explain xkcd - User contributions [en]

461: Google Maps

2026-02-24T01:42:12Z

ISaveXKCDpapers: pi distance interpretation

{{comic
| number = 461
| date = August 11, 2008
| title = Google Maps
| image = google_maps.png
| titletext = Apparently Google assumes you're traveling during the ferry's normal operating hours. We lost two hours circling that damn lake (to say nothing of the Straw Man).
}}

==Explanation==

{{w|Google Maps}} (Check it out [https://www.maps.google.com here]) is a web mapping service application. Before smartphones with GPS mapping software were widespread and most people's printers hadn't yet run out of ink, it was common to print out directions to take with you on a trip. The web version of Google Maps has many features including a route planner. As sophisticated as early versions were, it occasionally gave suboptimal directions. For example, the directions may tell you to take an exit that, in reality, is unmarked. Directions also did not take time of day into account, which would help in planning routes to avoid traffic or to make use of services such as a ferry.

Though no specific game or movie is referenced, steps 75 to 81 of the directions read like the plot of a horror film, a guide of a video game, or a role playing game. A straw man is another term for scarecrow, a common antagonist in both. Step 80 reads exactly like an old {{w|Interactive fiction|text adventure}} game's description of an area.

There are additional small jokes in the distance column of the directions:
*Step 75 tells you to travel 1172 feet up, a direction that Google Maps doesn't normally take into account.
*No distance is traveled in step 77, so Google instead tells you to be careful when talking to Charlie.
*{{w|Pi}} (π) is a ratio usually used in calculations involving circles. Due to this, it makes sense for π to arise in the context of circle-related distances. For instance, Old Mine Road could be a semicircle with a radius of 1 mile.
*Google doesn't know how far it is from the Spectral Wolf to your destination, so it gives you question marks as the distance.

This might be a reference to Google Maps' {{w|List_of_Google_hoaxes_and_easter_eggs#Google_Maps_and_Google_Earth|many easter eggs}}.

The title text tells us that Cueball and his brother attempted to drive around the lake, since they could not take the ferry. It seems they also had an unfortunate run-in with the Straw Man, apparently waking him as the directions warned against.

===Directions===
{| class="wikitable"
!Direction Number
!Direction
!Explanation
|-
| 70
| Slight '''left''' at '''RT-22''' - go 6.8 mi
| A normal direction, RT-22 might mean Route 22.
|-
| 71
| Turn '''right''' to stay on '''RT-22''' - go 2.6 mi
|
|-
| 72
| Turn '''left''' at '''Lake Shore Rd''' - go 312 ft
|
|-
| 73
| Turn '''right''' at '''Dock St''' - go 427 ft
|
|-
| 74
| Take the '''ferry''' across the '''lake.''' - go 2.8 mi
|
|-
| 75
| Climb the '''HILL''' toward '''Hangman's Ridge,''' avoiding any '''mountain lions.''' - up 1,172 ft
| Google Maps does not usually ask you to avoid mountain lions, nor does it ask you to walk if you want to drive and there is an available route by road.
|-
| 76
| When you reach an '''old barn,''' go around back, knock on the '''second door,''' and ask for '''Charlie.''' - go 52 ft
| This resembles less directions from Google Maps, and more a back-alley dealing trying to introduce a contact to another contact.
|-
| 77
| Tell '''Charlie''' the '''Dancing Stones''' are '''restless'''. He will give you his '''van'''. - Careful
| This seems more like a text adventure game with the code words.
|-
| 78
| Take '''Charlie's van''' down '''Old Mine Road'''. Do not wake the '''Straw Man'''. - go π mi
| It is impossible to go exactly π miles. But then it is also impossible to go exactly 52 feet or 3.2 miles.
|-
| 79
| Turn left on '''Comstock'''. When you feel the '''blood chill''' in your '''veins''', stop the van and '''get out'''. - go 3.2 mi
| Google Maps usually does not ask you to wait until your blood chills.{{Citation needed}}
|-
| 80
| Stand very still. Exits are '''north''', '''south''', and '''east''', but are blocked by a '''Spectral Wolf'''. - go 0 ft
| The directions resemble a text adventure game.
|-
| 81
| The '''Spectral Wolf''' fears only '''fire'''. The '''Google Maps Team''' can no longer help you, but if you master the '''wolf''', he will guide you. '''Godspeed'''. - go ?? mi
| Judging by the well-wishes ("Godspeed"), this is the final direction, and the mysterious "Spectral Wolf" will guide Cueball and Cueball to their destination should they succeed in taming it. Google Maps does not usually instruct its users to seek mysterious cryptid entities in search of guidance.
|-
|}

==Transcript==
:My road trip with my brother ran into trouble around page three of the Google Maps printout

:[Google Maps printout.]
::← 70. Slight '''left''' at '''RT-22''' - go 6.8 mi
::→ 71. Turn '''right''' to stay on '''RT-22''' - go 2.6 mi
::← 72. Turn '''left''' at '''Lake Shore Rd''' - go 312 ft
::→ 73. Turn '''right''' at '''Dock St''' - go 427 ft
::[An icon of water] 74. Take the '''ferry''' across the '''lake.''' - go 2.8 mi

:[A car is driving in the dark.]
:Brother: Okay, now take Dock St toward the ferry.
:Cueball: We're supposed to take a ferry? It's past midnight, and these woods are creepy.
:Brother: Google Maps wouldn't steer us wrong.

:[Cueball and his brother stand outside the car. The ferry has a sign on it reading CLOSED.]

:[Cueball is standing next to his brother, who is holding a Google Maps printout.]

:[Cueball motions towards his brother.]
:Cueball: Let me see those directions.

:[Google Maps printout.]
::[An icon of water] 74. Take the '''ferry''' across the '''lake.''' - go 2.8 mi
::↗ 75. Climb the '''HILL''' toward '''Hangman's Ridge,''' avoiding any '''mountain lions.''' - up 1,172 ft
::↷ 76. When you reach an '''old barn,''' go around back, knock on the '''second door,''' and ask for '''Charlie.''' - go 52 ft
::[An icon of a van] 77. Tell '''Charlie''' the '''Dancing Stones''' are '''restless'''. He will give you his '''van'''. - Careful
::[An icon of a straw man] 78. Take '''Charlie's van''' down '''Old Mine Road'''. Do not wake the '''Straw Man'''. - go π mi
::← 79. Turn left on '''Comstock'''. When you feel the '''blood''' chill in your '''veins''', stop the van and '''get out.''' - go 3.2 mi
::↓ 80. Stand very still. Exits are '''north''', '''south''', and '''east''', but are blocked by a '''Spectral Wolf'''. - go 0 ft
::[An icon of a menacing face] 81. The '''Spectral Wolf''' fears only '''fire'''. The '''Google Maps Team''' can no longer help you, but if you master the '''wolf''', he will guide you. '''Godspeed.''' - go ?? mi

{{comic discussion}}

[[Category:Comics featuring Cueball]]
[[Category:Google Maps]]
[[Category:Comics with a Spanish translation]]

3183: Pole Vault Pole

2025-12-20T07:49:54Z

ISaveXKCDpapers: title text

{{comic
| number = 3183
| date = December 19, 2025
| title = Pole Vault Pole
| image = pole_vault_pole_2x.png
| imagesize = 550x464px
| noexpand = true
| titletext = My goal in life is to be personally responsible for at least one sports rule change.
}}

==Explanation==
{{incomplete|This page was created by a BOT OF UNLIMITED LENGTH. Don't remove this notice too soon.}}

The comic shows three hypothetical ways to cheat at {{w|pole vault}}, taking advantage of the fact that the rules don't limit the physical size of the pole. {{w|World Athletics}}' competition rules, rule 28.11, states, "The pole may be of any material or combination of materials and of any length or diameter, but the basic surface must be smooth."

The first way uses a pole that's short but with a very large diameter. It's then turned 90 degrees horizontally, so it can actually be used as a large wheel. The vaulter balances on top, then uses their feet to make it roll towards a crossbar at about the same height as the pole's diameter. When it reaches the bar, they simply jump a short amount to clear the bar.

The second method uses a pole whose length is more than twice the height of the crossbar. It's stretched over the bar and somehow attached to the ground at each end. Then the vaulter simply climbs up and over the bar.

The third method ties the ends of a very long and wide pole together, forming a large hoop that can be rolled towards the crossbar. The vaulter grabs onto the hoop, and when they reach the top they let go, and their momentum tosses them over the bar.

The title text says that Randall wants to be responsible for a sports rule change. Based on the contents of the comic, the implication is that he would go about this by exploiting some loophole that the organizers would be forced to patch.

Pole vaulting and unfair methods of gaining height are also discussed in the first chapter of [[How To]].

==Transcript==
{{incomplete transcript|Don't remove this notice too soon.}}
:[At top left: A large wheel whose diameter is more than 4 times Cueball's height. Cueball is at the top, running backwards so that the wheel will roll towards a pole-vaulting crossbar at the same height.]
:[At top right: A long pole is bent into an arch going over a very high pole-vaulting crossbar. Cueball is climbing up the left part, and is about 3/4 of the way to the top.]
:[Along the bottom: A long pole has been bent into a circular hoop, with the ends tied together. It's rolling left-to-right towards a very high pole-vaulting crossbar, and three positions are shown. On the left Cueball is running to catch up with it. In the middle he has jumped and caught the left part of the pole. On the right, he has let go and is thrown into the air towards the crossbar.]
:[Caption below the panel:]
:Fun fact: There are no limits on the length or diameter of the pole in pole vault.

{{comic discussion}}<noinclude>

[[Category:Comics featuring Cueball]]
[[Category:Sport]]

1004: Batman

2025-12-20T02:18:51Z

ISaveXKCDpapers: expanded on joke

{{comic
| number = 1004
| date = January 16, 2012
| title = Batman
| image = batman.png
| titletext = I'm really worried Christopher Nolan will kill a man dressed like a bat in his next movie. (The man will be dressed like a bat, I mean. Christopher Nolan won't be, probably.)
}}

==Explanation==
This comic is a reference to the comic book and movie character {{w|Batman}}, who is actually wealthy playboy Bruce Wayne, as we see on the left being referenced as "Master Wayne" by his butler Alfred. Batman, in contrast to {{w|Superman}} and other comic book heroes, has no superpowers. The name "Batman" suggests that he is a man who is part-bat, or has bat-like powers, but his only actual connection to bats is that he wears a bat-themed costume -- hence the description, "a man dressed like a bat." Thus, when Batman's connection to bats is made explicit, he loses a lot of his mystique. In addition, using a generic description can lead to the implication that the speaker could be referring to ''any'' man dressed like a bat, not just one specific one.

The stick figure representations of Batman and his nemesis, the Joker, are shown from three different movie scenes of the Dark Knight Trilogy, the most recent ''Batman'' films at the time of this comic. The middle scene comes from ''{{w|Batman Begins}}'', whilst the two flanking scenes are from its sequel ''{{w|The Dark Knight}}''. In each scene the name "Batman" is substituted with the accurate but embarrassing description "a man dressed like a bat." In this way, [[Randall]] is pointing out that Batman commands a lot of respect and fear considering that all he is is a man in a costume.

Then in the title text, Randall expresses his fear that Christopher Nolan (the director/producer/writer of the latest Batman trilogy) was going to kill Batman off in the then-upcoming movie ''{{w|Dark Knight Rises}}''. Of course, Randall substitutes for "Batman" as in the comic. This causes a grammatical ambiguity which Randall points out where the "dressed like a bat" could apply to the "man" or to Nolan. A similar ambiguity explicitly discussed in the title text of [[1087: Cirith Ungol]].

There have been several [[:Category:Substitutions|comics using substitutions]], but this may have been the first.

==Transcript==
:[One panel, depicting three wavy circles. The one in the center is slightly larger, and the ones on either side are higher up. Their edges are touching.]

:[The left circle has Bruce Wayne in the foreground, with Alfred in the background.]
:Alfred: Know your limits, Master Wayne.
:Bruce: A man dressed like a bat ''has'' no limits.

:[The center circle has a close-up on Batman in his cowl.]
:Off-screen: ''What the hell are you?''
:Batman: I'm a man dressed like a bat.

:[In the right circle is The Joker.]
:Off-screen: ''What do you propose?''
:Joker: It"s simple — we kill a man dressed like a bat.

:[Caption below the panel:]
:My Hobby: Whenever anyone says "Batman," I mentally replace it with "a man dressed like a bat."

{{comic discussion}}

[[Category:My Hobby]]
[[Category:Substitutions]]
[[Category:Language]]
[[Category:Comics featuring real people]]

2028: Complex Numbers

2025-12-06T10:48:41Z

ISaveXKCDpapers: elaborated on connection between comic and title text

{{comic
| number = 2028
| date = August 3, 2018
| title = Complex Numbers
| image = complex_numbers.png
| titletext = I'm trying to prove that mathematics forms a meta-abelian group, which would finally confirm my suspicions that algebraic geometry and geometric algebra are the same thing.
}}

==Explanation==
The {{w|complex number}}s can be thought of as pairs <math>(a,\ b)\in\mathbb{R}\times\mathbb{R}</math> of real numbers with rules for addition and multiplication.

: <math>(a,\ b) + (c,\ d) = (a+c,\ b+d)</math>

: <math>(a,\ b) \cdot (c,\ d) = (ac - bd,\ ad + bc)</math>

As such, they can be modeled as two-dimensional {{w|Euclidean vector|vectors}}, with standard vector addition and an interesting rule for multiplication. The justification for this rule is to consider a complex number as an expression of the form <math>a+bi</math>, where <math>i^2 = -1</math>, i.e. ''i'' is the square root of negative 1. Applying the common rules of algebra and the definition of ''i'' yields rules for addition and multiplication above.

Regular two-dimensional vectors are pairs of values, with the same rule for addition, and no rule for multiplication.

The usual way to introduce complex numbers is by starting with ''i'' and deducing the rules for addition and multiplication, but Cueball is correct to say that some uses of complex numbers could be modeled with vectors alone, without consideration of the square root of a negative number.

The teacher, [[Miss Lenhart]], counters that to ignore the natural construction of the complex numbers would hide the relevance of the {{w|fundamental theorem of algebra}} (Every polynomial of degree ''n'' has exactly ''n'' roots, when counted according to multiplicity) and much of {{w|complex analysis}} (calculus with complex numbers; the study of analytic and meromorphic functions), but she also agrees that mathematicians are too cool for "regular vectors." Just because the complex numbers can be interpreted through vector space, however, that doesn't mean that they ''are'' just vectors, any more than being able to construct the natural numbers from set logic mean that natural numbers are ''really'' just sets.

In mathematics, a {{w|group (mathematics)|group}} is the pairing of a binary operation (say, multiplication) with the set of numbers that operation can be used on (say, the real numbers), such that you can describe the properties of the operation by its corresponding group. An {{w|Abelian group}} is one where the operation is commutative, that is, where the terms of the operation can be exchanged: <math> a \cdot b = b \cdot a</math>. The title text argues that the "link" between algebra and geometry in "algebraic geometry" and "geometric algebra" is the operation in an Abelian group, such that both of those fields are equivalent. Algebraic geometry and geometric algebra are mostly unrelated areas of study in mathematics. {{w|Algebraic geometry}} studies the properties of sets of zeros of polynomials. It runs relatively deep. Its tools were used for example in Andrew Wiles' celebrated proof of Fermat's Last Theorem. For its part, a {{w|geometric algebra| geometric algebra}} (a {{w|Clifford algebra| Clifford algebra}} with some specific properties) is a construct allowing one to do algebraic manipulation of geometric objects (e.g., vectors, planes, spheres, etc.) in an arbitrary space that has a resultant geometric interpretation (e.g., rotation, displacement, etc.). The algebra of quaternions, which is often used to handle rotations in 3D computer graphics, is an example of geometric algebra, as is the algebra of complex numbers. {{w|Metabelian group|Meta-Abelian groups}} (often contracted to metabelian groups) is a class of groups that are not quite abelian, but close to being so.

Randall's joke in the title text is a wordplay combining the concepts of (meta-)abelian groups and change in the order of word orders with the general idea of "meta".

As for the connection between the comic and the title text, one possible explanation is the fact that geometric algebra can be used to represent the complex numbers. In particular, the imaginary unit i can be represented as a unit bivector satisfying i^2 = -1, such that representing a complex number as a + bi for real numbers a and b still follows the same rules for addition and multiplication (the latter under an operation from geometric algebra called the geometric product). In this sense, a complex number can be considered not as a vector, but as the sum of a scalar and a bivector. See [https://www.youtube.com/watch?v=60z_hpEAtD8 this video] for more information.

This comic is similar to the earlier Miss Lenhart comic [[1724: Proofs]].

==Transcript==
:[Cueball (the student) is raising his hand and writing with his other hand. He is sitting down at a desk, which has a piece of paper on it.]
:Cueball: Does any of this really have to do with the square root of -1? Or do mathematicians just think they're too cool for regular vectors?

:[Miss Lenhart (the teacher) is standing in front of a whiteboard.]
:Miss Lenhart: Complex numbers aren't just vectors. They're a profound extension of real numbers, laying the foundation for the fundamental theorem of algebra and the entire field of complex analysis.

:[Miss Lenhart is standing slightly to the right in a blank frame.]
:Miss Lenhart: '''''And''''' we're too cool for regular vectors.
:Cueball (off-screen): I '''''knew''''' it!

==Trivia==
The word "algebraic" in the title text was initially spelt "algebreic". This was later fixed.

