https://www.explainxkcd.com/wiki/api.php?action=feedcontributions&user=173.245.48.24&feedformat=atomexplain xkcd - User contributions [en]2018-11-17T18:18:39ZUser contributionsMediaWiki 1.30.0https://www.explainxkcd.com/wiki/index.php?title=Talk:1635:_Birdsong&diff=110136Talk:1635: Birdsong2016-01-28T06:04:15Z<p>173.245.48.24: </p>
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<div>Cueball could instead be trying to capture it to figure out how what appears to be a regular bird can sing human lyrics, seeing as birds do not have anything resembling the human pharynx or diaphragm, as birds use a system of air sacs to push air into their lungs, analogous to how a mammalian heart moves blood. [[Special:Contributions/108.162.250.161|108.162.250.161]] 13:09, 27 January 2016 (UTC) Dom Vasta<br />
:However, birds split the actions of each side of their trachea to vocalise two notes at once, which gives those already capable of imitating human speech (or other anthropogenic sounds) ''more'' capability than a human to sing lyrics. That they lack understanding of what they are hearing (certain study parrots possibly excepted) deprives them of the ability to sing ''meaningful'' duets with themselves, but those capable of mimicry clearly have the basic ability to sing two independent voices at once, or a single distinctive voice with at least a simple musical accompaniment of an appropriate register, were they so inclined to separate the 'channels' and not just squish it as if into a mono 'recording' of composite sounds anyway. [[Special:Contributions/162.158.152.89|162.158.152.89]] 13:58, 27 January 2016 (UTC)<br />
::The interesting thing is that there's little strong evidence that even smaller birds /can't/ learn to speak in some degree. While researching the topic, I found a paper which makes a convincing mathematical argument that language formation is very unlikely in nature. http://rsif.royalsocietypublishing.org/content/10/88/20130520<br />
:That would make no sense together with the title text. Before reading this it more seems he would remove the bird from his perfect world, giving his look in the third panel! --[[User:Kynde|Kynde]] ([[User talk:Kynde|talk]]) 16:18, 27 January 2016 (UTC)<br />
<br />
::The study remarkably shows how language would evolve. On the other hand, I laughed quite hard when they tried, in a deus ex machina way, to pull out of a hat the conclusion that 'humans are unique because they already have the ability to process language'. Is that not an obvious sentiment that has nothing to do with their prior research? They proved how difficult language is for nature to develop, but not that it is limited to humans. Rather, they showed that any individual can evolve which has the capacity to understand language, but that the individual requires others with the same capacity in order to benefit enough not to be selected against by evolution. <br />
<br />
::So, if you could find a few birds with a predisposition to language, you may be able to make this happen. Getting them to understand the song on the other hand, would prove difficult. [[Special:Contributions/108.162.242.131|108.162.242.131]] 14:47, 27 January 2016 (UTC)<br />
<br />
Am I the only one who thought of {{w|Undertale}} from the first panel? [[Special:Contributions/108.162.216.47|108.162.216.47]] 13:56, 27 January 2016 (UTC)<br />
:I believe that you are experiencing {{w|Pareidolia}}. [[Special:Contributions/108.162.242.131|108.162.242.131]] 14:02, 27 January 2016 (UTC)<br />
::True to the pareidolia - see also [[1551: Pluto]] ;-) --[[User:Kynde|Kynde]] ([[User talk:Kynde|talk]]) 16:18, 27 January 2016 (UTC)<br />
<br />
The second panel immediately reminded me of that unusual phenomenon of birds mimicking ringtones and other incidental human sounds such as car alarms, substituting their natural and beautiful "songs" for our feeble attempts. [[User:Rotan|Rotan]] ([[User talk:Rotan|talk]]) 01:05, 28 January 2016 (UTC)<br />
<br />
The titletext could also be seen as referencing [http://amultiverse.com/comic/2014/09/10/bird-tools/ this episode] of Scenes From A Multiverse. Doubtful, as (to my knowledge) Randall has never expressed a particular love of that comic, but possible. [[Special:Contributions/162.158.253.18|162.158.253.18]] 20:35, 27 January 2016 (UTC)<br />
<br />
So, I'm not going to outright delete it since it is rather interesting, but those are some pretty serious Undertale spoilers in the explanation. Could we give that part of the description (especially the part about the ending of the Genocide Run) some kind of spoiler tag, if we don't want to just remove it? [[Special:Contributions/108.162.245.176|108.162.245.176]] 20:52, 27 January 2016 (UTC)<br />
<br />
Am I the only one to think that panel 3 represents that age-old truth that a bird in the hand is safer than one overhead? Helping to explain Cueball's irritation, and the butterfly net.--[[Special:Contributions/173.245.48.24|173.245.48.24]] 06:04, 28 January 2016 (UTC)<br />
\</div>173.245.48.24https://www.explainxkcd.com/wiki/index.php?title=388:_Fuck_Grapefruit&diff=97924388: Fuck Grapefruit2015-07-17T04:31:13Z<p>173.245.48.24: started moving comments down into the table</p>
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<div>{{comic<br />
| number = 388<br />
| date = February 25, 2008<br />
| title = Fuck Grapefruit<br />
| image = fuck_grapefruit.png<br />
| titletext = Coconuts are so far down to the left they couldn't be fit on the chart. Ever spent half an hour trying to open a coconut with a rock? Fuck coconuts.<br />
}}<br />
<br />
==Explanation==<br />
{{incomplete|A large portion of this explanation is waffle, and not even the tasty kind... Needs rewriting to be more succinct, perhaps moving the comments on individual fruits to an additional column in the table, or removing some of the less relevant ones completely.}}<br />
<br />
This comic consists of a chart where [[Randall]] has plotted {{w|fruits}} according to two criteria: ease/difficulty to eat, and tastiness. For instance, {{w|pineapples}} are deemed fairly tasty but very difficult to eat, whereas (seeded) {{w|grapes}} are very tasty and somewhat easy, and logically {{w|Grape#Seedless_grapes|seedless grapes}} are just about equally tasty while being far more easy at the same time.<br />
<br />
Obviously being easy to eat is preferable to being difficult, and being tasty is preferable to being untasty, so the "best" fruits (regarding these two aspects only) are in the top-right corner, and the worst in the bottom-left one; additionally, in the top-left corner are the "difficult-but-worthy" fruits, and in the bottom-right one, the "not-so-tasty-but-easy-anyway" ones.<br />
<br />
For health reasons, other factors should also be considered. For example, all of the{{w|citrus}} fruit appears in the least desirable part of the chart, but avoiding all citrus fruit can result in scurvy. <br />
<br />
===Foods not included===<br />
The {{w|cucumber}} is not included, although it could be considered a fruit. See, for instance, the description on this {{w|List of culinary fruits}}. <br />
<br />
Regarding citrus fruit, it would have been interesting to see where he would have placed the more sweet and often much more peel-able {{w|mandarin orange|mandarin}} and especially the hybrid {{w|clementine}}. (This could resemble the comparison of seeded vs. seedless grapes.)<br />
<br />
===Difficulty===<br />
Randall does not explain how he has decided how difficult or easy fruits are.<br />
<br />
For example, he considers a {{w|lemon}} easier to eat (or peel) than an {{w|orange}}, for no obvious reason.<br />
<br />
A pineapple could be considered difficult because of the need for a knife, but this is also true of a {{w|watermelon}}, which was considered easier than pineapples or oranges, which are fairly easy to peel. However, a watermelon is larger, so the amount of effort compared to the amount of fruit is lower.<br />
<br />
Many would also think that fruits containing a stone/pit such as peaches and {{w|cherries}} would be annoying (both because a person has to be careful not to swallow them and because of the need to dispose of them), but this clearly not the case for Randall.<br />
<br />
===Controversy===<br />
As evident from the above section, and according to [http://blog.xkcd.com/2008/02/25/fruit-opinions/ FRUIT OPINIONS!] on the [http://blog.xkcd.com/ Blag], this was the most controversial comic written to this point, ''beating out comics about cunnilingus, the Obama endorsement, and my making 4chan tiny on the map of the internet''.<br />
*{{w|Cunnilingus}} see [[136: Science Fair]].<br />
*{{w|List of Barack Obama presidential campaign endorsements, 2008|The Obama endorsement}} See [http://blog.xkcd.com/2008/01/28/obama/ Politics] also from the Blag. <br />
**The only comic mentioning {{w|Obama}} before this comic was [[360: Writers Strike]].<br />
*{{w|4chan}} is tiny on the map of the internet called [[256: Online Communities]]. (See the small island to the far right - left of "dragons" in the sentence ''Here there be anthromorphic dragons''.)<br />
**Not to be confused with the comic that is actually named [[195: Map of the Internet]].<br />
<br />
==Table==<br />
The comic shows an X-Y plot of fruit, showing tastiness on the vertical axis and difficulty-of-consumption on the horizontal axis. The Y-axis goes from "tasty" at the top, to "untasty" at the bottom. The X-axis goes from "easy" on the right to "difficult" on the left. <br />
<br />
Below is a table listing the coordinates for each fruit according to how tasty it is and assuming that the extremes of each axis goes from -100% to 100% (untasty = -tasty and difficult = -easy). Note that this does not agree with the title text since {{w|coconuts}} would be off the charts. However the extremes for both tasty and untasty as well as for easy and difficult are equally far from the main axis. <br />
<br />
The coordinated have been found by measuring each fruit from the center of the drawing (not center of mass, but center from left to right/top to bottom) to the two axes. The axes are hand drawn which is clearly visible. The numbers have been obtained be measuring to the nearest point of each axis, not taking into account that the axes are not perfect straight perpendicular lines. <br />
<br />
{| class="wikitable sortable"<br />
! Tastiness !! Easiness !! Fruit !! Comments<br />
|-<br />
| 100% || 68% || {{w|Peaches}} || Among Randall's favorite fruits, as it is the one deemed most tasty and it is far more tasty than the four fruits that are deemed easier to eat.<br />
|-<br />
| 85% || 13% || {{w|Grape|Seeded grapes}} || It is unclear why are the seeded grapes more tasty. <br />
|-<br />
| 75% || 75% || {{w|Strawberries}} || Actually not a berry but an {{w|accessory fruit}}, like many of the other fruits on the chart<br />
|-<br />
| 72% || 100% || {{w|Grape#Seedless_grapes|Seedless grapes}} || It is unclear why are the seeded grapes more tasty.<br />
|-<br />
| 68% || -100% || {{w|Pineapples}} ||<br />
|-<br />
| 58% || 87% || {{w|Blueberries}} ||<br />
|-<br />
| 38% || 42% || {{w|Cherries}} ||<br />
|-<br />
| 37% || 66% || {{w|Pear}}s ||<br />
|-<br />
| 22% || 79% || {{w|Green apples}} ||<br />
|-<br />
| 17% || 59% || {{w|Plums}} ||<br />
|-<br />
| 15% || -20% || {{w|Watermelons}} ||<br />
|-<br />
| -12% || 79% || {{w|Apple|Red apples}} ||<br />
|-<br />
| -18% || 16% || {{w|Bananas}} || Shown in the chart as difficult to eat, even though they are among the most easily peelable fruit. This could be because the eater is left with a rather large peel, which is a danger because someone can slip on it. Or he might have difficulty keeping monkeys from taking his bananas.<br />
|-<br />
| -20% || -85% || {{w|Pomegranates}} ||<br />
|-<br />
| -46% || -49% || {{w|Oranges}} ||<br />
|-<br />
| -74% || 26% || {{w|Tomatoes}} || Often claimed to be a {{w|vegetable}}. In the botanical sense however, it is actually a fruit (or a {{w|berry}}.<br />
|-<br />
| -86% || -75% || {{w|Grapefruit}} || According to the chart, the third hardest fruit to eat as well as the second most untasty fruit (from the ones listed at least). Eating one of them is like spending too much of one's time and energy without much reward. Hence Randall's quip in the title: "Fuck grapefruit".<br />
