has proven remarkably robust, and the comic extrapolates the rover's resilience to absurdity for comedic effect. As of the release date of this comic in 2015, the Opportunity rover {{w|Opportunity mission timeline|is still alive and moving}}. <br />
<br />
The comic depicts the two scientists [[Ponytail]] and [[Hairbun]] at ground control being amazed at this fact already in 2010, and (maybe the same two) scientists continue to debate this at present day in the second panel.<br />
<br />
They mention another Martian rover, {{w|Spirit (rover)|''Spirit''}} that was also sent to Mars on the same date as Opportunity. Unfortunately, it became stuck and a sand storm covered its solar panels. On March 22, 2010, it was thought that Spirit's batteries finally ran out, marking the end of its mission. This was covered in [[695: Spirit]], in which the Spirit rover is also portrayed with an anthropomorphic personality.<br />
<br />
However, in 2023, Opportunity has apparently become so powerful that it has become dangerous, presumably cannibalizing or destroying the {{w|Mars 2020|rover sent in 2020}}. [[Cueball]] and [[Megan]] note it even continues to operate without its original battery, suggesting it has developed a new power source. This evolution is similar to the stories of {{w|HAL 9000}} (from {{w|2001: A Space Odyssey (film)|''2001: A Space Odyssey''}}) and {{w|List of Star Trek characters (T–Z)#V'Ger|V'Ger}} (from ''{{w|Star Trek: The Motion Picture}}''), both of which became dangerous to human beings.<br />
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By 2450, humans have colonized and {{w|terraformed}} Mars. Maybe it is the 2023 Cueball and Megan's descendants that are looking out over their huge "kingdom" from the capital on Mars. However ''Opportunity'' is by now dominating half of the planet and will not allow humans to enter its dark reign. <br />
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"Everything the light touches" is a reference to a line by {{w|List of The Lion King characters#Mufasa|Mufasa}} in ''{{w|The Lion King}}''. Mufasa's son {{w|List of The Lion King characters#Simba|Simba}} then asks "What about that shadowy place?" and Mufasa tells him "That is beyond our borders. You must never go there". This was used again in [[1608: Hoverboard]], where [http://www.explainxkcd.com/wiki/images/a/a0/1608_0986x1076y_Our_kingdom_from_a_cliff.png Cueball tells the same line] to Ponytail in the left part of the world. In [http://www.what-if.xkcd.com/48 what-if xkcd], concerning the end of the sun shining on the British Empire, Cueball tells a child that everything the light touches is their kingdom, except for France, (which is covered in shadows,) to which Cueball replies, "That's France. We'll get it one of these days."<br />
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The title text forecasts the first words of the first astronauts on the surface of Mars. At first, the astronaut copies the first words of {{w|Neil Armstrong}} on the Moon ("That's one small step for [a] man, one giant leap for mankind") but it is interrupted by the ''Opportunity'' rover. On board the rover is a drill used for sampling rocks, but here it is heavily suggested that Opportunity uses the drill to attack the astronaut.<br />
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==Transcript==<br />
:[The year (or year and first sentence) for each panel is written in a small frame at the top of each panel. It breaks the top frame of the panels.]<br />
<br />
:[Ponytail is sitting at a computer, facing left. Hairbun stands behind her.]<br />
:2010:<br />
:Ponytail: After six years, ''Spirit'' is down, but ''Opportunity'' is still going strong.<br />
:Hairbun: Tough little rover!<br />
<br />
:[Opportunity traveling on Mars. Text is written in frames with zigzag lines]<br />
:2015:<br />
:Off-screen: Eleven years, wow.<br />
:Off-screen 2: Wasn't the original mission 90 days?<br />
:Off-screen: This is starting to get weird.<br />
<br />
:[Cueball and Megan sitting at a computer, facing right.]<br />
:2023:<br />
:Cueball: The battery is totally disconnected. How can it still be moving??<br />
:Megan: Given what it did to the Mars 2020 rover, we may never know.<br />
<br />
:[Two Martian inhabitants looking like Cueball and Megan stands on a cliff edge pointing towards a dark, mountainous region. Behind them are a tower and a hover car]<br />
:2450, terraformed Mars, Martian imperial capital:<br />
:Martian Cueball: Everything the light touches is our kingdom.<br />
:Martian Megan: What's that dark area?<br />
:Martian Cueball: That is ''Opportunity's'' half of the planet. We must never go there.<br />
<br />
{{comic discussion}}<br />
<br />
[[Category:Comics featuring Ponytail]]<br />
[[Category:Comics featuring Cueball]]<br />
[[Category:Comics featuring Megan]]<br />
[[Category:Comics featuring Hairbun]]<br />
[[Category:Multiple Cueballs]]<br />
[[Category:Mars rovers]]<br />
[[Category:Artificial Intelligence]]<br />
[[Category:The Lion King]]</div>108.162.215.89https://www.explainxkcd.com/wiki/index.php?title=Talk:765:_Dilution&diff=93479Talk:765: Dilution2015-05-17T00:07:04Z<p>108.162.215.89: Added comment</p>
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<div>Technically, however, homeopathy states that diluted semen should act as a contraceptive. To get pregnant, they would have to dilute a birth-control pill or something.{{unsigned ip|134.94.171.148}}<br />
:I remember someone complaining that homeopathic medicines are labeled similarly to the real thing on drug store shelves. Wouldn't this mean that, at least in some interpretations, it's the medicine which can be diluted into a cure?[[Special:Contributions/173.245.52.135|173.245.52.135]] 17:47, 13 April 2014 (UTC)<br />
::No, the previous user is correct. The "theory" of homeopathy is that to cure a symptom you provide something that would cause that symptom diluted into some ridiculously small dose. Any products that are marketed with the label "homeopathy" but are primarily an active ingredient that would have a detectable effect are using the label incorrectly. For example, there are two kinds of zinc in Zicam, both diluted at 2x, which means they're 1% of their original solution strength. The label doesn't indicate the original solution strength and doesn't indicate how much zinc is actually present in the medicine. Zinc is known to improve the immune system, so taking a diluted zinc supplement to stop a cold would, if homeopathy were true, would be the opposite of what you'd want to do. Contrast this with the homeopathic remedy Belladonna, prepared at 30X concentrations (preparation is 1x10^-30 of original concentration) supposedly cures the circulatory system, because in normal concentrations Belladonna causes tachycardia, among other things. [[User:Ioldanach|Ioldanach]] ([[User talk:Ioldanach|talk]]) 14:16, 17 September 2014 (UTC)<br />
:::Any marks for realising why scientists can't replicate homeopathy's results? :))<br />
::::Maybe Cueball is Sterile?[[Special:Contributions/108.162.215.89|108.162.215.89]] 00:07, 17 May 2015 (UTC)<br />
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[[User:Weatherlawyer| I used Google News BEFORE it was clickbait]] ([[User talk:Weatherlawyer|talk]]) 21:40, 26 January 2015 (UTC)<br />
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Can we conclude that the woman is Megan? [[Special:Contributions/67.188.195.182|67.188.195.182]] 21:32, 5 September 2013 (UTC)<br />
:The woman is in fact Megan, I did edit the transcript.--[[User:Dgbrt|Dgbrt]] ([[User talk:Dgbrt|talk]]) 22:38, 5 September 2013 (UTC)<br />
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I found this explanation of the wording helped: "Traits that cause greater reproductive success of an organism are said to be selected for, whereas those that reduce success are selected against." (From http://en.m.wikipedia.org/wiki/Natural_selection) --[[User:Insomniac|Insomniac]] ([[User talk:Insomniac|talk]]) 05:59, 26 September 2013 (UTC)</div>108.162.215.89https://www.explainxkcd.com/wiki/index.php?title=Talk:1464:_Santa&diff=92547Talk:1464: Santa2015-05-06T02:33:07Z<p>108.162.215.89: More comments</p>
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<div>Merry Christmas! --[[User:RenniePet|RenniePet]] ([[User talk:RenniePet|talk]]) 06:29, 24 December 2014 (UTC)<br />
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I think I have done the transcript... [[User:17jiangz1|17jiangz1]] ([[User talk:17jiangz1|talk]]) 06:38, 24 December 2014 (UTC)<br />
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And I did the explanation and put everything in there, I think. Looks like we're already pretty much done! [[Special:Contributions/173.245.56.174|173.245.56.174]] 06:44, 24 December 2014 (UTC)<br />
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I am not sure the word Puportedly fits here: "the omniscience he purportedly possesses". I think Purportedly means the character/caricature is carrying on the idea that.... {{unsigned|Weatherlawyer}}<br />
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It should maybe be mentioned that this comic is a turn of the screw on the many "The physics of Santa" joke articles - The usual chain e-mail or satire web page calculations that take it seriously the logistical calculations for Santa and end up concluding that he should beat the speed of light to deliver the presents. Randall doesn't settle on calculating the logistics for reindeer performance or route planning, he goes a step further and makes the calculations for the refuse. [[Special:Contributions/173.245.49.106|173.245.49.106]] 08:31, 24 December 2014 (UTC)<br />
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Well maybe all those calories are what allows him to move at the insane relativistic speeds needed to visit every house on Christmas eve. [[Special:Contributions/108.162.216.56|108.162.216.56]] 08:47, 24 December 2014 (UTC)<br />
: Exactly. And even if it wasn't, there'd still be a Mr Fusion for the poop. [[Special:Contributions/108.162.250.214|108.162.250.214]] 22:18, 24 December 2014 (UTC)<br />
:There's no need to move at relativistic or trans-light speeds when you can appear in multiple places simultaneously [https://comicdomwrecks.wordpress.com/2010/10/01/countdown-to-fables-100-jiminy-christmas/]. - Equinox [[Special:Contributions/199.27.128.117|199.27.128.117]] 17:41, 24 December 2014 (UTC)<br />
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We have video proof of how Santa handles this! http://youtu.be/b9TTz3R5SmI --[[User:Elipongo|Elipongo]] ([[User talk:Elipongo|talk]]) 09:03, 24 December 2014 (UTC)<br />
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No Christmas cat.? [[User:Halfhat|Halfhat]] ([[User talk:Halfhat|talk]]) 12:12, 24 December 2014 (UTC)<br />
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Santa's metabolic system is perfectly efficient at converting cookies to CO2 hang H2O which is exhaled. He needs to be efficient at converting cookie energy if he plans on being so active. He is also the main source of global warming. [[Special:Contributions/173.245.56.185|173.245.56.185]] 13:11, 24 December 2014 (UTC)<br />
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As a third option, what if Santa doesn't eat the cookies at all, but just redistributes them to other hungry children? --[[Special:Contributions/108.162.217.131|108.162.217.131]] 18:41, 24 December 2014 (UTC)<br />
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What part of the tradition(s) force an entity to eat more than he requires or wants?<br />
Since there is generally supposed to be a chimney involved in his manifestations, why would he have to eat them rather than deal with them in the more traditional way of dealing with things that get sacrificed. Or any other way he chooses. If he has just made romm in his sack, he could easily store a few offerings in it. Why has nobody considered the children?[[User:Weatherlawyer|Weatherlawyer]] ([[User talk:Weatherlawyer|talk]]) 18:13, 5 January 2015 (UTC)<br />
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Perhaps understandably, I first read the line in the explanation as " ... butt off the side of his sleigh." [[User:Miamiclay|Miamiclay]] ([[User talk:Miamiclay|talk]]) 22:27, 24 December 2014 (UTC)<br />