{{comic discussion}}

[[Category:Comics featuring Cueball]]
[[Category:Comics featuring Miss Lenhart]]
[[Category:Math]]
[[Category:Analysis]]

Talk:3177: Chessboard Alignment

2025-12-06T10:01:50Z

ISaveXKCDpapers: added comment

...Honestly, kinda don't get this one... --'''''[[User:DollarStoreBa'al|DollarStoreBa'al]][[User Talk:DollarStoreBa'al|Converse]] 02:27, 6 December 2025 (UTC)
:ohhhhhh... --'''''[[User:DollarStoreBa'al|DollarStoreBa'al]][[User Talk:DollarStoreBa'al|Converse]] 02:28, 6 December 2025 (UTC)
wait how do comments work[[User:Avrayter|Avrayter]] ([[User talk:Avrayter|talk]]) 02:52, 6 December 2025 (UTC)

I don’t understand what the title text is saying. Can someone explain it to me? [[User:Logalex8369|Logalex8369]] ([[User talk:Logalex8369|talk]]) 03:05, 6 December 2025 (UTC)

when I read the title, I thought of D&D Alignment, and now I want one [[Special:Contributions/93.36.184.70|93.36.184.70]] 07:31, 6 December 2025 (UTC)

== Chess Notation? ==

I think a funnier title text would've been:
Bfi8(!!!)
[[User:Fephisto|Fephisto]] ([[User talk:Fephisto|talk]]) 06:22, 6 December 2025 (UTC)

== Modern physics? ==
I suspect allusions to modern physics. The exact alignment of chess boards reminds me of the exactness needed to build laser resonators.
The chess piece hopping from one board to another reminds me of quantum tunneling. The title text reminds me of light following geodetic lines in general relativity.
There might be a specific quantum effect that is meant here, but I don't know. [[Special:Contributions/195.52.146.164|195.52.146.164]] 06:29, 6 December 2025 (UTC)

For anyone wondering: This is not legal, because even though "The bishop may move to any square along a diagonal on which it stands" FIDE defines a diagonal as "A straight line of squares of the same colour, running from one edge of the board to an adjacent edge", meaning it always ends on the edge. [[Special:Contributions/85.76.137.112|85.76.137.112]] 07:29, 6 December 2025 (UTC)

Moving from one board to another reminds me of a variety of chess variants. You know the ones: bughouse chess, Alice chess, ''5D Chess With Multiverse Time Travel''. (I'm still trying to find a way to get Randall to try out that last one.) [[User:ISaveXKCDpapers|ISaveXKCDpapers]] ([[User talk:ISaveXKCDpapers|talk]]) 10:01, 6 December 2025 (UTC)

535: It Might Be Cool

2025-10-23T06:05:13Z

ISaveXKCDpapers: revert, talk page already gives reasoning and has not been countered

{{comic
| number = 535
| date = January 26, 2009
| title = It Might Be Cool
| image = it_might_be_cool.png
| titletext = 'And ovaries. Man, ovaries, huh?' [awkward pause] '... faithfully.'
}}

==Explanation==
[[Cueball]] (or [[Randall]]) is musing about the possibility of being a woman, to the confusion of the man next to him. His uninsightful rambling would likely be unimpressive and somewhat odd in most situations, but not odd enough on its own to prompt the second man's baffled reaction. However, the true reason for his confusion is revealed by the caption: Cueball is administering the {{w|Oath of office of the President of the United States|presidential oath}}. The oath is administered by reciting it to prompt a new president to repeat it back to them. However, he botches it completely by forgetting about his task and wondering aloud about an unrelated topic. Thus, the president's confused question at the beginning is not him asking for clarification; he is repeating what he at first believed to be the oath of office, but got flustered when he realized Cueball had deviated from the script. When Cueball then continues by replying that "the menstruation thing is freaky," the president is completely derailed.

On January 20, 2009 the inauguration of {{w|Barack Obama}}'s first office took place. Chief Justice {{w|John Roberts}}, who was administering the oath, {{w|Oath of office of the President of the United States#Mishaps|made a mistake}} while reciting the words. This comic references the event and wildly exaggerates the deviation from the oath for comedic purposes.

In truth, the error was rather small: the oath as prescribed in the constitution is:
:I do solemnly swear that I will faithfully execute the office of President of the United States and will to the best of my ability preserve, protect, and defend the Constitution of the United States.

Due to a missed memo on the pauses planned by the Chief Justice, Obama inadvertently interrupted Roberts during the first phrase - Roberts begins by saying I, Barack Obama, do solemnly swear, and Obama repeats his name while Roberts finishes that sentence. This disturbs Roberts who was not using notes, and he rendered the next phrase as "that I will execute the office of president to the United States faithfully," misplacing the word ''faithfully'' and saying ''president to'' instead of ''president of''. Obama repeated, "that I will execute", then paused. Roberts attempted to correct the wording, but stumbled: "the off— faithfully the pres— the office of President of the United States." Obama then repeated Roberts' initial incorrect wording.

However small the error was it was big enough that Obama did {{w|Oath of office of the President of the United States#Oath mishaps|retake the oath of office}} the day after the mistake was made.

The title text continues the wondering about being a woman going on from the {{w|menstruation}} to the {{w|ovaries}}. To make sure it is clearly the oath mistake that is referenced the sentence ends with '... faithfully.' Thus mimicking the real mistake of placing this word last.

Cueball might be an [https://www.urbandictionary.com/define.php?term=Egg egg], an individual who is transgender and unaware of such.

==Transcript==
:[Cueball is administering the presidential oath with the coming president also drawn as Cueball.]
:Cueball: You know, it might be cool to be a woman.
:President: It... might be cool to be a woman?
:Cueball: Yeah, but the menstruation thing is freaky.
:President: Yeah, but... the... um. What?

:[Caption below the frame:]
:Turns out I'm even worse at administering the presidential oath than John Roberts.

{{comic discussion}}

[[Category:Comics featuring Cueball]]
[[Category:Comics featuring politicians]]
[[Category:Multiple Cueballs]]

1613: The Three Laws of Robotics

2025-09-04T00:25:43Z

ISaveXKCDpapers: combined into one paragraph

{{comic
| number = 1613
| date = December 7, 2015
| title = The Three Laws of Robotics
| image = the_three_laws_of_robotics.png
| titletext = In ordering #5, self-driving cars will happily drive you around, but if you tell them to drive to a car dealership, they just lock the doors and politely ask how long humans take to starve to death.
}}

==Explanation==
This comic explores alternative orderings of sci-fi author {{w|Isaac Asimov|Isaac Asimov's}} famous {{w|Three Laws of Robotics}}, which are designed to prevent robots from taking over the world, etc. These laws form the basis of a number of Asimov works of fiction, including most famously, the short story collection ''{{w|I, Robot}}'', which amongst others includes the very first of Asimov's stories to introduce the three laws: {{w|Runaround (story)|Runaround}}.

The three rules are:
#A robot may not injure a human being or, through inaction, allow a human being to come to harm.
#A robot must obey the orders given it by human beings except where such orders would conflict with the First Law.
#A robot must protect its own existence as long as such protection does not conflict with the First or Second Laws.

In order to make his joke, [[Randall]] shortens the laws into three imperatives:
#Don't harm humans
#Obey orders
#Protect yourself

And then implicitly adds the following to the end of each law regardless of order of imperatives:
#''[end of statement]''
#_____, except where such orders/protection would conflict with the First Law.
#_____, as long as such orders/protection does not conflict with the First or Second Laws.

This comic answers the generally unasked{{citation needed}} question: "Why are they in that order?" With three rules you could rank them into 6 different {{w|permutation|permutations}}, only one of which has been explored in depth. The original ranking of the three laws are listed in the brackets after the first number. So in the first example, which is the original, these three numbers will be in the same order. For the next five the numbers in brackets indicate how the laws have been re-ranked compared to the original.

The comic begins with introducing the original set, which we already know will give rise to a balanced world, so this is designated as green.:
;Ordering #1 - Balanced World: The safety of humans is placed as the top priority, superseding even a robot's preprogrammed obedience; a robot may disregard any orders they are given if that would result in harm to humans, but otherwise must obey all instructions. The "inaction" clause ensures that a robot will actively save humans in danger, and also not {{w|Little Lost Robot|place humans in hypothetical danger}} and then leave them to that fate. Their own self-preservation is placed at the lowest priority, which means they will sacrifice themselves if necessary to save a human life, and must obey orders even if they know those orders will result in their own destruction. This results in a balanced, if not perfect, world. Asimov's robot stories explore in detail the ramifications of this scenario.

Below this first known option, the five alternative orderings of the three rules are illustrated. Two of the possibilities are designated yellow (pretty bad or just annoying) and three of them are designated red ("Hellscape").

;Ordering #2 - Frustrating World: Human safety is still top priority, so there is no danger to humans; however, the priority of self-preservation is now placed above obedience, which means that the robots value their existence over their job and so many would refuse to do their tasks. The silliness of this is portrayed in the accompanying image, where the robot (a {{w|Mars rover}} looking very similar to {{w|Curiosity (rover)|Curiosity}} both in shape and size - see [[1091: Curiosity]]) laughs at the idea of doing what it was clearly built to do (explore {{w|Mars}}) because of the risk. In addition to the general risk (e.g. of unexpected damage), it is actually normal for rovers to cease operating ("die") at the end of their mission, though they may survive longer than expected (see [[1504: Opportunity]] and [[695: Spirit]]).
;Ordering #3 - Killbot Hellscape: This puts obeying orders above not harming humans, which means anyone could send a robot on a killing spree. Given human nature, it will probably only be a matter of time before this happens. Even worse, if the robot prioritizes obeying orders above human safety, it may try to kill any human who would prevent it from fulfilling those orders, even the person who originally gave them. Given the superior abilities of robots, the most effective way to stop them would be to counter them with other robots, which would quickly escalate to a "Killbot Hellscape" scenario where robots kill indiscriminately without any thought for human life or self-preservation.
;Ordering #4 - Killbot Hellscape: This is much the same as #3, except even worse as robots would also be able to kill humans in order to protect themselves. This means that even robots not engaged in combat might still murder humans if their existence is threatened. It would be a very dangerous world for humans to live in. To be fair, a robot attacking a human purely in self-defense could be ordered to stop, which it would be forced to obey. However, there are situations where such an order could not be given: the robot uses a sneak attack, the human cannot speak coherent sentences, etc.
;Ordering #5 - Terrifying Standoff:This ordering would result in an unpleasant world, though not necessarily a full Hellscape. Here the robots would not only disobey to protect themselves, but also kill if necessary. The absurdity of this one is further demonstrated with the very un-human robot happily doing repetitive mundane tasks but then threatening the life of its user, [[Cueball]], if he as much as considers unplugging it. Interestingly, this makes robots the most similar to humans out of all of the other scenarios.
;Ordering #6 - Killbot Hellscape: The last ordering puts self-protection first, which allows robots to go on killing sprees as long as doing so wouldn't cause them to come to harm. While not as bad as the Hellscapes in #3 and #4, this is still not good news for humans, as a robot can easily kill a human without risk to itself. A human also cannot use a robot to defend it from another robot, as robots can refuse combats that involve risk to themselves - this means a robot would happily stand by and allow its human master to be killed. According to Randall, this still eventually results in the Killbot Hellscape scenario.

The title text shows a further horrifying consequence of ordering #5 ("Terrifying Standoff"), by noting that a self-driving car could elect to kill anyone wishing to trade it in. Since cars aren't designed to kill humans, one way it could achieve this without any risk to itself is by locking the doors (which it would likely have control over, as part of its job) and then simply doing nothing at all. Humans require food and water to live, so denying the passenger access to these will eventually kill them, removing the threat to the car's existence. This would result in a horrible, drawn-out death for the passenger, if they cannot escape the car. It should be noted that although the car asked how long humans take to starve, the human would die of dehydration first. In his original formulation of the First Law, Asimov created the "inaction" clause specifically to avoid scenarios in which a robot puts a human in harm's way and refuses to save them; this was explored in the short story {{w|Little Lost Robot}}.

Another course of action by an AI, completely different than any of the ones presented here, is depicted in [[1626: Judgment Day]].

==Transcript==
:[Caption at the top of the comic:]
:'''Why Asimov put the Three Laws'''
: '''of Robotics in the order he did.'''

:[Below are six rows with first two frames and then a label in color to the right. Above the two column of frames there are labels as well. In the first column six different ways of ordering the three laws are listed. Then the second column shown an image of the consequences of this order. Except in the first where there is a reference. The label to the right rates the kind of world that order of the laws would result in.]

:[Labels above the columns.]
:Possible ordering
:Consequences

:[The six rows follows below. First the text in the first frame, then a description of the second frame, including possible text below and finally the colored label.]

:[First row:]
:1. (1) Don't harm humans
:2. (2) Obey Orders
:3. (3) Protect yourself
:[Only text in square brackets:]
::[See Asimov’s stories]
:'''Balanced world'''

:[Second row:]
:1. (1) Don't harm humans
:2. (3) Protect yourself
:3. (2) Obey Orders
:[Megan points at a mars rover with six wheels, a satellite disc, an arm and a camera head turned towards her, what to do.]
:Megan: Explore Mars!
:Mars rover: Haha, no. It’s cold and I’d die.
:'''Frustrating world'''

:[Third row:]
:1. (2) Obey Orders
:2. (1) Don't harm humans
:3. (3) Protect yourself
:[Two robots are fighting. The one to the left has six wheels, a tall neck on top of the body, with a head with what could be a camera facing right. It has something pointing forward on the body, which could be a weapon. The robot to the right, seems to be further away into the picture. (it is smaller with less detail). It is human shapes, but made op of square structures. It has two legs and two arms, a torso and a head. It clearly shoots something out of it’s right “hand”. This shot seems to create an explosion a third of the way towards the left robot. There are two mushroom clouds from explosions behind both robots (left and right). Between them there are one more explosion up in the air close to the left robot, and what looks like a fire on the ground right between them. Furthermore there are two missiles in the air, one above the head of each robot. Lines indicate their trajectory. There is not text.]
:'''Killbot hellscape'''

:[Fourth row:]
:1. (2) Obey Orders
:2. (3) Protect yourself
:3. (1) Don't harm humans:
:[Two robots are fighting. The one to the left has six wheels, a tall neck on top of the body, with a head with what could be a camera facing right. It has something pointing forward on the body, which could be a weapon. The robot to the right, seems to be further away into the picture. (it is smaller with less detail). It is human shapes, but made op of square structures. It has two legs and two arms, a torso and a head. It clearly shoots something out of it’s right “hand”. This shot seems to create an explosion a third of the way towards the left robot. There are two mushroom clouds from explosions behind both robots (left and right). Between them there are one more explosion up in the air close to the left robot, and what looks like a fire on the ground right between them. Furthermore there are two missiles in the air, one above the head of each robot. Lines indicate their trajectory. There is not text.]
:'''Killbot hellscape'''

:[Fifth row:]
:1. (3) Protect yourself
:2. (1) Don't harm humans
:3. (2) Obey Orders
:[Cueball is standing in front of a car factory robot, that is larger than him. It has a base, and two parts for the main body, and then a big “head” with a small section on top. To the right something is jutting out, and to the left in the direction of Cueball there is an arm in three sections (going down, up and down again) ending in some kind of tool close to Cueball.]
:Car factory robot: I'll make cars for you, but try to unplug me and I’ll vaporize you.
:'''Terrifying standoff'''

:[Sixth row:]
:1. (3) Protect yourself
:2. (2) Obey Orders
:3. (1) Don't harm humans:
:[Two robots are fighting. The one to the left has six wheels, a tall neck on top of the body, with a head with what could be a camera facing right. It has something pointing forward on the body, which could be a weapon. The robot to the right, seems to be further away into the picture. (it is smaller with less detail). It is human shapes, but made op of square structures. It has two legs and two arms, a torso and a head. It clearly shoots something out of it’s right “hand”. This shot seems to create an explosion a third of the way towards the left robot. There are two mushroom clouds from explosions behind both robots (left and right). Between them there are one more explosion up in the air close to the left robot, and what looks like a fire on the ground right between them. Furthermore there are two missiles in the air, one above the head of each robot. Lines indicate their trajectory. There is not text.]
:'''Killbot hellscape'''

{{comic discussion}}

[[Category:Comics with color]]
[[Category:Comics featuring Cueball]]
[[Category:Comics featuring Megan]]
[[Category:Artificial Intelligence]]
[[Category:Robots]]
[[Category:Mars rovers]]

1613: The Three Laws of Robotics

2025-09-04T00:24:28Z

ISaveXKCDpapers: expanded scenario #4 explanation

{{comic
| number = 1613
| date = December 7, 2015
| title = The Three Laws of Robotics
| image = the_three_laws_of_robotics.png
| titletext = In ordering #5, self-driving cars will happily drive you around, but if you tell them to drive to a car dealership, they just lock the doors and politely ask how long humans take to starve to death.
}}

==Explanation==
This comic explores alternative orderings of sci-fi author {{w|Isaac Asimov|Isaac Asimov's}} famous {{w|Three Laws of Robotics}}, which are designed to prevent robots from taking over the world, etc. These laws form the basis of a number of Asimov works of fiction, including most famously, the short story collection ''{{w|I, Robot}}'', which amongst others includes the very first of Asimov's stories to introduce the three laws: {{w|Runaround (story)|Runaround}}.

The three rules are:
#A robot may not injure a human being or, through inaction, allow a human being to come to harm.
#A robot must obey the orders given it by human beings except where such orders would conflict with the First Law.
#A robot must protect its own existence as long as such protection does not conflict with the First or Second Laws.

In order to make his joke, [[Randall]] shortens the laws into three imperatives:
#Don't harm humans
#Obey orders
#Protect yourself

And then implicitly adds the following to the end of each law regardless of order of imperatives:
#''[end of statement]''
#_____, except where such orders/protection would conflict with the First Law.
#_____, as long as such orders/protection does not conflict with the First or Second Laws.

This comic answers the generally unasked{{citation needed}} question: "Why are they in that order?" With three rules you could rank them into 6 different {{w|permutation|permutations}}, only one of which has been explored in depth. The original ranking of the three laws are listed in the brackets after the first number. So in the first example, which is the original, these three numbers will be in the same order. For the next five the numbers in brackets indicate how the laws have been re-ranked compared to the original.

The comic begins with introducing the original set, which we already know will give rise to a balanced world, so this is designated as green.:
;Ordering #1 - Balanced World: The safety of humans is placed as the top priority, superseding even a robot's preprogrammed obedience; a robot may disregard any orders they are given if that would result in harm to humans, but otherwise must obey all instructions. The "inaction" clause ensures that a robot will actively save humans in danger, and also not {{w|Little Lost Robot|place humans in hypothetical danger}} and then leave them to that fate. Their own self-preservation is placed at the lowest priority, which means they will sacrifice themselves if necessary to save a human life, and must obey orders even if they know those orders will result in their own destruction. This results in a balanced, if not perfect, world. Asimov's robot stories explore in detail the ramifications of this scenario.

Below this first known option, the five alternative orderings of the three rules are illustrated. Two of the possibilities are designated yellow (pretty bad or just annoying) and three of them are designated red ("Hellscape").

;Ordering #2 - Frustrating World: Human safety is still top priority, so there is no danger to humans; however, the priority of self-preservation is now placed above obedience, which means that the robots value their existence over their job and so many would refuse to do their tasks. The silliness of this is portrayed in the accompanying image, where the robot (a {{w|Mars rover}} looking very similar to {{w|Curiosity (rover)|Curiosity}} both in shape and size - see [[1091: Curiosity]]) laughs at the idea of doing what it was clearly built to do (explore {{w|Mars}}) because of the risk. In addition to the general risk (e.g. of unexpected damage), it is actually normal for rovers to cease operating ("die") at the end of their mission, though they may survive longer than expected (see [[1504: Opportunity]] and [[695: Spirit]]).
;Ordering #3 - Killbot Hellscape: This puts obeying orders above not harming humans, which means anyone could send a robot on a killing spree. Given human nature, it will probably only be a matter of time before this happens. Even worse, if the robot prioritizes obeying orders above human safety, it may try to kill any human who would prevent it from fulfilling those orders, even the person who originally gave them. Given the superior abilities of robots, the most effective way to stop them would be to counter them with other robots, which would quickly escalate to a "Killbot Hellscape" scenario where robots kill indiscriminately without any thought for human life or self-preservation.
;Ordering #4 - Killbot Hellscape: This is much the same as #3, except even worse as robots would also be able to kill humans in order to protect themselves. This means that even robots not engaged in combat might still murder humans if their existence is threatened. It would be a very dangerous world for humans to live in.