|-<br />
| -100% || -15% || {{w|Lemons}} ||<br />
|-<br />
| <-100% || <-100% || {{w|Coconuts}} || Mentioned only in the title text. As Randall puts it, they are far more difficult to eat (especially to open) than the usual mainstream fruits such as the ones plotted here, and once open Randall doesn't even like the taste. Therefore, they could not even fit on the chart because they are too far in the bottom-left corner. Having spent half an hour trying (in vain?) to open a coconut, Randall has superseded the previous "worst fruit" in the title: "Fuck coconuts".<br />
<br />
Although it has "{{w|nut}}" in its name, the coconut is actually a {{w|Drupe|stone fruit}} and thus belongs on a chart of fruit. <br />
|}<br />
<br />
==Transcript==<br />
:[An X-Y plot of fruit where both axises have arrows in both ends. At the end of each arrow is a label.] <br />
:[The X-axis from left to right:]<br />
:Difficult<br />
:Easy<br />
:[The Y-axis from top to bottom:]<br />
:Tasty<br />
:Untasty<br />
<br />
:[The fruit names are listed here below from top to bottom according to the how tasty the fruit is, not necessarily in the same order that the names are written if one fruit is tall/large and the other low:]<br />
:Peaches<br />
:Seeded grapes<br />
:Strawberries<br />
:Seedless grapes<br />
:Pineapples<br />
:Blueberries<br />
:Cherries<br />
:Pears<br />
:Green apples<br />
:Plums<br />
:Watermelons<br />
:Red apples<br />
:Bananas<br />
:Pomegranates<br />
:Oranges<br />
:Tomatoes<br />
:Grapefruit<br />
:Lemons<br />
<br />
{{comic discussion}}<br />
[[Category:Comics with color]]<br />
[[Category:Charts]]</div>173.245.48.24https://www.explainxkcd.com/wiki/index.php?title=1526:_Placebo_Blocker&diff=967211526: Placebo Blocker2015-06-30T04:38:15Z<p>173.245.48.24: /* Explanation */</p>
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<div>{{comic<br />
| number = 1526<br />
| date = May 18, 2015<br />
| title = Placebo Blocker<br />
| image = placebo blocker.png<br />
| titletext = They work even better if you take them with our experimental placebo booster, which I keep in the same bottle.<br />
}}<br />
<br />
==Explanation==<br />
{{Incomplete|Grammar correction required, work on making the explanation more simple, as well as a summary of the article: Genetics and the placebo effect}}<br />
This comic is a joke about the difficulty of testing a drug that is supposed to block the placebo effect.<br />
<br />
In a classic experiment there is a test group and a control. The control group is used to isolated a variable that the experiment can not be devised to avoid. For example, when people are treated for a illness they generally show improvement relative to an untreated patient, regardless of the effectiveness of the treatment. This is called the {{w|placebo}} effect. Because scientists wish to determine the effectivity of the treatment, they wish to isolate it from the placebo effect. Therefore, in many drug trials one group is given a placebo (an ineffective treatment) and one group is given the real treatment. This comic specifically refers to a study published in May 2015, the same month in which the comic was released, about possible mechanisms for the placebo effect. <br />
<br />
[[Cueball]] states to [[Hair Bun Girl]], with a citation from the real world, that his team created a Placebo Blocker, a drug designed to prevent the placebo effect. Cueball begins to design a test for this new drug. Following the typical experimental design, patients would be split into two groups a control group, and the group that receives the treatment. The first group would receive the ''Placebo Blocker'' drug, while the second would get a placebo pill. Cuball knows that the treatment given to the control group is supposed to be designed so that it is not influenced by the variable trying to be isolated. As the placebo is the treatment that will have an effect, it can not be used as a control treatment against a placebo blocker. Cueball tries to design around this by having the first group receive the Placebo Blocker drug and the placebo, while the second receives placebo pill and something else that counteracts the placebo effect, and fails as that would also be the Placebo Blocker. Cueball and Hair Bun Girl think about this trial until they both develop headache from frustration. Cueball then kindly offers Hair Bun Girl a sugar pill. While this might have helped cure the headache via the placebo effect had he told her it was a headache treatment, by revealing the pill as merely a sugar pill, it may suppress or reduce the effect.<br />
<br />
In title text, Cueball mentions that his sugar pills against headache works even better together with the new experimental placebo ''boosters''. Incidentally, he indicates that he keeps those in the same bottle with his sugar pills. Assuming he or anyone that knows the pills are sugar pills believes cebo ''boosters'' are in the jar this this would allow them to take the sugar pills in the jar and receive the placebo effect, and would therefore boost the placebo effect regardless of whether there are placebo ''boosters'' in the jar.<br />
<br />
It is possible but unlikely that:<br />
* Cueball's sugar pills are, in fact, the Placebo Blockers themselves and that, seeing Hair Bun Girl has a headache, Cueball is inspired to somehow use the opportunity as an experiment to test the Blockers<br />
* Cueball is suggesting Hair Bun Girl take a "placebo booster" which is really a "placebo blocker", thus testing the blocker he mentioned earlier in the comic.<br />
<br />
Questionable neuroscience research is also discussed in [[1453: fMRI]].<br />
<br />
===Placebos===<br />
The {{w|placebo}} effect refers to the phenomenon in which patients given an inactive treatment such as a sugar pill can still show improvement relative to an untreated patient. The placebo effect is thus very important to consider when testing new drugs, since even ineffective drugs can have a positive effect on the patients due to the placebo effect. Modern drug experiments are hence conducted as {{w|Blind experiment#Double-blind trials|double-blind trials}}, where the patients are randomly given either the treatment or a placebo without either they or the administering doctors knowing who receives the new drug and who received the placebo pill.<br />
<br />
Generally the patients need to believe that they are receiving an active treatment, but one [http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0015591 study] showed that the effect can occur even if the patients are told that they are receiving a placebo pill. The key factor seems to be that the patients most believe that a positive effect will occur. For example, (1) patients experience a greater effect if they believe that the treatment is expensive and (2) patients who know that they have not been given an active treatment will experience the effect if they are told that placebos can have a positive effect through the power of the mind.<br />
<br />
Several reasons for the placebo effect have been proposed, from study artifacts - such as under-reporting of negative outcomes by patients who think they are being treated, to neurological explanations for how mental state can translate into physical outcomes.<br />
<br />
Placebo-blockers do actually already exist. A side-effect of the opiate antagonist {{w|Naloxone}} is that it [http://healthland.time.com/2009/08/26/a-neurological-explanation-for-the-placebo-effect/ blocks the placebo effect].<br />
<br />
===Mechanisms of the placebo effect===<br />
Kathryn T. Hall, Joseph Loscalzo, and Ted J. Kaptchuk. (2015) ''[https://dx.doi.org/10.1016/J.MOLMED.2015.02.009 Genetics and the placebo effect: the placebome.]'' Trends in Mol Medicine. Volume 21, Issue 5, May 2015, Pages 285–294<br />
<br />
It is possible to test the placebo blocker by giving the test group a placebo blocker and one control group that receives no treatment what so ever, as a lack of treatment is the variable that an actual placebo is designed to control for, and another control group a placebo, to show the strength of the placebo effect in the experiment. Still it might be hard to determine if the pills are having a negative effect or blocking the placebo effect, so multiple trials with multiple diseases may have to be done.<br />
<br />
==Transcript==<br />
:[Hair Bun Girl is standing in front of Cueball who does all the talking. Below them is a footnote.]<br />
:Cueball: Some researchers* are starting to figure out the mechanism behind the placebo effect.<br />
:Cueball: We've used their work to create a new drug: A ''placebo effect blocker''.<br />
:Footnote: * Hall et al, DOI: 10.1016/J.MOLMED.2015.02.009<br />
<br />
:[Zoom in on Cueball who now holds his arms out.]<br />
:Cueball: Now we just need to run a trial! We'll get two groups, give them both placebos, then give one the ''real'' placebo blocker, and the other a...<br />
:Cueball: ...wait.<br />
<br />
:[Hair Bun Girl holds her chin, while Cueball just stand there for a beat panel.]<br />
<br />
:[Hair Bun looks again at Cueball who begins to take the lid off of a medicine bottle.]<br />
:Hair Bun Girl: ...My head hurts.<br />
:Cueball: Mine too.<br />
:Cueball: Here, want a sugar pill?<br />
<br />
{{comic discussion}}<br />
[[Category:Comics featuring Cueball]]<br />
[[Category:Comics featuring Hair Bun Girl]]<br />
[[Category:Science]]</div>173.245.48.24https://www.explainxkcd.com/wiki/index.php?title=1047:_Approximations&diff=674981047: Approximations2014-05-17T21:47:14Z<p>173.245.48.24: /* Explanation */ revised gravitational constant (G and g) entries</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°) = <sup>3</sup>√/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°) = <sup>3</sup>√/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>173.245.48.24https://www.explainxkcd.com/wiki/index.php?title=1047:_Approximations&diff=674971047: Approximations2014-05-17T21:33:12Z<p>173.245.48.24: /* Explanation */ fixed heading</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°) = <sup>3</sup>√/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 {{w|gravitational constant}} relates to, uh, gravity.<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"|{{math|''&#x0261;''<sub>0</sub>}} or {{math|''&#x0261;''<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 or the symbol {{math|''&#x0261;''}} 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 (see [[Earth's gravity]]). However, in context, he obviously intended the standard value.<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°) = <sup>3</sup>√/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>173.245.48.24https://www.explainxkcd.com/wiki/index.php?title=1047:_Approximations&diff=674961047: Approximations2014-05-17T21:31:41Z<p>173.245.48.24: /* Explanation */ added more values</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"|Name and formula proposed:<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°) = <sup>3</sup>√/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 {{w|gravitational constant}} relates to, uh, gravity.<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"|{{math|''&#x0261;''<sub>0</sub>}} or {{math|''&#x0261;''<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 or the symbol {{math|''&#x0261;''}} 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 (see [[Earth's gravity]]). However, in context, he obviously intended the standard value.<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°) = <sup>3</sup>√/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>173.245.48.24https://www.explainxkcd.com/wiki/index.php?title=1047:_Approximations&diff=674951047: Approximations2014-05-17T21:01:41Z<p>173.245.48.24: /* Explanation */ added parenthesis</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"|Name and formula proposed:<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"|9.46x10<sup>15</sup><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"| <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"|<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"|<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"|1.602x10<sup>-19</sup><br />