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As a native English speaker, I'd like to say that "few" is relative, and defies any hard limits. A "few" transistors, for me, might mean dozens (as opposed to hundreds or more). A "few"lightyears might be 10 or so. The use of the word "few" can't be analyzed with such hard limits as 2 to 5. [[Special:Contributions/108.162.221.169|108.162.221.169]] 17:47, 25 December 2014 (UTC)<br />
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There are only a few countries where kids leave cookies for Santa (I'm guessing North America and some European countries). That would mean fewer than 100 million kids. [[Special:Contributions/103.22.200.195|103.22.200.195]] 05:36, 26 December 2014 (UTC)<br />
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: 32% of the world population are christians, and even some non-christians celebrate Christmas. [[User:17jiangz1|17jiangz1]] ([[User talk:17jiangz1|talk]]) 09:56, 26 December 2014 (UTC)<br />
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:: Yes, but leaving cookies for Santa is not integral part of christian Christmas celebration, not speaking about non-christian. There are more children who believe Santa is bringing them gifts than children who are leaving cookies for Santa. -- [[User:Hkmaly|Hkmaly]] ([[User talk:Hkmaly|talk]]) 13:53, 27 December 2014 (UTC)<br />
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::: Some children may be leaving cookies "for Santa", but suspect that their parents (or older siblings) are actually consuming them. It may be possible that Santa is not consuming as many cookies as this comic estimates. {{unsigned ip|108.162.237.186}}<br />
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:::: Maybe, but he's gotta be eating most of them. There's no way parents can eat all those cookies in one night! [[Special:Contributions/108.162.221.169|108.162.221.169]] 07:24, 28 December 2014 (UTC)<br />
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::: Cookies aren't (traditionally) left in the UK. Even with the usual translation of "cookie" as "biscuit". A glass of sherry and a mince pie is our variant (at least locally to me), and of course a carrot for Rudolph. Or one or other of whichever of the on-duty reindeer is next due a nibble, which would somewhat mitigate the problem of "carrot throughput" similar to the cookie one, although ultimately mid-air ejection is probably the answer in their case anyway. [[Special:Contributions/141.101.98.191|141.101.98.191]] 20:42, 28 December 2014 (UTC)<br />
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And despite the chill of the upper atmosphere where Santa would be flying, no one has yet made the joke that, even in this post-Cold War era, there's still reason to be concerned about Icy BMs. {{unsigned ip|173.245.52.88}}<br />
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Santa, instead of eating and flushing the cookie excrement, he could just flush the cookies down the toilet. Why do all of the cookies have to go through santa's digestive system? [[Special:Contributions/173.245.56.159|173.245.56.159]] 06:03, 17 January 2015 (UTC)<br />
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What if Santa just converted the mass of the cookies to energy to power his sleigh? [[Special:Contributions/108.162.215.89|108.162.215.89]] 02:33, 6 May 2015 (UTC)</div>108.162.215.89https://www.explainxkcd.com/wiki/index.php?title=647:_Scary&diff=68611647: Scary2014-06-01T02:45:29Z<p>108.162.215.89: /* Explanation */ fixed the English, no change in meaning</p>
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<div>{{comic<br />
| number = 647<br />
| date = October 9, 2009<br />
| title = Scary<br />
| image = scary.png<br />
| titletext = I'm teaching every 8-year-old relative to say this, and every 14-year-old to do the same thing with Toy Story. Also, Pokemon hit the US over a decade ago and kids born after Aladdin came out will turn 18 next year.<br />
}}<br />
<br />
==Explanation==<br />
{{incomplete|The first part is still bad. "Uncle Rob tlks about a lame story" isn't explained as well.}}<br />
No hidden meaning here, but this sure is scary. What's being implied here is that time is moving really quickly and we're getting older faster than we think. Events that seem like they "just happened" have happened long enough ago for a whole other person to come into existence, grow up, and learn to carry on a conversation.<br />
<br />
{{w|9/11}} was a major disaster in 2001 to the United States — maybe also topping the {{w|Attack on Pearl Harbor}} in 1941 — but the boy here knows much about that attack even when he was born after. Nevertheless he states that he can share scary discussions with his adult counterpart even without own experiences on this incident.<br />
<br />
The title text mentions that [[Randall]] is teaching his 8 year old relatives to say the same as in the comic — presumably to the annoyance of his older relatives who will be reminded of the fast passage of time. He does not stop here, but teaches the 14 year old's to say they are born after {{w|Toy Story}} — a major block buster hit from {{w|Pixar}} which came out in 1995. A movie many people will remember fondly and feel just came out the other day... He continues with these scary thoughts by mentioning that {{w|Pokémon}} (1996) came out over a decade ago and that kids born after the big {{w|Disney}} hit movie {{w|Aladdin_(1992_Disney_film)|Aladdin}} from 1992 will turn 18 a year after this comic was published.<br />
<br />
==Transcript==<br />
:[Rob and his nephew are sitting on the ground. Rob is holding a flashlight up to his face.]<br />
:Rob: But they ''never found the ghost's head!''<br />
:Nephew: Lame story, Uncle Rob.<br />
:Rob: And you could do scarier?<br />
:Nephew: Sure.<br />
<br />
:Rob: Try me.<br />
:Nephew: 9/11 happened before I was born, yet I'm old enough to have this conversation with you.<br />
<br />
:[Rob has dropped the flashlight.]<br />
<br />
:[Rob has curled up and wrapped his arms around himself.]<br />
<br />
{{comic discussion}}<br />
[[Category:Comics featuring Rob]]</div>108.162.215.89https://www.explainxkcd.com/wiki/index.php?title=1047:_Approximations&diff=675811047: Approximations2014-05-20T03:06:55Z<p>108.162.215.89: /* Explanation */ added proof for the cosine identity</p>
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<div>{{comic<br />
| number = 1047<br />
| date = April 25, 2012<br />
| title = Approximations<br />
| image = approximations.png<br />
| titletext = Two tips: 1) 8675309 is not just prime, it's a twin prime, and 2) if you ever find yourself raising log(anything)^e or taking the pi-th root of anything, set down the marker and back away from the whiteboard; something has gone horribly wrong.<br />
}}<br />
{{incomplete|The layout is still bad. And the cos(pi/7) + cos(3pi/7) + cos(5pi/7) issue is still not explained. See discussion.}}<br />
==Explanation==<br />
This comic lists some approximations for numbers, most of them mathematical and physical constants. All of them work astonishingly well. There are reoccurring math jokes along the lines of, “3/5 + π/(7 – π) – √2 = 0, but your calculator is probably not good enough to compute this correctly”, which are mainly used to troll geeks.<br />
<br />
Furthermore, there are some useful approximations (which were even more useful in times before calculators) such as “pi is approximately equal to 22/7”.<br />
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[[Randall]] makes fun of both of these, using rather strange approximations (honestly: you may handle 22/7, but who can calculate in a sensible way with 99^8, let alone 30^(pi^e)?) to calculate some constants that are easy enough to handle in the decimal system, and stating such “slightly wrong” trick equations, one of which ''is'' actually correct (which may astonish only those who are not familiar with cosines):<br> cos(π/7)+cos(3π/7)+cos(5π/7)<br>= (cos(π/7)+cos(3π/7)+cos(5π/7))*2sin(π/7)/(2sin(π/7)) <br>= (2cos(π/7)sin(π/7)+2cos(3π/7)sin(π/7)+2cos(5π/7)sin(π/7))/(2sin(π/7)) <br>= ( 2cos(π/7)sin(π/7) + (sin(3π/7)*cos(π/7)+cos(3π/7)sin(π/7)) - (sin(3π/7)*cos(π/7)-cos(3π/7)sin(π/7)) + (sin(5π/7)*cos(π/7)+cos(5π/7)sin(π/7)) - (sin(5π/7)*cos(π/7)-cos(5π/7)sin(π/7)) ) / (2sin(π/7)) <br>= (sin(2π/7)+sin(3π/7+π/7)-sin(3π/7-π/7)+sin(5π/7+π/7)-sin(5π/7-π/7))/(2sin(π/7))<br><br />
= (sin(2π/7)+sin(4π/7)-sin(2π/7)+sin(π-π/7)-sin(4π/7))/(2sin(π/7))<br><br />
= sin(π-π/7)/(2sin(π/7))<br><br />
= (sin(π)cos(-π/7)+cos(π)sin(-π/7))/(2sin(π/7))<br><br />
= (0*cos(-π/7)+(-1*sin(-π/7))/(2sin(π/7))<br><br />
= -sin(-π/7)/(2sin(π/7))<br><br />
= sin(π/7)/(2sin(π/7))<br><br />
= 1/2<br><br><br />
<br />
:{| class="wikitable"<br />
|-<br />
|align="center"|Thing to be approximated:<br />
|align="center"|Formula proposed:<br />
|align="center"|Resulting approximate value:<br />
|align="center"|Correct value:<br />
|align="center"|Discussion:<br />
|-<br />
|align="center"|One light year(m)<br />
|align="center"|99<sup>8</sup><br />
|align="center"|9,227,446,944,279,201<br />
|align="center"|9,460,730,472,580,800 (exact)<br />
|align="left"|99<sup>8</sup> and 69<sup>8</sup> are sexual references.<br />
|-<br />
|align="center"|Earth Surface(m<sup>2</sup>)<br />
|align="center"|69<sup>8</sup><br />
|align="center"|513,798,374,428,641<br />
|align="center"|5.10072*10<sup>14</sup><br />
|align="left"|99<sup>8</sup> and 69<sup>8</sup> are sexual references.<br />
|-<br />
|align="center"|Ocean's volume(m<sup>3</sup>)<br />
|align="center"|9<sup>19</sup><br />
|align="center"|1,350,851,717,672,992,089<br />
|align="center"|1,332*10<sup>18</sup><br />
|align="left"|<br />
|-<br />
|align="center"|Seconds in a year<br />
|align="center"|75<sup>4</sup><br />
|align="center"|31,640,625<br />
|align="center"|31,557,600 (Julian calendar) 31,556,952 (Gregorian calendar)<br />
|align="left"|After this comic was released [[Randall]] got many responses by viewers. So he did add this statement to the top of the comic page:<br />
"Lots of emails mention the physicist favorite, 1 year = pi x 10<sup>7</sup> seconds. 75<sup>4</sup> is a hair more accurate, but it's hard to top 3,141,592's elegance." Pi x 10<sup>7</sup> is nearly equal to 31,415,926.536, and 75<sup>4</sup> is exactly 31,640,625. Randall's elegance belongs to the number pi, but it should be multiplied by the factor of ten.<br><br><br />
Using the traditional definitions that a second is 1/60th of a minute, a minute is 1/60th of an hour, and an hour is 1/24th of a day, a 365-day year is exactly 31,536,000 seconds (the "rent method approximation). Until the calendar was reformed by Pope Gregory, there was one leap year in every four years, making the average year 365.25 days, or 31,557,600. On the current calendar system, there are only 97 leap years in every 400 years, making the average year 365.2425 days, or 31,556,952 seconds. In technical usage, a "second" is now defined based on physical constants, even though the length of a day varies inversely with the changing angular velocity of the earth. To keep the official time synchronized with the rotation of the earth, a "leap second" is occasionally added, resulting in a slightly longer year.<br />
|-<br />
|align="center"|Seconds in a year (rent method)<br />
|align="center"|525,600 x 60<br />
|align="center"|31,536,000<br />
|align="center"|31,557,600 (Julian calendar) 31,556,952 (Gregorian calendar)<br />
|align="left"|“Rent Method” refers to the song “Seasons of Love” from the musical “{{w|Rent (musical)|Rent}}.” The song asks, “How do you measure a year?” One line says “525,600 minutes” while most of the rest of the song suggests the best way to measure a year is moments shared with a loved one.<br />
|-<br />
|align="center"|Age of the universe (seconds)<br />
|align="center"|15<sup>15</sup><br />
|align="center"|437,893,890,380,859,375<br />
|align="center"|4.354±0.012*10<sup>17</sup> (best estimate; exact value unknown)<br />
|align="left"|<br />
|-<br />
|align="center"|Planck's constant<br />
|align="center"|1/(30<sup>π<sup>e</sup></sup>)<br />
|align="center"|6.68499014108082*10<sup>-34</sup> (rounded)<br />
|align="center"|6.62606957*10<sup>-34</sup><br />