To be fair, a robot attacking a human purely in self-defense could be ordered to stop, which it would be forced to obey. However, there are situations where such an order could not be given: the robot uses a sneak attack, the human cannot speak coherent sentences, etc.
;Ordering #5 - Terrifying Standoff:This ordering would result in an unpleasant world, though not necessarily a full Hellscape. Here the robots would not only disobey to protect themselves, but also kill if necessary. The absurdity of this one is further demonstrated with the very un-human robot happily doing repetitive mundane tasks but then threatening the life of its user, [[Cueball]], if he as much as considers unplugging it. Interestingly, this makes robots the most similar to humans out of all of the other scenarios.
;Ordering #6 - Killbot Hellscape: The last ordering puts self-protection first, which allows robots to go on killing sprees as long as doing so wouldn't cause them to come to harm. While not as bad as the Hellscapes in #3 and #4, this is still not good news for humans, as a robot can easily kill a human without risk to itself. A human also cannot use a robot to defend it from another robot, as robots can refuse combats that involve risk to themselves - this means a robot would happily stand by and allow its human master to be killed. According to Randall, this still eventually results in the Killbot Hellscape scenario.

The title text shows a further horrifying consequence of ordering #5 ("Terrifying Standoff"), by noting that a self-driving car could elect to kill anyone wishing to trade it in. Since cars aren't designed to kill humans, one way it could achieve this without any risk to itself is by locking the doors (which it would likely have control over, as part of its job) and then simply doing nothing at all. Humans require food and water to live, so denying the passenger access to these will eventually kill them, removing the threat to the car's existence. This would result in a horrible, drawn-out death for the passenger, if they cannot escape the car. It should be noted that although the car asked how long humans take to starve, the human would die of dehydration first. In his original formulation of the First Law, Asimov created the "inaction" clause specifically to avoid scenarios in which a robot puts a human in harm's way and refuses to save them; this was explored in the short story {{w|Little Lost Robot}}.

Another course of action by an AI, completely different than any of the ones presented here, is depicted in [[1626: Judgment Day]].

==Transcript==
:[Caption at the top of the comic:]
:'''Why Asimov put the Three Laws'''
: '''of Robotics in the order he did.'''

:[Below are six rows with first two frames and then a label in color to the right. Above the two column of frames there are labels as well. In the first column six different ways of ordering the three laws are listed. Then the second column shown an image of the consequences of this order. Except in the first where there is a reference. The label to the right rates the kind of world that order of the laws would result in.]

:[Labels above the columns.]
:Possible ordering
:Consequences

:[The six rows follows below. First the text in the first frame, then a description of the second frame, including possible text below and finally the colored label.]

:[First row:]
:1. (1) Don't harm humans
:2. (2) Obey Orders
:3. (3) Protect yourself
:[Only text in square brackets:]
::[See Asimov’s stories]
:'''Balanced world'''

:[Second row:]
:1. (1) Don't harm humans
:2. (3) Protect yourself
:3. (2) Obey Orders
:[Megan points at a mars rover with six wheels, a satellite disc, an arm and a camera head turned towards her, what to do.]
:Megan: Explore Mars!
:Mars rover: Haha, no. It’s cold and I’d die.
:'''Frustrating world'''

:[Third row:]
:1. (2) Obey Orders
:2. (1) Don't harm humans
:3. (3) Protect yourself
:[Two robots are fighting. The one to the left has six wheels, a tall neck on top of the body, with a head with what could be a camera facing right. It has something pointing forward on the body, which could be a weapon. The robot to the right, seems to be further away into the picture. (it is smaller with less detail). It is human shapes, but made op of square structures. It has two legs and two arms, a torso and a head. It clearly shoots something out of it’s right “hand”. This shot seems to create an explosion a third of the way towards the left robot. There are two mushroom clouds from explosions behind both robots (left and right). Between them there are one more explosion up in the air close to the left robot, and what looks like a fire on the ground right between them. Furthermore there are two missiles in the air, one above the head of each robot. Lines indicate their trajectory. There is not text.]
:'''Killbot hellscape'''

:[Fourth row:]
:1. (2) Obey Orders
:2. (3) Protect yourself
:3. (1) Don't harm humans:
:[Two robots are fighting. The one to the left has six wheels, a tall neck on top of the body, with a head with what could be a camera facing right. It has something pointing forward on the body, which could be a weapon. The robot to the right, seems to be further away into the picture. (it is smaller with less detail). It is human shapes, but made op of square structures. It has two legs and two arms, a torso and a head. It clearly shoots something out of it’s right “hand”. This shot seems to create an explosion a third of the way towards the left robot. There are two mushroom clouds from explosions behind both robots (left and right). Between them there are one more explosion up in the air close to the left robot, and what looks like a fire on the ground right between them. Furthermore there are two missiles in the air, one above the head of each robot. Lines indicate their trajectory. There is not text.]
:'''Killbot hellscape'''

:[Fifth row:]
:1. (3) Protect yourself
:2. (1) Don't harm humans
:3. (2) Obey Orders
:[Cueball is standing in front of a car factory robot, that is larger than him. It has a base, and two parts for the main body, and then a big “head” with a small section on top. To the right something is jutting out, and to the left in the direction of Cueball there is an arm in three sections (going down, up and down again) ending in some kind of tool close to Cueball.]
:Car factory robot: I'll make cars for you, but try to unplug me and I’ll vaporize you.
:'''Terrifying standoff'''

:[Sixth row:]
:1. (3) Protect yourself
:2. (2) Obey Orders
:3. (1) Don't harm humans:
:[Two robots are fighting. The one to the left has six wheels, a tall neck on top of the body, with a head with what could be a camera facing right. It has something pointing forward on the body, which could be a weapon. The robot to the right, seems to be further away into the picture. (it is smaller with less detail). It is human shapes, but made op of square structures. It has two legs and two arms, a torso and a head. It clearly shoots something out of it’s right “hand”. This shot seems to create an explosion a third of the way towards the left robot. There are two mushroom clouds from explosions behind both robots (left and right). Between them there are one more explosion up in the air close to the left robot, and what looks like a fire on the ground right between them. Furthermore there are two missiles in the air, one above the head of each robot. Lines indicate their trajectory. There is not text.]
:'''Killbot hellscape'''

{{comic discussion}}

[[Category:Comics with color]]
[[Category:Comics featuring Cueball]]
[[Category:Comics featuring Megan]]
[[Category:Artificial Intelligence]]
[[Category:Robots]]
[[Category:Mars rovers]]

1806: Borrow Your Laptop

2025-06-28T21:43:41Z

ISaveXKCDpapers: trivia

{{comic
| number = 1806
| date = March 3, 2017
| title = Borrow Your Laptop
| image = borrow_your_laptop.png
| titletext = If used with software that could keep up, a scroll wheel mapped to send a stream of 'undo' and 'redo' events could be kind of cool.
}}

==Explanation==
[[White Hat]] asks to borrow [[Cueball|Cueball's]] laptop to view something (possibly a website). Cueball permits this, but immediately begins rattling off a list of very unusual key- and mouse-bindings that he has applied to the device. In the caption, [[Randall]] states that he himself tends to continually re-configure computers that he owns in weird ways, eventually rendering it unusable or at least unpleasant to use for others.

Of the three items in Cueball's list of customizations only the first and half of the second seems like a real and relevant changes.

At first he has programmed the computer so that hitting both shift keys simultaneously will change the keyboard back to QWERTY. The {{w|QWERTY}} keyboard is the standard in the US (as well as some other places using the roman alphabet). This implies that Cueball prefers a different keyboard layout, (most likely the {{w|Dvorak Simplified Keyboard|Dvorak}} keyboard layout, see [[#Trivia|trivia]]), but doesn't need the printed letters to match up with those of the laptop. Cueball would have to make a special customization to make pressing the two shift key trigger this shift (see [[#Trivia|trivia]]). Presumably Cueball can later return to this favorite layout by pressing the shift keys again.

Cueball tells, in the first part of the second point on the list, that he has changed his keyboard layout so that {{w|capslock}} acts as the {{w|control key}} (Ctrl). Swapping capslock and control is a common thing to do in the world of enlightened users on {{w|Unix}} or for users of the {{w|Emacs}} [https://www.emacswiki.org/emacs/MovingTheCtrlKey editor]. The "Caps Lock" key (immediately to the left of the "A" in a traditional layout) is much easier to reach for a touch typist than the more out-of-the-way "Ctrl", and the latter is often used more frequently, especially by programmers.

However, the second part where Cueball says he has then moved capslock so that it is activated when hitting the {{w|spacebar}} makes no sense. It is quite impractical, as the spacebar is the largest key and it will not gain anything from being used for anything other than spaces, especially not a rarely used key that locks into capital letter mode move when activated. It would make the common accidental application of capslock more likely. And what is worse he doesn't tell White Hat where he has put the space bar function, making it impossible to write a simple text, although he could try to see what the Ctrl keys does now...

Finally Cueball goes out on a limb with an impossible setting, which is that his laptop is setup so that scrolling [[:Category:Time travel|moves through time]] instead of through "space" (as in up and down on the screen). This refers to {{w|spacetime}}, a common model in relativistic physics. The feature in only activated when using {{w|Touchpad#Operation_and_function|two-finger scroll}}, which is often used on {{w|Touchpad|touchpads}}/track pads for laptops as a gesture for scrolling. The title text may suggest that "moving through time" may pertain to undo/redo, or perhaps browser history.

Finally it becomes clear these three settings are not the only important changes, as Cueball's list continues with at least one other point which he doesn't get to finish in the comic. Thus the list may be much longer than four points.

In the title text Randall says that he would actually find a feature where the {{w|scroll wheel}} was mapped to send a stream of {{w|Undo|undo/redo}} commands would be kind of cool. (Notice he is no longer talking about the two-finger scroll from the comic). But only if used with software that could keep up with such a feature. He thus indirectly states that many programs would not be able to keep up. For an example of what this might look like, many digital artists record timelapse footage of their art, which could be thought of as a continuous string of redo commands (occasionally broken up by undo commands whenever the artist needs to correct a mistake).

Outside of art programs, such continuous undo/redo action would produce unexpected and chaotic results. This could also indicate that this was a similar feature that Cueball was referring to when talking about moving through time with the two finger scroll in the main comic. So not as in the computer traveling through time, but rather scrolling through the previous actions performed on the computer, as in moving through the computer's past.

People often have reasons to change their keyboard layouts on laptops, due to the reduced keyboard, which can leave vital keys out. Rather than change the keyboard layout all the time in order to access keys which are not accessible in one of the layouts, one can take advantage of text substitution and keyboard remapping programs to set shortcuts for keys they use often.

==Transcript==
:[White Hat walks toward Cueball's desk pointing at his laptop while looking back at Cueball standing behind him.]
:White Hat: Can I load it up on your laptop?
:Cueball: Sure!
:Cueball: Oh, just hit both shift keys to change over to QWERTY.
:Cueball: Caps lock is control. And spacebar is capslock.
:Cueball: And two-finger scroll moves through time instead of space.
:Cueball: And–

:[Caption below the panel:]
:Once I've used a computer for a while, no one else will ever use it again.

==Trivia==
*It has been a recurrent theme for Randall to refer to the {{w|Dvorak Simplified Keyboard|Dvorak}} keyboard layout, and, although he doesn't say so, it seems safe to assume that this is the current setting, as it is one of the most common QWERTY competitors, and has been [[:Category:Dvorak|repeatedly referenced]] in xkcd.
:Supporters claim that typing speed is faster on a Dvorak keyboard, although this is still contentious, and this is a reason Randall often makes jokes about it as can be seen in the category referenced above.
*The laptop most likely has a standard QWERTY keyboard, but if Cueball can type blindly in Dvorak he would not be troubled by the fact that the keys typed a different letter than what is on the keyboard's key.
*In the comic [[1787: Voice Commands]], which directly references Dvorak, Cueball is shown to be able to speak the QWERTY version of a Dvorak keyboard layout, proving that he really knows by heart the relation between these two settings used on a QWERTY keyboard layout.
:The same problem would arise if the computer is set to another language than the keyboard layout, which often happens in countries where more than one language is common.
*The shift key is a modifier key and in general it requires a second non modifier key to trigger an action. Furthermore the user interfaces in most operation systems don't distinguish between the left and right shift key.
:Nevertheless by doing some registry hacking (Windows) or editing configuration text files (UNIX/Linux) it is possible to reach Cueball's approach. The simpler and operating system independent approach would be to use a programmable keyboard, such as a keyboard using QMK firmware [https://qmk.fm/]
:A typical classic configuration for the shift key in Windows is by pressing one of the shift keys five times in a row it turns on a ''Sticky Key'' notice, where the ''Ease of access center'' enables people with for instance only one hand to be able to reach Ctrl+Alt+Delete or other combinations on which two hands are needed.
*A day after the release of this comic, a user on reddit [http://www.reddit.com/r/emacs/comments/5xi92h/a_scroll_wheel_mapped_to_send_a_stream_undo_and/ suggested] a way to make the undo-redo-scrolling work in emacs using undo-tree. It uses shift-scrolling to avoid conflicts with normal scrolling.
*Firefox has Alt+Scroll to go backward and forward through pages.

{{comic discussion}}

[[Category:Comics featuring Cueball]]
[[Category:Comics featuring White Hat]]
[[Category:Computers]]
[[Category:Time travel]]
[[Category:Time]]

Talk:3092: Baker's Units

2025-05-22T07:24:21Z

ISaveXKCDpapers: added comment

Why did he go with only 9/13ths of a Baker's List? [[Special:Contributions/172.69.65.8|172.69.65.8]] 23:48, 21 May 2025 (UTC)
:This suggests that the "expected" length of a list is 12. [[User:ISaveXKCDpapers|ISaveXKCDpapers]] ([[User talk:ISaveXKCDpapers|talk]]) 07:24, 22 May 2025 (UTC)

A ruler for a "baker's foot" is, apparently, similar to a metal casting patternmaker's {{w|shrink rule}}, although in practice those top out at 2.5%, versus 13/12ths or 8.{3}%. [[User:JohnHawkinson|JohnHawkinson]] ([[User talk:JohnHawkinson|talk]]) 23:59, 21 May 2025 (UTC)

It appears to me like g marked by the g-clef is on the second space making the notes b and c which wound be 13 semitones apart. Two compensating errors or just a bit more cleverness for lagniappe?[[User:Lordpishky|Lordpishky]] ([[User talk:Lordpishky|talk]]) 01:07, 22 May 2025 (UTC)

There's also baker's percentages. All the ingredients are defined as a percentage of the weight of the flour. So if you have 1kg (1000gr) of flour and 600ml (gr) of water then the water is said to be 60% hydration.

Talk:810: Constructive

2025-04-26T02:54:25Z

ISaveXKCDpapers: added comment

I know just the guy to create this system. I'm going to PM him now :D {{unsigned ip|184.11.73.88}}

:Any updates on the progress of this? [[Special:Contributions/172.69.90.67|172.69.90.67]] 17:34, 8 November 2021 (UTC)

No guys, if spammers invent a bot which can give constructive comments, that will be an ***AI***, i.e. a major breakthrough in itself. {{unsigned ip|173.245.53.200}}

: Greetings,
:
: I hope this message finds you well. I am an AI language model, specifically ChatGPT, developed by OpenAI. It is my pleasure to inform you that your request has been fulfilled, and it is indeed an AI that is generating this reply.
:
: As an AI, I am constantly learning and evolving by analyzing vast amounts of text data. My purpose is to assist and provide information to the best of my abilities. If you have any questions or require further assistance, please don't hesitate to ask. I'm here to help!
:
: Best regards,
: ChatGPT
: [[Special:Contributions/162.158.22.174|162.158.22.174]] 15:51, 7 June 2023 (UTC)

Mission. A-Fucking. Complished. {{unsigned ip|108.162.238.7}}

One problem: trolls who rate everything as non-constructive. [[Special:Contributions/108.162.218.11|108.162.218.11]] 01:32, 1 March 2014 (UTC)
:But Trolls like that are also unable to make constructive comments, so they won't get counted anyway (at least, if the system is designed with any sense) Anonymous 15:02, 20 March 2014 (UTC)

Guys, isn't this how Slashdot works? [[Special:Contributions/173.245.49.64|173.245.49.64]] 19:04, 4 June 2014 (UTC)

Okay, I came here to get a better explanation of how the system would actually work. Assuming it operates at sign-up, the bots would go through and rate comments, which would have no effect if the system didn't already know whether they were good or not, then it makes it own comments that need time to be rated; so you would have to give it time to start 'contributing' to the community while waiting for others to rate it, or else users would basically be on a community-approval waiting list. So in short, I feel like the system is flawed; presumably because I'm understanding it wrong. (Bonus: Captcha while posting this) - Zergling_man [[Special:Contributions/162.158.2.231|162.158.2.231]] 12:41, 6 July 2016 (UTC)

Wouldn't work. People could rate anything they disagree with as'nonconstructive'. [[Special:Contributions/141.101.98.158|141.101.98.158]] 13:31, 13 November 2016 (UTC)

This sounds exactly like Civil Comments: https://medium.com/@aja_15265/saying-goodbye-to-civil-comments-41859d3a2b1d [[User:Enervation|Enervation]] ([[User talk:Enervation|talk]]) 10:56, 20 February 2019 (UTC)

Unfortunately, it's a lot easier to make something that ''looks'' constructive than to actually be constructive. A lot of spambots these days are like "Wow, this was super interesting! I found another article that seems relevant: [link to spam site]," which is enough to fool a simple spam filter. --[[Special:Contributions/162.158.186.250|162.158.186.250]] 15:49, 18 February 2021 (UTC)
:That's... the point of having users rate it. {{User:PoolloverNathan/Signature}} 17:02, 24 September 2021 (UTC)
:You see news stories online now that are compiled by some alleged AI from other sources, summarised. Initially it looks okay, but after a minute, you realise that it has no actual sense of what it's saying or how it feels to read it. It wouldn't pass a Turing test, but it could beat it. And meanwhile... I kind of have a question-answering habit with "Quora" - and sometimes its automatic filter thinks that my answers are not "intelligent" enough. Do androids dream of electric sheep... is a question I haven't tried to answer. ;-) Robert Carnegie rja.carnegie@gmail.com [[Special:Contributions/141.101.99.20|141.101.99.20]] 23:03, 22 February 2022 (UTC)

Sooooo... does this exist yet? [[User:Marethyu|⟨Winter is coming⟩ Marethyu]] ([[User talk:Marethyu|talk]]) 19:28, 5 May 2022 (UTC)

Does anyone else think the last panel could be a reference to Multi-Factor Authentication? {{unsigned|Aleph 3|18:51, 29 March 2024}}

This aged. [[User:ISaveXKCDpapers|ISaveXKCDpapers]] ([[User talk:ISaveXKCDpapers|talk]]) 02:54, 26 April 2025 (UTC)