|align="center"|<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°) = <sup>3</sup>√/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 {{w|gravitational constant}} relates to, uh, gravity.<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<br />
|align="center"|6 + ln(45)<br />
|align="center"|9.8066624898 (rounded)<br />
|align="center"|<br />
|align="left"|The constant "g" is not a fundamental constant, but rather the local value of the acceleration caused the earth's gravity on the surface of the earth. The actual acceleration caused by earth's gravity varies with altitude, latitude, and local geologic features, making the "constant" somewhat arbitrary.<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"|<br />
|align="left"|The {{w|Earth radios#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.<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°) = <sup>3</sup>√/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>173.245.48.24https://www.explainxkcd.com/wiki/index.php?title=1047:_Approximations&diff=674931047: Approximations2014-05-17T20:58:45Z<p>173.245.48.24: /* Explanation */ looked up actual values for many constants and fixed some errors</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"|Name and formula proposed:<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"|9.46x10<sup>15</sup><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"| <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"|<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"|<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"|1.602x10<sup>-19</sup><br />
|align="center"|<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°) = <sup>3</sup>√/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 {{w|gravitational constant}} relates to, uh, gravity.<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<br />
|align="center"|6 + ln(45)<br />
|align="center"|9.8066624898 (rounded)<br />
|align="center"|<br />
|align="left"|The constant "g" is not a fundamental constant, but rather the local value of the acceleration caused the earth's gravity on the surface of the earth. The actual acceleration caused by earth's gravity varies with altitude, latitude, and local geologic features, making the "constant" somewhat arbitrary.<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"|<br />
|align="left"|The {{w|Earth radios#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.<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°) = <sup>3</sup>√/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>173.245.48.24https://www.explainxkcd.com/wiki/index.php?title=1047:_Approximations&diff=674921047: Approximations2014-05-17T20:57:45Z<p>173.245.48.24: /* Transcript */ added parenthesis in (e<sup>8</sup> - 10) / ϕ</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"|Name and formula proposed:<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"|9.46x10<sup>15</sup><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"| <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"|<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"|<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"|6.626x10<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"|Approximately 0.007297, 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"|1.602x10<sup>-19</sup><br />
|align="center"|<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<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<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"|<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"|<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°) = <sup>3</sup>√/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"|<br />
|align="left"|<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"|<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"|<br />
|align="left"|The {{w|gravitational constant}} relates to, uh, gravity.<br />
|-<br />
|align="center"|R(gas constant)<br />
|align="center"|(e+1) √5<br />
|align="center"|8.3143309279 (rounded)<br />
|align="center"|<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"|<br />
|align="left"|<br />
|-<br />
|align="center"|Liters in a gallon<br />
|align="center"|3 + π/4<br />
|align="center"|3.7853981634 (rounded)<br />
|align="center"|<br />
|align="left"|<br />
|-<br />
|align="center"|g<br />
|align="center"|6 + ln(45)<br />
|align="center"|9.8066624898 (rounded)<br />
|align="center"|<br />
|align="left"|The constant "g" is not a fundamental constant, but rather the local value of the acceleration caused the earth's gravity on the surface of the earth. The actual acceleration caused by earth's gravity varies with altitude, latitude, and local geologic features, making the "constant" somewhat arbitrary.<br />
|-<br />
|align="center"|Proton-electron mass ratio<br />
|align="center"|e<sup>8</sup> - 10 / ϕ<br />
|align="center"|2974.7776471542 (rounded)<br />
|align="center"|<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"|<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"|<br />
|align="left"|The {{w|Earth radios#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.<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"|<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°) = <sup>3</sup>√/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>173.245.48.24https://www.explainxkcd.com/wiki/index.php?title=1047:_Approximations&diff=674911047: Approximations2014-05-17T20:22:31Z<p>173.245.48.24: /* Explanation */ calculated the rest of the estimated values</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"|Name and formula proposed:<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"|9.46x10<sup>15</sup><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"| <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"|<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"|<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"|6.626x10<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"|Approximately 0.007297, 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"|1.602x10<sup>-19</sup><br />
|align="center"|<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<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<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"|<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"|<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°) = <sup>3</sup>√/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"|<br />
|align="left"|<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"|<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"|<br />
|align="left"|The {{w|gravitational constant}} relates to, uh, gravity.<br />
|-<br />
|align="center"|R(gas constant)<br />
|align="center"|(e+1) √5<br />
|align="center"|8.3143309279 (rounded)<br />
|align="center"|<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"|<br />
|align="left"|<br />
|-<br />
|align="center"|Liters in a gallon<br />
|align="center"|3 + π/4<br />
|align="center"|3.7853981634 (rounded)<br />
|align="center"|<br />
|align="left"|<br />
|-<br />
|align="center"|g<br />
|align="center"|6 + ln(45)<br />
|align="center"|9.8066624898 (rounded)<br />
|align="center"|<br />
|align="left"|The constant "g" is not a fundamental constant, but rather the local value of the acceleration caused the earth's gravity on the surface of the earth. The actual acceleration caused by earth's gravity varies with altitude, latitude, and local geologic features, making the "constant" somewhat arbitrary.<br />
|-<br />
|align="center"|Proton-electron mass ratio<br />
|align="center"|e<sup>8</sup> - 10 / ϕ<br />
|align="center"|2974.7776471542 (rounded)<br />
|align="center"|<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"|<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"|<br />
|align="left"|The {{w|Earth radios#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.<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"|<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°) = <sup>3</sup>√/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>173.245.48.24https://www.explainxkcd.com/wiki/index.php?title=1047:_Approximations&diff=674891047: Approximations2014-05-17T19:51:51Z<p>173.245.48.24: /* Transcript */ added parenthesis in √5 = (13 + 4π) / (24 - 4π)</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"|Name and formula proposed:<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"|9.46x10<sup>15</sup><br />
|align="center"|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"| <br />
|align="center"|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"|<br />
|align="center"|<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="center"|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="center"|“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"|<br />
|align="center"|<br />
|-<br />
|align="center"|Planck's constant<br />
|align="center"|1/(30<sup>π<sup>e</sup></sup>)<br />
|align="center"|6.685x10<sup>-34</sup><br />
|align="center"|6.626x10<sup>-34</sup><br />
|align="center"|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"|Approximately 0.007297, close to 1/137<br />
|align="center"|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"|1.599x10<sup>-19</sup><br />
|align="center"|1.602x10<sup>-19</sup><br />
|align="center"|<br />
|align="center"|<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="center"|<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"|approximately 867.530901981685.<br />
|align="center"|8675309<br />
|align="center"|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<br />
|align="center"|Equivalent to 6+((3/4 Year + 1/4 (Year mod 4) - 1499)/10) billion<br />
|align="center"|<br />
|align="center"|<br />
|align="center"|<br />
|-<br />
|align="center"|U.S. Population Estimate<br />
|align="center"|Equivalent to 310+3*(Year - 2010) million<br />
|align="center"|<br />
|align="center"|<br />
|align="center"|<br />
|-<br />
|align="center"|Electron rest energy<br />
|align="center"|e/7<sup>16</sup> Joules<br />
|align="center"|<br />
|align="center"|<br />
|align="center"|<br />
|-<br />
|align="center"|Light-year(miles)<br />
|align="center"|2<sup>(42.42)</sup><br />
|align="center"|5884267614436.97 (rounded)<br />
|align="center"|<br />
|align="center"|{{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°) = <sup>3</sup>√/2<br />
|align="center"|e/π<br />
|align="center"|0.8652559794 (rounded)<br />
|align="center"|0.8660254038 (rounded)<br />
|align="center"|<br />
|-<br />
|align="center"|√3<br />
|align="center"|2e/π<br />
|align="center"|1.7305119589 (rounded)<br />
|align="center"|1.7320508076 (rounded)<br />
|align="center"|<br />
|-<br />
|align="center"|gamma(Euler's gamma constant)<br />
|align="center"|1/√3<br />
|align="center"|0.5773502692 (rounded)<br />
|align="center"|<br />
|align="center"|<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="center"|<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="center"|<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"|<br />
|align="center"|<br />
|-<br />
|align="center"|Gravitational constant G<br />
|align="center"|1 / e<sup>(pi - 1)<sup>(pi + 1)</sup></sup><br />
|align="center"|<br />
|align="center"|<br />
|align="center"|The {{w|gravitational constant}} relates to, uh, gravity.<br />
|-<br />
|align="center"|R(gas constant)<br />
|align="center"|(e+1) √5<br />
|align="center"|<br />
|align="center"|<br />
|align="center"|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"|<br />
|align="center"|<br />
|-<br />
|align="center"|Liters in a gallon<br />
|align="center"|3 + π/4<br />
|align="center"|3.7853981634 (rounded)<br />
|align="center"|<br />
|align="center"|<br />
|-<br />
|align="center"|g<br />
|align="center"|6 + ln(45)<br />
|align="center"|9.8066624898 (rounded)<br />
|align="center"|<br />
|align="center"|The constant "g" is not a fundamental constant, but rather the local value of the acceleration caused the earth's gravity on the surface of the earth. The actual acceleration caused by earth's gravity varies with altitude, latitude, and local geologic features, making the "constant" somewhat arbitrary.<br />
|-<br />