|align="left"|Informally, the {{w|Planck constant}} is the smallest action possible in quantum mechanics.<br />
|-<br />
|align="center"|Fine structure constant<br />
|align="center"|1/140<br />
|align="center"|0.00714285717142857171428571, etc. (repeating 71428571)<br />
|align="center"|0.00729735257 (accepted value as of 2011), close to 1/137<br />
|align="left"|The {{w|fine structure constant}} indicates the strength of electromagnetism. It is unitless and around 0.007297, close to 1/137. At one point it was believed to be exactly the reciprocal of 137, and many people have tried to find a simple formula explaining this (with a pinch of {{w|numerology}} thrown in at times), including the infamous {{w|Arthur Eddington|Sir Arthur Adding-One}}.<br />
|-<br />
|align="center"|Fundamental charge<br />
|align="center"|3/(14 * π<sup>π<sup>π</sup></sup>)<br />
|align="center"|1.59895121062716*10<sup>-19</sup> (rounded)<br />
|align="center"|1.602176565*10<sup>-19</sup> (rounded)<br />
|align="left"|<br />
|-<br />
|align="center"|Telephone number for the White House Switchboard<br />
|align="center"|1/<br /><br />
<sup>π</sup>√(e<sup>(1 + <sup>(e-1)</sup>√8</sup>)<br />
|align="center"|.2024561414 (truncated)<br />
|align="center"|2024561414<br />
|align="left"|<br />
|-<br />
|align="center"|Jenny's Constant<br />
|align="center"|(7<sup>(e/1 - 1/e)</sup> - 9) * π<sup>2</sup><br />
|align="center"|867.530901981685 (approximately)<br />
|align="center"|8675309<br />
|align="left"|Jenny's constant comes from Tommy Tutone's tune {{w|867-5309/Jenny}}. The number 8675309 at the title text refers to the song 867-5309/Jenny as mentioned above, causing a fad of people dialing this number and asking for "Jenny". The number is in fact a {{w|twin prime}} because 8675311 is also a prime. Twin primes have always been a subject of interest, because they are comparatively rare, and because it is not yet known whether there are infinitely many of them.<br />
|-<br />
|align="center"|World Population Estimate (billions)<br />
|align="center"|Equivalent to 6+((3/4 Year + 1/4 (Year mod 4) - 1499)/10) billion<br />
|align="center"|2005 6.5<br />
2006 6.6<br />
2007 6.7<br />
2008 6.7<br />
2009 6.8<br />
2010 6.9<br />
2011 7<br />
2012 7<br />
2013 7.1<br />
2014 7.2<br />
2015 7.3<br />
2016 7.3<br />
2017 7.4<br />
2018 7.5<br />
2019 7.6<br />
2020 7.6<br />
2021 7.7<br />
2022 7.8<br />
2023 7.9<br />
2024 7.9<br />
2025 8<br />
2026 8.1<br />
2027 8.2<br />
2028 8.2<br />
2029 8.3<br />
2030 8.4<br />
2031 8.5<br />
2032 8.5<br />
|align="center"|<br />
|align="left"|<br />
|-<br />
|align="center"|U.S. Population Estimate (millions)<br />
|align="center"|Equivalent to 310+3*(Year - 2010) million<br />
|align="center"|2000 280<br />
2001 283<br />
2002 286<br />
2003 289<br />
2004 292<br />
2005 295<br />
2006 298<br />
2007 301<br />
2008 304<br />
2009 307<br />
2010 310<br />
2011 313<br />
2012 316<br />
2013 319<br />
2014 322<br />
2015 325<br />
2016 328<br />
2017 331<br />
2018 334<br />
2019 337<br />
2020 340<br />
2021 343<br />
2022 346<br />
2023 349<br />
2024 352<br />
2025 355<br />
2026 358<br />
2027 361<br />
2028 364<br />
2029 367<br />
2030 370<br />
2031 373<br />
2032 376<br />
|align="center"|<br />
|align="left"|<br />
|-<br />
|align="center"|Electron rest energy<br />
|align="center"|e/7<sup>16</sup> Joules<br />
|align="center"|8.17948276564429*10<sup>-14</sup><br />
|align="center"|8.18710438*10<sup>-14</sup> (rounded)<br />
|align="left"|<br />
|-<br />
|align="center"|Light-year(miles)<br />
|align="center"|2<sup>(42.42)</sup><br />
|align="center"|5884267614436.97 (rounded)<br />
|align="center"|9460730472580800 (meters in a light-year, by definition) / 1609.344 (meters in a mile) = 8212439646337500/1397 (exact) = 5878625373183.61 (rounded)<br />
|align="left"|{{w|42 (number)|42}} is, according to Douglas Adams' ''The Hitchhiker's Guide to the Galaxy'', the Answer to the Ultimate Question of Life, the Universe, and Everything.<br />
|-<br />
|align="center"|sin(60°) = √3/2<br />
|align="center"|e/π<br />
|align="center"|0.8652559794 (rounded)<br />
|align="center"|0.8660254038 (rounded)<br />
|align="left"|<br />
|-<br />
|align="center"|√3<br />
|align="center"|2e/π<br />
|align="center"|1.7305119589 (rounded)<br />
|align="center"|1.7320508076 (rounded)<br />
|align="left"|<br />
|-<br />
|align="center"|gamma(Euler's gamma constant)<br />
|align="center"|1/√3<br />
|align="center"|0.5773502692 (rounded)<br />
|align="center"|0.5772156649015328606065120900824024310421...<br />
|align="left"|In {{w|mathematics}}, the {{w|Euler-Mascheroni constant}} (Euler gamma constant) is a mysterious number describing the relationship between the {{w|Harmonic series (mathematics)|harmonic series}} and the {{w|natural logarithm}}.<br />
|-<br />
|align="center"|Feet in a meter<br />
|align="center"|5/(<sup>e</sup>√π)<br />
|align="center"|3.2815481951<br />
|align="center"|1/.3048 (exact) = 3.280839895 (rounded)<br />
|align="left"|<br />
|-<br />
|align="center"|√5<br />
|align="center"|2/e + 3/2<br />
|align="center"|2.2357588823 (rounded)<br />
|align="center"|2.2360679775 (rounded)<br />
|align="left"|<br />
|-<br />
|align="center"|Avogadro's number<br />
|align="center"|69<sup>π<sup>√5</sup></sup><br />
|align="center"|6.02191201246329*10<sup>23</sup> (rounded)<br />
|align="center"|6.02214129*10<sup>23</sup> (rounded)<br />
|align="left"|<br />
|-<br />
|align="center"|Gravitational constant G<br />
|align="center"|1 / e<sup>(pi - 1)<sup>(pi + 1)</sup></sup><br />
|align="center"|6.67361106850561*10<sup>-11</sup> (rounded)<br />
|align="center"|6.67385*10<sup>-11</sup> (rounded)<br />
|align="left"|The universal {{w|gravitational constant}} G is equal to F*r<sup>2</sup>/Mm, where F is the gravitational force between two objects, r is the distance between them, and M and m are their masses.<br />
|-<br />
|align="center"|R(gas constant)<br />
|align="center"|(e+1) √5<br />
|align="center"|8.3143309279 (rounded)<br />
|align="center"|8.3144622 (rounded)<br />
|align="left"|The {{w|gas constant}} relates energy to temperature in physics, as well as a gas's volume, pressure, temperature and {{w|mole (unit)|molar amount}} (hence the name).<br />
|-<br />
|align="center"|Proton-electron mass ratio<br />
|align="center"|6*π<sup>5</sup><br />
|align="center"|1836.1181087117 (rounded)<br />
|align="center"|1836.15267246 (rounded)<br />
|align="left"|<br />
|-<br />
|align="center"|Liters in a gallon (U.S. liquid gallon, defined by law as 231 cubic inches)<br />
|align="center"|3 + π/4<br />
|align="center"|3.7853981634 (rounded)<br />
|align="center"|3.785411784 (exact)<br />
|align="left"|<br />
|-<br />
|align="center"|''g''<sub>0</sub> or ''g''<sub>n</sub><br />
|align="center"|6 + ln(45)<br />
|align="center"|9.8066624898 (rounded)<br />
|align="center"|9.80665 (standard)<br />
|align="left"|Standard gravity, or standard acceleration due to free fall is the nominal gravitational acceleration of an object in a vacuum near the surface of the Earth. It is defined by standard as 9.80665 m/s2, which is exactly 35.30394 (km/h)/s (about 32.174 ft/s2, or 21.937 mph/s). This value was established by the 3rd CGPM (1901, CR 70) and used to define the standard weight of an object as the product of its mass and this nominal acceleration. The acceleration of a body near the surface of the Earth is due to the combined effects of gravity and centrifugal acceleration from rotation of the Earth (but which is small enough to be neglected for most purposes); the total (the apparent gravity) is about 0.5 percent greater at the poles than at the equator.<br><br>Randall used a letter g without a suffix, which can also mean the local acceleration due to local gravity and centrifugal acceleration, which varies depending on one's position on Earth.<br />
|-<br />
|align="center"|Proton-electron mass ratio<br />
|align="center"|(e<sup>8</sup> - 10) / ϕ<br />
|align="center"|1836.1530151398 (rounded)<br />
|align="center"|1836.15267246 (rounded)<br />
|align="left"|ϕ is the {{w|golden ratio}}, or (1 + √5)/2. It has many interesting geometrical properties.<br />
|-<br />
|align="center"|Ruby laser wavelength<br />
|align="center"|1 / (1200<sup>2</sup>)<br />
|align="center"|.00000069444444444444... (repeating decimal)<br />
|align="center"|694.3 nm<br />
|align="left"|The ruby laser wavelength varies because “ruby” is not clearly defined.<br />
|-<br />
|align="center"|Mean Earth Radius<br />
|align="center"|(5<sup>8</sup>)*6e<br />
|align="center"|2343750e (exact), 6,370,973.035450887 (6370 km, 973 m, 3 cm, 5 mm, 450,887 nm) (rounded)<br />
|align="center"|6,371,008.7 (International Union of Geodesy and Geophysics definition)<br />
|align="left"|The {{w|Earth radius#mean radii|mean earth radius}} varies because there is not one single way to make a sphere out of the earth. Randall's value lies within the actual variation of Earth's radius. The International Union of Geodesy and Geophysics (IUGG) defines the mean radius as 2/3 of the equatorial radius (6,378,137.0 m) plus 1/3 of the polar radius (6,356,752.3 m).<br />
|-<br />
|align="center"|√2<br />
|align="center"|3/5 + π/(7-π)<br />
|align="center"|1.4142200581 (rounded)<br />
|align="center"|1.4142135624 (rounded)<br />
|align="left"|<br />
|-<br />
|align="center"|cos(π/7) + cos(3π/7) + cos(5π/7)<br />
|align="center"|1/2<br />
|align="center"|0.5<br />
|align="center"|0.5 (exact)<br />
|align="left"|The correct equation in the "Pro tip - Not all of these are wrong" section is cos(pi/7) + cos(3pi/7) + cos(5pi/7) = 1/2 as [http://math.stackexchange.com/questions/140388/how-can-one-prove-cos-pi-7-cos3-pi-7-cos5-pi-7-1-2 shown here]. If you're still confused, the functions use {{w|radians}}, not {{w|degrees (angle)|degrees}}.<br />
|-<br />
|align="center"|γ(Euler's gamma constant)<br />
|align="center"|e/3<sup>4</sup> + e/5<br />
|align="center"|0.5772154006 (rounded)<br />
|align="center"|0.5772156649015328606065120900824024310421...<br />
|align="left"|In {{w|mathematics}}, the {{w|Euler-Mascheroni constant}} (Euler gamma constant) is a mysterious number describing the relationship between the {{w|Harmonic series (mathematics)|harmonic series}} and the {{w|natural logarithm}}.<br />
|-<br />
|align="center"|√5<br />
|align="center"|(13 + 4π) / (24 - 4π)<br />
|align="center"|2.2360678094 (rounded)<br />
|align="center"|2.2360679775 (rounded)<br />
|align="left"|<br />
|-<br />
|align="center"|Σ 1/n<sup>n</sup><br />
|align="center"|ln(3)<sup>e</sup><br />
|align="center"|1.2912987577 (rounded)<br />
|align="center"|1.2912859971 (rounded)<br />
|align="left"|<br />
|}<br />
<br />
{{w|Pi}} is a natural constant that arises in describing circles or ellipses. As such, useful as it may be, it doesn't usually occur anywhere in an exponent. When it does, such as with complex numbers, taking the pi-th root is rarely helpful. For example, if we try to derive:<br />
<br />
''e''<sup>π''i''</sup> + 1 = 0<br />
<br />
''e''<sup>π''i''</sup> = -1<br />
<br />
(''e''<sup>''i''</sup>)<sup>π</sup> = -1<br />
<br />
''e''<sup>''i''</sup> = <sup>π</sup>√(-1)<br />
<br />
We get nowhere.<br />
<br />
Same goes for the e-th power: e typically appears in the basis of a power (forming the {{w|exponential function}}), not in the exponent. (This is later referenced in [http://what-if.xkcd.com/73/ Lethal Neutrinos]).<br />
<br />
The software referred to in the comic is [http://mrob.com/pub/ries/ ries], a 'reverse calculator' which forms equations matching a given number.<br />
<br />
==Transcript==<br />
:'''A table of slightly wrong equations and identities useful for approximations and/or trolling teachers.'''<br />
:(Found using a mix of trial-and-error, ''Mathematica'', and Robert Munafo's ''Ries'' tool.)<br />
: All units are SI MKS unless otherwise noted.<br />
<br />
:{| class="wikitable"<br />
|-<br />
|colspan="2" align="center" | Relation:<br />
|align="center" | Accurate to within:<br />
|-<br />
|align="center" | One light year(m)<br />
|align="center" | 99<sup>8</sup><br />
|align="center" | one part in 40<br />
|-<br />
|align="center" | Earth Surface(m<sup>2</sup>)<br />
|align="center" | 69<sup>8</sup><br />
|align="center" | one part in 130<br />