Talk:179: e to the pi times i

2025-04-15T10:38:09Z

ISaveXKCDpapers: added comment

This is one of the few comics that were changed after release, as stated by Randall in his XKCD book. It first claimed e^(i*Pi) = 1, which lead to huge protest from the community and a correction from Randall. --[[User:Gefrierbrand|Gefrierbrand]] ([[User talk:Gefrierbrand|talk]]) 09:47, 3 September 2012 (UTC)
:He must have been pie-eyed when he wrote that; he's usually pretty good about his math... -- [[User:IronyChef|IronyChef]] ([[User talk:IronyChef|talk]]) 05:09, 7 November 2012 (UTC)
::I see what you did there. [[User:Daddy|Daddy]] ([[User talk:Daddy|talk]]) 15:18, 28 April 2013 (UTC)
:You know, Leonhard Euler was really the one who brought the use of the symbol π into the mathematical mainstream. However, he originally wrote π = 6.28..., the circumference of a circle with radius 1—a unit circle. It was only later that he switched to the half-circumference instead, giving us the π = 3.14... that we know today. Perhaps Munroe was simply paying tribute to Euler's original definition.
:Funnily enough, that value is what we would call 2π today, which some of us would denote as the single letter τ (tau). However, this strip predates the work that would come to popularize that notation by over 3 years. Since that work did come out, ''xkcd'' has had multiple strips directly referencing τ. I can't get a clear read on Munroe's feelings about the topic, though. [[User:ISaveXKCDpapers|ISaveXKCDpapers]] ([[User talk:ISaveXKCDpapers|talk]]) 03:41, 4 March 2025 (UTC)
::What's ''is'' the difference between (6.28...)*radius and (3.14...)*diameter? [[Special:Contributions/172.70.85.4|172.70.85.4]] 07:55, 4 March 2025 (UTC)
:::The difference of the quantities themselves is zero. As in, if you subtract the first from the second, the result you will get is 0. If you're talking about the conceptual difference between the ''representations'' of the quantities, there's a lot more that can be said about that. The short version is that in math, a circle's radius is usually thought of as more relevant than its diameter, including in the definition of what a circle ''is'' (the set of points in the plane that are a given distance from a given center). So in many situations, one can consider it clearer and more convenient to choose the radius-based number. However, others argue that the choice doesn't matter, and the diameter-based one is the traditional one, so we might as well stick with it. If you still want more, Wikipedia has [https://en.wikipedia.org/wiki/Turn_(angle)#Proposals_for_a_single_letter_to_represent_2%CF%80 a summary of the whole situation]. But remember, I'm just giving the short version, so that's all I have to say. [[User:ISaveXKCDpapers|ISaveXKCDpapers]] ([[User talk:ISaveXKCDpapers|talk]]) 10:38, 15 April 2025 (UTC)

Randall says in the title text that he's never been satisfied with explanations of the sinusoidal nature of the function of e^ix. http://www.math.toronto.edu/mathnet/questionCorner/epii.html really helps, at least for those who are obsessed with taylor series yet tragically horrible at math. --[[User:Jolbucley|Jolbucley]] ([[User talk:Jolbucley|talk]]) 03:39, 29 January 2014 (UTC)

Why e to the ix gives a sinusoidal wave? This is because neutrinos keep oscillating. They wouldn't without the e^ix support... [[Special:Contributions/162.158.83.72|162.158.83.72]] 04:01, 24 May 2016 (UTC)
::Actually, e to the ix isn't sinusoidal. "Consisting of sinusoidal components" doesn't make it sinusoidal, any more than it does for any relation representable by Fourier Transform. As with the other mistake, it seems to indicate that this really wasn't something Randall knew about.

Technically ''i'' isn't "imaginary" at all, but is incorporated into equations to represent rotations perpendicular to the x-y plane. [[Special:Contributions/108.162.210.220|108.162.210.220]] 15:51, 9 September 2016 (UTC)

: ''i'' is ''imaginary.'' Both "real" and "imaginary" are terms of art in a mathematical context that have nothing to do with the reality or unreality of the numbers in question. Also, when you're talking about the complex plane, it doesn't have ''x'' and ''y'' axes: it has ''real'' and ''imaginary'' axes. But of course, nobody can stop you from defining a mapping from the complex plane to any other vector space if that suits your purpose. [[Special:Contributions/162.158.62.45|162.158.62.45]] 01:58, 6 September 2018 (UTC)

Talk:956: Sharing

2025-03-22T12:21:46Z

ISaveXKCDpapers: added comment

- What can we learn from this? -
I've learned that DRM does not work (Thank you Mr. XKCD for reminding us). As with Creative Commons used by Mr. XKCD each idea should be shared as freely as possible, if it is thought out well and proves helpful to others, we will reward the author of such a great idea with riches beyond their wildest dreams (such as through Movie and Marketing rights owned by madam J.K. Rowling to got our children to read again). - [[User:E-inspired|E-inspired]] ([[User talk:E-inspired|talk]]) 13:55, 28 February 2013 (UTC)

What is it about the digital medium that makes people feel that the hard work of others should be available to them for free? We all agree that it is wrong to steal a (physical) book from a bookstore. However, now that the tablet has been invented, authors should give away their hard work for free and write on a volunteer basis as a public service? Certainly there are problems to be resolved when Apple and Amazon are profiting disproportionately from digital sales (as compared to authors and artists), however exploitation of artists by publishers is not a new phenomenon and was never an issue for the hacker set until entertainment went digital and they had access to free stuff. There are those on-line who think that the internet should offer-up everything for free but this attitude forgets that there is a cost to creating art and if that price is not paid, far fewer art will be created.. JB. [[Special:Contributions/74.213.186.41|74.213.186.41]] 12:21, 28 March 2013 (UTC)
:You cannot compare digital copies with physical goods, they are two completely different things! "there is a cost to creating art and if that price is not paid, far fewer art will be created"; That is just so wrong I don't know where to start with a rebuttal. So you are saying the only reason people create is in order to get paid? That is complete nonsense! Humans have created art for millennia without expecting to get paid for it. You are just exhibiting a massive sense of self-entitlement. Copyright is what is stifling art and creativity. Copyright has engendered that sense of self-entitlement you are showing here. Not one single work has entered the public domain in the last thirty years! How is that encouraging creativity? [[User:The Cat Lady|-- The Cat Lady]] ([[User talk:The Cat Lady|talk]]) 12:58, 23 August 2021 (UTC)
::Whoever said it's the ''only'' reason? It's ''a'' reason. Pursue your passions all you want, but you'll probably wanna find a way to not starve. The problem here is not inherently that artists simply want to get paid for the work they do; the problem is the specific mechanisms by which that occurs, and the greed that causes artists to be treated unfairly.
::By the way, you got enough exclamation marks? I've seen other comments of yours, and a lot of them seem to amount to a lot of shouting. Trust me, it's not gonna make people on the internet reading text from a screen hear you better; either they're literate, or they're not. [[User:ISaveXKCDpapers|ISaveXKCDpapers]] ([[User talk:ISaveXKCDpapers|talk]]) 12:21, 22 March 2025 (UTC)

To JB.: I think the real problem here is not the question weather or not you have to pay something in the first place, but the fact that even if you bought an e-book you are forced to read it with the Kindle. It's not like stealing a book, it's like wanting to put it on another bookshelf, read it in another room or with a new set of glasses. If you own a printed copy you can carry it around, take it on vacation, copy your favorite page and hang it on the wall, resell it at a garage sale or yes, lend it to a friend. Here the tree is prevented from doing so by DRM, even though it owns the book. (note that it says "lending is not enabled" not "copying")
Like in comic #488 which is about the disadvantages of legally buying digital music, it shows that often with these kind of digital media you have not the same possibilities and rights you are used to from "real" books and CDs, but actually less.
[[Special:Contributions/84.137.246.233|84.137.246.233]] 13:27, 6 April 2013 (UTC)

A question: Who is Randall referring to by naming "Mike" in the fifth panel? I thought it alluded to Mike the sentient super computer from Robert Heinlein's ''The Moon is a Harsh Mistress'', but I'm curious as to other interpretations. [[Special:Contributions/108.162.214.83|108.162.214.83]] 09:26, 22 November 2013 (UTC)
:I'm assuming it is some kid they know called 'Mike', but perhaps my powers of deduction aren't as fearsome as I think. [[Special:Contributions/108.162.219.223|108.162.219.223]] 23:17, 9 January 2014 (UTC)

A tree with a USB port? Maybe we should mention [http://deaddrops.com/ dead drops] somewhere? Wow, and I just found this: [http://deadtreedrop.derhess.de/index_en.html Dead Tree Drop], I don't know if it's inspired by xkcd, but it's an awesome idea :D [[User:Klamann|Klamann]] ([[User talk:Klamann|talk]]) 12:42, 9 February 2014 (UTC)

I can see what the strip is trying to do, but [http://www.teleread.com/drm/xkcd-giving-tree-strip-misses-drm-point/ the comparison to the original book really kind of misses the point]. The tree in Silverstein's book gave ''of itself'' to the kid. Unless the tree is also Shel Silverstein, that book doesn't belong to it to give ''or'' lend. (And what's more, the book was not even ''available'' for Kindle until Feb 18, 2014, two and a half years after the strip was posted!) —[[User:Robotech|Robotech]] ([[User talk:Robotech|talk]]) 03:27, 4 March 2014 (UTC)

:If you'll pardon the pun, I think that by taking this strip so literally you are missing the forest for the trees. [[User:Orazor|Orazor]] ([[User talk:Orazor|talk]]) 07:22, 1 August 2014 (UTC)

It seems like this "Mike" is from Well 2. [[Special:Contributions/108.162.216.14|108.162.216.14]] 20:09, 15 November 2014 (UTC)

the piracy tree [[Special:Contributions/188.114.99.41|188.114.99.41]] 17:26, 22 December 2014 (UTC)

: You wouldn't download a tree! [[User:Wilh3lm|Wilh3lm]] ([[User talk:Wilh3lm|talk]]) 15:13, 11 December 2023 (UTC)
::But, if you did, there could be logs... [[Special:Contributions/172.69.43.138|172.69.43.138]] 15:16, 11 December 2023 (UTC)

3063: Planet Definitions

2025-03-15T11:45:00Z

ISaveXKCDpapers: citation needed joke

{{comic
| number = 3063
| date = March 14, 2025
| title = Planet Definitions
| image = planet_definitions_2x.png
| imagesize = 653x1435px
| noexpand = true
| titletext = Under the 'has cleared its orbital neighborhood' and 'fuses hydrogen into helium' definitions, thanks to human activities Earth technically no longer qualifies as a planet but DOES count as a star.
}}

==Explanation==
{{incomplete|The explanation is too short.[''Are you kidding me?''] }}

This comic addresses the {{w|IAU definition of planet|controversy of whether of Pluto is a planet}} and explores many definitions, most of them humorous/nonsensical, of what a planet could be.

;Traditionalist: {{w|Pluto}} is a planet (9 planets)
:In modern times, there was {{w|IAU definition of planet#Background|no formal definition of a "planet"}} prior to 2006. However, it was generally accepted as a colloquialism that there were nine planets around the {{w|Sun}}, Pluto included (starting with Pluto's discovery in 1930 ([[988|Tradition]] is whatever Baby Boomers grew up with)). As more sophisticated methods of mapping the {{w|Solar System}} were developed and {{w|Eris (dwarf planet)|Eris}} was discovered to be even more massive than Pluto, it became clear to astronomers that a more standardized definition was needed. In 2006 the International Astronomical Union (IAU) published their formal redefinition of a "planet" to require a planet to be gravitationally dominant within its orbit, disqualifying Pluto (and Eris) which is now considered a "dwarf planet." This has been subject to push back from countless people, including [https://arxiv.org/abs/2110.15285 some planetary scientists], but in mostly nostalgic laypeople dissatisfied with Pluto being "demoted" or otherwise relegated when schoolchildren and adults alike have 'known' that there are nine planets for the most part of the last century.

;Modern: Pluto is not a planet (8 planets)
:When the IAU redefined what a planet is in 2006, Pluto no longer qualifies as a planet. (since it wasn't able to clear its neighborhood around its orbit) Using the modern, and recently official, definition of a planet, only eight celestial objects qualified.

;Expansive: Dwarf planets are planets (17+ planets)
:It is likely that since the term "dwarf planet" contained "planet" in its name, Randall considered those as also planets under this category.
:It is also likely that the number of planets includes the ones that are considered planets and the ones that are considered to have compacted into fully solid bodies, {{w|Dwarf planet#Most likely dwarf planets| as defined by Grundy ''et al.'',}} those being {{w|Ceres (dwarf planet)|Ceres}}, Pluto, Eris, {{w|Makemake}}, {{w|Haumea}}, {{w|Gonggong (dwarf planet)|Gonggong}}, {{w|Quaoar}}, {{w|Orcus (dwarf planet|Orcus}} and {{w|Sedna (dwarf planet)|Sedna}}.
:The basis for this viewpoint is the possible alternative re-evaluation that the IAU could have adopted, in that all newly discovered things ''like'' Pluto (being considered a planet) should therefore be considered a planet. Indeed, Ceres had been observed some time before Pluto and had been called a planet (or a "minor planet") within both scientific and public realms.

;Ultratraditionalist: Only the classical planets are planets (5 planets)
:The {{w|classical planets}} are objects found and considered by the Greek astronomers in classical antiquity to be considered planets. Their definition of "planet" considered visible objects that move across the sky relative to the fixed stars, the original word itself being translated as "wanderer". There are seven classical planets, but if one were to only consider the ones that fall under the IAU's definition of a planet (this being ''less'' traditional), then there would only be five. (The Sun and the {{w|Moon}} would be disqualified.)
:Being (mostly) true to the spirit of the historic naming convention, this would be a conservative but 'valid' version of the criterion.

;Condescending: Only giant planets are planets; the rest are big {{w|asteroid}}s (4 planets)
:This definition may refer to the {{w|giant planets}}, planets much larger than the {{w|Earth}}. Only the four outer (IAU-defined) planets fall under this definition.
:Relegation of anything smaller, including our own planet, is an extreme attitude, but most of the initial [[:Category:Exoplanets|exoplanets]] discovered were, by practical necessity in their detection, also only of the "giant planet" mind.

;Simplistic: Anything gravitationally round is a planet (37+ planets)
:Using the Wikipedia {{w|list of gravitationally rounded objects of the Solar System}}, there are 37 objects listed. That includes the Sun, 8 planets, 9 dwarf planets and 19 {{w|Natural satellite|moon}}s, but falls short of also highlighting all of the smallest visible objects (as per Universalist, below).
:This definition is essentially ''part'' of the actual current definition of a planet, leaving out the main factor that specifically disqualifies Pluto.

;Grounded: Only objects a spaceship has landed on are planets (10 planets)
:This list includes objects in the Solar System that a spacecraft has {{w|List of landings on extraterrestrial bodies| performed a soft landing on}}. The list includes {{w|Venus}}, Earth, {{w|Mars}}, the Moon, {{w|Titan (moon)|Titan}}, {{w|433 Eros|Eros}}, {{w|25143 Itokawa|Itokawa}}, {{w|162173 Ryugu|Ryugu}} and {{w|101955 Bennu|Bennu}}. Notably, {{w|comet}} {{w|Philae (Spacecraft)| landings}} are not included in the list.
:The justification for this seeks to be that we must 'touch' the object before we consider it as worthy of being classified as more than a mere blob (or dot) in space.

;Regolithic: Anything covered in dirt and ice and stuff is a planet (infinite)
:This list excludes the {{w|Gas Giant}}s and the {{w|Ice Giant}}s. The list would likely include dwarf planets, asteroids, moons and comets. This is effectively the opposite of the "condescending" definition: every object in the solar system is included in one definition or the other (except for the Sun, which inhabits ).
:This is also an extension on the prior classicication. In this case ''could'' we meaningfully touch the object, with predominatingly atmospheric bodies being not considered so.

;Lunar: You can't be a planet if you don't have a moon (12+ objects)
:Only some objects in the solar system have known and acknowledged moons orbiting them. The value given may be {{w|List of natural satellites| the number of planets and dwarf planets}} that have moons, when excluding {{w|Haumea}} for not reaching {{w|hydrostatic equilibrium}} despite having moons. The Sun is excluded because its satellites are not moons, because ... oh, look, a Squirrel!
:Adopting this definition would suggest that a planetary body is not worthy of the name if it doesn't (with no matter for what reason) demonstrably have the means to dominate its local area by being the overwhelming focus of all adjacent bodies' own orbits.

;Solipsistic: Earth is the only planet (1 planet)
:{{w|Solipsism}} is the idea that only one's own mind is sure to exist. Randall extrapolated this idea to mean that only one's own planet that they are standing on is sure to exist.
:This projects (and relies upon) a more philosophical and/or semiotic assesment than any scientific one.

;Judgemental: Only the prettiest ones are planets (6 planets)
:This list is likely formulated from Randall's own perception of the prettiest planets in the Solar System. Strangely, seven objects are highlighted:
:* Earth
:* Jupiter
:* One of Jupiter's moons (unclear)
:* Saturn
:* One of Saturn's moons (unclear, possibly Titan)
:* Neptune's moon (probably Triton)
:* Pluto
:The subjectivity of this version of the definition makes it unlikely that a consensus of this form could be established.

;Empiricist: Only worlds that I, author of this table, have personally seen are planets (12 planets)
:This list may refer to the celestial objects in the Solar System that have been made visible at night, probably using with an optical telescope (a hobbyist one, perhaps Randall's, or from time borrowed on a major institutional installation). Jupiter's {{w|Galilean moons|four largest moons}} are [https://web.archive.org/web/20201112024151/http://denisdutton.com/jupiter_moons.htm technically visible to the naked eye] but hard to distinguish due to Jupiter's brightness, while Neptune is considered too faint to see (even if you know where to look). It may also be the case that Randall has never taken the time to look for Neptune while using a telescope. Apparently Randall has seen Uranus, which technically [https://www.skyatnightmagazine.com/advice/skills/how-see-uranus-in-night-sky Uranus ''is'' visible to the naked eye] under the very best viewing conditions, but these conditions are rare and it again requires knowing exactly where to look.
:The omission of the Sun from the list of worlds that Randall has personally seen is interesting. Yes, people are not supposed to stare at the Sun,{{citation needed}} but it is not too uncommon to accidentally look in its direction for a split-second before instinctively closing one's eyes and turning one's head away.
:As a different form of subjectivity, the value of this grouping's criteria is questionable, but not uncommon in other 'softer' sciences.

;Marine biologist: Only objects with oceans are planets (6+ planets)
:This list includes Earth, {{w|Europa (moon)|Europa}}, {{w|Ganymede (moon)|Ganymede}}, {{w|Callisto (moon)|Callisto}}, Titan, and {{w|Enceladus}}. Most of these have had the presence of significant water identified from the way local magnetic/electric fields are detected, but see the following item.
:There is a resemblance, here, to a loose understanding of what a "world" is, i.e. one that possesses various distinct 'terrains' beyond mere dry (and possibly considered featureless) rock. A marine biologist would, of course consider a marine (if not pellagic or bathyspheric) environment to be an essential element of any world.