|align="center"|Proton-electron mass ratio<br />
|align="center"|e<sup>8</sup> - 10 / ϕ<br />
|align="center"|<br />
|align="center"|<br />
|align="center"|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"|<br />
|align="center"|<br />
|align="center"|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"|<br />
|align="center"|<br />
|align="center"|The {{w|Earth radios#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.<br />
|-<br />
|align="center"|√2<br />
|align="center"|3/5 + π/(7-π)<br />
|align="center"|<br />
|align="center"|<br />
|align="center"|<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="center"|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"|<br />
|align="center"|<br />
|align="center"|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"|<br />
|align="center"|<br />
|align="center"|<br />
|-<br />
|align="center"|Σ 1/n<sup>n</sup><br />
|align="center"|ln(3)<sup>e</sup><br />
|align="center"|<br />
|align="center"|<br />
|align="center"|<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°) = <sup>3</sup>√/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>173.245.48.24https://www.explainxkcd.com/wiki/index.php?title=1047:_Approximations&diff=674861047: Approximations2014-05-17T19:26:37Z<p>173.245.48.24: /* Explanation */ Created a table and moved much of the text into it.</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"|Name and formula proposed:<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"|9.46x10<sup>15</sup><br />
|align="center"|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"| <br />
|align="center"|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"|<br />
|align="center"|<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="center"|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="center"|“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"|<br />
|align="center"|<br />
|-<br />
|align="center"|Planck's constant<br />
|align="center"|1/(30<sup>π<sup>e</sup></sup>)<br />
|align="center"|6.685x10<sup>-34</sup><br />
|align="center"|6.626x10<sup>-34</sup><br />
|align="center"|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"|Approximately 0.007297, close to 1/137<br />
|align="center"|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"|1.599x10<sup>-19</sup><br />
|align="center"|1.602x10<sup>-19</sup><br />
|align="center"|<br />
|align="center"|<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="center"|<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"|approximately 867.530901981685.<br />
|align="center"|8675309<br />
|align="center"|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<br />
|align="center"|Equivalent to 6+((3/4 Year + 1/4 (Year mod 4) - 1499)/10) billion<br />
|align="center"|<br />
|align="center"|<br />
|align="center"|<br />
|-<br />
|align="center"|U.S. Population Estimate<br />
|align="center"|Equivalent to 310+3*(Year - 2010) million<br />
|align="center"|<br />
|align="center"|<br />
|align="center"|<br />
|-<br />
|align="center"|Electron rest energy<br />
|align="center"|e/7<sup>16</sup> Joules<br />
|align="center"|<br />
|align="center"|<br />
|align="center"|<br />
|-<br />
|align="center"|Light-year(miles)<br />
|align="center"|2<sup>(42.42)</sup><br />
|align="center"|5884267614436.97 (rounded)<br />
|align="center"|<br />
|align="center"|{{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°) = <sup>3</sup>√/2<br />
|align="center"|e/π<br />
|align="center"|0.8652559794 (rounded)<br />
|align="center"|0.8660254038 (rounded)<br />
|align="center"|<br />
|-<br />
|align="center"|√3<br />
|align="center"|2e/π<br />
|align="center"|1.7305119589 (rounded)<br />
|align="center"|1.7320508076 (rounded)<br />
|align="center"|<br />
|-<br />
|align="center"|gamma(Euler's gamma constant)<br />
|align="center"|1/√3<br />
|align="center"|0.5773502692 (rounded)<br />
|align="center"|<br />
|align="center"|<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="center"|<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="center"|<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"|<br />
|align="center"|<br />
|-<br />
|align="center"|Gravitational constant G<br />
|align="center"|1 / e<sup>(pi - 1)<sup>(pi + 1)</sup></sup><br />
|align="center"|<br />
|align="center"|<br />
|align="center"|The {{w|gravitational constant}} relates to, uh, gravity.<br />
|-<br />
|align="center"|R(gas constant)<br />
|align="center"|(e+1) √5<br />
|align="center"|<br />
|align="center"|<br />
|align="center"|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"|<br />
|align="center"|<br />
|-<br />
|align="center"|Liters in a gallon<br />
|align="center"|3 + π/4<br />
|align="center"|3.7853981634 (rounded)<br />
|align="center"|<br />
|align="center"|<br />
|-<br />
|align="center"|g<br />
|align="center"|6 + ln(45)<br />
|align="center"|9.8066624898 (rounded)<br />
|align="center"|<br />
|align="center"|The constant "g" is not a fundamental constant, but rather the local value of the acceleration caused the earth's gravity on the surface of the earth. The actual acceleration caused by earth's gravity varies with altitude, latitude, and local geologic features, making the "constant" somewhat arbitrary.<br />
|-<br />
|align="center"|Proton-electron mass ratio<br />
|align="center"|e<sup>8</sup> - 10 / ϕ<br />
|align="center"|<br />
|align="center"|<br />
|align="center"|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"|<br />
|align="center"|<br />
|align="center"|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"|<br />
|align="center"|<br />
|align="center"|The {{w|Earth radios#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.<br />
|-<br />
|align="center"|√2<br />
|align="center"|3/5 + π/(7-π)<br />
|align="center"|<br />
|align="center"|<br />
|align="center"|<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="center"|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"|<br />
|align="center"|<br />
|align="center"|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"|<br />
|align="center"|<br />
|align="center"|<br />
|-<br />
|align="center"|Σ 1/n<sup>n</sup><br />
|align="center"|ln(3)<sup>e</sup><br />
|align="center"|<br />
|align="center"|<br />
|align="center"|<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°) = <sup>3</sup>√/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>173.245.48.24https://www.explainxkcd.com/wiki/index.php?title=1047:_Approximations&diff=674811047: Approximations2014-05-17T17:15:00Z<p>173.245.48.24: /* Explanation */ expanded on length of a year calculations</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.}}<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 />
There are a few cultural references in this comic:<br />
<br />
*99<sup>8</sup> and 69<sup>8</sup> are sexual references.<br />
*“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 />
*(202) 456-1414 is the phone number for the White House switchboard. Truncated, Randall's formula yields 0.2024561414.<br />
*Jenny's constant comes from Tommy Tutone's tune {{w|867-5309/Jenny}}. Randall's formula gives approximately 867.530901981685.<br />
*{{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 />
And here are some of the mathematical and physical ones, with Wikipedia links.<br />
<br />
*Informally, the {{w|Planck constant}} is the smallest action possible in quantum mechanics.<br />
*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 />
*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 />
*The {{w|gravitational constant}} relates to, uh, gravity.<br />
*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 />
*ϕ is the {{w|golden ratio}}, or (1 + √5)/2. It has many interesting geometrical properties.<br />
*The ruby laser wavelength varies because “ruby” is not clearly defined.<br />
*The {{w|Earth radios#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.<br />
<br />
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 />
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 />
{{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 />
:{| class="wikitable"<br />
|-<br />
|colspan="2" align="center"|Actual<br />
|colspan="2" align="center"|Approximation<br />
|-<br />
|align="center"|One light year(m)<br />
|align="center"|9.46x10<sup>15</sup><br />
|align="center"|99<sup>8</sup><br />
|align="center"|9.23x10<sup>15</sup><br />
|-<br />
|align="center"|Earth Surface(m<sup>2</sup>)<br />
|align="center"|<br />
|align="center"|69<sup>8</sup><br />
|align="center"|<br />
|-<br />
|align="center"|Ocean's volume(m<sup>3</sup>)<br />
|align="center"|<br />
|align="center"|9<sup>19</sup><br />
|align="center"|<br />
|-<br />
|align="center"|Seconds in a year<br />
|align="center"|31557600<br />
|align="center"|75<sup>4</sup><br />
|align="center"|31640625<br />
|-<br />
|align="center"|Seconds in a year (rent method)<br />
|align="center"|31557600<br />
|align="center"|525,600 x 60<br />
|align="center"|31536000<br />
|-<br />
|align="center"|Age of the universe (seconds)<br />
|align="center"|<br />
|align="center"|15<sup>15</sup><br />
|align="center"|4.379x10<sup>17</sup><br />
|-<br />
|align="center"|Planck's constant<br />
|align="center"|6.626x10<sup>-34</sup><br />
|align="center"|1/(30<sup>π<sup>e</sup></sup>)<br />
|align="center"|6.685x10<sup>-34</sup><br />
|-<br />
|align="center"|Fine structure constant<br />
|align="center"|7.297x10<sup>-3</sup><br />
|align="center"|1/140<br />
|align="center"|7.143x10<sup>-3</sup><br />
|-<br />
|align="center"|Fundamental charge<br />
|align="center"|1.602x10<sup>-19</sup><br />
|align="center"|3/(14π<sup>π<sup>π</sup></sup>)<br />
|align="center"|1.599x10<sup>-19</sup><br />
|}<br />
<br />
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."<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. On the current calendar system, there are 97 leap years in every 400 years, making the average year 365.2425 days, or 31,556,952 second. 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. 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 />
==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°) = <sup>3</sup>√/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>173.245.48.24https://www.explainxkcd.com/wiki/index.php?title=1354:_Heartbleed_Explanation&diff=653081354: Heartbleed Explanation2014-04-12T19:08:25Z<p>173.245.48.24: </p>
<hr />
<div>{{comic<br />
| number = 1354<br />
| date = April 11, 2014<br />
| title = Heartbleed Explanation<br />
| image = heartbleed_explanation.png<br />
| titletext = Are you still there, server? It's me, Margaret.<br />
}}<br />
<br />
==Explanation==<br />
{{incomplete|PLEASE don't not remove this TAG that fast!!! Many issues still have to be solved here.}}<br />
The {{w|Heartbleed bug}} has received a lot of news coverage recently and was also the topic of the previous comic [[1353: Heartbleed]]. This comic explains how the bug may have been discovered and can be exploited to reveal a server's memory contents. A hypothetical cracker [[Megan|Meg]] sends heartbeat requests to the server, the server responds to the heartbeat request by returning the contents of the body of the request up to the number of letters requested. The first two requests are well formed, requesting exactly the number of characters in the request body. The server's memory is showing Meg's request with many other requests going on at the same time.<br />
<br />
The last request asks for "HAT" but requests that it be 500 letters long; the server&#8202;—&#8202;not checking if or simply unaware that 500 letters is larger than the request body&#8202;—&#8202;returns "HAT" plus 497 letters that happened to be next to the word "HAT" in its memory. Included are many sensitive bits of information, including a master key and user passwords. One of the passwords shown is "CoHoBaSt", a reference to [[936: Password Strength]], which suggests using "<u>co</u>rrect <u>ho</u>rse <u>ba</u>ttery <u>st</u>aple" as a password.<br />
<br />
Often popular explanations of security bugs require the issue to be simplified a lot and to leave out a lot of details. In this case Randall didn't have to do much simplifying; the bug is actually that simple. Also, it should be noted that any client which can connect to the server typically can exploit this bug in the underlying OpenSSL software&#8202;—&#8202;the use of the term "User Meg" does not imply that Meg had to authenticate first. <br />