|-<br />
|align="center" | Ocean's volume(m<sup>3</sup>)<br />
|align="center" | 9<sup>19</sup><br />
|align="center" | one part in 70<br />
|-<br />
|align="center" | Seconds in a year<br />
|align="center" | 75<sup>4</sup><br />
|align="center" | one part in 400<br />
|-<br />
|align="center" | Seconds in a year (rent method)<br />
|align="center" | 525,600 x 60<br />
|align="center" | one part in 1400<br />
|-<br />
|align="center" | Age of the universe (seconds)<br />
|align="center" | 15<sup>15</sup><br />
|align="center" | one part in 70<br />
|-<br />
|align="center" | Planck's constant<br />
|align="center" | 1/(30<sup>π<sup>e</sup></sup>)<br />
|align="center" | one part in 110<br />
|-<br />
|align="center" | Fine structure constant<br />
|align="center" | 1/140<br />
|align="center" | [I've had enough of this 137 crap]<br />
|-<br />
|align="center" | Fundamental charge<br />
|align="center" | 3/(14 * π<sup>π<sup>π</sup></sup>)<br />
|align="center" | one part in 500<br />
|-<br />
|align="center"|White House Switchboard<br />
|colspan="2" align="center"|1/<br /><br />
<sup>π</sup>√(e<sup>(1 + <sup>(e-1)</sup>√8</sup>)<br />
|-<br />
|align="center"|Jenny's Constant<br />
|colspan="2" align="center"|(7<sup>(e/1 - 1/e)</sup> - 9) * π<sup>2</sup><br />
|-<br />
|colspan="3" align="center"|Intermission:<br /> World Population Estimate<br /> which should stay current<br /> for a decade or two:<br /><br />
Take the last two digits of the current year<br />
<br />
Example: 20[14] <br />
<br />
Subtract the number of leap years since hurricane Katrina<br />
<br />
Example:14 (minus 2008 and 2012) is 12<br />
<br />
Add a decimal point<br />
<br />
Example: 1.2<br />
<br />
Add 6<br />
<br />
Example: 6 + 1.2<br />
<br />
7.2 = World population in billions.<br />
<br />
Version for US population:<br />
<br />
Example: 20[14]<br />
<br />
Subtract 10<br />
<br />
Example: 4<br />
<br />
Multiply by 3<br />
<br />
Example: 12<br />
<br />
Add 10<br />
<br />
Example: 3[22] million<br />
|-<br />
|align="center"|Electron rest energy<br />
|align="center"|e/7<sup>16</sup> Joules<br />
|align="center"|one part in 1000<br />
|-<br />
|align="center"|Light-year(miles)<br />
|align="center"|2<sup>(42.42)</sup><br />
|align="center"|one part in 1000<br />
|-<br />
|colspan="2" align="center"|sin(60°) = √3/2 = e/π<br />
|align="center"|one part in 1000<br />
|-<br />
|colspan="2" align="center"|√3 = 2e/π<br />
|align="center"|one part in 1000<br />
|-<br />
|align="center"|gamma(Euler's gamma constant)<br />
|align="center"|1/√3<br />
|align="center"|one part in 4000<br />
|-<br />
|align="center"|Feet in a meter<br />
|align="center"|5/(<sup>e</sup>√π)<br />
|align="center"|one part in 4000<br />
|-<br />
|colspan="2" align="center"|√5 = 2/e + 3/2<br />
|align="center"|one part in 7000<br />
|-<br />
|align="center"|Avogadro's number<br />
|align="center"|69<sup>π<sup>√5</sup></sup><br />
|align="center"|one part in 25,000<br />
|-<br />
|align="center"|Gravitational constant G<br />
|align="center"|1 / e<sup>(pi - 1)<sup>(pi + 1)</sup></sup><br />
|align="center"|one part in 25,000<br />
|-<br />
|align="center"|R(gas constant)<br />
|align="center"|(e+1) √5<br />
|align="center"|one part in 50,000<br />
|-<br />
|align="center"|Proton-electron mass ratio<br />
|align="center"|6*π<sup>5</sup><br />
|align="center"|one part in 50,000<br />
|-<br />
|align="center"|Liters in a gallon<br />
|align="center"|3 + π/4<br />
|align="center"|one part in 500,000<br />
|-<br />
|align="center"|g<br />
|align="center"|6 + ln(45)<br />
|align="center"|one part in 750,000<br />
|-<br />
|align="center"|Proton-electron mass ratio<br />
|align="center"|(e<sup>8</sup> - 10) / ϕ<br />
|align="center"|one part in 5,000,000<br />
|-<br />
|align="center"|Ruby laser wavelength<br />
|align="center"|1 / (1200<sup>2</sup>)<br />
|align="center"|[within actual variation]<br />
|-<br />
|align="center"|Mean Earth Radius<br />
|align="center"|(5<sup>8</sup>)*6e<br />
|align="center"|[within actual variation]<br />
|-<br />
|colspan="3" align="center"|Protip - not all of these are wrong:<br />
|-<br />
|colspan="2" align="center"|√2 = 3/5 + π/(7-π)<br />
|align="center"|cos(π/7) + cos(3π/7) + cos(5π/7) = 1/2<br />
|-<br />
|align="center"|γ(Euler's gamma constant) = e/3<sup>4</sup> + e/5<br />
|align="center"|√5 = (13 + 4π) / (24 - 4π)<br />
|align="center"|Σ 1/n<sup>n</sup> = ln(3)<sup>e</sup><br />
|}<br />
<br />
{{comic discussion}}<br />
[[Category:Charts]]<br />
[[Category:Math]]<br />
[[Category:Physics]]<br />
[[Category:Protip]]</div>108.162.215.89https://www.explainxkcd.com/wiki/index.php?title=Talk:1047:_Approximations&diff=67580Talk:1047: Approximations2014-05-20T02:34:43Z<p>108.162.215.89: added: Dgbrt, yes, sin(6π/7)=sin(π/7). Simple proof: sin(6π/7)=sin(π-π/7)=sin(π)cos(-π/7)+cos(π)sin(-π/7)=0*cos(-π/7)+(-1)*(-sin(π/7))=0+sin(π/7)=sin(π/7)</p>
<hr />
<div>They're actually quite accurate. I've used these in calculations, and they seem to give close enough answers. '''[[User:Davidy22|<span title="I want you."><u><font color="purple" size="2px">David</font><font color="green" size="3px">y</font></u><sup><font color="indigo" size="1px">22</font></sup></span>]]'''[[User talk:Davidy22|<tt>[talk]</tt>]] 14:03, 8 January 2013 (UTC)<br />
<br />
I only see a use for the liters in a gallon one. The rest are for trolling or simple amusement. The cosine identity bit our math team in the butt at a competition. It was painful. --[[User:Quicksilver|Quicksilver]] ([[User talk:Quicksilver|talk]]) 05:27, 17 August 2013 (UTC)<br />
<br />
Annoyingly this explanation does not cover 42 properly, it does not say that Douglas Adams got the number 42 from Lewis Carroll, who is more relevant to the page because he was a mathematician named Charles Lutwidge Dodgson. He was obsessed with the number forty-two. The original plate illustrations of Alice in Wonderland drawn by him numbered forty-two. Rule Forty-Two in Alice in Wonderland is "All persons more than a mile high to leave the court", There is also a Code of Honour in the preface of The Hunting of the Snark, an extremely long poem written by him when he was 42 years old, in which rule forty-two is "No one shall speak to the Man at the Helm". The queens in Alice Through the Looking Glass the White Queen announces her age as "one hundred and one, five months and a day", which - if the best possible date is assumed for the action of Through the Looking-Glass - gives a total of 37,044 days. With the further (textually unconfirmed) assumption that both Queens were born on the same day their combined age becomes 74,088 days, which is 42 x 42 x 42. --[[Special:Contributions/139.216.242.254|139.216.242.254]] 02:43, 29 August 2013 (UTC)<br />
<br />
<br />
:: This explanation covers 42 adequately, and would probably be made slightly worse if such information were added. The very widely known cultural reference is to Adams's interpretation, not Dodgson's original obsession. Adding it would be akin to introducing the MPLM into the explanation for the hijacking of Renaissance artists' names by the TMNT. I definitely concede that it does not cover 42 exhaustively, but I think it can be considered complete and in working order without such an addition. If it really irks you, be bold and add it! --[[User:Quicksilver|Quicksilver]] ([[User talk:Quicksilver|talk]]) 00:37, 30 August 2013 (UTC)<br />
<br />
"sqrt(2) is not even algebraic in the quotient field of Z[pi]" is not correct. Q is part of the quotient field of Z[pi] and sqrt(2) is algebraic of it. The needed facts are that pi is not algebraic, but the formula implies it is in Q(sqrt(2)). --DrMath 06:47, 7 September 2013 (UTC)<br />
<br />
13/15 is a better approximation to sqrt(3)/2 than is e/pi. Continued fraction approximations are great! --DrMath 07:23, 7 September 2013 (UTC)<br />
<br />
<br />
How could he forget 1 gallon ≈ 0.1337 ft³?! [[Special:Contributions/67.188.195.182|67.188.195.182]] 00:51, 8 September 2013 (UTC)<br />
<br />
Worth mentioning that Wolfram Alpha now officially recognizes the [http://www.wolframalpha.com/input/?i=e%5E-%28%281%2B8%5E%281%2F%28e-1%29%29%29%5E%281%2Fpi%29%29 White House switchboard constant] and the [http://www.wolframalpha.com/input/?i=%287%5E%28e-1%2Fe%29-9%29*pi%5E2 Jenny constant]. [[Special:Contributions/86.164.243.91|86.164.243.91]] 18:28, 8 October 2013 (UTC)<br />
<br />
Maybe we should add the [Extension:LaTeXSVG LaTeX extension] to make it easier to transcribe these equations. -- [[Special:Contributions/108.162.219.220|108.162.219.220]] 23:02, 16 December 2013 (UTC)<br />
<br />
<br />
;Protip - Does anyone see the correct equation?<br />
Maybe this is just an other Wolfram Alpha error, like we recently have had here: [[1292: Pi vs. Tau]]. All equations still look invalid to me.<br />
*''√2 = 3/5 + π/(7-π)'': is impossible because √2 is an irrational number and no equation can match.<br />
*''cos(π/7) + cos(3π/7) + cos(5π/7) = 1/2'': could only match if ''cos(x) + cos(3x) + cos(5x) = 1/2'' would be valid, because ''π/7'' is also an irrational number.<br />
*''γ = e/3<sup>4</sup> + e/5 or γ = e/54 + e/5'': would mean that a sum of two irrational numbers do fit to the Gamma Constant. Impossible.<br />
*''√5 = 13 + 4π / 24 - 4π'': √5 and π are irrational numbers, there is no way to match them in any equation like this.<br />
*''Σ 1/n<sup>n</sup> = ln(3)<sup>e</sup>'': doesn't make any sense either.<br />
Maybe [[:Category:Comics featuring Miss Lenhart|Miss Lenhart]] can help.<br />
--[[User:Dgbrt|Dgbrt]] ([[User talk:Dgbrt|talk]]) 21:41, 17 December 2013 (UTC)<br />
<br />
cos(π/7) + cos(3π/7) + cos(5π/7) = 1/2 is exactly correct. <br />
<br />
Let a=π/7, b=3π/7, and c=5π/7, then <br />
(cosa+cosb+cosc)⋅2sina=2cosasina+2cosbsina+2coscsina=sin2a+sin(b+a)−sin(b−a)+sin(c+a)−sin(c−a)=sin(2π/7)+sin(4π/7)−sin(2π/7)+sin(6π/7)−sin(4π/7)=sin(6π/7)=sin(π/7)=sina<br />
<br />
Hence, cos(π/7) + cos(3π/7) + cos(5π/7) = sin(π/7) / 2sin(π/7) = 1/2<br />
[[Special:Contributions/108.162.216.74|108.162.216.74]] 01:57, 16 January 2014 (UTC)<br />
<br />
:What is this: sin(6π/7)=sin(π/7) ? A new math is born... --[[User:Dgbrt|Dgbrt]] ([[User talk:Dgbrt|talk]]) 20:49, 16 January 2014 (UTC)<br />
<br />
::Actually it does. My proof is geometric: the sines of two supplementary angles (angle a + angle b = π (in radians)) are equivalent because they necessarily have the same x height in a Cartesian plane. Look on a unit circle, or even a sine function. Also, Calculus and most other mathematics use radians over degrees because they make the functions simpler and eliminate irrationality when a trig function shows up, but physics uses degrees because it's easier to understand and taught first. Anonymous 01:27, 13 February 2014 (UTC)<br />
::As an aside, just how far along in math are you? Radian measure is taught in high school (at least the good ones). Anonymous 13:24, 13 February 2014 (UTC)<br />
:::Sure, I was wrong at my last statement. sin(6π/7)=sin(π/7) is correct by using the radian measure. But just change π/7 to π/77 would give a very different result on that formular here. I still can't figure out why PI divided by the number 7 should be that unique, PI divided by 77 should be the same. My fault is: I still can't find the Nerd Sniping here. And we all do know that Randall did use wrong WolframAlpha results here. According to the last question: I'm very well on Math, that's because I want to understand this. This is like 0.999=1. --[[User:Dgbrt|Dgbrt]] ([[User talk:Dgbrt|talk]]) 22:01, 13 February 2014 (UTC)<br />
::::Ah, I see. I think it has to do with the way e^i*π breaks down, as one of the answers shown in the corresponding link explains, but other answers rely on various angle identities (including the supplementary sines one in the proof above). Anonymous 03:10, 14 February 2014 (UTC) (PS, have you checked [[545]] lately? I answered your question there, too)<br />
:::::As per the derivation from january 16 , you can use any a,b,c that satisfies this set of equations: 2 a = b - a, a + b = c- a, c + a = π - a. This is due to the fact that sin(x) = sin(π-x), and what was derived the 16th. [[Special:Contributions/173.245.53.199|173.245.53.199]] 12:38, 21 February 2014 (UTC)<br />