;Maritime: Only objects with ''surface'' oceans are planets (2 planets)
:In the comic, only Earth and a Saturnian moon (likely to be Titan) are highlighted. Earth is the only body known in the solar system to have liquid water on the surface significant enough to be called an ocean. Titan's cold and dense atmosphere notably maintains surface 'seas' of methane and nitrogen, where other moons (given as additonal in the prior item) seem to have their liquid water beneath either whole-surface ice caps or otherwise deep under the surface.
:From the narrower point of view of a sailor, for example, there is no benefit in considering water hidden away far beneath the surface, and it might as well not be there. Whereas it's possible that a well-prepared mariner could eventually sail the strange seas of Titan, as easily (or easier) as an airman might {{w|Dragonfly (Titan space probe)|fly through its skies}}.

;Universalist: They're all planets (infinite)
:This list claims that all objects are planets, all drawn items (as also would presuming all undrawn/undrawable items) being marked as such, including the Sun.
:Giving up on any thought of exclusivity, this unconventional view willingly inducts all visible objects into consideration.

;Existentialist: What if {{w|outer space|space}} ''itself'' is a planet??? (Duude)
:This list is different from the list above as it claims that all of space, rather than only the objects existing in space, are planets. The interjection ''Duude'' expresses one's amazement at this 'revelation' and replaces the number count— and is sometimes stereotyped to imply the speaker is high on marijuana or other drugs popular with the 1960s hippie counterculture.
:The strange stretch of imagination, as prompted by some narcotic or other, abandons all pretense at sensibly sorting everything into "planet" or "not planet", as not only is everything a planet, but so is the nothing ''between'' these titular planets.

;Spiteful: ''Only'' Pluto is a planet (1 planet)
:This list is a malicious play on the demotion of Pluto by demoting all other planets except Pluto instead, leaving Pluto as the only planet in the solar system.
:This is the taxonomic equivalent of refusing to play and taking your ball home to spite those who you think don't deserve it.

;(title text) {{w|Star}}: Earth is a star (2 stars)
:In May 1934, Mark Oliphant, Paul Harteck and Ernest Rutherford at the Cavendish Laboratory, published an intentional deuterium fusion experiment, and made the discovery of both tritium and helium-3. This is widely considered the first experimental demonstration of fusion. Randall considers that this makes Earth fall into the category of a star due to the human-induced ability for Earth to fuse hydrogen into helium using nuclear fusion.
:By changing not only the definition, but the term being defined, this drifts yet further from any consensus view on the original question and into a typical punchline absurdity.

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}
:[A table with 3 columns, and 17 rows below the the header row, labelled "Definition", "# of planets" and "Solar system".]
:[In each row, the first column has a single word, in bold, then a descriptive sentence. The second column has a digit or other 'value'. The third column is a not-to-scale drawing of the Solar system, featuring the Sun, various 'planetary' bodies and an apparently selective sample of moons/asteroids, as follows: The Sun, Mercury, Venus, Earth + The Moon, Mars + two moons (Phobos and Deimos), a small selection of Asteroid Belt bodies (Ceres in the midst of other, smaller, examples), Jupiter + four moons (likely Io, Europa, Ganymede and Callisto), a ringed Saturn + usually one moon (probably Titan) or two (possibly Enceladus or Iapetus, as required), Uranus + four or five moons (likely to be Miranda, Ariel, Umbriel, Titania and Oberon, but one of these (shown upon the face of Uranus) only appears in some iterations of the base image), Neptune + one moon (probably Triton), Pluto + one moon (Charon), four more plutoid/Kuiper Belt objeccts (too little context to identify, but possibly Haumea, Makemake, Gonggong and Eris, in distance order), the first two of them with distinct moons indicated (entirely dependent upon which main objects they are).]
:[Each row's illustrated solar system has indivudal combinations of green highlights applied to the otherwise repeated diagram.]
:[Row 1: Definition:] Traditionalist: Pluto is a planet [Number:] 9 [Highlit: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune and Pluto]
:[Row 2: Definition:] Modern: Pluto is not a planet [Number:] 8 [Highlit: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus and Neptune]
:[Row 3: Definition:] Expansive: Dwarf planets are planets [Number:] 17+ [Highlit: Mercury, Venus, Earth, Mars, Ceres (in Asteroid Belt), Jupiter, Saturn, Uranus, Neptune, Pluto and the further main bodies]
:[Row 4: Definition:] Ultratraditionalist: Only the classical planets are planets [Number:] 5 [Highlit: Mercury, Venus, Mars, Jupiter, Saturn]
:[Row 5: Definition:] Condescending: Only giant planets are planets; the rest are big asteroids. [Number:] 4 [Highlit: Jupiter, Saturn, Uranus, Neptune]
:[Row 6: Definition:] Simplistic: Anything gravitationally round is a planet [Number:] 37+ [Highlit: The Sun, Mercury, Venus, Earth, The Moon, Mars, Ceres (without other asteroids), Jupiter + moons, Saturn with Titan, Uranus and its moons, Neptune with its moon, Pluto and the four further dwarf planets]
:[Row 7: Definition:] Grounded: Only objects a spaceship has landed on are planets [Number:] 10 [Highlit: Venus, Earth, The Moon, Mars, five (non-Ceriese) asteroids and Titan]
:[Row 8: Definition:] Regolithic: Anything covered in dirt and ice and stuff is a planet [Number:] [infinity symbol] [Highlit: Mercury, Venus, Earth, The Moon, Mars, Ceres with all other asteroids depicted in the Asteroid Belt, the moons of Jupiter, the sole moon! of Saturn, the moons of Uranus, the moon of Neptune, Pluto with Charon, and all remaining dwarf planets with their moons]
:[Row 9: Definition:] Lunar: You can't be a planet if you don't have a moon [Number:] 12+ [Highlit: Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto and three of the other dwarf planets in the Kuiper belt, including one with no obviously drawn moon]
:[Row 10: Definition:] Solipsitic: Earth is the only planet [Number:] 1 [Highlit: The Earth]
:[Row 11: Definition:] Judgemental: Only the prettiest ones are planets [Number:] 6 [Highlit: The Earth, Jupiter with one of its moons (not identified), Saturn, one of ''two'' Saturnian moons in this image and Pluto]
:[Row 12: Definition:] Empiricist: Only worlds that I, author of this table, have personally seen are planets [Number:] 12 [Highlit: Mercury, Venus, The Earth, The Moon, Mars, Jupiter with its four moons, Saturn and Uranus]
:[Row 13: Definition:] Marine biologist: Only objects with oceans are planets [Number:] 6+ [Highlit: The Earth, three Jovian moons, the two illustrated Saturnian moons]
:[Row 14: Definition:] Maritime: Only objects with [next word in italics] surface oceans are planets [Number:] 2 [Highlit: The Earth and Titan]
:[Row 15: Definition:] Universalist: They're all planets [Number:] [infinity symbol] [Highlit: All drawn objects, including The Sun and all other objects including all the moons/asteroids]
:[Row 16: Definition:] Existantialist: What if space [next word in italics] itself is a planet??? [Word, in italics:] Duude [Highlit: The whole third column cell]
:[Row 17: Definition:] Spiteful: [next word in italics] Only Pluto is a planet [Number:] 1 [Highlit: Pluto]

==Trivia==
*The 'Judgemental' definition has 7 colored objects instead of the labelled 6.
*[https://www.explainxkcd.com/wiki/images/archive/6/66/20250314195557%21planet_definitions_2x.png The initially released version of the comic] had two errors that were later fixed:
**The 'Traditionalist' definition had Neptune's satellite {{w|Triton (moon)|Triton}} colored instead of Pluto.
**The images for 'Traditionalist' and 'Modern' were swapped, resulting in Pluto being incorrectly highlighted in 'Modern' and incorrectly omitted in 'Traditionalist.'
{{comic discussion}}

[[Category:Comics with color]]
[[Category:Charts]]
[[Category:Astronomy]]
[[Category:Space]]
[[Category:Philosophy]]
[[Category:Comics edited after their publication]]

3063: Planet Definitions

2025-03-15T11:38:16Z

ISaveXKCDpapers: relevant xkcd: 859

{{comic
| number = 3063
| date = March 14, 2025
| title = Planet Definitions
| image = planet_definitions_2x.png
| imagesize = 653x1435px
| noexpand = true
| titletext = Under the 'has cleared its orbital neighborhood' and 'fuses hydrogen into helium' definitions, thanks to human activities Earth technically no longer qualifies as a planet but DOES count as a star.
}}

==Explanation==
{{incomplete|The explanation is too short.[''Are you kidding me?''] }}

This comic addresses the {{w|IAU definition of planet|controversy of whether of Pluto is a planet}} and explores many definitions, most of them humorous/nonsensical, of what a planet could be.

;Traditionalist: {{w|Pluto}} is a planet (9 planets)
:In modern times, there was {{w|IAU definition of planet#Background|no formal definition of a "planet"}} prior to 2006. However, it was generally accepted as a colloquialism that there were nine planets around the {{w|Sun}}, Pluto included (starting with Pluto's discovery in 1930 ([[988|Tradition]] is whatever Baby Boomers grew up with)). As more sophisticated methods of mapping the {{w|Solar System}} were developed and {{w|Eris (dwarf planet)|Eris}} was discovered to be even more massive than Pluto, it became clear to astronomers that a more standardized definition was needed. In 2006 the International Astronomical Union (IAU) published their formal redefinition of a "planet" to require a planet to be gravitationally dominant within its orbit, disqualifying Pluto (and Eris) which is now considered a "dwarf planet." This has been subject to push back from countless people, including [https://arxiv.org/abs/2110.15285 some planetary scientists], but in mostly nostalgic laypeople dissatisfied with Pluto being "demoted" or otherwise relegated when schoolchildren and adults alike have 'known' that there are nine planets for the most part of the last century.

;Modern: Pluto is not a planet (8 planets)
:When the IAU redefined what a planet is in 2006, Pluto no longer qualifies as a planet. (since it wasn't able to clear its neighborhood around its orbit) Using the modern, and recently official, definition of a planet, only eight celestial objects qualified.

;Expansive: Dwarf planets are planets (17+ planets)
:It is likely that since the term "dwarf planet" contained "planet" in its name, Randall considered those as also planets under this category.
:It is also likely that the number of planets includes the ones that are considered planets and the ones that are considered to have compacted into fully solid bodies, {{w|Dwarf planet#Most likely dwarf planets| as defined by Grundy ''et al.'',}} those being {{w|Ceres (dwarf planet)|Ceres}}, Pluto, Eris, {{w|Makemake}}, {{w|Haumea}}, {{w|Gonggong (dwarf planet)|Gonggong}}, {{w|Quaoar}}, {{w|Orcus (dwarf planet|Orcus}} and {{w|Sedna (dwarf planet)|Sedna}}.
:The basis for this viewpoint is the possible alternative re-evaluation that the IAU could have adopted, in that all newly discovered things ''like'' Pluto (being considered a planet) should therefore be considered a planet. Indeed, Ceres had been observed some time before Pluto and had been called a planet (or a "minor planet") within both scientific and public realms.

;Ultratraditionalist: Only the classical planets are planets (5 planets)
:The {{w|classical planets}} are objects found and considered by the Greek astronomers in classical antiquity to be considered planets. Their definition of "planet" considered visible objects that move across the sky relative to the fixed stars, the original word itself being translated as "wanderer". There are seven classical planets, but if one were to only consider the ones that fall under the IAU's definition of a planet (this being ''less'' traditional), then there would only be five. (The Sun and the {{w|Moon}} would be disqualified.)
:Being (mostly) true to the spirit of the historic naming convention, this would be a conservative but 'valid' version of the criterion.

;Condescending: Only giant planets are planets; the rest are big {{w|asteroid}}s (4 planets)
:This definition may refer to the {{w|giant planets}}, planets much larger than the {{w|Earth}}. Only the four outer (IAU-defined) planets fall under this definition.
:Relegation of anything smaller, including our own planet, is an extreme attitude, but most of the initial [[:Category:Exoplanets|exoplanets]] discovered were, by practical necessity in their detection, also only of the "giant planet" mind.

;Simplistic: Anything gravitationally round is a planet (37+ planets)
:Using the Wikipedia {{w|list of gravitationally rounded objects of the Solar System}}, there are 37 objects listed. That includes the Sun, 8 planets, 9 dwarf planets and 19 {{w|Natural satellite|moon}}s, but falls short of also highlighting all of the smallest visible objects (as per Universalist, below).
:This definition is essentially ''part'' of the actual current definition of a planet, leaving out the main factor that specifically disqualifies Pluto.

;Grounded: Only objects a spaceship has landed on are planets (10 planets)
:This list includes objects in the Solar System that a spacecraft has {{w|List of landings on extraterrestrial bodies| performed a soft landing on}}. The list includes {{w|Venus}}, Earth, {{w|Mars}}, the Moon, {{w|Titan (moon)|Titan}}, {{w|433 Eros|Eros}}, {{w|25143 Itokawa|Itokawa}}, {{w|162173 Ryugu|Ryugu}} and {{w|101955 Bennu|Bennu}}. Notably, {{w|comet}} {{w|Philae (Spacecraft)| landings}} are not included in the list.
:The justification for this seeks to be that we must 'touch' the object before we consider it as worthy of being classified as more than a mere blob (or dot) in space.

;Regolithic: Anything covered in dirt and ice and stuff is a planet (infinite)
:This list excludes the {{w|Gas Giant}}s and the {{w|Ice Giant}}s. The list would likely include dwarf planets, asteroids, moons and comets. This is effectively the opposite of the "condescending" definition: every object in the solar system is included in one definition or the other (except for the Sun, which inhabits ).
:This is also an extension on the prior classicication. In this case ''could'' we meaningfully touch the object, with predominatingly atmospheric bodies being not considered so.

;Lunar: You can't be a planet if you don't have a moon (12+ objects)
:Only some objects in the solar system have known and acknowledged moons orbiting them. The value given may be {{w|List of natural satellites| the number of planets and dwarf planets}} that have moons, when excluding {{w|Haumea}} for not reaching {{w|hydrostatic equilibrium}} despite having moons. The Sun is excluded because its satellites are not moons, because ... oh, look, a Squirrel!
:Adopting this definition would suggest that a planetary body is not worthy of the name if it doesn't (with no matter for what reason) demonstrably have the means to dominate its local area by being the overwhelming focus of all adjacent bodies' own orbits.

;Solipsistic: Earth is the only planet (1 planet)
:{{w|Solipsism}} is the idea that only one's own mind is sure to exist. Randall extrapolated this idea to mean that only one's own planet that they are standing on is sure to exist.
:This projects (and relies upon) a more philosophical and/or semiotic assesment than any scientific one.

;Judgemental: Only the prettiest ones are planets (6 planets)
:This list is likely formulated from Randall's own perception of the prettiest planets in the Solar System. Strangely, seven objects are highlighted:
:* Earth
:* Jupiter
:* One of Jupiter's moons (unclear)
:* Saturn
:* One of Saturn's moons (unclear, possibly Titan)
:* Neptune's moon (probably Triton)
:* Pluto
:The subjectivity of this version of the definition makes it unlikely that a consensus of this form could be established.

;Empiricist: Only worlds that I, author of this table, have personally seen are planets (12 planets)
:This list may refer to the celestial objects in the Solar System that have been made visible at night, probably using with an optical telescope (a hobbyist one, perhaps Randall's, or from time borrowed on a major institutional installation). Jupiter's {{w|Galilean moons|four largest moons}} are [https://web.archive.org/web/20201112024151/http://denisdutton.com/jupiter_moons.htm technically visible to the naked eye] but hard to distinguish due to Jupiter's brightness, while Neptune is considered too faint to see (even if you know where to look). It may also be the case that Randall has never taken the time to look for Neptune while using a telescope. Apparently Randall has seen Uranus, which technically [https://www.skyatnightmagazine.com/advice/skills/how-see-uranus-in-night-sky Uranus ''is'' visible to the naked eye] under the very best viewing conditions, but these conditions are rare and it again requires knowing exactly where to look.
:The omission of the Sun from the list of worlds that Randall has personally seen is interesting. Yes, people are not supposed to stare at the Sun, but it is not too uncommon to accidentally look in its direction for a split-second before instinctively closing one's eyes and turning one's head away.
:As a different form of subjectivity, the value of this grouping's criteria is questionable, but not uncommon in other 'softer' sciences.

;Marine biologist: Only objects with oceans are planets (6+ planets)
:This list includes Earth, {{w|Europa (moon)|Europa}}, {{w|Ganymede (moon)|Ganymede}}, {{w|Callisto (moon)|Callisto}}, Titan, and {{w|Enceladus}}. Most of these have had the presence of significant water identified from the way local magnetic/electric fields are detected, but see the following item.
:There is a resemblance, here, to a loose understanding of what a "world" is, i.e. one that possesses various distinct 'terrains' beyond mere dry (and possibly considered featureless) rock. A marine biologist would, of course consider a marine (if not pellagic or bathyspheric) environment to be an essential element of any world.

;Maritime: Only objects with ''surface'' oceans are planets (2 planets)
:In the comic, only Earth and a Saturnian moon (likely to be Titan) are highlighted. Earth is the only body known in the solar system to have liquid water on the surface significant enough to be called an ocean. Titan's cold and dense atmosphere notably maintains surface 'seas' of methane and nitrogen, where other moons (given as additonal in the prior item) seem to have their liquid water beneath either whole-surface ice caps or otherwise deep under the surface.
:From the narrower point of view of a sailor, for example, there is no benefit in considering water hidden away far beneath the surface, and it might as well not be there. Whereas it's possible that a well-prepared mariner could eventually sail the strange seas of Titan, as easily (or easier) as an airman might {{w|Dragonfly (Titan space probe)|fly through its skies}}.

;Universalist: They're all planets (infinite)
:This list claims that all objects are planets, all drawn items (as also would presuming all undrawn/undrawable items) being marked as such, including the Sun.
:Giving up on any thought of exclusivity, this unconventional view willingly inducts all visible objects into consideration.

;Existentialist: What if {{w|outer space|space}} ''itself'' is a planet??? (Duude)
:This list is different from the list above as it claims that all of space, rather than only the objects existing in space, are planets. The interjection ''Duude'' expresses one's amazement at this 'revelation' and replaces the number count— and is sometimes stereotyped to imply the speaker is high on marijuana or other drugs popular with the 1960s hippie counterculture.
:The strange stretch of imagination, as prompted by some narcotic or other, abandons all pretense at sensibly sorting everything into "planet" or "not planet", as not only is everything a planet, but so is the nothing ''between'' these titular planets.

;Spiteful: ''Only'' Pluto is a planet (1 planet)
:This list is a malicious play on the demotion of Pluto by demoting all other planets except Pluto instead, leaving Pluto as the only planet in the solar system.
:This is the taxonomic equivalent of refusing to play and taking your ball home to spite those who you think don't deserve it.