<br />
The title text is a reference to ''{{w|Are You There God? It's Me, Margaret.|Are you there God? It's me, Margaret.}}'' a novel by Judy Blume, and plays off of the "server, are you still there?" line in every panel where she did start a request. ''Meg'' can be a nickname for ''Margaret'' as well as ''Megan'', which perhaps explains why the character's usual name, Megan, is abbreviated here.<br />
<br />
==Transcript==<br />
<br />
:How the Heartbleed bug works:<br />
<br />
:Megan: Server, are you still there? If so, reply "POTATO" (6 letters).<br />
:The server's memory is shown: ...<tt>wants pages about boats. User Erica requests secure connection using key "4538538374224". '''User Meg wants these 6 letters: POTATO.''' User Ada wants pages about "irl games". Unlocking secure records with key 5130985733435. Maggie (chrome user) sends this message: "Hi</tt>...<br />
<br />
:Server shows the same memory content but POTATO is highlighted.<br />
:Server: <tt>POTATO</tt><br />
<br />
:Megan: Server, are you still there? If so, reply "BIRD" (4 letters).<br />
:The server's memory is shown: ...<tt>User Olivia from London wants pages about "mad bees in car why". Note: Files for IP 375.381.283.17 are in /tmp/files-3843. '''User Meg wants these 4 letters: BIRD.''' There are currently 346 connections open. User Brendan uploaded the file selfie.jpg (contents: 834ba962e2ceb9ff89bd3bff8c</tt>...<br />
<br />
:Server shows the same memory content but now with BIRD highlighted.<br />
:Server: <tt>BIRD</tt><br />
:Megan: ''Hmm...''<br />
<br />
:Megan: Server, are you still there? If so, reply "HAT" (500 letters).<br />
:Server memory: ...<tt>a connection. Jake requested pictures of deer. '''User Meg wants these 500 letters: HAT.''' Lucas requests the "missed connections" page. Eve (administrator) wants to set server's key to "14835038534". Isabel wants pages about "snakes but not too long". User Karen wants to change account password to "CoHoBaSt". User</tt>...<br />
<br />
:Server shows the same memory content, highlighting the first 500 letters of the memory beginning at HAT.<br />
:Server: <tt>HAT. Lucas requests the "missed connections" page. Eve (administrator) wants to set server's key to "14835038534". Isabel wants pages about "snakes but not too long". User Karen wants to change account password to "CoHoBaSt". User Amber requests pages</tt>...<br />
:Megan writes this all down.<br />
<br />
{{comic discussion}}<br />
[[Category:Comics with color]]<br />
[[Category:Comics featuring Megan]]<br />
[[Category:Computers]]</div>173.245.48.24https://www.explainxkcd.com/wiki/index.php?title=Talk:1352:_Cosmologist_on_a_Tire_Swing&diff=64785Talk:1352: Cosmologist on a Tire Swing2014-04-07T16:32:50Z<p>173.245.48.24: </p>
<hr />
<div>See this TED talk for clue: http://www.ted.com/talks/allan_adams_the_discovery_that_could_rewrite_physics<br />
[[Special:Contributions/108.162.218.101|108.162.218.101]] 07:54, 7 April 2014 (UTC)<br />
<br />
The question "what lies outside our observable universe?" is pretty easily answered with a "the same stuff as inside it, we just can't observe it". The more poignant question is whether the universe as a whole (not just its observable part) has an edge and if so, what lies beyond it. --[[User:Koveras|Koveras]] ([[User talk:Koveras|talk]]) 08:09, 7 April 2014 (UTC)<br />
<br />
:The parts of universe which are not observable due to speed of light looks the same as the ones we can observe, sure. Just bigger. But there is nothing in physics saying there can't be something even more "outside". In fact, some theories consider it probable. And what can be THERE? Anything. Dragons. Possibly literally. Unfortunately, according to current physic, we can't PROVE something outside exists, much less look at it. -- [[User:Hkmaly|Hkmaly]] ([[User talk:Hkmaly|talk]]) 10:21, 7 April 2014 (UTC)<br />
<br />
What's the setting of the first panel? Given the cosmological context, could it be a reference to the {{w|Wood between the Worlds}} from the ''Narnia'' series? [[Special:Contributions/108.162.219.29|108.162.219.29]] 10:57, 7 April 2014 (UTC)<br />
<br />
Someone please rewrite my shunted in assertion about the aforementioned wood. There's a reason for the setting in the first panel, I just can't think of a better place to put it in the explanation.<br />
[[Special:Contributions/108.162.244.6|108.162.244.6]] 11:43, 7 April 2014 (UTC)<br />
<br />
The tire swing also evokes the cosmology field in itself; accepted theories keep changing like the swing, endlessly being replaced by the next one. [[User:Ralfoide|Ralfoide]] ([[User talk:Ralfoide|talk]]) 14:19, 7 April 2014 (UTC)<br />
<br />
Seems to me Randall is referring to the the A and B theories of time. ''&mdash; [[User:Tbc|tbc]] ([[User talk:Tbc|talk]]) 14:24, 7 April 2014 (UTC)''<br />
<br />
It's possible the tire swing might also be a reference to the idea that the universe is "shaped like a doughnut". [[Special:Contributions/173.245.48.24|173.245.48.24]] 16:32, 7 April 2014 (UTC)</div>173.245.48.24https://www.explainxkcd.com/wiki/index.php?title=Talk:901:_Temperature&diff=63724Talk:901: Temperature2014-04-01T04:10:13Z<p>173.245.48.24: </p>
<hr />
<div>Apparently, male pregnancy is a thing. It requires surgery and artificial implantation, but it's a legitimate thing that yields live babies. '''[[User:Davidy22|<u>{{Color|#707|David}}<font color=#070 size=3>y</font></u><font color=#508 size=4>²²</font>]]'''[[User talk:Davidy22|<tt>[talk]</tt>]] 02:00, 17 April 2013 (UTC)<br />
:Link? That's really difficult to believe. [[User:Theo|Theo]] ([[User talk:Theo|talk]]) 18:13, 22 August 2013 (UTC)<br />
<br />
:: http://en.wikipedia.org/wiki/Junior_%28film%29 <-- reference [[Special:Contributions/184.66.160.91|184.66.160.91]] 03:07, 26 August 2013 (UTC)<br />
<br />
Just a bit of trivia: there's a photo online of a pregnancy test where the control line is not (or faintly) visible and the test line is very visible. Someone said that it can happen if there was so much of that hormone that the test line drained ink from the control line. [[Special:Contributions/173.245.48.24|173.245.48.24]] 04:10, 1 April 2014 (UTC)</div>173.245.48.24https://www.explainxkcd.com/wiki/index.php?title=66:_Abusive_Astronomy&diff=6361466: Abusive Astronomy2014-03-29T20:53:10Z<p>173.245.48.24: /* Explanation */ typo</p>
<hr />
<div>{{comic<br />
| number = 66<br />
| date = February 20, 2006<br />
| title = Abusive Astronomy<br />
| image = abusive_astronomy.jpg<br />
| titletext = Medium: Pencil on paper<br />
}}<br />
==Explanation==<br />
A constellation is a pattern of stars which form some sort of perceived shape in the night sky. Modern astronomy recognizes 88 unique constellations, but different cultures saw different patterns in the same night sky, going back at least as far as the Babylonians. The {{w|Pleiades}}, {{w|Orion's belt}} and the {{w|Big Dipper}} (formally known as Ursa Major) are among the most common constellations that we recognize today and are among the first taught to people with an interest in astronomy. The Big Dipper is only visible from the {{w|Northern Hemisphere}} and it can be used to help find the north pole star {{w|Polaris}} which is an aid to night-time navigation.<br />
<br />
During planetarium tours, the tour guide will point out popular constellations and stars, sometimes they will ask a question to get the audience involved in the presentation. Usually these people are big on showing the wonder of the galaxy and are all smiles, but people have bad days. The comic is presenting an especially aggressive way of introducing the constellations. A sidenote is that the Pleiades, while a famous night-sky object, is not a constellation at all; it is an open star cluster.<br />
<br />
When astronomers in the Northern Hemisphere are showing stars to people, there will frequently be someone who points to the Pleiades and says, "There's the Big Dipper!" This gets frustrating about the 100th time that you encounter this error. So, this comic shows someone releasing their frustration on the mis-informed public by pointing out that what they just pointed at is actually the Pleiades. Then, pointing out that you can always locate the Pleiades by following the line of the stars in the belt of Orion. Then, pointing out the REAL Big Dipper.<br />
<br />
In the title text, Randall explains that he drew this comic as a line drawing on white paper, using only a pencil. The image was later inverted for publication.<br />
<br />
==Transcript==<br />
:Identifying star clusters:<br />
:[Image of a star cluster.]<br />
:This is the <u>Pleiades</u>, asshole.<br />
<br />
:Orion's Belt:<br />
:[Image of Orion's Belt.]<br />
:Only a moron couldn't find it.<br />
<br />
:This is the <u>Big</u> <u>Dipper</u>:<br />
:[Image of the Big Dipper.]<br />
:What the hell is <u>wrong</u> with you?<br />
<br />
{{comic discussion}}<br />
[[Category:Comics with color]]<br />
[[Category:Astronomy]]</div>173.245.48.24https://www.explainxkcd.com/wiki/index.php?title=253:_Highway_Engineer_Pranks&diff=63610253: Highway Engineer Pranks2014-03-29T19:24:45Z<p>173.245.48.24: /* Explanation */ changed this to these</p>
<hr />
<div>{{comic<br />
| number = 253<br />
| date = April 25, 2007<br />
| title = Highway Engineer Pranks<br />
| image = highway_engineer_pranks.png<br />
| titletext = Prank #11: Boston<br />
}}<br />
<br />
==Explanation==<br />
Some classical but inaccurate {{w|Interchange (road)|interchanges}} are shown. In general, these interchanges are designed to allow the traffic to flow without directly crossing any other traffic stream. But here we can see some different approaches:<br />
<br />
*The first design resembles a {{w|cloverleaf interchange}}, but has no way to merge back onto the highway once you enter it, making it inescapable.<br />
<br />
*The second interchange has off-ramps that you would normally use to change to the other highway, but in this design they simply merge back to the original highway, so you don't really have a choice in where to go.<br />
<br />
*The {{w|roundabout|rotary}} has a path that puts cars from opposite sides of the rotary onto a collision course. This is a humorous reference to {{w|particle accelerators}} (such as the {{w|Superconducting Super Collider}}) which are designed to put particles on a collision course.<br />
<br />
In the title text {{w|Boston}} is mentioned, a slightly more complicated prank in itself. A common fiction is that the streets evolved from old cowpaths; but in the 17th century they avoided swamps and marshes and followed shorelines before the original peninsula comprising the city was expanded with landfill in the 19th century. {{w|Transportation in Boston#Road infrastructure|Boston's road infrastructure}} in general lacks a {{w|street grid}} like most other US-cities have. On top of that, roads change names and lose and add lanes seemingly at random.<br />
<br />
==Transcript==<br />
:[Each panel depicts a highway intersection.]<br />
:The Inescapable Cloverleaf:<br />
:[Roads lead onto the rings for each leaf, but then are trapped in the circles. Minor roads also allow travel between the rings.]<br />
<br />
:The Zero-Choice Interchange:<br />
:[On and off-ramps exist, but they lead back to the same lane they disconnected from.]<br />
<br />
:The Rotary Supercollider:<br />
:[The roads lead into a traffic circle, and then a loop reverses the direction of flow so all the roads run into each other.]<br />
<br />
{{comic discussion}}</div>173.245.48.24https://www.explainxkcd.com/wiki/index.php?title=Talk:1340:_Unique_Date&diff=63607Talk:1340: Unique Date2014-03-29T18:42:03Z<p>173.245.48.24: DST, leap second</p>
<hr />
<div>What about Daylight Saving Time adjustments and leap seconds? Don't they bring duplicates of the same time or is there a way to account for that in the current system? --[[User:Muskar|Muskar]] ([[User talk:Muskar|talk]]) 10:06, 28 March 2014 (UTC)<br />
<br />