:::: Dgbrt: If not convinced by the proofs linked to in the "explanation" part, you might want to try this: [http://www.wolframalpha.com/input/?i=sum_%28k%3D0%29%5E38+cos%281%2F77+%282+k%2B1%29+pi%29]. I'm sure you'll find inspiration for similar formulas using PI over [any odd integer]. Your assumption that Randall used WolframAlpha for this very identity is probably wrong. This is a very well-known formula that appears in many high school books, and I am pretty sure it is part of Randall's culture. And this has nothing to do with 1=1. As for your original post,<br />
::::*√2 = (√2-1)/((4-2)π/2-π)+1 : Is this what you call "matching an equation" to √2?<br />
::::*So what you mean is that if an equation is true for an irrational number, then it must be for any real number? Like, (√2)^2 = 2, but because √2 is irrational, then x^2=2 (for all x?)<br />
::::*This one's a bit tough. You will probably agree that γ-√2 is irrational. And so is √2. What about their sum?<br />
::::*Well, maybe it doesn't to you. But is Σ n<sup>-2</sup> = π^2/6 any better? Well, this one is true (using Fourier's expansion of the rectangular function). <br />
::::Finally,<br />
::::*√2 = 3/5 + π/(7-π) is false because it would imply that π is an algebraic number<br />
::::*cos(π/7) + cos(3π/7) + cos(5π/7) = 1/2 is true, and proven by many<br />
::::*γ = e/3<sup>4</sup> + e/5 seems false. But there doesn't seem to be a quick way to disprove.<br />
::::*Σ 1/n<sup>n</sup> = ln(3)<sup>e</sup> seems false, but I can't see why. [[Special:Contributions/108.162.210.234|108.162.210.234]] 09:15, 11 May 2014 (UTC)<br />
::Dgbrt, yes, sin(6π/7)=sin(π/7). Simple proof: sin(6π/7)=sin(π-π/7)=sin(π)cos(-π/7)+cos(π)sin(-π/7)=0*cos(-π/7)+(-1)*(-sin(π/7))=0+sin(π/7)=sin(π/7) [[Special:Contributions/108.162.215.89|108.162.215.89]] 02:34, 20 May 2014 (UTC) <br />
;So, still incomplete?<br />
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Where's our (in)complete judge? [[Special:Contributions/199.27.128.186|199.27.128.186]] 19:21, 18 December 2013 (UTC)<br />
:The protip is still a mystery. I'm calling for help a few lines above. --[[User:Dgbrt|Dgbrt]] ([[User talk:Dgbrt|talk]]) 21:16, 18 December 2013 (UTC)<br />
::The cosine one, in radians, is correct [[Special:Contributions/141.101.88.225|141.101.88.225]] 12:54, 28 April 2014 (UTC)<br />
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The 'Seconds in a year' ones remind me of one of my favorite quotes: "How many seconds are there in a year? If I tell you there are 3.155 x 10^7, you won't even try to remember it. On the other hand, who could forget that, to within half a percent, pi seconds is a nanocentury" -- Tom Duff, Bell Labs. [[User:Beolach|Beolach]] ([[User talk:Beolach|talk]]) 19:14, 17 April 2014 (UTC)<br />
:Please do not change former discussions. --[[User:Dgbrt|Dgbrt]] ([[User talk:Dgbrt|talk]]) 23:57, 17 April 2014 (UTC)<br />
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;cos(pi/7) + cos(3pi/7) + cos(5pi/7) = 1/2 ???<br />
Why the hell the divider seven makes the difference?<br />
*cos(pi) + cos(3*pi) + cos(5*pi) = -3<br />
*cos(pi/8) + cos(3*pi/8) + cos(5*pi/8) = 0.92387953251128675612818318939678828682241662586364...<br />
So why the "magic" prime number seven produces this exact result? I know radians and π/7 is just a small part of a circle which is 2π. One prove claims that sin(6π/7) equals to sin(π/7); my best calculator can't show a difference. Of course sin(6π) equals to sin(π), in radians, BUT sin(6π/8) is NOT equal to sin(π/8). So if the number 7 plays a magic rule here this would be "one of the", no... the BIGGEST mystery in mathematics forever. --[[User:Dgbrt|Dgbrt]] ([[User talk:Dgbrt|talk]]) 23:03, 16 May 2014 (UTC)<br />
:Dgbrt, please see my answer from 11 May 2014 up there. Any odd integer will do, as long as you sum enough of cos(pi/[thing]). <br />
:*Let's try with 5 : cos(pi/5) + cos (3pi/5) = 1/2.<br />
:*With 9 : cos(pi/9)+ cos(3pi/9) + cos (5pi/9) + cos(7pi/9) = 1/2<br />
: No big mystery around here. Just a beautiful formula :) I think there are similar formulas with cosines and even integers. I'll post them here if I have time. [[User:Varal7|Varal7]] ([[User talk:Varal7|talk]]) 09:56, 17 May 2014 (UTC)<br />
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::You mixing up to different equations and even not prove them. If there is any prove to a mathematician I would accept and include a proper explain for non math people here. We still have to find a prove. And I do not trust my calculators, we just have to explain why even cos(pi/5) + cos (3pi/5) is also nearly the same. This issue is still not explained. So please give us a explain. And a PROTIP: This does not work with Integers, PI is infinite--[[User:Dgbrt|Dgbrt]] ([[User talk:Dgbrt|talk]]) 17:55, 17 May 2014 (UTC)<br />
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:::Okay. If I understood what you said.<br />
:::* I mix up different topics. -> True. From now on, we'll just focus on the cosine one.<br />
:::* You ask for a proof/explanation. -> My opinion is those are two different requests. Maybe that's why you use the distinction between math people/not math people. For a proof, please read further. What I exposed above are just other "fun experiments" we could do. e.g : [http://www.wolframalpha.com/input/?i=cos%28pi%2F11%29%2Bcos+%283pi%2F11%29+%2B+cos+%285pi%2F11%29+%2B+cos+%287pi%2F11%29%2Bcos+%289pi%2F11%29].<br />
:::* You do not trust your calculators -> Great. I don't either. (Well more accurately, I trust mine to 10^-8, so I would definitely not use it to prove any of the discussed equations in PROTIP). That's why we'll prove the formulas we assert.<br />
:::* "This does not work with integers" -> Well, I got myself misunderstood. It would probably have been better if I had said: the following formula is true for all integer n. sum_{k=0}^{n-1}{cos((2k+1)*pi/(2n+1)). But It's harder to read, so just say. Choose any odd integer, say N=2n+1. Then start the following sum. cos(pi/N) + cos(3pi/N) + … and stop when the numerator is cos((N-2)pi/N). Then the result is 1/2. And that's what we'll prove, a few lines down from here.<br />
:::*"Pi is infinite" -> That's a common misconception. What you mean is, Pi is irrational. (Fun fact: Pi is a transcendental number. Quite difficult theorem. Lindeman proved it in 1882. Hence, if we identify the real number x with the Q-vector space Q[x], it would make sense to say that "x is infinite" because, the Q-vector space Q[x] is indeed of infinite dimension. But then, that's not what mathematicians do). I think Vi Hart made a video where she addresses this issue (or was it someone else?). Anyway, I might come to that point some other time in the future.<br />
:::Okay, so now let's first prove the protip formula. Well first, here is the link that the explainxkcd wiki points to: [http://math.stackexchange.com/questions/140388/how-can-one-prove-cos-pi-7-cos3-pi-7-cos5-pi-7-1-2]. Most of them are correct. Some are more ugly than others. I'll adapt the last one.<br />
::: We need a [http://en.wikipedia.org/wiki/Complex_number complex numbers]. (I choosed this because I think explainxkcd readers are fine with this. See comic [http://explainxkcd.com/179/ 179]). I will be using dots to show the steps of my proof. Please allow me an extra level of indent for clarity's sake.<br />
:::'''Proof'''<br />
:::: *Let z be a primitive 14-th [http://en.wikipedia.org/wiki/Root_of_unity root of unity] (the reader doesn't need to understand the 3 last words). Just say z = exp(i*pi/7) = cos(pi/7) + i sin(pi/7). Using [http://en.wikipedia.org/wiki/Euler%27s_formula Euler's formula].<br />
:::: *We have z^14-1 = (exp(i*pi/7))^14-1 = exp(i*2pi) - 1 = 0. Using exponential law for integer powers, as seen in this article: [http://en.wikipedia.org/wiki/De_Moivre%27s_formula De Moivre's formula].<br />
:::: *Now let's factor: z^14-1 = (z^7-1)(z^7+1) = (z^7-1)(z+1)(z^6-z^5+z^4-z^3+z^2-z+1) = (z^7-1)(z+1)*Phi_14(z). where Phi_14(X)= X^6-X^5+X^4-X^3+X^2-X+1, (see [http://en.wikipedia.org/wiki/Cyclotomic_polynomial cycltomic polynomial]). Now, because z^7-1 = (exp(i*pi/7))^7-1 = exp(i*pi)-1 = -2. And because z is not -1, the two first factors are not 0 so, Phi_14(z) = 0, which is already a pretty awesome equality.<br />
:::: *Note that exp(i*pi/7)*exp(i*6pi/7)= exp(i*pi)=-1. So the inverse of z is -exp(i*6pi/7). But we also know that it is exp(-i*pi/7). Well. That was just a fancy way to prove that exp(-i*pi/7) = - exp(i*6pi/7). Good enough. The same holds for exp(-i*3pi/7) = exp(i*14pi/7)*exp(-i*3pi/7)=exp(i*11pi/7)=exp(i*7pi/7)*exp(i*4pi/7)=-exp(i*4pi/7). And the exact same calculation shows that exp(-i*5pi/7)=-exp(i*2pi/7). Alright.<br />
:::: *Now, use that for any x, we have cos(x) = (exp(ix)+exp(-ix))/2. See [http://en.wikipedia.org/wiki/Euler%27s_formula#Relationship_to_trigonometry here]. Let's calculate twice the sum of the left hand side. 2(cos(pi/7)+cos(3pi/7)+cos(5pi/7))= exp(i*pi/7) + expi(-i*pi/7) + exp(3pi/7) + exp(-3pi/7) + exp(5pi/7) +exp(-5pi/7) = exp(i*pi/7)-exp(i*2pi/7)+exp(i*3pi/7)-exp(i*4pi/7)+exp(i*5pi/7)-exp(i*6pi/7) = -Phi_14(z) +1 = 1.<br />
:::: * So dividing both sides by 2, we get what we want. Pfew. <br />
::: '''Why is 7 so special? Well it isn't.''' Let's prove it for 9. <br />
::::* Let z = exp(i*pi/9) = cos(pi/9) + i sin(pi/9). We have z^18-1 = 0, and z^9-1 and z+1 are not 0, so using the same factorisation, Phi_18(z) = z^8-z^7+z^6-z^5+z^4-z^3+z^2-z+1 = 0. <br />
::::* Hence, the conclusion follow from: 2(cos(pi/9) + cos(3pi/9) + cos(5pi/9) + cos(7pi/9)) = exp(i*pi/9) + exp(-i*pi/9) + exp(i*3pi/9) + exp(-i*3pi/9) + exp(i*5pi/9) + exp(-i*5pi/9) + exp(i*7pi/9) + exp(-i*7pi/9) = -Phi_18(z)+1 = 1. <br />
::: Well, well. I hope you kinda see the pattern. Dgbrt, I know you hate typos, and I'm pretty sure that in this long text lay many of them. So I apologize, and I will correct them later. The following paragraph was posted after I started my text but before I finished mine. It wasn't signed so I will just leave it down there. It's another valid straightforward proof. Oh. And Friendly TIP: Don't say protip when you're not pro. [[User:Varal7|Varal7]] ([[User talk:Varal7|talk]]) 21:50, 17 May 2014 (UTC)<br />
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The valid identity cos(pi/7)+cos(3pi/7)+cos(5pi/7)=1/2 was correctly proved by the writer at 108.162.216.74 above. For a different proof, consider the complex number z = cos(pi/7)+i sin(pi/7) corresponding to rotation of the complex plane by pi/7 radians, i.e., 1/14th of a full rotation. It satisfies z^{14} -1 = 0 (z to the fourteenth is one). Dividing by z-1 gives z^{13} + z^{12} + ... + z + 1 = 0. The same argument, starting with z^2 corresponding to 1/7th of a full rotation, gives z^{12} + z^{10} + ... z^2 + 1 = 0. Taking the difference, we get z^{13} + z^{11} + ... + z^3 + z = 0. Looking only at the real parts, we get cos(13pi/7) + cos(11pi/7) + cos(9pi/7) + cos(7pi/7) + cos(5pi/7) + cos(3pi/7) + cos(pi/7) = 0. Here cos(13pi/7) = cos(pi/7), cos(11pi/7) = cos(3pi/7) and cos(9pi/7) = cos(5pi/7), since cos is even and 2pi-periodic. Finally cos(7pi/7) = -1, so 2(cos(pi/7) + cos(3pi/7) + cos(5pi/7)) - 1 = 0, which you can rewrite as the desired identity. All of this can be clearly visualized using a regular 14-gon, so a proof with pictures is possible. {{unsigned ip|141.101.81.216}}<br />
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;99 is sexual reference?<br />
<br />
In first explanation it says: "99^8 and 69^8 are sexual references". 69 I understand, but what would 99 refer too? <br />
--[[Special:Contributions/173.245.53.167|173.245.53.167]] 17:38, 18 May 2014 (UTC)</div>108.162.215.89https://www.explainxkcd.com/wiki/index.php?title=1047:_Approximations&diff=675311047: Approximations2014-05-18T22:54:10Z<p>108.162.215.89: /* Transcript */ transcript showed cube root of 2, should be square root of 3, divided by 2.; made an "o" lowercase</p>
<hr />
<div>{{comic<br />
| number = 1047<br />
| date = April 25, 2012<br />
| title = Approximations<br />
| image = approximations.png<br />
| titletext = Two tips: 1) 8675309 is not just prime, it's a twin prime, and 2) if you ever find yourself raising log(anything)^e or taking the pi-th root of anything, set down the marker and back away from the whiteboard; something has gone horribly wrong.<br />