;(title text) {{w|Star}}: Earth is a star (2 stars)
:In May 1934, Mark Oliphant, Paul Harteck and Ernest Rutherford at the Cavendish Laboratory, published an intentional deuterium fusion experiment, and made the discovery of both tritium and helium-3. This is widely considered the first experimental demonstration of fusion. Randall considers that this makes Earth fall into the category of a star due to the human-induced ability for Earth to fuse hydrogen into helium using nuclear fusion.
:By changing not only the definition, but the term being defined, this drifts yet further from any consensus view on the original question and into a typical punchline absurdity.

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}
:[A table with 3 columns, and 17 rows below the the header row, labelled "Definition", "# of planets" and "Solar system".]
:[In each row, the first column has a single word, in bold, then a descriptive sentence. The second column has a digit or other 'value'. The third column is a not-to-scale drawing of the Solar system, featuring the Sun, various 'planetary' bodies and an apparently selective sample of moons/asteroids, as follows: The Sun, Mercury, Venus, Earth + The Moon, Mars + two moons (Phobos and Deimos), a small selection of Asteroid Belt bodies (Ceres in the midst of other, smaller, examples), Jupiter + four moons (likely Io, Europa, Ganymede and Callisto), a ringed Saturn + usually one moon (probably Titan) or two (possibly Enceladus or Iapetus, as required), Uranus + four or five moons (likely to be Miranda, Ariel, Umbriel, Titania and Oberon, but one of these (shown upon the face of Uranus) only appears in some iterations of the base image), Neptune + one moon (probably Triton), Pluto + one moon (Charon), four more plutoid/Kuiper Belt objeccts (too little context to identify, but possibly Haumea, Makemake, Gonggong and Eris, in distance order), the first two of them with distinct moons indicated (entirely dependent upon which main objects they are).]
:[Each row's illustrated solar system has indivudal combinations of green highlights applied to the otherwise repeated diagram.]
:[Row 1: Definition:] Traditionalist: Pluto is a planet [Number:] 9 [Highlit: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune and Pluto]
:[Row 2: Definition:] Modern: Pluto is not a planet [Number:] 8 [Highlit: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus and Neptune]
:[Row 3: Definition:] Expansive: Dwarf planets are planets [Number:] 17+ [Highlit: Mercury, Venus, Earth, Mars, Ceres (in Asteroid Belt), Jupiter, Saturn, Uranus, Neptune, Pluto and the further main bodies]
:[Row 4: Definition:] Ultratraditionalist: Only the classical planets are planets [Number:] 5 [Highlit: Mercury, Venus, Mars, Jupiter, Saturn]
:[Row 5: Definition:] Condescending: Only giant planets are planets; the rest are big asteroids. [Number:] 4 [Highlit: Jupiter, Saturn, Uranus, Neptune]
:[Row 6: Definition:] Simplistic: Anything gravitationally round is a planet [Number:] 37+ [Highlit: The Sun, Mercury, Venus, Earth, The Moon, Mars, Ceres (without other asteroids), Jupiter + moons, Saturn with Titan, Uranus and its moons, Neptune with its moon, Pluto and the four further dwarf planets]
:[Row 7: Definition:] Grounded: Only objects a spaceship has landed on are planets [Number:] 10 [Highlit: Venus, Earth, The Moon, Mars, five (non-Ceriese) asteroids and Titan]
:[Row 8: Definition:] Regolithic: Anything covered in dirt and ice and stuff is a planet [Number:] [infinity symbol] [Highlit: Mercury, Venus, Earth, The Moon, Mars, Ceres with all other asteroids depicted in the Asteroid Belt, the moons of Jupiter, the sole moon! of Saturn, the moons of Uranus, the moon of Neptune, Pluto with Charon, and all remaining dwarf planets with their moons]
:[Row 9: Definition:] Lunar: You can't be a planet if you don't have a moon [Number:] 12+ [Highlit: Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto and three of the other dwarf planets in the Kuiper belt, including one with no obviously drawn moon]
:[Row 10: Definition:] Solipsitic: Earth is the only planet [Number:] 1 [Highlit: The Earth]
:[Row 11: Definition:] Judgemental: Only the prettiest ones are planets [Number:] 6 [Highlit: The Earth, Jupiter with one of its moons (not identified), Saturn, one of ''two'' Saturnian moons in this image and Pluto]
:[Row 12: Definition:] Empiricist: Only worlds that I, author of this table, have personally seen are planets [Number:] 12 [Highlit: Mercury, Venus, The Earth, The Moon, Mars, Jupiter with its four moons, Saturn and Uranus]
:[Row 13: Definition:] Marine biologist: Only objects with oceans are planets [Number:] 6+ [Highlit: The Earth, three Jovian moons, the two illustrated Saturnian moons]
:[Row 14: Definition:] Maritime: Only objects with [next word in italics] surface oceans are planets [Number:] 2 [Highlit: The Earth and Titan]
:[Row 15: Definition:] Universalist: They're all planets [Number:] [infinity symbol] [Highlit: All drawn objects, including The Sun and all other objects including all the moons/asteroids]
:[Row 16: Definition:] Existantialist: What if space [next word in italics] itself is a planet??? [Word, in italics:] Duude [Highlit: The whole third column cell]
:[Row 17: Definition:] Spiteful: [next word in italics] Only Pluto is a planet [Number:] 1 [Highlit: Pluto]

==Trivia==
*The 'Judgemental' definition has 7 colored objects instead of the labelled 6.
*[https://www.explainxkcd.com/wiki/images/archive/6/66/20250314195557%21planet_definitions_2x.png The initially released version of the comic] had two errors that were later fixed:
**The 'Traditionalist' definition had Neptune's satellite {{w|Triton (moon)|Triton}} colored instead of Pluto.
**The images for 'Traditionalist' and 'Modern' were swapped, resulting in Pluto being incorrectly highlighted in 'Modern' and incorrectly omitted in 'Traditionalist.'
{{comic discussion}}

[[Category:Comics with color]]
[[Category:Charts]]
[[Category:Astronomy]]
[[Category:Space]]
[[Category:Philosophy]]
[[Category:Comics edited after their publication]]

2966: Exam Numbers

2025-03-11T18:12:09Z

ISaveXKCDpapers: removed unnecessary parentheses

{{comic
| number = 2966
| date = July 31, 2024
| title = Exam Numbers
| image = exam_numbers_2x.png
| imagesize = 553x400px
| noexpand = true
| titletext = Calligraphy exam: Write down the number 37, spelled out, nicely.
}}

==Explanation==
The comic depicts various similarly formatted examination questions that might appear on test papers at various points in a student's potential academic career. While they all share a similar feel, they are asking for different things, some of which might be considered more serious and examinable proofs of study than others. The joke is that the first and last questions are the same, but have very different interpretations based on the context.
{| class="wikitable"
|+ Exam numbers
! Exam level !! Question !! Answer
|-
| {{w|Kindergarten}} math || Write down the biggest number you can think of || At a kindergarten-level education, it is assumed that an individual might write down a relatively small number like 300, depending upon whether they have encountered the concept of hundreds. It might also be interpreted as "what's the highest number that you (think you) can count up to". Given the nature of a child's exuberant glee at learning about ''really'' big (but otherwise normal) numbers, they may even try an answer something like "a million billion squillion gazillion... and seven!". However, a kindergarten student is not likely to be able to write small enough to fit that number in the space provided; the largest numeral that will fit in a box of arbitrary size is consists of repeated ones, which are narrower than other digits, thereby allowing more digits to fit. It is not certain what criteria would be used to mark this question correct or otherwise, it may actually be a stealth question in child psychology or a question that everybody "gets right" so long as they answer it.

Although an advanced concept taught years later, infinity is not out of the question here, as employing it is often the only way to explain a substantial number of very simple math questions.
|-
| Pre-{{w|algebra}} || Write down the value of x if x=3x-8 ||

''3x'' refers to the multiplication of 3 and the originally unknown number ''x'', as a convenient shorthand.

By subtracting ''3x'' from both sides, ''-2x = -8''. Divide both sides by -2 to find ''x''=4. Alternately, subtract ''x'' from both sides to give ''0 = 2x - 8'', and as taking 8 from two ''x''s makes it zero, one ''x'' is half that

(Verify by plugging ''x''=4 into the original equation. '''4''' = (3*'''4''') - 8 -> '''4''' = 12 - 8 -> '''4''' = 4.)

4 is also the subject of [[221: Random Number]].

|-
| {{w|Calculus}} || Write down the value of <pre style="line-height: initial;">⌠π
│ x sin²x dx
⌡0</pre>||
The integral can be solved using {{w|List_of_trigonometric_identities#Power-reduction_formulae|a trigonometric identity}} and [https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/07%3A_Techniques_of_Integration/7.01%3A_Integration_by_Parts integration by parts]:

<pre style="line-height: initial;">⌠π
│ x sin²x dx
⌡0

⌠π 1 − cos 2x
= │ x ―――――――――― dx
⌡0 2

1 ⌠π 1 ⌠π
= ― │ x dx − ― │ x cos 2x dx
2 ⌡0 2 ⌡0

1 │π 1 ⌠π
= ― x² │ - ― │ x (sin 2x)′ dx
4 │0 4 ⌡0

1 │π 1 │π 1 ⌠π
= ― x² │ - ― x sin 2x │ + ― │ (x)′ sin 2x dx
4 │0 4 │0 4 ⌡0

1 │π 1 │π 1 ⌠π
= ― x² │ - ― x sin 2x │ + ― │ sin 2x dx
4 │0 4 │0 4 ⌡0

1 │π 1 │π 1 │π
= ― x² │ - ― x sin 2x │ - ― cos 2x │
4 │0 4 │0 8 │0

1
= ― π² = 2.4674…
4
</pre>

|-
| PhD {{w|Cosmology}} || Write down the Hubble constant to within 1% || The Hubble constant is a component of {{w|Hubble's law}}, which describes the relationship between the distance between galaxies and their speed of separation. Its exact value is not known to this level of precision; it is ''about'' 70 (km/s)/Mpc. Different methods of measuring it have come up with significantly different values, with their own error bars, generally of 2% to 5% (both plus and minus, or asymmetrically, but with a similar overall range) and resolving this difference (the {{w|Hubble's law#Hubble tension|Hubble tension}}) is one of the great challenges of modern cosmology. Getting a PhD involves advancing the field, so it seems this particular PhD student has completed a thesis solving this problem. This question might have a different acceptable answer if asked again of the student, depending upon further developments in cosmology, and the 'constant' itself is supposed to change over cosmological time (the nature of this change being yet another subject requiring further study, calculation and observation).

What would a correct answer look like? Because of how {{w|significant digits}} are interpreted, 69 (km/s)/Mpc could be correct to within 1% while 70 is not.
|-
| {{w|Game Theory}} || Write down 10 more than the average of the class's answers || Game Theory studies "games" (and 'game-like' situations) in which two or more participants take actions that will succeed or fail based on other participants' decisions. In this case, all students' answers will be averaged (likely a simple {{w|arithmetical mean}}), and the best answer would be one that is 10 more than that average. For instance, if the answers end up being 30, 40, 50, 60, and 70, the mean would be 50, making 60 the best possible answer. As an individual test taker who doesn't know what your classmates are going to write down, the objective is to answer 10 more than your estimate of the average including your own guess, ''knowing that everyone else will be doing the same.'' Attempts at numerical solutions, including {{w|Monte-Carlo method}}s, will not converge because the problem as stated is unstable. The best answer will always be ten more than what the other students think the best answer is, and they have the same information as everyone else, which results in a contradiction.

However, the average of any set of numbers that includes infinity is infinity, and ∞ + 10 is still ∞. Something similar to this question is found in the title text of [[2385: Final Exam]].
|-
| Postgraduate Math || Write down the biggest number you can think of || This question echoes the very first example, but would be expected to be answered very differently (unlike a revisiting of most of the others).
Postgraduate math students can probably think of ''very'' large numbers. While a tempting answer could be "{{w|infinity}}", most mathematicians do not consider infinity to be a number,{{acn}} but rather a class of numbers. (Writing down "Infinity" in this context would be as wrong as writing down "Primes" or "Positive integers"). Even if infinity is an acceptable answer, some infinities are bigger than others. Students familiar with the field of [https://googology.fandom.com/wiki/Googology#History Googology] or [https://sites.google.com/view/transcology/lists/numbers Transcology] may give an answer such as {{w|Rayo's number}}, which was the winning entry in the [https://googology.fandom.com/wiki/Big_Number_Duel Big Number Duel].

This might heavily depend upon the branch of mathematics you are studying. Named (finite) numbers, or ones with specific and useful notations, might satisfy some questioning contexts, whilst the existence of a whole further set of trans-finite numbers (i.e. increasingly large types of "infinity") would be important considerations in others. For those associated with more computational mathematics, any infinity would be {{w|NaN|Not a Number}}, and their answer might instead be the ceiling of some binary representation (typically ''28n-1'' for some value of ''n''), the largest unsigned value reliably storable in a given byte form for an integer (e.g. a {{w|Integer (computer science)|double quadword}}). On the other end of the spectrum, many abstract algebraists might answer with some variation of "What ring are we working in, and is it even well ordered?" It also might be a trick question: if you can envision a real number greater than one, are you even doing real math (in a given field)?

As with the kindergarten question, there may be no previously anticipated "correct" answer. It could be another "correct just so long as you answer it" (or perhaps "sensibly" so) or the mark goes only to those giving the greatest valid number across all submissions.
|-
| {{w|Calligraphy}} (title text) || Write down the number 37, spelled out, nicely || Calligraphy is the art of artistic writing. The title text expands the joke outside the realm of math and points out that since calligraphy does not require any math skills, the only way a calligraphy exam would even mention numbers is if one had to write them out in such a way as to showcase their calligraphic skill and aesthetic judgement (choosing a form and adornment of script that is "nice", which may be a highly subjective choice). In this case, it could be rendered as "thirty-seven" or "thirty seven", or possibly, "one score and seventeen" in old-fashioned writing. The subject may choose to render it in a language other than English — for example "dau ar bymtheg ar hugain" would provide significant scope to show off calligraphic skill. {{w|37_(number)|37}} is a number that some people believe [http://thirty-seven.org mysteriously appears more often than it should]; this was a subject of a [https://www.youtube.com/watch?v=d6iQrh2TK98 recent Veritasium video].
|}

Note that "PhD Cosmology" and "Postgraduate Math", unlike the other items, aren't specific courses but instead refer to areas of study, and as such wouldn't have a exam (unless a PhD thesis defense is considered an "exam").

This style of final exam question, un-numbered and therefore possibly the ''only'' question upon the whole of each final paper, in some ways (for some instances) echoes the question "What is your name?" that Randall will be aware was the sole question given to Discworld's {{w|List of Discworld characters#Victor Tugelbend|Victor Tugelbend}} in an attempt to ensure he comprehensively passed (or utterly failed) his final student-wizard's exam, after many prior times of deliberately not-quite-passing.

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}
:[6 different math test questions.]

:[The first panel:]
:Kindergarten math final exam
:Q. Write down the biggest number you can think of
:A. [empty box]

:[The second panel:]
:Pre-algebra final exam
:Q. Write down the value of x if x=3x-8
:A. [empty box]

:[The third panel:]
:Calculus final exam
:Q. Write down the value of [integral sign, from 0 to pi] x sin^2 x dx
:A. [empty box]

:[The fourth panel:]
:PhD cosmology final exam
:Q. Write down the Hubble constant to within 1%
:A. [empty box]

:[The fifth panel:]
:Game theory final exam
:Q. Write down 10 more than the average of the class's answers
:A. [empty box]

:[The sixth panel:]
:Postgraduate math final exam
:Q. Write down the biggest number you can think of
:A. [empty box]

{{comic discussion}}

[[Category:Math]]
[[Category:Cosmology]]
[[Category:Comics with lowercase text]]

2966: Exam Numbers

2025-03-11T18:10:49Z

ISaveXKCDpapers: fixed sign error, formatting overhaul

{{comic
| number = 2966
| date = July 31, 2024
| title = Exam Numbers
| image = exam_numbers_2x.png
| imagesize = 553x400px
| noexpand = true
| titletext = Calligraphy exam: Write down the number 37, spelled out, nicely.
}}

==Explanation==
The comic depicts various similarly formatted examination questions that might appear on test papers at various points in a student's potential academic career. While they all share a similar feel, they are asking for different things, some of which might be considered more serious and examinable proofs of study than others. The joke is that the first and last questions are the same, but have very different interpretations based on the context.
{| class="wikitable"
|+ Exam numbers
! Exam level !! Question !! Answer
|-
| {{w|Kindergarten}} math || Write down the biggest number you can think of || At a kindergarten-level education, it is assumed that an individual might write down a relatively small number like 300, depending upon whether they have encountered the concept of hundreds. It might also be interpreted as "what's the highest number that you (think you) can count up to". Given the nature of a child's exuberant glee at learning about ''really'' big (but otherwise normal) numbers, they may even try an answer something like "a million billion squillion gazillion... and seven!". However, a kindergarten student is not likely to be able to write small enough to fit that number in the space provided; the largest numeral that will fit in a box of arbitrary size is consists of repeated ones, which are narrower than other digits, thereby allowing more digits to fit. It is not certain what criteria would be used to mark this question correct or otherwise, it may actually be a stealth question in child psychology or a question that everybody "gets right" so long as they answer it.

Although an advanced concept taught years later, infinity is not out of the question here, as employing it is often the only way to explain a substantial number of very simple math questions.
|-
| Pre-{{w|algebra}} || Write down the value of x if x=3x-8 ||

''3x'' refers to the multiplication of 3 and the originally unknown number ''x'', as a convenient shorthand.

By subtracting ''3x'' from both sides, ''-2x = -8''. Divide both sides by -2 to find ''x''=4. Alternately, subtract ''x'' from both sides to give ''0 = 2x - 8'', and as taking 8 from two ''x''s makes it zero, one ''x'' is half that

(Verify by plugging ''x''=4 into the original equation. '''4''' = (3*'''4''') - 8 -> '''4''' = 12 - 8 -> '''4''' = 4.)