:One hour is duplicated each year at the end of DST. Not much happens during that hour, because it's the middle of the night. A poorly written computer program that instructs the computer to set back the clock one hour whenever the clock reaches a specific time would get caught in a recursive loop (never advancing beyond that time). Properly, clocks are set back one hour when that time is first reached, but are allowed to advance after the duplicate hour concludes.<br />
<br />
Leap second does not result in a duplicate. The additional second is allowed by increasing the number of seconds in a minute. Normally, the 60 seconds of 11:59 are numbered from 11:59:00 to 11:59:59, which is followed by 12:00:00. When there is a leap second, 11:59 has 61 seconds, numbered from 11:59:00 to 11:59:60 (61 total seconds) and then 11:59:60 is followed by 12:00:00.[[Special:Contributions/173.245.48.24|173.245.48.24]] 18:42, 29 March 2014 (UTC)<br />
<br />
My first thought was that he makes fun of people that consider dates like the 12.12.12 as important. As any other date they occur only once and are thus not more special. [[Special:Contributions/108.162.254.66|108.162.254.66]] 04:37, 10 March 2014 (UTC)<br />
<br />
:Good point, I have added something about that. [[Special:Contributions/108.162.246.117|108.162.246.117]] 04:49, 10 March 2014 (UTC)<br />
<br />
Possibly related to the upcoming Pi Day. Also, next year's Pi Day will be 03-14-(20)15, which a few images going around on the Internet have made an annoyingly big deal about. [[Special:Contributions/108.162.237.64|108.162.237.64]] 06:24, 10 March 2014 (UTC)<br />
<br />
So - Maybe I suck at searching (I do), but I can't find any information about us being limited to 4 digits in our calendar system...?[[Special:Contributions/173.245.53.107|173.245.53.107]] 08:38, 10 March 2014 (UTC)<br />
<br />
:Most of the computer software that handles dates would have problems with more (or less) than four digits. Why bother with variable year length when you can just take the first four characters of "2014-03-10" and it works for the next 8 thousand years? [[Special:Contributions/103.22.200.103|103.22.200.103]] 09:42, 10 March 2014 (UTC)<br />
:Also, most digital displays are limited to four digits for the year. [[Special:Contributions/103.22.200.103|103.22.200.103]] 09:43, 10 March 2014 (UTC)<br />
<br />
::And I don't think we actually start address that sooner that in September 9999. It will be Y2K over again! .... not sure where will people of 9999 get {{w|Fortran}} and {{w|Cobol}} programmers, though. Maybe we should freeze some before we run out of them. :-) -- [[User:Hkmaly|Hkmaly]] ([[User talk:Hkmaly|talk]]) 10:20, 10 March 2014 (UTC)<br />
<br />
:::Check [http://en.wikipedia.org/wiki/Unix_billennium#In_literature this] out.--[[User:Rael|Rael]] ([[User talk:Rael|talk]]) 21:38, 10 March 2014 (UTC)<br />
<br />
:I'm with you. I suppose there may be places where leading zeros are used (somewhere in software where memory space has been set aside, I suppose) but I can't think of '''any''' common system where one has to use five digits when using a four digit number.<br />
:When we get to December 31, 9999 (assuming he Gregorian calendar is still in use (BIG assumption)) the next day will simply be January 1, 10000 because, as you said, the Gregorian calendar isn't limited to four-digit years. And, as I say, anyone who think there is some problem with writing years as four digit numbers is simply demonstrating that they are not someone to take seriously. [[Special:Contributions/199.27.128.84|199.27.128.84]] 16:32, 10 March 2014 (UTC)<br />
<br />
After visiting the website for the "Long Now Foundation", I find I'm left wondering - why, oh why, would they stop at using a five digit year? why not six? eight? ten? sixteen? thirty-two? [[User:Brettpeirce|Brettpeirce]] ([[User talk:Brettpeirce|talk]]) 12:06, 10 March 2014 (UTC)<br />
<br />
I think the point in the comic title is that writing years always with 5 digits is as significant as the zero to the left it will take to do so for most of the next 8000 years. [[User:FlavianusEP|FlavianusEP]] ([[User talk:FlavianusEP|talk]]) 12:25, 10 March 2014 (UTC)<br />
<br />
My first thought was that the comic was about date formats and yyyy-mm-dd being better than yy-mm-dd or dd.mm.yy. [[Special:Contributions/173.245.53.138|173.245.53.138]] 12:40, 10 March 2014 (UTC)<br />
<br />
;Dynamic?<br />
<br />
:It isn't, but I've made a dynamic one (based on UTC): https://voidptr.de/xkcd-1340 [[User:N.st|n.st]] ([[User talk:N.st|talk]]) 19:36, 12 March 2014 (UTC)<br />
<br />
Wanna bet that this comic always shows the current date?--[[User:Henke37|Henke37]] ([[User talk:Henke37|talk]]) 10:23, 10 March 2014 (UTC)<br />
<br />
:Haha, that's a great observation! I wish it were so, I'll check again tomorrow. If it's not, someone email Mr. Munroe to make it so, great idea. {{unsigned|Adityarajbhatt}}<br />
<br />
:It's 00:07 (11th of March) right now in China where I am currently located and it still shows 10th of March...just for the record [[Special:Contributions/108.162.225.191|108.162.225.191]] 16:13, 10 March 2014 (UTC)<br />
<br />
::It's March 15th now, and it still says the 10th. It's not dynamic. [[Special:Contributions/199.27.128.76|199.27.128.76]] 20:47, 15 March 2014 (UTC)<br />
<br />
It's funny that Randall seems to have never heard of [http://tools.ietf.org/html/rfc2550 RFC 2550], which goes than the Long Now Foundation in expanding the representable date range. [[Special:Contributions/173.245.53.161|173.245.53.161]] 15:05, 10 March 2014 (UTC)<br />
<br />
:Technically, there will be another 2014-03-10; on October 3rd. - [[Special:Contributions/108.162.219.65|108.162.219.65]] 16:01, 10 March 2014 (UTC)<br />
::It would actually be 2014-10-03 "under our system" as stated in the comic. Technically. [[Special:Contributions/108.162.237.64|108.162.237.64]] 17:14, 10 March 2014 (UTC)<br />
::It's like me saying that there will be another 2014-03-10 on March 14th. [[Special:Contributions/173.245.50.63|173.245.50.63]] 19:45, 10 March 2014 (UTC)<br />
<br />
I wonder if this is also somehow related to the [http://en.wikipedia.org/wiki/Interesting_number_paradox Interesting number paradox]. [[Special:Contributions/199.27.128.29|199.27.128.29]] 18:48, 10 March 2014 (UTC)<br />
<br />
<br />
The problem of the date rolling back is partially mitigated by storing the year as an integer instead of as characters, such as how certain Spreadsheet programs, such as OpenOffice Calc, stores years as a 16-bit signed integer. This doesn't solve the issue, only pushing it back to be the year 32768 problem. This is even less of an issue for 64 bit Unix time, which expire on 15:30:08 UTC on Sun, 4 December 292,277,026,596. It's also important to note that the dates, such as 99, or 00 should not be seen as digits, they should be seen as characters (unless, of course, they are BCD digits, which entirely defeats the purpose of shortening the date to 2 characters length). This might seem trivial, but I think it's an important difference.[[Special:Contributions/108.162.216.41|108.162.216.41]] 02:46, 11 March 2014 (UTC)<br />
: 3rd of October won't happen for another seven months. {{unsigned ip|173.245.53.125}}<br />
:: As a (culturally) dd/mm/[yy]yy person (and ignoring, for brevity, the different options for delimiter), I find yyyy-dd-mm as illogical as mm/dd/yyyy... Why should anybody switch 'precision direction', mid-way? Still, as someone who went through the Y2K process ''and'' worked with colleagues across the Atlantic, I tend to use dd/Mmm/yyyy habitually in "for humans" systems (giving the abbreviated month spelling to avoid all ambiguity, as well as full year-number), or my own "yyyymmdd[-hhmm[ss[.ddd...]]]" format in (informal and internal) programming situations, with comments attached to any conversion routines (inwards and outwards). ((And, yes, there ''are'' ISO/other standards, but I find converting from/to them and internally working with my own long-practiced format works best, for me. YMMV. But be aware of how'd you deal with (or ignore) Leap Seconds!)) [[Special:Contributions/141.101.98.47|141.101.98.47]] 14:58, 12 March 2014 (UTC)<br />
::: I once toyed with the notation 0y20140310, with the "0y" prefix (a pun on C's "0x") distinguishing it from the eight-digit integer 20140310. I later decided that 0y20140310.175959 would be a good way to extend it to specify both date and time, and it still parses as a single C token if that property is useful. (And it sorts properly, of course.) [[Special:Contributions/199.27.128.68|199.27.128.68]] 04:15, 24 March 2014 (UTC)</div>173.245.48.24https://www.explainxkcd.com/wiki/index.php?title=690:_Semicontrolled_Demolition&diff=63604690: Semicontrolled Demolition2014-03-29T17:33:55Z<p>173.245.48.24: /* Explanation */ The video was not taken by a fire fighter, it was a guy shooting a documentary about the fire department who had gone with them that day</p>
<hr />
<div>{{comic<br />
| number = 690<br />
| date = January 18, 2010<br />
| title = Semicontrolled Demolition<br />
| image = semicontrolled demolition.png<br />
| titletext = I believe the truth always lies halfway between the most extreme claims.<br />
}}<br />
<br />
==Explanation==<br />
The {{w|World Trade Center}} towers were destroyed on September 11, 2001 (9/11 in American date notation). The planned attack was for two planes to collide with the north and south towers simultaneously, but what ended up happening was that [http://www.youtube.com/watch?v=7g6V8KZE3GA&t=35s plane 1 hit the north tower at 8:46 am], and [http://www.youtube.com/watch?v=7KYE4zViAAg&t=11m44s the second plane hit the south tower a little less than 20 minutes later]. In the ensuing investigation many people raised questions that didn't seem to get a satisfactory answer for several months, if not years. Many people, who called themselves {{w|9/11 Truthers}}, began to claim that the whole thing was a government conspiracy, in what has come to be known as the "controlled demolition plot" (referenced by the title of this comic), which alleges that the towers were brought down not by the fires caused by the planes but by demolition charges intentionally placed there by the government.<br />
<br />
[[Randall]] proposes a compromise to make both those who believe in the conspiracy and those who don't happy. Since there is only a government related video of a plane flying into the north tower — done by a man who was with {{w|NYFD}} fire fighters — that was a government conspiracy. But it just so happened that the government decided to demolish the north tower on the same day that terrorists decided to demolish the south tower.<br />
<br />
The title text is a restatement of the {{w|Argument_to_moderation|Golden Mean fallacy}}: that the truth can be found in a compromise between two opposite positions. In this comic, one of the positions is a fanciful conspiracy theory and the other is a sober fact-based conclusion. The error of this fallacy is apparent here, as it can lead to even more ridiculous conclusions. In this case, the compromise theory would make no one happy because both sides would have to concede claims which they have already dismissed as bogus, as well as accept an incredibly unlikely coincidence.<br />
<br />
==Transcript==<br />
:[Cueball is holding up a pointer to a screen with an image of the World Trade Center towers mid-disaster.]<br />
:Cueball: Based on my analysis, I believe the government faked the plane crash and demolished the WTC north tower with explosives.<br />
:Cueball: The south tower, in a simultaneous but unrelated plot, was brought down by actual terrorists.<br />
<br />
:The 9/11 truthers responded poorly to my compromise theory.<br />
<br />
{{comic discussion}}<br />
[[Category:Comics featuring Cueball]]<br />
[[Category:Public speaking]]<br />
[[Category:Compromise]]<br />
[[Category:Conspiracy theory]]</div>173.245.48.24https://www.explainxkcd.com/wiki/index.php?title=966:_Jet_Fuel&diff=63603966: Jet Fuel2014-03-29T17:23:36Z<p>173.245.48.24: /* Explanation */ added clarification of why 50% reduction in strength is enough to cause catostrophic failure</p>
<hr />
<div>{{comic<br />
| number = 966<br />
| date = October 19, 2011<br />
| title = Jet Fuel<br />