}}<br />
{{incomplete|The layout is still bad. And the cos(pi/7) + cos(3pi/7) + cos(5pi/7) issue is still not explained. See discussion.}}<br />
==Explanation==<br />
This comic lists some approximations for numbers, most of them mathematical and physical constants. All of them work astonishingly well. There are reoccurring math jokes along the lines of, “3/5 + π/(7 – π) – √2 = 0, but your calculator is probably not good enough to compute this correctly”, which are mainly used to troll geeks.<br />
<br />
Furthermore, there are some useful approximations (which were even more useful in times before calculators) such as “pi is approximately equal to 22/7”.<br />
<br />
[[Randall]] makes fun of both of these, using rather strange approximations (honestly: you may handle 22/7, but who can calculate in a sensible way with 99^8, let alone 30^(pi^e)?) to calculate some constants that are easy enough to handle in the decimal system, and stating such “slightly wrong” trick equations, one of which ''is'' actually correct (which may astonish only those who are not familiar with cosines).<br />
<br />
<br />
:{| class="wikitable"<br />
|-<br />
|align="center"|Thing to be approximated:<br />
|align="center"|Formula proposed:<br />
|align="center"|Resulting approximate value:<br />
|align="center"|Correct value:<br />
|align="center"|Discussion:<br />
|-<br />
|align="center"|One light year(m)<br />
|align="center"|99<sup>8</sup><br />
|align="center"|9,227,446,944,279,201<br />
|align="center"|9,460,730,472,580,800 (exact)<br />
|align="left"|99<sup>8</sup> and 69<sup>8</sup> are sexual references.<br />
|-<br />
|align="center"|Earth Surface(m<sup>2</sup>)<br />
|align="center"|69<sup>8</sup><br />
|align="center"|513,798,374,428,641<br />
|align="center"|5.10072*10<sup>14</sup><br />
|align="left"|99<sup>8</sup> and 69<sup>8</sup> are sexual references.<br />
|-<br />
|align="center"|Ocean's volume(m<sup>3</sup>)<br />
|align="center"|9<sup>19</sup><br />
|align="center"|1,350,851,717,672,992,089<br />
|align="center"|1,332*10<sup>18</sup><br />
|align="left"|<br />
|-<br />
|align="center"|Seconds in a year<br />
|align="center"|75<sup>4</sup><br />
|align="center"|31,640,625<br />
|align="center"|31,557,600 (Julian calendar) 31,556,952 (Gregorian calendar)<br />
|align="left"|After this comic was released [[Randall]] got many responses by viewers. So he did add this statement to the top of the comic page:<br />
"Lots of emails mention the physicist favorite, 1 year = pi x 10<sup>7</sup> seconds. 75<sup>4</sup> is a hair more accurate, but it's hard to top 3,141,592's elegance." Pi x 10<sup>7</sup> is nearly equal to 31,415,926.536, and 75<sup>4</sup> is exactly 31,640,625. Randall's elegance belongs to the number pi, but it should be multiplied by the factor of ten.<br><br><br />
Using the traditional definitions that a second is 1/60th of a minute, a minute is 1/60th of an hour, and an hour is 1/24th of a day, a 365-day year is exactly 31,536,000 seconds (the "rent method approximation). Until the calendar was reformed by Pope Gregory, there was one leap year in every four years, making the average year 365.25 days, or 31,557,600. On the current calendar system, there are only 97 leap years in every 400 years, making the average year 365.2425 days, or 31,556,952 seconds. In technical usage, a "second" is now defined based on physical constants, even though the length of a day varies inversely with the changing angular velocity of the earth. To keep the official time synchronized with the rotation of the earth, a "leap second" is occasionally added, resulting in a slightly longer year.<br />
|-<br />
|align="center"|Seconds in a year (rent method)<br />
|align="center"|525,600 x 60<br />
|align="center"|31,536,000<br />
|align="center"|31,557,600 (Julian calendar) 31,556,952 (Gregorian calendar)<br />
|align="left"|“Rent Method” refers to the song “Seasons of Love” from the musical “{{w|Rent (musical)|Rent}}.” The song asks, “How do you measure a year?” One line says “525,600 minutes” while most of the rest of the song suggests the best way to measure a year is moments shared with a loved one.<br />
|-<br />
|align="center"|Age of the universe (seconds)<br />
|align="center"|15<sup>15</sup><br />
|align="center"|437,893,890,380,859,375<br />
|align="center"|4.354±0.012*10<sup>17</sup> (best estimate; exact value unknown)<br />
|align="left"|<br />
|-<br />
|align="center"|Planck's constant<br />
|align="center"|1/(30<sup>π<sup>e</sup></sup>)<br />
|align="center"|6.68499014108082*10<sup>-34</sup> (rounded)<br />
|align="center"|6.62606957*10<sup>-34</sup><br />
|align="left"|Informally, the {{w|Planck constant}} is the smallest action possible in quantum mechanics.<br />
|-<br />
|align="center"|Fine structure constant<br />
|align="center"|1/140<br />
|align="center"|0.00714285717142857171428571, etc. (repeating 71428571)<br />
|align="center"|0.00729735257 (accepted value as of 2011), close to 1/137<br />
|align="left"|The {{w|fine structure constant}} indicates the strength of electromagnetism. It is unitless and around 0.007297, close to 1/137. At one point it was believed to be exactly the reciprocal of 137, and many people have tried to find a simple formula explaining this (with a pinch of {{w|numerology}} thrown in at times), including the infamous {{w|Arthur Eddington|Sir Arthur Adding-One}}.<br />
|-<br />
|align="center"|Fundamental charge<br />
|align="center"|3/(14 * π<sup>π<sup>π</sup></sup>)<br />
|align="center"|1.59895121062716*10<sup>-19</sup> (rounded)<br />
|align="center"|1.602176565*10<sup>-19</sup> (rounded)<br />
|align="left"|<br />
|-<br />
|align="center"|Telephone number for the White House Switchboard<br />
|align="center"|1/<br /><br />
<sup>π</sup>√(e<sup>(1 + <sup>(e-1)</sup>√8</sup>)<br />
|align="center"|.2024561414 (truncated)<br />
|align="center"|2024561414<br />
|align="left"|<br />
|-<br />
|align="center"|Jenny's Constant<br />
|align="center"|(7<sup>(e/1 - 1/e)</sup> - 9) * π<sup>2</sup><br />
|align="center"|867.530901981685 (approximately)<br />
|align="center"|8675309<br />
|align="left"|Jenny's constant comes from Tommy Tutone's tune {{w|867-5309/Jenny}}. The number 8675309 at the title text refers to the song 867-5309/Jenny as mentioned above, causing a fad of people dialing this number and asking for "Jenny". The number is in fact a {{w|twin prime}} because 8675311 is also a prime. Twin primes have always been a subject of interest, because they are comparatively rare, and because it is not yet known whether there are infinitely many of them.<br />
|-<br />
|align="center"|World Population Estimate (billions)<br />
|align="center"|Equivalent to 6+((3/4 Year + 1/4 (Year mod 4) - 1499)/10) billion<br />
|align="center"|2005 6.5<br />
2006 6.6<br />
2007 6.7<br />
2008 6.7<br />
2009 6.8<br />
2010 6.9<br />
2011 7<br />
2012 7<br />
2013 7.1<br />
2014 7.2<br />
2015 7.3<br />
2016 7.3<br />
2017 7.4<br />
2018 7.5<br />
2019 7.6<br />
2020 7.6<br />
2021 7.7<br />
2022 7.8<br />
2023 7.9<br />
2024 7.9<br />
2025 8<br />
2026 8.1<br />
2027 8.2<br />
2028 8.2<br />
2029 8.3<br />
2030 8.4<br />
2031 8.5<br />
2032 8.5<br />
|align="center"|<br />
|align="left"|<br />
|-<br />
|align="center"|U.S. Population Estimate (millions)<br />
|align="center"|Equivalent to 310+3*(Year - 2010) million<br />
|align="center"|2000 280<br />
2001 283<br />
2002 286<br />
2003 289<br />
2004 292<br />
2005 295<br />
2006 298<br />
2007 301<br />
2008 304<br />
2009 307<br />
2010 310<br />
2011 313<br />
2012 316<br />
2013 319<br />
2014 322<br />
2015 325<br />
2016 328<br />
2017 331<br />
2018 334<br />
2019 337<br />
2020 340<br />
2021 343<br />
2022 346<br />
2023 349<br />
2024 352<br />
2025 355<br />
2026 358<br />
2027 361<br />
2028 364<br />
2029 367<br />
2030 370<br />
2031 373<br />
2032 376<br />
|align="center"|<br />
|align="left"|<br />
|-<br />
|align="center"|Electron rest energy<br />
|align="center"|e/7<sup>16</sup> Joules<br />
|align="center"|8.17948276564429*10<sup>-14</sup><br />
|align="center"|8.18710438*10<sup>-14</sup> (rounded)<br />
|align="left"|<br />
|-<br />
|align="center"|Light-year(miles)<br />
|align="center"|2<sup>(42.42)</sup><br />
|align="center"|5884267614436.97 (rounded)<br />
|align="center"|9460730472580800 (meters in a light-year, by definition) / 1609.344 (meters in a mile) = 8212439646337500/1397 (exact) = 5878625373183.61 (rounded)<br />
|align="left"|{{w|42 (number)|42}} is, according to Douglas Adams' ''The Hitchhiker's Guide to the Galaxy'', the Answer to the Ultimate Question of Life, the Universe, and Everything.<br />
|-<br />
|align="center"|sin(60°) = √3/2<br />
|align="center"|e/π<br />
|align="center"|0.8652559794 (rounded)<br />
|align="center"|0.8660254038 (rounded)<br />
|align="left"|<br />
|-<br />
|align="center"|√3<br />
|align="center"|2e/π<br />
|align="center"|1.7305119589 (rounded)<br />
|align="center"|1.7320508076 (rounded)<br />
|align="left"|<br />
|-<br />
|align="center"|gamma(Euler's gamma constant)<br />
|align="center"|1/√3<br />
|align="center"|0.5773502692 (rounded)<br />
|align="center"|0.5772156649015328606065120900824024310421...<br />
|align="left"|In {{w|mathematics}}, the {{w|Euler-Mascheroni constant}} (Euler gamma constant) is a mysterious number describing the relationship between the {{w|Harmonic series (mathematics)|harmonic series}} and the {{w|natural logarithm}}.<br />
|-<br />
|align="center"|Feet in a meter<br />
|align="center"|5/(<sup>e</sup>√π)<br />
|align="center"|3.2815481951<br />
|align="center"|1/.3048 (exact) = 3.280839895 (rounded)<br />
|align="left"|<br />
|-<br />
|align="center"|√5<br />
|align="center"|2/e + 3/2<br />
|align="center"|2.2357588823 (rounded)<br />
|align="center"|2.2360679775 (rounded)<br />
|align="left"|<br />
|-<br />
|align="center"|Avogadro's number<br />
|align="center"|69<sup>π<sup>√5</sup></sup><br />
|align="center"|6.02191201246329*10<sup>23</sup> (rounded)<br />
|align="center"|6.02214129*10<sup>23</sup> (rounded)<br />
|align="left"|<br />
|-<br />
|align="center"|Gravitational constant G<br />
|align="center"|1 / e<sup>(pi - 1)<sup>(pi + 1)</sup></sup><br />
|align="center"|6.67361106850561*10<sup>-11</sup> (rounded)<br />
|align="center"|6.67385*10<sup>-11</sup> (rounded)<br />
|align="left"|The universal {{w|gravitational constant}} G is equal to F*r<sup>2</sup>/Mm, where F is the gravitational force between two objects, r is the distance between them, and M and m are their masses.<br />
|-<br />
|align="center"|R(gas constant)<br />
|align="center"|(e+1) √5<br />
|align="center"|8.3143309279 (rounded)<br />
|align="center"|8.3144622 (rounded)<br />
|align="left"|The {{w|gas constant}} relates energy to temperature in physics, as well as a gas's volume, pressure, temperature and {{w|mole (unit)|molar amount}} (hence the name).<br />
|-<br />
|align="center"|Proton-electron mass ratio<br />
|align="center"|6*π<sup>5</sup><br />
|align="center"|1836.1181087117 (rounded)<br />
|align="center"|1836.15267246 (rounded)<br />
|align="left"|<br />
|-<br />
|align="center"|Liters in a gallon (U.S. liquid gallon, defined by law as 231 cubic inches)<br />
|align="center"|3 + π/4<br />
|align="center"|3.7853981634 (rounded)<br />
|align="center"|3.785411784 (exact)<br />
|align="left"|<br />
|-<br />
|align="center"|''g''<sub>0</sub> or ''g''<sub>n</sub><br />
|align="center"|6 + ln(45)<br />
|align="center"|9.8066624898 (rounded)<br />
|align="center"|9.80665 (standard)<br />
|align="left"|Standard gravity, or standard acceleration due to free fall is the nominal gravitational acceleration of an object in a vacuum near the surface of the Earth. It is defined by standard as 9.80665 m/s2, which is exactly 35.30394 (km/h)/s (about 32.174 ft/s2, or 21.937 mph/s). This value was established by the 3rd CGPM (1901, CR 70) and used to define the standard weight of an object as the product of its mass and this nominal acceleration. The acceleration of a body near the surface of the Earth is due to the combined effects of gravity and centrifugal acceleration from rotation of the Earth (but which is small enough to be neglected for most purposes); the total (the apparent gravity) is about 0.5 percent greater at the poles than at the equator.<br><br>Randall used a letter g without a suffix, which can also mean the local acceleration due to local gravity and centrifugal acceleration, which varies depending on one's position on Earth.<br />