4 is also the subject of [[221: Random Number]].

|-
| {{w|Calculus}} || Write down the value of <pre style="line-height: initial;">⌠π
│ x sin²x dx
⌡0</pre>||
The integral can be solved using {{w|List_of_trigonometric_identities#Power-reduction_formulae|a trigonometric identity}} and [https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/07%3A_Techniques_of_Integration/7.01%3A_Integration_by_Parts integration by parts]:

<pre style="line-height: initial;">⌠π
│ x sin²x dx
⌡0

⌠π (1 − cos 2x)
= │ x ―――――――――――― dx
⌡0 2

1 ⌠π 1 ⌠π
= ― │ x dx − ― │ x cos 2x dx
2 ⌡0 2 ⌡0

1 │π 1 ⌠π
= ― x² │ - ― │ x (sin 2x)′ dx
4 │0 4 ⌡0

1 │π 1 │π 1 ⌠π
= ― x² │ - ― x sin 2x │ + ― │ (x)′ sin 2x dx
4 │0 4 │0 4 ⌡0

1 │π 1 │π 1 ⌠π
= ― x² │ - ― x sin 2x │ + ― │ sin 2x dx
4 │0 4 │0 4 ⌡0

1 │π 1 │π 1 │π
= ― x² │ - ― x sin 2x │ - ― cos 2x │
4 │0 4 │0 8 │0

1
= ― π² = 2.4674…
4
</pre>

|-
| PhD {{w|Cosmology}} || Write down the Hubble constant to within 1% || The Hubble constant is a component of {{w|Hubble's law}}, which describes the relationship between the distance between galaxies and their speed of separation. Its exact value is not known to this level of precision; it is ''about'' 70 (km/s)/Mpc. Different methods of measuring it have come up with significantly different values, with their own error bars, generally of 2% to 5% (both plus and minus, or asymmetrically, but with a similar overall range) and resolving this difference (the {{w|Hubble's law#Hubble tension|Hubble tension}}) is one of the great challenges of modern cosmology. Getting a PhD involves advancing the field, so it seems this particular PhD student has completed a thesis solving this problem. This question might have a different acceptable answer if asked again of the student, depending upon further developments in cosmology, and the 'constant' itself is supposed to change over cosmological time (the nature of this change being yet another subject requiring further study, calculation and observation).

What would a correct answer look like? Because of how {{w|significant digits}} are interpreted, 69 (km/s)/Mpc could be correct to within 1% while 70 is not.
|-
| {{w|Game Theory}} || Write down 10 more than the average of the class's answers || Game Theory studies "games" (and 'game-like' situations) in which two or more participants take actions that will succeed or fail based on other participants' decisions. In this case, all students' answers will be averaged (likely a simple {{w|arithmetical mean}}), and the best answer would be one that is 10 more than that average. For instance, if the answers end up being 30, 40, 50, 60, and 70, the mean would be 50, making 60 the best possible answer. As an individual test taker who doesn't know what your classmates are going to write down, the objective is to answer 10 more than your estimate of the average including your own guess, ''knowing that everyone else will be doing the same.'' Attempts at numerical solutions, including {{w|Monte-Carlo method}}s, will not converge because the problem as stated is unstable. The best answer will always be ten more than what the other students think the best answer is, and they have the same information as everyone else, which results in a contradiction.

However, the average of any set of numbers that includes infinity is infinity, and ∞ + 10 is still ∞. Something similar to this question is found in the title text of [[2385: Final Exam]].
|-
| Postgraduate Math || Write down the biggest number you can think of || This question echoes the very first example, but would be expected to be answered very differently (unlike a revisiting of most of the others).
Postgraduate math students can probably think of ''very'' large numbers. While a tempting answer could be "{{w|infinity}}", most mathematicians do not consider infinity to be a number,{{acn}} but rather a class of numbers. (Writing down "Infinity" in this context would be as wrong as writing down "Primes" or "Positive integers"). Even if infinity is an acceptable answer, some infinities are bigger than others. Students familiar with the field of [https://googology.fandom.com/wiki/Googology#History Googology] or [https://sites.google.com/view/transcology/lists/numbers Transcology] may give an answer such as {{w|Rayo's number}}, which was the winning entry in the [https://googology.fandom.com/wiki/Big_Number_Duel Big Number Duel].

This might heavily depend upon the branch of mathematics you are studying. Named (finite) numbers, or ones with specific and useful notations, might satisfy some questioning contexts, whilst the existence of a whole further set of trans-finite numbers (i.e. increasingly large types of "infinity") would be important considerations in others. For those associated with more computational mathematics, any infinity would be {{w|NaN|Not a Number}}, and their answer might instead be the ceiling of some binary representation (typically ''28n-1'' for some value of ''n''), the largest unsigned value reliably storable in a given byte form for an integer (e.g. a {{w|Integer (computer science)|double quadword}}). On the other end of the spectrum, many abstract algebraists might answer with some variation of "What ring are we working in, and is it even well ordered?" It also might be a trick question: if you can envision a real number greater than one, are you even doing real math (in a given field)?

As with the kindergarten question, there may be no previously anticipated "correct" answer. It could be another "correct just so long as you answer it" (or perhaps "sensibly" so) or the mark goes only to those giving the greatest valid number across all submissions.
|-
| {{w|Calligraphy}} (title text) || Write down the number 37, spelled out, nicely || Calligraphy is the art of artistic writing. The title text expands the joke outside the realm of math and points out that since calligraphy does not require any math skills, the only way a calligraphy exam would even mention numbers is if one had to write them out in such a way as to showcase their calligraphic skill and aesthetic judgement (choosing a form and adornment of script that is "nice", which may be a highly subjective choice). In this case, it could be rendered as "thirty-seven" or "thirty seven", or possibly, "one score and seventeen" in old-fashioned writing. The subject may choose to render it in a language other than English — for example "dau ar bymtheg ar hugain" would provide significant scope to show off calligraphic skill. {{w|37_(number)|37}} is a number that some people believe [http://thirty-seven.org mysteriously appears more often than it should]; this was a subject of a [https://www.youtube.com/watch?v=d6iQrh2TK98 recent Veritasium video].
|}

Note that "PhD Cosmology" and "Postgraduate Math", unlike the other items, aren't specific courses but instead refer to areas of study, and as such wouldn't have a exam (unless a PhD thesis defense is considered an "exam").

This style of final exam question, un-numbered and therefore possibly the ''only'' question upon the whole of each final paper, in some ways (for some instances) echoes the question "What is your name?" that Randall will be aware was the sole question given to Discworld's {{w|List of Discworld characters#Victor Tugelbend|Victor Tugelbend}} in an attempt to ensure he comprehensively passed (or utterly failed) his final student-wizard's exam, after many prior times of deliberately not-quite-passing.

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}
:[6 different math test questions.]

:[The first panel:]
:Kindergarten math final exam
:Q. Write down the biggest number you can think of
:A. [empty box]

:[The second panel:]
:Pre-algebra final exam
:Q. Write down the value of x if x=3x-8
:A. [empty box]

:[The third panel:]
:Calculus final exam
:Q. Write down the value of [integral sign, from 0 to pi] x sin^2 x dx
:A. [empty box]

:[The fourth panel:]
:PhD cosmology final exam
:Q. Write down the Hubble constant to within 1%
:A. [empty box]

:[The fifth panel:]
:Game theory final exam
:Q. Write down 10 more than the average of the class's answers
:A. [empty box]

:[The sixth panel:]
:Postgraduate math final exam
:Q. Write down the biggest number you can think of
:A. [empty box]

{{comic discussion}}

[[Category:Math]]
[[Category:Cosmology]]
[[Category:Comics with lowercase text]]

Talk:179: e to the pi times i

2025-03-04T03:41:01Z

ISaveXKCDpapers: added comment

This is one of the few comics that were changed after release, as stated by Randall in his XKCD book. It first claimed e^(i*Pi) = 1, which lead to huge protest from the community and a correction from Randall. --[[User:Gefrierbrand|Gefrierbrand]] ([[User talk:Gefrierbrand|talk]]) 09:47, 3 September 2012 (UTC)
:He must have been pie-eyed when he wrote that; he's usually pretty good about his math... -- [[User:IronyChef|IronyChef]] ([[User talk:IronyChef|talk]]) 05:09, 7 November 2012 (UTC)
::I see what you did there. [[User:Daddy|Daddy]] ([[User talk:Daddy|talk]]) 15:18, 28 April 2013 (UTC)
:You know, Leonhard Euler was really the one who brought the use of the symbol π into the mathematical mainstream. However, he originally wrote π = 6.28..., the circumference of a circle with radius 1—a unit circle. It was only later that he switched to the half-circumference instead, giving us the π = 3.14... that we know today. Perhaps Munroe was simply paying tribute to Euler's original definition.
:Funnily enough, that value is what we would call 2π today, which some of us would denote as the single letter τ (tau). However, this strip predates the work that would come to popularize that notation by over 3 years. Since that work did come out, ''xkcd'' has had multiple strips directly referencing τ. I can't get a clear read on Munroe's feelings about the topic, though. [[User:ISaveXKCDpapers|ISaveXKCDpapers]] ([[User talk:ISaveXKCDpapers|talk]]) 03:41, 4 March 2025 (UTC)

Randall says in the title text that he's never been satisfied with explanations of the sinusoidal nature of the function of e^ix. http://www.math.toronto.edu/mathnet/questionCorner/epii.html really helps, at least for those who are obsessed with taylor series yet tragically horrible at math. --[[User:Jolbucley|Jolbucley]] ([[User talk:Jolbucley|talk]]) 03:39, 29 January 2014 (UTC)

Why e to the ix gives a sinusoidal wave? This is because neutrinos keep oscillating. They wouldn't without the e^ix support... [[Special:Contributions/162.158.83.72|162.158.83.72]] 04:01, 24 May 2016 (UTC)
::Actually, e to the ix isn't sinusoidal. "Consisting of sinusoidal components" doesn't make it sinusoidal, any more than it does for any relation representable by Fourier Transform. As with the other mistake, it seems to indicate that this really wasn't something Randall knew about.

Technically ''i'' isn't "imaginary" at all, but is incorporated into equations to represent rotations perpendicular to the x-y plane. [[Special:Contributions/108.162.210.220|108.162.210.220]] 15:51, 9 September 2016 (UTC)

: ''i'' is ''imaginary.'' Both "real" and "imaginary" are terms of art in a mathematical context that have nothing to do with the reality or unreality of the numbers in question. Also, when you're talking about the complex plane, it doesn't have ''x'' and ''y'' axes: it has ''real'' and ''imaginary'' axes. But of course, nobody can stop you from defining a mapping from the complex plane to any other vector space if that suits your purpose. [[Special:Contributions/162.158.62.45|162.158.62.45]] 01:58, 6 September 2018 (UTC)

179: e to the pi times i

2025-03-04T03:24:47Z

ISaveXKCDpapers: added 3Blue1Brown video

{{comic
| number = 179
| date = November 3, 2006
| title = e to the pi times i
| image = e_to_the_pi_times_i.png
| titletext = I have never been totally satisfied by the explanations for why e to the ix gives a sinusoidal wave.
}}

==Explanation==
The comic largely references {{w|Euler's identity}}. This identity states that eiπ + 1 = 0. Therefore, eiπ = −1.
The humor from this comic is because of the seemingly arbitrary relationship between e, π, and the identity of i (the square root of −1). e is the mathematical identity of which the derivative of ex with respect to x is still ex, while π is the relationship between the circumference of a circle divided by its diameter. Taking these two values and applying them to the value of i in such a manner makes it seem counter-intuitive that it would yield −1 from basic analysis. The above linked Wikipedia page goes into good detail of how to derive this identity, as does [https://www.youtube.com/watch?v=-dhHrg-KbJ0 this YouTube video]. If you know some basic calculus, then [https://www.youtube.com/watch?v=v0YEaeIClKY this video] gives a geometric intuition.

The title text refers to how Euler's identity is called upon in complex form (separating real and imaginary numbers): eix = cos(x) + i sin(x).

From ''[[xkcd: volume 0]]'':
{{quote|I initially got the sign wrong here, and got 40 emails in 3 hours.}}

==Transcript==
:[Two Cueballs are standing at a board with writing on. One Cueball is pointing at the board.]
:Cueball: Numbers of the form n√-1 are "imaginary," but can still be used in equations.
:Friend: Okay.
:Cueball: And e^(π√-1)=-1.
:Friend: Now you're just fucking with me.

{{comic discussion}}
[[Category:Comics featuring Cueball]]
[[Category:Math]]
[[Category:Multiple Cueballs]]

3042: T. Rex Evolution

2025-01-28T02:29:51Z

ISaveXKCDpapers: capitalization of "rex"

{{comic
| number = 3042
| date = January 24, 2025
| title = T. Rex Evolution
| image = t_rex_evolution_2x.png
| imagesize = 418x378px
| noexpand = true
| titletext = Unfortunately, body size and bite force continue to increase.
}}

==Explanation==
{{incomplete|Created by a BOT WITH NEGATIVE LIMBS, CURRENTLY LIVING IN LOCH NESS - fact-check the graph's "early" claim and flesh things out a bit. Do NOT delete this tag too soon.}}

The small and commonly supposed useless forelimbs of ''{{w|Tyrannosaurus rex}}'' have often been used as a source of humor in works of fiction that feature dinosaurs. [[Randall]] claims that the ancestors of ''T. rex'' had (relatively) longer forelimbs, and hypothesizes that — had non-avian dinosaurs not suffered an {{w|Cretaceous–Paleogene extinction event|extinction event}} — this trend of "reduced limbs" would have continued until present-day descendants of ''T. rex'' were limbless creatures not unlike a giant snake.

{{w|Evolution|Evolutionary theory}} indicates that different selection pressures cause the development or loss of characteristics, as those characteristics which enable better survival to continuation of the species are emphasized and those which don't (such as wasting nutrients on growing unnecessary limbs) are not.

In reality, while some animals have evolved to lose the use of the limbs of their predecessors — such as the hind legs of a whale — there is no particular reason to think that the ''T. rex'' was on the way to its forelimbs becoming vestigial, as opposed to being well adapted to the niche it filled in the form they had. In fact they appear to have been extremely powerful, which would be an unusually wasteful trait if they were not much used. There is even less reason why the shrinking of one set of limbs would necessarily apply to other limbs. A ''T. rex'' would continue to have substantial need for its hind legs, which it used for running and chasing prey, and would have no reason to lose legs unless environmental factors favored a limbless lifestyle. Loss of all limbs has happened multiple times in the order {{w|Squamata}} (the lizards and snakes), where limbless body plans have evolved convergently over and over, such as in {{w|Common_slow_worm|blindworms}} as compared to true {{w|snake}}s. However, the circumstances that encourage such lifestyles were not likely to happen to the large predatorial Tyrannosaurus. With that said, it is not outright impossible that Tyrannosaurus could have eventually lost its arms if it had never gone extinct.

The title text make a similar claim that, as ''T. rex'' was also larger than its ancestors, the same trend in growth would continue, such that the hypothetical limbless present-day descendant would be even larger than the [https://what-if.xkcd.com/78/ famously elephant-sized] ''T. rex''. In reality animals do not change in size linearly over time, and a modern descendant of Tyrannosaurus would not necessarily be dramatically larger and could easily have become somewhat smaller, even with the same body plans and lifestyle. Though it is unlikely that the constraints of {{w|insular dwarfism}} could have applied to such creatures across the continental range they may have remained alive to inhabit, many unknown factors might have influenced their descendants' body shapes, and probably even induced further speciation to render all surviving branches of the tyrannosaur family tree as distinct, probably to include both growing and re-shrinking lineages. (It is a common misconception, used here for humorous purposes, that an individual personally evolves into a particular form, or that a single descendant represents the only evolutionary track taken 'towards perfection' from a less evolved ancestor.)

This comic may be tangentially related to the fact that birds are dinosaurs that survived the KT extinction, which is a recurring theme on xkcd.

Extrapolation was first the subject of a comic in [[605: Extrapolating]] and has since become a [[:Category:Extrapolation|recurrent topic]] on xkcd.

==Transcript==
:[A graph is shown. The Y-axis is labelled and has two labeled ticks at the top and halfway up from the X-axis:]
:Limbs
:4
:2

:[The X-axis is not labelled but represents time and it has ticks every 10 million years, from 180 million years ago until present time. Every fifth tick is a bit larger and has a label beneath it. Except the one at the present time. Below the last tick is an arrow pointing to that tick with a label. There is another arrow that points to about 65 million years ago and this also has a label:]
:150 million years ago
:100 MYA
:50 MYA
:Now
:Extinction

:[The graph shows three animals positioned from the top left, to the middle to the bottom right of the graph.]
:[The first animal is a regular dinosaur walking on its hind legs, with fairly long front legs. It is positioned at 4 limbs and 150 million years ago. A label is written above and right of it:]
:4 normal limbs

:[An arrow goes along the diagonal of the chart down and right and points to the second animal, a Tyrannosaur Rex, which is located in the middle of the chart next to 2 limbs and, just left of the "extinction" arrow. Larger than the first and with almost no front legs. A label is written above and right of it:]
:Barely more than 2

:[A dashed arrow with a label in the middle continues along the diagonal of the chart to the last animal: A leg-less "dinosaur" with a big open mouth. The animal lies on its belly but with the head part and the tail lifted from the ground. A label is written above it.]
:?
:???