| image = jet_fuel.png<br />
| titletext = The 'controlled demolition' theory was concocted by the government to distract us. '9/11 was an inside job' was an inside job!<br />
}}<br />
<br />
==Explanation==<br />
This comic is a reference to the "{{w|9/11}} {{w|9/11 truther|Was An Inside Job}}" theory that the {{w|World Trade Center}} in {{w|New York City}} was blown up by a "controlled demolition". The guy with the weird hair only on the back of his head (balding?) uses the typical argument and [[Cueball]] one ups the craziness and of course the balding guy eats it up. The {{w|Chemtrail}}s conspiracy theory is a completely different conspiracy theory which says that (exactly as Cueball does) the US Government puts chemicals and mind control agents in jets and airliners to subtly have US citizens ingest the agents.<br />
<br />
And the title text is the natural "double down" on a theory which says that the conspiracy theory itself was concocted by the government and was supposed to distract from the truth.<br />
<br />
For those wondering: it is true that kerosene does not burn hot enough to ''melt'' steel, but it does burn hot enough to cut the steel's supporting strength roughly in half, which is more than enough to collapse a building weighing thousands of tons. (Although standard engineering practice is to use a safety factor of three, and a safety factor of two is sufficient to allow for a 50% reduction in strength, several columns were severed in the initial impact, increasing the stress on the remaining columns.)<br />
<br />
Cueball messing with 9/11 truther conspiracy theorists was also the subject of [[690: Semicontrolled Demolition]].<br />
<br />
==Transcript==<br />
:[Two people are having a conversation.]<br />
:Person: 9/11 was an inside job! Jet fuel can't burn hot enough to melt steel!<br />
:Cueball: Well, remember — jet fuel wasn't the only thing on those planes. They would've also carried tanks full of the mind-control agents airliners use to make chemtrails. Who ''knows'' what temperature that stuff burns at!<br />
:Person: Whoa. Good point!<br />
<br />
:My Hobby: Playing conspiracy theories off against each other.<br />
<br />
{{comic discussion}}<br />
[[Category:Comics featuring Cueball]]<br />
[[Category:My Hobby]]<br />
[[Category:Conspiracy theory]]</div>173.245.48.24https://www.explainxkcd.com/wiki/index.php?title=1204:_Detail&diff=636021204: Detail2014-03-29T16:56:53Z<p>173.245.48.24: /* Explanation */ removed the absurd claim that today's technology allows scanning electron microscopy of a house</p>
<hr />
<div>{{comic<br />
| number = 1204<br />
| date = April 26, 2013<br />
| title = Detail<br />
| image = detail.png<br />
| titletext = 2031: Google defends the swiveling roof-mounted scanning electron microscopes on its Street View cars, saying they 'don't reveal anything that couldn't be seen by any pedestrian scanning your house with an electron microscope.'<br />
}}<br />
<br />
==Explanation==<br />
{{w|Google Earth}} is a mapping software service provided by Google that allows people to view the Earth from above. If you zoom in to maximum magnification, you can obtain clear views of individual streets and homes.<br />
<br />
{{w|Optical resolution|Resolution}} is a term meaning the smallest length detectable in an image. In this context, this corresponds to the real-life size of a single pixel in a aerial image. [[Randall]] points out that the level of detail in images used in Google Earth has been improved exponentially since its introduction. This occurs as aerial imaging technology improves and better ways of collecting the data are found. Each tick in the scale represents a resolution improvement by 1000 times.<br />
<br />
The {{w|Planck length}} is considered to be the smallest meaningful length in {{w|quantum mechanics}}. In the graph, it is used to denote the actual "resolution" of the universe, as indicated by the horizontal line labeled "Earth". It is defined as approximately 1.6×10<sup>−35</sup> meters, or around 10<sup>20</sup> times smaller than the diameter of a proton.<br />
<br />
An extrapolation of the trend of increasing resolution of Google Earth implies that it could reveal details at levels approaching the Planck length at or around the year 2120. Obviously this idea is fanciful to an extraordinary degree. Even in the laboratory, the Planck length cannot be directly observed by any current or likely future instruments; it is a theoretical construct only. Current microscopes are not even able to resolve at the level of the atom.<br />
<br />
Like [[605: Extrapolating]], this comic deals with unwarranted extrapolation (see also [[1007: Sustainable]] and [[1281: Minifigs]]). <br />
<br />
The title text refers to the fact that the trendline predicts an available resolution in the nanometer range by 2031, which Randall implies would be possible (using today's technology) only with the use of {{w|scanning electron microscope}}s. (In reality, current scanning electron microscopes are lab equipment used with small specimens at very close range, and not suitable for observing something as large as a house or for observations from a passing car.) It also refers to some heat that Google received before about its vehicle-mounted {{w|Google Street View|Street View}} cameras being an invasion of privacy. Google responded by saying that its cameras see nothing more than could be seen by a pedestrian walking by. Or, put another way, Street View gives the power to anybody in the world to virtually stand outside your home.<br />
<br />
==Transcript==<br />
:My Neghborhood's Resolution in:<br />
:[A chart showing the Resolution of Google Earth increasing on a logarithmic scale towards the Planck Length, with resolution on the y-axis and time in years on the x-axis.]<br />
<br />
{{comic discussion}}<br />
[[Category:Charts]]<br />
[[Category:Extrapolation]]</div>173.245.48.24https://www.explainxkcd.com/wiki/index.php?title=715:_Numbers&diff=63601715: Numbers2014-03-29T16:50:24Z<p>173.245.48.24: /* Explanation */ fixed some assumptions related to the genders of those studied</p>
<hr />
<div>{{comic<br />
| number = 715<br />
| date = March 17, 2010<br />
| title = Numbers<br />
| image = numbers.png<br />
| titletext = The typical internet user (who wants to share) has an IQ of 147 and a 9-inch penis. Better than the reverse, I guess}}<br />
<br />
==Explanation==<br />
The graphics use the popular search engine Google to show how many hits (or web pages) are returned as relevant based on a given search replacing <X> by different numbers.<br />
<br />
;Bottles of beer<br />
:The top one is of the popular children's song {{w|99 Bottles of Beer on the Wall}}. In this song, the singers begin with 99 bottles and with each repeat of the verse, decrease the bottles of beer by one. The graph shows a slowdown at 66 bottles of beer, something highlighted. A spike occurs at 49 bottles of beer, which seems to be a popular variant (possibly due to 49 bottles taking about half the time that 99 would).<br />
<br />
;Boy/girfriends<br />
:On the second row, the left graph represents how many girl or boy friends someone has had. They seem pretty similar, though the logarithmic chart may be working on that. To the right is how old (in grade) Internet users seem to be. Going purely by grade, the average is at 7th grade. However, using the notation of Freshman (9)/Sophomore (10)/Junior (11)/Senior (12), there's a notable resurgence.<br />
<br />
;Penis length<br />
:The graph on the far right of the second row describes Internet users talking about the lengths of penises that they "have". (This would include both males discussing their own penises and persons of both genders saying that they "have" their boyfriend's penis, or, less plausibly, a severed penis.) Several websites report 5-6 inches is considered average, but it doesn't appear that way on the Internet.<br />
<br />
;Cup size<br />
:The third row contains four graphs. The far left is the breast size of the Internet user. (This presumably refers to female users, since male breasts are not typically measured in cup size.) The actual breast size is generally considered a bell curve around a B or C cup, yet the hits on Google describe almost an exact opposite trend. Taken with the above male penis length and this describes a trend where either the "average" person posting information seems to embellish or the majority stay quiet.<br />
<br />
;I have never had a boyfriend<br />
:Next to it is the number of hits per (mostly) female Internet users talking about how old they are without having a boyfriend. There's a spike at 18. The comic was written in 2010; as of 2014 the spike does not exist. Google behaves very strangely in this case, as it shows two very different numbers for each search.<br />
{|class="wikitable"<br />
!Age!!First number reported by Google!!Actual number of pages after advancing to the last one!!First number reported by Google after clicking on "More results from answers.yahoo.com"!!Actual number of pages after advancing to the last one (Yahoo Answers)<br />
|-<br />
|15||4||4||56,000||35<br />
|-<br />
|16||29,200||16||66,300||51<br />
|-<br />
|17||25,900||11||62,900||49<br />
|-<br />
|18||22,600||18||33,900||31<br />
|-<br />
|19||16,600||16||25,300||30<br />
|-<br />
|20||13,700||18||8,110||25<br />
|-<br />
|21||13,600||18||11,700||46<br />
|}<br />
<br />
;Glasses of water a day<br />
:Third from the left is the number of glasses drunk per day. {{w|Drinking water}} has a commonly accepted standard of 8 glasses a day of being the most healthy.<br />
<br />
;Number of lights<br />
:On the far right is a description of the number of lights. The spike at four is due to a famous scene from Star Trek: The Next Generation, episode Chain of Command Part 2 where Captain Picard answers that there are four lights, despite pressure to answer that there are five. This is itself a reference to George Orwell's novel 1984, where Winston Smith is tortured until he "learns" to be unsure of the number of fingers being held up by his torturer, despite him only holding up four.<br />
<br />
;Number of problems<br />
:Bottom left is a reference to the popular Jay-Z rap song {{w|99 Problems}}.<br />
<br />
;IQ<br />
:Bottom right describes the IQ of the Internet goer compared to the average. By the definition of the test, when the test scale was developed, the average was 100 with a standard deviation of 15. However, the comic implies that the average claimed IQ closes in on 145, more than 3 standard deviations above the real average! (In reality, many studies have shown that the real average is now higher than 100.) <br />
<br />
The title text refers to the searches. In this, it humorously states that having a 9-inch penis and 147 IQ is better than having a 147-inch penis and 9 IQ. While this is true for humans, it may not be true for all species.<br />
<br />
==Transcript==<br />
:Google Result for Various Phrases:<br />
:{Each panel is a scatterplot of the described X against the number of Google hits, with trend lines. The scales vary.}<br />
<br />
:<X> Bottles of Beer on the Wall<br />
:[There are peaks at 1, 49, 73, and 99. A dip in the middle is marked &quot;They lose steam at 66.&quot; After 99 is a steep dropoff. The largest peak is around 100,000 hits.]<br />
<br />
:I've Had <X> Boy/Girlfriends<br />
:[Both lines descend at roughly the same rate from 1 to 10, although the boyfriend graph is smoother; the girlfriend graph has a small peak at 4 and a small dip at 6. The peaks are between 100,000 and 1,000,000 hits.]<br />
<br />
:I'm in <X>st/nd/rd/th Grade<br />
:[The curve is a bell peaking at 7th grade and about 500,000 hits. A second line labeled &quot;Including Junior, Senior, etc.&quot; follows the bell curve until the peak, then dips only slightly for 10th grade and resumes climbing.]<br />
<br />
:I Have a/an <X>-Inch Penis<br />
:[The line ascends shallowly from 100,000 hits for 3 inches to a peak of 180,000 for 9 inches, then descends steeply to 20,000 for 13 inches.]<br />
<br />
:I'm a/an <X>-Cup<br />
:[A has a few hundred thousand hits; the graph dips to a few thousand for C, peaks again around 100,000 for E, and then tails off.]<br />