|-<br />
|align="center"|Proton-electron mass ratio<br />
|align="center"|(e<sup>8</sup> - 10) / ϕ<br />
|align="center"|1836.1530151398 (rounded)<br />
|align="center"|1836.15267246 (rounded)<br />
|align="left"|ϕ is the {{w|golden ratio}}, or (1 + √5)/2. It has many interesting geometrical properties.<br />
|-<br />
|align="center"|Ruby laser wavelength<br />
|align="center"|1 / (1200<sup>2</sup>)<br />
|align="center"|.00000069444444444444... (repeating decimal)<br />
|align="center"|694.3 nm<br />
|align="left"|The ruby laser wavelength varies because “ruby” is not clearly defined.<br />
|-<br />
|align="center"|Mean Earth Radius<br />
|align="center"|(5<sup>8</sup>)*6e<br />
|align="center"|2343750e (exact), 6,370,973.035450887 (6370 km, 973 m, 3 cm, 5 mm, 450,887 nm) (rounded)<br />
|align="center"|6,371,008.7 (International Union of Geodesy and Geophysics definition)<br />
|align="left"|The {{w|Earth radius#mean radii|mean earth radius}} varies because there is not one single way to make a sphere out of the earth. Randall's value lies within the actual variation of Earth's radius. The International Union of Geodesy and Geophysics (IUGG) defines the mean radius as 2/3 of the equatorial radius (6,378,137.0 m) plus 1/3 of the polar radius (6,356,752.3 m).<br />
|-<br />
|align="center"|√2<br />
|align="center"|3/5 + π/(7-π)<br />
|align="center"|1.4142200581 (rounded)<br />
|align="center"|1.4142135624 (rounded)<br />
|align="left"|<br />
|-<br />
|align="center"|cos(π/7) + cos(3π/7) + cos(5π/7)<br />
|align="center"|1/2<br />
|align="center"|0.5<br />
|align="center"|0.5 (exact)<br />
|align="left"|The correct equation in the "Pro tip - Not all of these are wrong" section is cos(pi/7) + cos(3pi/7) + cos(5pi/7) = 1/2 as [http://math.stackexchange.com/questions/140388/how-can-one-prove-cos-pi-7-cos3-pi-7-cos5-pi-7-1-2 shown here]. If you're still confused, the functions use {{w|radians}}, not {{w|degrees (angle)|degrees}}.<br />
|-<br />
|align="center"|γ(Euler's gamma constant)<br />
|align="center"|e/3<sup>4</sup> + e/5<br />
|align="center"|0.5772154006 (rounded)<br />
|align="center"|0.5772156649015328606065120900824024310421...<br />
|align="left"|In {{w|mathematics}}, the {{w|Euler-Mascheroni constant}} (Euler gamma constant) is a mysterious number describing the relationship between the {{w|Harmonic series (mathematics)|harmonic series}} and the {{w|natural logarithm}}.<br />
|-<br />
|align="center"|√5<br />
|align="center"|(13 + 4π) / (24 - 4π)<br />
|align="center"|2.2360678094 (rounded)<br />
|align="center"|2.2360679775 (rounded)<br />
|align="left"|<br />
|-<br />
|align="center"|Σ 1/n<sup>n</sup><br />
|align="center"|ln(3)<sup>e</sup><br />
|align="center"|1.2912987577 (rounded)<br />
|align="center"|1.2912859971 (rounded)<br />
|align="left"|<br />
|}<br />
<br />
{{w|Pi}} is a natural constant that arises in describing circles or ellipses. As such, useful as it may be, it doesn't usually occur anywhere in an exponent. When it does, such as with complex numbers, taking the pi-th root is rarely helpful. For example, if we try to derive:<br />
<br />
''e''<sup>π''i''</sup> + 1 = 0<br />
<br />
''e''<sup>π''i''</sup> = -1<br />
<br />
(''e''<sup>''i''</sup>)<sup>π</sup> = -1<br />
<br />
''e''<sup>''i''</sup> = <sup>π</sup>√(-1)<br />
<br />
We get nowhere.<br />
<br />
Same goes for the e-th power: e typically appears in the basis of a power (forming the {{w|exponential function}}), not in the exponent. (This is later referenced in [http://what-if.xkcd.com/73/ Lethal Neutrinos]).<br />
<br />
The software referred to in the comic is [http://mrob.com/pub/ries/ ries], a 'reverse calculator' which forms equations matching a given number.<br />
<br />
==Transcript==<br />
:'''A table of slightly wrong equations and identities useful for approximations and/or trolling teachers.'''<br />
:(Found using a mix of trial-and-error, ''Mathematica'', and Robert Munafo's ''Ries'' tool.)<br />
: All units are SI MKS unless otherwise noted.<br />
<br />
:{| class="wikitable"<br />
|-<br />
|colspan="2" align="center" | Relation:<br />
|align="center" | Accurate to within:<br />
|-<br />
|align="center" | One light year(m)<br />
|align="center" | 99<sup>8</sup><br />
|align="center" | one part in 40<br />
|-<br />
|align="center" | Earth Surface(m<sup>2</sup>)<br />
|align="center" | 69<sup>8</sup><br />
|align="center" | one part in 130<br />
|-<br />
|align="center" | Ocean's volume(m<sup>3</sup>)<br />
|align="center" | 9<sup>19</sup><br />
|align="center" | one part in 70<br />
|-<br />
|align="center" | Seconds in a year<br />
|align="center" | 75<sup>4</sup><br />
|align="center" | one part in 400<br />
|-<br />
|align="center" | Seconds in a year (rent method)<br />
|align="center" | 525,600 x 60<br />
|align="center" | one part in 1400<br />
|-<br />
|align="center" | Age of the universe (seconds)<br />
|align="center" | 15<sup>15</sup><br />
|align="center" | one part in 70<br />
|-<br />
|align="center" | Planck's constant<br />
|align="center" | 1/(30<sup>π<sup>e</sup></sup>)<br />
|align="center" | one part in 110<br />
|-<br />
|align="center" | Fine structure constant<br />
|align="center" | 1/140<br />
|align="center" | [I've had enough of this 137 crap]<br />
|-<br />
|align="center" | Fundamental charge<br />
|align="center" | 3/(14 * π<sup>π<sup>π</sup></sup>)<br />
|align="center" | one part in 500<br />
|-<br />
|align="center"|White House Switchboard<br />
|colspan="2" align="center"|1/<br /><br />
<sup>π</sup>√(e<sup>(1 + <sup>(e-1)</sup>√8</sup>)<br />
|-<br />
|align="center"|Jenny's Constant<br />
|colspan="2" align="center"|(7<sup>(e/1 - 1/e)</sup> - 9) * π<sup>2</sup><br />
|-<br />
|colspan="3" align="center"|Intermission:<br /> World Population Estimate<br /> which should stay current<br /> for a decade or two:<br /><br />
Take the last two digits of the current year<br />
<br />
Example: 20[14] <br />
<br />
Subtract the number of leap years since hurricane Katrina<br />
<br />
Example:14 (minus 2008 and 2012) is 12<br />
<br />
Add a decimal point<br />
<br />
Example: 1.2<br />
<br />
Add 6<br />
<br />
Example: 6 + 1.2<br />
<br />
7.2 = World population in billions.<br />
<br />
Version for US population:<br />
<br />
Example: 20[14]<br />
<br />
Subtract 10<br />
<br />
Example: 4<br />
<br />
Multiply by 3<br />
<br />
Example: 12<br />
<br />
Add 10<br />
<br />
Example: 3[22] million<br />
|-<br />
|align="center"|Electron rest energy<br />
|align="center"|e/7<sup>16</sup> Joules<br />
|align="center"|one part in 1000<br />
|-<br />
|align="center"|Light-year(miles)<br />
|align="center"|2<sup>(42.42)</sup><br />
|align="center"|one part in 1000<br />
|-<br />
|colspan="2" align="center"|sin(60°) = √3/2 = e/π<br />
|align="center"|one part in 1000<br />
|-<br />
|colspan="2" align="center"|√3 = 2e/π<br />
|align="center"|one part in 1000<br />
|-<br />
|align="center"|gamma(Euler's gamma constant)<br />
|align="center"|1/√3<br />
|align="center"|one part in 4000<br />
|-<br />
|align="center"|Feet in a meter<br />
|align="center"|5/(<sup>e</sup>√π)<br />
|align="center"|one part in 4000<br />
|-<br />
|colspan="2" align="center"|√5 = 2/e + 3/2<br />
|align="center"|one part in 7000<br />
|-<br />
|align="center"|Avogadro's number<br />
|align="center"|69<sup>π<sup>√5</sup></sup><br />
|align="center"|one part in 25,000<br />
|-<br />
|align="center"|Gravitational constant G<br />
|align="center"|1 / e<sup>(pi - 1)<sup>(pi + 1)</sup></sup><br />
|align="center"|one part in 25,000<br />
|-<br />
|align="center"|R(gas constant)<br />
|align="center"|(e+1) √5<br />
|align="center"|one part in 50,000<br />
|-<br />
|align="center"|Proton-electron mass ratio<br />
|align="center"|6*π<sup>5</sup><br />
|align="center"|one part in 50,000<br />
|-<br />
|align="center"|Liters in a gallon<br />
|align="center"|3 + π/4<br />
|align="center"|one part in 500,000<br />
|-<br />
|align="center"|g<br />
|align="center"|6 + ln(45)<br />
|align="center"|one part in 750,000<br />
|-<br />
|align="center"|Proton-electron mass ratio<br />
|align="center"|(e<sup>8</sup> - 10) / ϕ<br />
|align="center"|one part in 5,000,000<br />
|-<br />
|align="center"|Ruby laser wavelength<br />
|align="center"|1 / (1200<sup>2</sup>)<br />
|align="center"|[within actual variation]<br />
|-<br />
|align="center"|Mean Earth Radius<br />
|align="center"|(5<sup>8</sup>)*6e<br />
|align="center"|[within actual variation]<br />
|-<br />
|colspan="3" align="center"|Protip - not all of these are wrong:<br />
|-<br />
|colspan="2" align="center"|√2 = 3/5 + π/(7-π)<br />
|align="center"|cos(π/7) + cos(3π/7) + cos(5π/7) = 1/2<br />
|-<br />
|align="center"|γ(Euler's gamma constant) = e/3<sup>4</sup> + e/5<br />
|align="center"|√5 = (13 + 4π) / (24 - 4π)<br />
|align="center"|Σ 1/n<sup>n</sup> = ln(3)<sup>e</sup><br />
|}<br />
<br />
{{comic discussion}}<br />
[[Category:Charts]]<br />
[[Category:Math]]<br />
[[Category:Physics]]<br />
[[Category:Protip]]</div>108.162.215.89https://www.explainxkcd.com/wiki/index.php?title=899:_Number_Line&diff=66033899: Number Line2014-04-25T03:22:39Z<p>108.162.215.89: /* Explanation */ 8 isn't prime</p>
<hr />
<div>{{comic<br />
| number = 899<br />
| date = May 16, 2011<br />
| title = Number Line<br />
| image = number line.png<br />
| titletext = The Wikipedia page List of Numbers opens with "This list is incomplete; you can help by expanding it."<br />
}}<br />
<br />
== Explanation ==<br />
<br />
Once again, Randall seems to be just messing around, this time with a number line.<br />
<br />
*'''Negative numbers''' have the same magnitude as positive numbers but can only be used to represent the removal of that same magnitude (hence the term "difference" being used for subtraction).<br />
<br />
*'''0.<span style="text-decoration: overline;">99</span>'''.... is {{w|0.999...|equal to 1}} because there is no number between 0.<span style="text-decoration: overline;">99</span>.... and 1. 1 - '''0.0000000372''' is 1 bit less than the IEEE 754 32-bit floating-point representation of 1.<br />
<br />
*The '''{{w|golden ratio}}''' or "phi" is the number (1 + sqrt(5)) / 2, about 1.61803. It has many interesting mathematical properties, mostly relating to geometry, and has occasional appearances in nature, such as spirals formed by the seeds in sunflowers. It is also subject to many less credible claims, such as the belief that phi appears in {{w|Parthenon}} (a well-disputed claim) or that rectangles proportioned after phi are more aesthetically pleasing.<br />
<br />
* The approximate range from 2.1 to 2.3 is marked as '''The Forbidden Region'''. Why Randall marked this range as forbidden is really anyone's guess; it seems to be an entirely arbitrary designation.<br />
<br />
*'''{{w|e (mathematical constant)|e}}''' (Euler's number) is 2.71828... and '''π''' (pi) is 3.14159265...<br />
<br />
*'''2.9299372''' is a President's Day reference. It is the average of e and pi just as the American Presidents' Day is always observed on the 3rd Monday of February (between {{w|George Washington}} and {{w|Abraham Lincoln}}'s birthdays). (For non-US residents, Washington and Lincoln were the 1st and 16th Presidents of the USA, respectively. Each has a celebrated place in American history.)<br />
<br />
*'''{{w|Gird}}''' is a purely fictional number. (The glyph that Randall uses seems to resemble an older shape of the digit 4, such as seen on [http://www.bl.uk/learning/images/mappinghist/large2296.html archaic maps].). Canon and orthodox are references to organised religions. Gird could be a reference to any or all of:<br />
**[http://www.strangehorizons.com/2000/20001120/secret_number.shtml Bleem] - a fictional integer between 3 and 4<br />
**iCarly's [http://icarly.wikia.com/wiki/Derf Derf] - a fictional integer between 5 and 6<br />
**George Carlin's [http://www.urbandictionary.com/define.php?term=bleen Bleen] - a fictional integer between 6 and 7<br />
**[http://www.scp-wiki.net/scp-033 SCP-033] - a fictional number that causes freaky things to happen<br />
<br />