<br />
:I'm <X> and Have Never Had a Boyfriend<br />
:[The graph is mostly a simple bell, starting and ending around 300,000 hits for 13 or 21, but there is a sharp peak of 700,000 at 18 (well above the trend line).]<br />
<br />
:Drink <X> Glasses of Water a Day<br />
:[There are barely any hits below 4 or above 12; between the two it rises steeply to about 1,000 hits, with a steep, narrow peak of 10,000 at 8.]<br />
<br />
:There Are <X> Lights<br />
:[The graph descends smoothly from several hundred thousand hits for 1 to about 10,000 for 10, except for a peak of about 1,000,000 for 4.]<br />
<br />
:I Got <X> Problems<br />
:[The plot is extremely jagged, with the largest peak of 10,000,000 hits at 99, another of 10,000 at 96, and 100 and 88.]<br />
<br />
:My IQ Is <X><br />
:[A smooth curve starts and ends at a few thousand hits for around 85 and around 170, with the peak at several tens of thousands for 140, but there are several prominent outliers: 100, 110, 133, and 142 are all around 100,000 hits, and 147 is around 1,000,000.]<br />
<br />
{{comic discussion}}<br />
[[Category:Statistics]]<br />
[[Category:Charts]]<br />
[[Category:Penis]]</div>173.245.48.24https://www.explainxkcd.com/wiki/index.php?title=216:_Romantic_Drama_Equation&diff=63600216: Romantic Drama Equation2014-03-29T16:36:17Z<p>173.245.48.24: /* Explanation */ added line breaks</p>
<hr />
<div>{{comic<br />
| number = 216<br />
| date = January 29, 2007<br />
| title = Romantic Drama Equation<br />
| image = romantic drama equation.png<br />
| titletext = Real-life prospective-pairing curves over things like age can get depressing.<br />
}}<br />
<br />
==Explanation==<br />
{{incomplete|Even since Randall himself is stuck by wolfram-alpha, a better solution has to be done here.}}<br />
The equations in the comic and the graph show how many different love pairs can be made if you know the number of females and males in a group. The text explains that it was inspired by TV Romantic Drama (in this case, the gay drama {{w|Queer as Folk (U.S. TV series)|Queer as Folk}}), but of course the formula is valid for any group of people. There are two graphs and equations - gay option is the case when we are looking for pairs with same gender, straight option is for heterosexual equations. The interesting/funny part about the results is that in most cases there are more possibilities when we consider the homosexual option. Also it is interesting to observe what is kind of obvious - in the heterosexual case the "best" case is if both genders are present equally and the possibilities drop very fast if there is substantial difference between genders.<br />
<br />
The graph makes a note that it only holds true for large casts. Case in point, with a cast of only four people: a two-to-two female-to-male ratio will have four straight pairings to two gay pairings, while a three-to-one female-to-male ratio will have three straight pairings and three gay pairings.<br />
<br />
The title-text mentions that Randall made a chart of his own prospective dating pool as he gets older, and was depressed by the results.<br />
<br />
'''The formulas may be derived as follows:'''<br />
<br />
Each straight couple needs to include one of the x males and one of the (n-x) females so there are x(n-x) possible ways of combining one of each. E.g., if there are n=5 people, of whom x=2 are male, then there will be 3 possible pairings involving the first male, and three possible pairings involving the second yielding 2(5-2)=6 possible pairings.<br />
<br />
Each gay couple needs to include either two males or two females. To choose two males, we can start with any of the x males and choose any of the (x-1) remaining males. However, that counts each possible pairing twice. E.g., Adam&Steve got counted when we chose Adam first and Steve second, and again when we chose Steve first and Adam second. To avoid double counting the possible couples, we therefore need to divide that total by 2. So there are x(x-1)/2 possible male-male pairings. Similar reasoning involving the (n-x) females tells us that there are (n-x)(n-x-1)/2 possible female-female pairings. Multiplying these out and combining the male and lesbian couples together, we get the total number of possible gay couples is [x^2 - x + n^2 - nx - n - xn + x^2 + x]/2. That simplifies to [n^2 - n + 2 x^2 - 2 xn]/2. The left two terms can be combined together as n(n-1) and the right two terms can be combined together as -2x(n-x) or 2x(x-n) [which is negative, because x-n<0]. Since the sum of these terms was divided by 2, we get that the total number of possible same-sex pairs is n(n-1)/2 - x(n-x) or n(n-1)/2 + x(x-n), which is what the cartoon says.<br />
<br />
'''Equations and links to wolfram-alpha:'''<br />
<br />
* Formula for Gay pairing: [https://www.wolframalpha.com/input/?i=n*%28n-1%29%2F2%2Bx*%28x-n%29 n*(n-1)/2+x*(x-n)]<br />
<br />
* Formula for Straight pairing: [https://www.wolframalpha.com/input/?i=x*%28x-n%29 x*(x-n)]<br />
<br />
'''Assumptions'''<br />
<br />
The chart and the calculations assume that:<br />
<br />
1. No one is bisexual (able to be paired with both genders) or a hermaphrodite (person with the sexual organs of both genders, which is a real medical condition, but very rare in humans).<br />
<br />
2. The ENTIRE cast, male AND female, will ALL be of the same sexuality (all homosexual OR all heterosexual).<br />
<br />
3. All genders are constant. (If sex-change operations are allowed, then the number of possible pairings is n(n-1)/2, because any two individuals can undergo surgery to become a possible pair.)<br />
<br />
==Transcript==<br />
{{incomplete transcript}}<br />
:TV Romantic Drama Equation (Derived during a series of "Queer as Folk" episodes)<br />
:[A table shows equations for possible romantic pairings in a TV show. The equation under "gay" is n(n-1)/2+x(x-n); the equation under "straight" is x(n-x).]<br />
:x: Number of male (or female) cast members.<br />
:n: total number of cast members.<br />
:[A graph plots pairings (for large casts) against cast makeup. Each of the above equations forms a curve. "Gay cast" starts high for an all male cast, dips down at 50/50 cast makeup, and then rises again for all female. "Straight cast" starts at zero for an all male cast, peaks at 50/50 cast makeup, and then drops to zero again for an all female cast. The two curves intersect at two points close to the middle.]<br />
<br />
{{comic discussion}}<br />
[[Category:Charts]]<br />
[[Category:Math]]<br />
[[Category:Romance]]</div>173.245.48.24https://www.explainxkcd.com/wiki/index.php?title=216:_Romantic_Drama_Equation&diff=63599216: Romantic Drama Equation2014-03-29T16:35:05Z<p>173.245.48.24: /* Explanation */ expanded on the assumptions and the math</p>
<hr />
<div>{{comic<br />
| number = 216<br />
| date = January 29, 2007<br />
| title = Romantic Drama Equation<br />
| image = romantic drama equation.png<br />
| titletext = Real-life prospective-pairing curves over things like age can get depressing.<br />
}}<br />
<br />
==Explanation==<br />
{{incomplete|Even since Randall himself is stuck by wolfram-alpha, a better solution has to be done here.}}<br />
The equations in the comic and the graph show how many different love pairs can be made if you know the number of females and males in a group. The text explains that it was inspired by TV Romantic Drama (in this case, the gay drama {{w|Queer as Folk (U.S. TV series)|Queer as Folk}}), but of course the formula is valid for any group of people. There are two graphs and equations - gay option is the case when we are looking for pairs with same gender, straight option is for heterosexual equations. The interesting/funny part about the results is that in most cases there are more possibilities when we consider the homosexual option. Also it is interesting to observe what is kind of obvious - in the heterosexual case the "best" case is if both genders are present equally and the possibilities drop very fast if there is substantial difference between genders.<br />
<br />
The graph makes a note that it only holds true for large casts. Case in point, with a cast of only four people: a two-to-two female-to-male ratio will have four straight pairings to two gay pairings, while a three-to-one female-to-male ratio will have three straight pairings and three gay pairings.<br />
<br />
The title-text mentions that Randall made a chart of his own prospective dating pool as he gets older, and was depressed by the results.<br />
<br />
'''The formulas may be derived as follows:'''<br />
<br />
Each straight couple needs to include one of the x males and one of the (n-x) females so there are x(n-x) possible ways of combining one of each. E.g., if there are n=5 people, of whom x=2 are male, then there will be 3 possible pairings involving the first male, and three possible pairings involving the second yielding 2(5-2)=6 possible pairings.<br />
<br />
Each gay couple needs to include either two males or two females. To choose two males, we can start with any of the x males and choose any of the (x-1) remaining males. However, that counts each possible pairing twice. E.g., Adam&Steve got counted when we chose Adam first and Steve second, and again when we chose Steve first and Adam second. To avoid double counting the possible couples, we therefore need to divide that total by 2. So there are x(x-1)/2 possible male-male pairings. Similar reasoning involving the (n-x) females tells us that there are (n-x)(n-x-1)/2 possible female-female pairings. Multiplying these out and combining the male and lesbian couples together, we get the total number of possible gay couples is [x^2 - x + n^2 - nx - n - xn + x^2 + x]/2. That simplifies to [n^2 - n + 2 x^2 - 2 xn]/2. The left two terms can be combined together as n(n-1) and the right two terms can be combined together as -2x(n-x) or 2x(x-n) [which is negative, because x-n<0]. Since the sum of these terms was divided by 2, we get that the total number of possible same-sex pairs is n(n-1)/2 - x(n-x) or n(n-1)/2 + x(x-n), which is what the cartoon says.<br />
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'''Equations and links to wolfram-alpha:'''<br />
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* Formula for Gay pairing: [https://www.wolframalpha.com/input/?i=n*%28n-1%29%2F2%2Bx*%28x-n%29 n*(n-1)/2+x*(x-n)]<br />
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* Formula for Straight pairing: [https://www.wolframalpha.com/input/?i=x*%28x-n%29 x*(x-n)]<br />
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'''Assumptions'''<br />
The chart and the calculations assume that:<br />
1. No one is bisexual (able to be paired with both genders) or a hermaphrodite (person with the sexual organs of both genders, which is a real medical condition, but very rare in humans).<br />
2. The ENTIRE cast, male AND female, will ALL be of the same sexuality (all homosexual OR all heterosexual).<br />
3. All genders are constant. (If sex-change operations are allowed, then the number of possible pairings is n(n-1)/2, because any two individuals can undergo surgery to become a possible pair.)<br />
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==Transcript==<br />
{{incomplete transcript}}<br />
:TV Romantic Drama Equation (Derived during a series of "Queer as Folk" episodes)<br />
:[A table shows equations for possible romantic pairings in a TV show. The equation under "gay" is n(n-1)/2+x(x-n); the equation under "straight" is x(n-x).]<br />
:x: Number of male (or female) cast members.<br />
:n: total number of cast members.<br />
:[A graph plots pairings (for large casts) against cast makeup. Each of the above equations forms a curve. "Gay cast" starts high for an all male cast, dips down at 50/50 cast makeup, and then rises again for all female. "Straight cast" starts at zero for an all male cast, peaks at 50/50 cast makeup, and then drops to zero again for an all female cast. The two curves intersect at two points close to the middle.]<br />
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{{comic discussion}}<br />
[[Category:Charts]]<br />
[[Category:Math]]<br />
[[Category:Romance]]</div>173.245.48.24