*'''Site of the Battle of 4.108''' is another map joke, implying that 4.108 is an actual location, where an eponymous battle was previously fought. It may be a reference (or homage) to the {{w|Battle of Wolf 359}}, a famous military conflict in the fictional universe of Star Trek.<br />
<br />
*An '''Unexplored''' region obscures the line approximately ranging all values from from 4.5 to 6.7, indicating that the integers 5 and 6 are in uncharted territory, and thus completely unknown. This is, of course, patently ridiculous, and the humor seems to derive solely from how nonsensical and unbelievable it is. This is also a map joke as unexplored regions where labeled as such on maps before the earth became effectively explored.<br />
<br />
*It is often the case in the media that "It has been 7 years..." or "In the last 7 years..." etc. It is made to seem like a believable statistic but cannot always be true. Alternatively, it is intended as an absurd joke that the number 7 is just "not to be believed".<br />
<br />
*'''8''' is not the largest even prime, nor is it a prime at all. The largest even prime is 2. A joke intended for those who clearly know that the claim is false.<br />
<br />
*The last entry seems to be a reference to {{w|discrete mathematics}}, which rarely deals with numbers higher than 9. It finishes off the tone of the comic that seems to be shaping the number line terms of what is commonly useful to certain areas of applied mathematics, rather than a complete, accurate version of the number line.<br />
<br />
The title text is a literalism joke implying that Wikipedia would like its "{{w|List of numbers}}" page to include every number from negative infinity to infinity. It could also be a reference to {{w|Gödel's incompleteness theorems}}, which [[Randall]] has used as comic fodder before [[468: Fetishes]]. Gödel's theorems roughly assert that a number theory could never be fully complete. The equivalent for just a sheer list of numbers is {{w|Cantor's diagonal argument}}, which is a "proof of the uncountability of the real numbers." Therefore, if Wikipedia ever did have a "List of numbers", it would perforce forever be incomplete, no matter how much it was expanded. Both Gödel's incompleteness theorems and Cantor's diagonal argument feature prominently in {{w|Gödel, Escher, Bach}} by {{w|Douglas Hofstadter}}, to whom Randall devoted later comic [[917: Hofstadter]].<br />
<br />
== Transcript ==<br />
:[Number line ranging from -1 to 10.]<br />
:[Arrow pointing left, towards negative numbers] Negative "imitator" numbers (do not use)<br />
:[Line right before the number one] 0.99... (actually 0.0000000372 less than 1)<br />
:[Line at the golden ratio.] Φ - Parthenon; sunflowers; golden ratio; wait, come back, I have facts!<br />
:[Line at a region between two and 2.2] forbidden region<br />
:[Line at Euler's number.] e<br />
:[Line a bit before 3] 2.9299372 (e and pi, observed)<br />
:[Line at π.] π<br />
:[Line at 3.5 with a ribbon as the numeral] Gird - accepted as canon by orthodox mathematicians <br />
:[Line a bit after 4.] site of battle of 4.108<br />
:[Blob between 4.5 and 6.5 labeled unexplored.]<br />
:[Line at seven.] Number indicating a factoid is made up ("every 7 years...", "science says there are 7...", etc)<br />
:[Line at eight.] Largest even prime<br />
:[Line at 8.75.] If you encounter a number higher than this, you"re not doing real math<br />
<br />
{{comic discussion}}<br />
[[Category:Math]]<br />
[[Category:Wikipedia]]</div>108.162.215.89https://www.explainxkcd.com/wiki/index.php?title=Talk:1351:_Metamaterials&diff=65096Talk:1351: Metamaterials2014-04-10T07:00:56Z<p>108.162.215.89: /* Linking to Photonic Metamaterial for the Physics or Engineering-inclined */ new section</p>
<hr />
<div>You should also note the reference to https://en.wikipedia.org/wiki/Roses_are_red - which should be quite obvious though.<br />
--[[Special:Contributions/108.162.229.61|108.162.229.61]] 06:34, 4 April 2014 (UTC)<br />
:See also http://en.m.wikipedia.org/wiki/Roses_are_red Sebastian :--[[Special:Contributions/173.245.53.194|173.245.53.194]] 06:28, 4 April 2014 (UTC)<br />
<br />
Is there any guidelines for making a transcript? In my opinion any explanation of the comic should stop at a note of who deliver the line. Only when it is not clear in which order a text should be read or for special comics should there be anything else than written text from the comic. [[User:Kynde|Kynde]] ([[User talk:Kynde|talk]]) 11:50, 4 April 2014 (UTC)<br />
:Take a look [http://explainxkcd.com/wiki/index.php/explain_xkcd:Community_portal/Proposals#Transcripts here]. It was a (short) discussion about transcripts. {{User:Grep/signature|12:20, 04 April 2014}}<br />
::Thanks - from that I would say loose the explanation of what is in each image, and just write Megan or Megan off screen. I will. Change to that.[[User:Kynde|Kynde]] ([[User talk:Kynde|talk]]) 10:13, 5 April 2014 (UTC)<br />
If I'm not terrible mistaken, this is not how metamaterials work. Can anybody link me a work about wavelength-shifting metamaterials?<br />
[[Special:Contributions/108.162.210.249|108.162.210.249]] 16:35, 4 April 2014 (UTC) <span style="font-size: 0.7em;">(whoever put this on grep's talk page... it's supposed to go here)</span><br />
:I have a feeling this cartoon would be funnier if it ''was'' how metamaterials work. But what do I know about metamaterials? The reference to Instagram was pretty funny, though. Why people want to wreck fine photos is beyond me. [[Special:Contributions/108.162.225.157|108.162.225.157]] 05:08, 5 April 2014 (UTC)<br />
<br />
For red to turn into blue, you still need a nonlinear medium and a lot of red. Or maybe a temporally modulated medium with a modulation similiar to the frequency of visible light...?<br />
(This needs to go in the discussion. Not the explanation. FTFY. You are welcome. ;) [[Special:Contributions/173.245.55.67|173.245.55.67]] 20:25, 4 April 2014 (UTC)BK201<br />
<br />
;The title text<br />
I understood it as a filter that go online into peoples instagram pictures and turning them back to the original version. That is also a kind of a meta material - ie. not a real material. Of course(?) Randall will not apply a filter in front of every users camera when the picture is taken and before people use a filter on this "meta filtered" picture when they post it on instagram...? The reason I wish someone else to make the change is that I do not use instagram or filters and hope someone else who can do this better than me now dare to change this wrong explanation :-) [[User:Kynde|Kynde]] ([[User talk:Kynde|talk]]) 10:09, 5 April 2014 (UTC)<br />
<br />
:I really think it's a sheet that Megan holds in front of the flowers, and not a box. And I think that Randall does indeed mean to hold a sheet in front of peoples cameras when they take a picture. The phrase photobombing is used for the act of intruding into the camera view when someone is taking a photo in order to disturb the picture (for example jumping out in front of a group photo, obscuring the group). {{unsigned ip|141.101.80.205}}<br />
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::Well good thing I didn't correct then :-) But then the explanation is not good enough for someone not using these things. Is it so that you apply the filter before you take the photo directly into instagram - so you never see the real photo. Then the title would make some sence! [[User:Kynde|Kynde]] ([[User talk:Kynde|talk]]) 20:25, 5 April 2014 (UTC)<br />
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It's a box. If you look closely you can see the side in panel 3 and the bottom in panel 4. [[Special:Contributions/108.162.219.44|108.162.219.44]] 05:35, 7 April 2014 (UTC)<br />
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== Linking to Photonic Metamaterial for the Physics or Engineering-inclined ==<br />
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This article explains the challenges and the current approaches for metamaterials that operate at the frequencies of light.<br />
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http://en.wikipedia.org/wiki/Photonic_metamaterials<br />
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[[Special:Contributions/108.162.215.89|108.162.215.89]] 07:00, 10 April 2014 (UTC)</div>108.162.215.89https://www.explainxkcd.com/wiki/index.php?title=1342:_Ancient_Stars&diff=627491342: Ancient Stars2014-03-15T04:14:15Z<p>108.162.215.89: /* Explanation */ added explanation of the 14 Bright Star Catalogue items that are not stars</p>
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<div>{{comic<br />
| number = 1342<br />
| date = March 14, 2014<br />
| title = Ancient Stars<br />
| image = ancient_stars.png<br />
| titletext = 'The light from those millions of stars you see is probably many thousands of years old' is a rare example of laypeople substantially OVERestimating astronomical numbers.<br />
}}<br />
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==Explanation==<br />
[[Cueball]] makes the common observation that many of the visible stars in the sky are so distant that it takes thousands years for light from that star to reach Earth. However, the brightest star {{W|Sirius}} is one of the nearest at a mere 8.6 {{W|Light-year|light-years}} distance. In other words, the light that was arriving from Sirius in March 2014, when the comic was posted, was emitted some time around August 2005. The previous US president, {{W|George W. Bush}}, was in office from 2001 to 2009 and [[Megan]] notes that this isn't a terribly impressive observation.<br />
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The title text references the fact that most normal people have a hard time imagining the large scale of astronomical numbers. For example, the distance between astronomical bodies or the size of the Sun are hard to imagine; they typically underestimating them by many orders of magnitude smaller than they actually are. See [http://tvtropes.org/pmwiki/pmwiki.php/Main/SciFiWritersHaveNoSenseOfScale this TV Tropes article] for more detail. ([[609|We'll notify your next of kin.]])<br />
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In this case, however, people instead overestimate both the number of visible stars and their distance by quite a bit. It's frequently cited that about 5,000 to 10,000 stars are visible in the sky by the naked eye. The {{W|Bright Star Catalogue}} is a star catalogue that lists all stars of {{W|apparent magnitude}} 6.5 or brighter, which is roughly every star visible to the naked eye from Earth. The catalog contains 9,110 objects, of which 9,096 are stars (the others are ten novae or supernovae, two globular clusters, and two open clusters). To see most of these you need pretty good eyes and a very dark night. Also at any point on Earth you will of course only be able to see less than half of these as the rest are blocked by the Earth!<br />
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This list shows the {{W|Visible stars|91 brightest stars}}. Of these 59 are more than 100 light years away and only 6 are more than 1,000 light years away. The farthest on this list, {{W|Eta Canis Majoris|Aludra}}, is "only" 3,200 light years away. Our entire {{w|Milky Way}} contains up to 400 Billion (400x10<sup>9</sup>) stars and has a diameter of 100,000 light years.<br />
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There are visible objects much further away, like the {{w|Andromeda Galaxy}} which is 2.5 million light years away and made up of billions of stars.<br />
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==Transcript==<br />
:[Cueball pointing up into the night sky.]<br />
:Cueball: Just think - the light from that star was emitted thousands of years ago. It could be long gone.<br />
:Megan: That's Sirius. It's eight light-years away.<br />
:Cueball: Oh.<br />
:Cueball: Just think - the light from that star was emitted in the previous presidential administration.<br />
:Megan: Mmm, doesn't pack quite the punch.<br />
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{{comic discussion}}<br />
[[Category:Comics featuring Cueball]]<br />
[[Category:Comics featuring Megan]]<br />
[[Category:Astronomy]]</div>108.162.215.89