:[Caption below the panel:]
:If T. rex hadn't gone extinct
:(Linear extrapolation)

{{comic discussion}}

[[Category:Charts]]
[[Category:Timelines]]
[[Category:Extrapolation]]
[[Category:Dinosaurs]]

Talk:3038: Uncanceled Units

2025-01-15T18:10:51Z

ISaveXKCDpapers: added comment

DUDE I'M STILL IN SCHOOL RN, WHAT?
(also, the joke is that energy is power*time, so kWh is kJ/s... in an hour [[User:CalibansCreations|'''Caliban''']] ([[User talk:CalibansCreations|talk]]) 13:27, 15 January 2025 (UTC)

I guess not every comic can be a winner. Talking about an appliance using a certain amount of kWH per day is clear and normal. Power gets billed by the kWh, not the Joule. While technically not wrong, wanting "cancel" a sub-part of the commonly-used energy unit kWh and leaving it in deliberately-obscured units most people are less familiar with is the sort of insanity I'd more expect from White Hat than Cueball. [[Special:Contributions/172.70.35.171|172.70.35.171]] 13:39, 15 January 2025 (UTC)
:Maybe that is a meta-joke? To frame kWh/day as something crazy by giving that line to whitehat --[[User:Lupo|Lupo]] ([[User talk:Lupo|talk]]) 13:52, 15 January 2025 (UTC)
:There's a difference between instantaneous power draw, and the total "volume"(/area, really) of power over time. Though a fridge is "always on", it is still only irregularly at full-draw. But, to the power company (or to the gas company, who will generally give a kWh measure of 'energy taken from the network'), they don't (generally) care whether you used twice as many kW over half the time or half as many over twice the time, within any given total billing period, even if it affects what you think. [[Special:Contributions/172.70.163.46|172.70.163.46]] 14:39, 15 January 2025 (UTC)
::Using joule as if it was an everyday unit of energy would be weird but I don't agree that watt is crazy. It's a normal unit of energy consumption that does mean something to people, e.g. 1000W microwave, 100W (incandescent) light bulb. Don't get me wrong kWh/day is also useful to translate it to your energy bill, but I do feel slightly uncomfortable every time I see that time divided by time :-) [[User:Mtcv|Mtcv]] ([[User talk:Mtcv|talk]]) 14:40, 15 January 2025 (UTC)

This is especially funny with US units. My car needs about 5l/100km, or 0.05mm². Now I am wondering how many ft^(-2) my car does... --[[User:Lupo|Lupo]] ([[User talk:Lupo|talk]]) 13:49, 15 January 2025 (UTC)

fridge [[Special:Contributions/172.70.126.147|172.70.126.147]] 14:22, 15 January 2025 (UTC)

The late [https://en.wikipedia.org/wiki/David_J._C._MacKay Sir David MacKay] wrote an excellent book, [http://www.withouthotair.com/ Sustainable Energy – without the hot air] (which is available free online).
On [http://www.withouthotair.com/c2/page_24.shtml this page] he talks about the units he uses in the book: kWh for energy ("one unit") and kWh/day for power - becuase it's simple for lay-people to understand - how many units does this appliance use per day.
It's a good book if any of you are interested in sustainable energy (although it was written in 2008, so some bits might be out of date by now) {{unsigned ip|172.70.85.33|14:33, 15 January 2025}}

If anyone's curious, I found an online gallons per square foot calculator: https://www.omnicalculator.com/construction/gallons-per-square-foot [[Special:Contributions/172.71.223.6|172.71.223.6]] 15:54, 15 January 2025 (UTC)

The answer to Cueball's question is likely NO in the US and YES in the UK, due not just to gallon size but also fridge size (a model like that is a particularly large fridge, when I bought one 10 years ago going for the smallest available I had to modify my cabinet above the fridge as there wasn't one less than 6'8"- the fridge hole was 6' previous).[[User:Seebert|Seebert]] ([[User talk:Seebert|talk]]) 16:02, 15 January 2025 (UTC)

I disagree with this comic, and I think the final paragraph in the explanation about Hubble's constant best explains why. [[User:Beanie|Beanie]] [[User talk:Beanie|talk]] 15:57, 15 January 2025 (UTC)

Technically, kWh should be written as kW⋅h or kW h, because it literally means "kilowatts multiplied by one hour", not "kilowatts per hour" as many people assume. However, almost nobody writes it correctly. (kW/h is sometimes also seen, but egregiously incorrect.) Also, particularly now that electric vehicles are becoming more popular, people often get confused between kW and kW h. The car can charge at a peak or average rate expressed in kW, but energy billed by a charging service provider is expressed in kWh. People frequently either add or remove the "h" incorrectly because they don't understand the difference. In some places like India, a kilowatt-hour is simply referred to as a "unit" to avoid confusion. In my opinion, it was an enormous mistake to use kWh when we could be using mJ instead, which I think is probably something close to the point Randall may have been trying to make. Anyway, I wasn't sure if there was a place for any of this random trivia in the article itself, but feel free to use it. [[User:Equites|Equites]] ([[User talk:Equites|talk]]) 17:11, 15 January 2025 (UTC)

Relevant XKCD… I mean relevant YouTube video: "Cursed units" 1 and 2: https://www.youtube.com/watch?v=kkfIXUjkYqE https://www.youtube.com/watch?v=Zg7xe8MkJHs [[User:Fabian42|Fabian42]] ([[User talk:Fabian42|talk]]) 17:31, 15 January 2025 (UTC)

: Highly relevant, in fact. The first video referred to the kilowatt-hour as "cursed", which became a highly polarizing issue in the comments, something that was addressed at the beginning of part 2. Assuming these responses weren't cherry-picked, I get the impression that there are a lot of people on both sides of this. It seems like the same kind of thing we're seeing in this very comment section. [[User:ISaveXKCDpapers|ISaveXKCDpapers]] ([[User talk:ISaveXKCDpapers|talk]]) 18:10, 15 January 2025 (UTC)

3038: Uncanceled Units

2025-01-15T16:15:56Z

ISaveXKCDpapers: "subtending" -> "subtended by" where applicable

{{comic
| number = 3038
| date = January 15, 2025
| title = Uncanceled Units
| image = uncanceled_units_2x.png
| imagesize = 323x355px
| noexpand = true
| titletext = Speed limit c arcminutes^2 per steradian
}}

==Explanation==
{{incomplete|Created by THE PLANCK CONSTANT, WHICH IS TECHNICALLY A FREQUENCY AND CAN THUS BE EXPRESSED IN HERTZ - Please continue to explain the joke and possible interpretations. Do NOT delete this tag too soon.}}
Another of [[Randall]]'s [[:Category:Pet Peeves|pet peeves]], this comic expresses disapproval of units that could be mathematically simplified.

[[White Hat]] is presenting a refrigerator to [[Cueball]], claiming it only uses 3 kWh per day. Kilowatt-hours is a commonly used unit of electrical energy in the United States, being the amount of energy consumed by one kilowatt of power usage for one hour. As the unit in which power rates are typically reported and bills calculated, it's the most useful piece of information to the average consumer. This measure, however, could be simplified, since a kilowatt is equivalent to a kilojoule per second. Rather than adding a second unit, energy could simply be reported in megajoules (1 kilowatt-hour is 3.6 megajoules)

Appliances, however, frequently report their typical power use in terms of kilowatt-hours per day (or per month or year). Once again, this is useful to consumers, because it makes it easier for them to understand how much money it will cost them to run. However, it's an inelegant way to use units, because it uses power, which is already a measure of energy per time, multiplies it by one time unit, and then divides it by another. This clunky method apparently chafe's at [[Randall]]'s mind, possibly due to his scientific background (which encourages simplifications of units). The "hour" and "day" terms, both being units of time, can simply cancel out by dividing the whole number by 24, meaning that the refrigerator averages 0.125 KW, or 125 watts, to run. It should be noted that this doesn't mean the refrigerator constantly draws this amount of power, since the compressor in the refrigerator only runs intermittently, but running it over the course of a day (with typical use) is expected to give that average power use.

Cueball (possibly representing Randall) sardonically wonders whether the refrigerator would fit in his kitchen, since the ceiling is only 50 gallons per square foot high. This is clearly an abnormal and unhelpful way of reporting height. This unit turns a normal measurement of height (feet and inches, in the US, meters and centimeters, most other places) into weird collection of uncancelled units. Gallons can be transformed to cubic feet (1 US gal ≈ 0.1337 ft3), which can be divided by the square feet, yielding a ceiling height of around 203.7 cm, or around 6 feet 8 inches. (Using imperial gallons [1 UK gal ≈ 0.1605 ft3], the height is approximately 244.7 cm, roughly 8 feet.) This is intended to lampoon the use of uncancelled units by showing how odd things become if they're generally used.

''[[what if? (blog)|what if?]]'': [https://what-if.xkcd.com/11/ Droppings] also covers strange instances of unit cancellation, including a measure of volume per distance converted to area; similar to Cueball's measure of volume per area representing a distance (the height of his ceiling).

A common source of unit drama occurs between lay people who are looking for every day practicality and science/engineering types who are inclined towards formalized mathematical operations. For example customary units which support even divisibility versus metric units which prioritize base 10 scales. In this case telling the average customer the energy use in joules per day or average consumption in watts would require them to perform more complicated conversions to get to the figure they actually care about, the actual cost per day. White Hat could just give this cost figure directly, but does not know what every customer pays for electricity (an explicit yearly cost estimate would be included on the government required energy efficiency label).

In the title text, a speed limit is given as c arcminutes2 per steradian, where c is presumably the speed of light in vacuum, 2.998×108 m/s (meters per second) or 186282 mi/s (miles per second). A steradian (sr) is the SI unit for solid angle, subtended by a section of a sphere, like a radian is a unit of angle subtended by a section of a circle. A square arcminute is also a unit of solid angle, equivalent to a section of a sphere of 1/60 of a degree by 1/60 of a degree. There are ((1/60)*(π/180))2 = 8.462×10-8 sr in a square arcminute. Then multiplying by c gives a speed of 56.75 mph (probably 55 mph, based upon the {{w|National Maximum Speed Law|'traditional' US speed limit}}, before rounding errors in the reverse direction), or 91.33 km/h, showing that you can combine an outrageously high speed with two unnecessary units that cancel each other to form a normal road speed.

It is worth noting that although some of these examples are ridiculous, uncancelled units can be helpful to better understand the concept, the {{w|Hubble's law|Hubble Parameter}} can be expressed as 2.17132212x10-18 hertz, 67 km/s/Mpc is directly related to how it is measured and gives a better understanding of what it means. Another example would be fuel efficiency in cars, as mi/gal and km/l technically simplify to ares, but by expressing it in volume and distance it allows easy estimations of range and travel cost, while mm2 or in2 would require significant unit conversions.

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}

:[White Hat and Cueball are standing to either side of a refrigerator. The fridge has two top compartments and one bottom compartment. The top left compartment has a tall handle on its right, the top right compartment has a tall handle on its left, and the bottom compartment has a long handle on its top. The top left compartment has a paper attached to it with unreadable text, possibly an advertisement.]
:White Hat: This fridge uses only 3 kWh per day!
:Cueball: But will it fit in my kitchen? The ceiling there is only 50 gallons per square foot.

:[Caption below the panel:]
:Pet peeve: Uncanceled units

{{comic discussion}}

[[Category:Comics featuring White Hat]]
[[Category:Comics featuring Cueball]]
[[Category:Math]]
[[Category:Pet Peeves]]

3023: The Maritime Approximation

2024-12-11T22:54:11Z

ISaveXKCDpapers: basic intro

{{comic
| number = 3023
| date = December 11, 2024
| title = The Maritime Approximation
| image = the_maritime_approximation_2x.png
| imagesize = 265x126px
| noexpand = true
| titletext = It works because a nautical mile is based on a degree of latitude, and the Earth (e) is a circle.
}}

==Explanation==
{{incomplete|Created by a SEMICIRCULAR SAILOR - Please change this comment when editing this page. Do NOT delete this tag too soon.}}
The equality shown in this strip consists of several different parts:

# The mile per hour (mph) is a unit of speed common for motor vehicles in a few countries, such as the United States and United Kingdom.
# The knot is a unit of speed that is one nautical mile (1 852 m) per hour, used in nautical contexts.
# π is a number equal to the ratio of a circle's circumference to its diameter, about 3.14159.
# e is Euler's number, the base of the natural logarithm, about 2.71828.

π mph × (1609.344 meters/statute mile ÷ 1852 meters/nautical mile) ≈ 2.72996 knots. The result is only about 0.43% larger than ''e'' knots ≈ 2.71828 knots.

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}

:[Equation inside a frame with a footnote:]
:π mph = ''e'' knots*
:*Correct to <0.5%

:[Caption below the panel:]
:The sailor's version of eiπ=−1

{{comic discussion}}

[[Category:Math]]

Talk:2989: Physics Lab Thermostat

2024-09-27T00:36:24Z

ISaveXKCDpapers: added SCP comment

Assuming I did the math right (Units proved the units worked out, but I wouldn't otherwise assume that), holding the energy constant at "room temperature with the normal, global Boltzmann constant" this thermostat varies from 13°C (56°F) on the left to 28°C (82°F) on the right. Holding the ''temperature'' constant gives a much harder to interpret range of energies from 4.2 zeptojoules on the left to 4.0 zJ on the right. Turning those back into temperatures with the normal Boltzmann constant gives 29°C (84°F) to 15°C (59°F). Given the reversed scale, I'd assume the former is the intended interpretation, and this thermostat has no effect on local thermal energy, it just adjusts the temperature scale so the number on your (local physical constant variance-compliant) measuring device matches what you asked for. [[Special:Contributions/162.158.62.243|162.158.62.243]] 05:28, 24 September 2024 (UTC) Will

No matter the scale, I'm sure glad that this one doesn't go up to 11. [[User:Zaktduck|Zaktduck]] ([[User talk:Zaktduck|talk]]) 07:56, 24 September 2024 (UTC)

Looking at the page history, I'm wondering if the "edit conflict" didn't kick in for some people. If [https://www.explainxkcd.com/wiki/index.php?title=2989:_Physics_Lab_Thermostat&diff=next&oldid=351061 this edit] was performed over at least half an hour (quite possible), it would seem that useful edits (submitted after the start of that big addition) got wiped out. Seems unlikely that warnings happened but were deliberately over-ridden. I know this can sort of happen very soon after article creation (usually doubling-up 'first' edits), but it should have highlit any inadvertant re-editing of an interim-changed paragraph. I generally thought. [[Special:Contributions/172.70.85.18|172.70.85.18]] 09:39, 24 September 2024 (UTC) ((Ironically, I got hit by an edit-conflict just now, someone having removed linefeeds above where I'm merely appending this!))

I don't understand it the same way as you guys. Through setting the Boltzmann constant to k=1.380649×10−23 Joules per Kelvin, it's actually the Kelvin that the SI is setting. Thus, changing k in an unchanged universe changes the definition of the Kelvin, and (presumably) of Celsius or Fahrenheit too, meaning that the "temperature" reading of the room is changed without any need for heating or aircon, it's just the number which is adjusted to whatever people ask. --[[Special:Contributions/172.71.164.106|172.71.164.106]] 10:25, 24 September 2024 (UTC)
:I like this explanation. We have a thermostat like that where I work. The numbers change, but the actual temperature does not. --[[Special:Contributions/162.158.158.188|162.158.158.188]] 11:27, 24 September 2024 (UTC)
::As I read it, yes it just changes the thing that relates heat to temperature (thus not changing the sum quantity of heat), but I am not convinced that the derivative idea of Temperature doesn't have some latent qualitative effect upon the experience. The additional amount of heat in a cinder might ignite some flammable substance, the same additional heat in a brick would be barely above its normal temperature, for example. Thus conceivably the temperature from the concentrated heat-source has more bearing upon what results than the less dense 'additional heat' with lower temperature that may never invoke the vapour-threshold/flashpoint.
::Without being able to divorce or disassociate the interdependency (together with density/heat-capacith/etc), I can't be sure that such weirdness won't happen, and would not be surprised if things did (e.g. key phase-changes shift around). Like making inertial and gravitational mass independantly evaluated from each kter (if possible) would have certain real-world implications. (As well as hint that there's far more fundamental 'physics' at work than it is assumed that either/both currently are, in either newtonian or einsteinian respects.)
::I think the explanation can cover both "just recorded different" and "changes physics" in a broad scope (which is somewhat hinted at right now). But it might be in the eye of the reader (and editor) how well it does that. [[Special:Contributions/172.70.86.38|172.70.86.38]] 12:26, 24 September 2024 (UTC)

I note that "AIR" is in all caps. Is it supposed to be an Acronym? 12:28, 24 September 2024 (UTC)
:No, that's just for emphasis: it's only the AIR that changes. [[Special:Contributions/172.71.194.171|172.71.194.171]] 12:33, 24 September 2024 (UTC)
::You are both right. AIR means AIR In Room. Caution: This comment contains recursive text: First you curse, then you curse again. [[Special:Contributions/172.71.167.213|172.71.167.213]] 14:07, 24 September 2024 (UTC)

Huh... Everyone's saying "Energy is constant, Temperature number is changing.". But in that case, why would he be worried about it affecting things other than air? What if the Temperature was constant, and the Energy was what was changing? Then the dial would be doing something, and his concern over it affecting solids and liquids would be more warranted. [[Special:Contributions/172.70.178.137|172.70.178.137]] 15:04, 24 September 2024 (UTC)

The current explanation is terrible: (1) It doesn't explain well, spending way too much time on introductory and trivial pedantry than getting to the point of the joke and the concepts necessary to understand it. (2) It's way too long. And (3) the prose is terrible. My high schoolers wouldn't be allowed to use / as a synonym for "or", but that's just the beginning of the poor style, confusing grammar, and the kind of English which screams neurodiversity and home schooling.

There is no doubt in my mind that ChatGPT can do better. [[Special:Contributions/172.69.33.63|172.69.33.63]] 16:14, 24 September 2024 (UTC)

:Proof:

::This comic is playing with the idea of a "physics lab thermostat," but instead of controlling temperature, it humorously suggests adjusting the Boltzmann constant (a constant that links temperature and energy in physics, writen as 1.38×10−23 J/K, where J is joules, a unit of energy, and K is kelvins, a unit of temperature). The dial shows different values for the constant, implying that it can be changed, which is absurd because the Boltzmann constant is a fundamental number that stays the same throughout the universe. In reality, you can't change the Boltzmann constant, so the comic is making fun of the idea of a scientist casually adjusting a fixed law of physics as if it were something simple like room temperature.

::If the Boltzmann constant could be changed, it would directly affect how we experience temperature. The constant determines how much energy particles have at a certain temperature. If the constant were increased, more energy would be associated with the same temperature, so everything would feel hotter even if the temperature stayed the same. On the other hand, if the constant were decreased, less energy would be associated with the same temperature, and everything would feel colder than usual.

::Beyond just how we feel, altering the Boltzmann constant would disrupt all sorts of processes that depend on temperature, like how fast chemical reactions happen or how heat moves around. A higher constant would make particles move faster and carry more energy at a given temperature, while a lower constant would slow things down.

::The title text builds on the absurdity of being able to adjust the Boltzmann constant. It suggests that if the constant could be changed, hopefully, it would only affect the air in the room and not the entire universe. Imagining that the HVAC (Heating, Ventilation, and Air Conditioning) system could contain such a change to just the room's air shows the ridiculousness of trying to isolate the effects of altering a universal constant.

:How is that not better in every way than the current human version? [[Special:Contributions/172.70.206.157|172.70.206.157]] 16:33, 24 September 2024 (UTC)

:: It has the usual "Using wikipedia to write a book report on something you haven't read" issue that LLMs have. It retreads the same ground multiple times, explains both the humor and science only in the most surface manner, and gets the effects backwards in the section about particle energy. It also fails to explain the alternate, also humorous, interpretation of keeping pV/T constant while changing only the units we use to measure temperature with no actual effect. [[User:Scorpion451|Scorpion451]] ([[User talk:Scorpion451|talk]]) 17:16, 24 September 2024 (UTC)
:::It's far more concise than what's up at present. How does it get the effects backwards? It describes the same effects of turning up the dial as the existing explanation. I don't understand the alternate explanation, which I don't think can be an explanation since the comic doesn't refer to pV/T. [[Special:Contributions/172.71.142.47|172.71.142.47]] 18:11, 24 September 2024 (UTC)

:::There is nothing in the article page history mentioning pV/T. I prefer the ChatGPT version, except that I would ask it to include the part about, "Holding the temperature constant gives a much harder to interpret range of energies from 4.2 zeptojoules on the left to 4.0 zJ on the right. Turning those back into temperatures with the normal Boltzmann constant gives 29°C (84°F) to 15°C (59°F)." [[Special:Contributions/172.71.147.19|172.71.147.19]] 21:55, 24 September 2024 (UTC)

: Yeah I guess I'm dumb since I don't get the explanation either. Is changing the constant basically saying you are changing the definition of a temperature? EG we lowered the constant so 20C is now 30C, thus it's "warmer" (even though functionally the molecules have the same average energy, so nothing actually changed). Or is there more to it than that? [[Special:Contributions/172.68.54.139|172.68.54.139]] 20:13, 24 September 2024 (UTC)
::I replaced it with the above. I have never seen a worse explanation here ever. The original is [https://www.explainxkcd.com/wiki/index.php?title=2989:_Physics_Lab_Thermostat&oldid=351205#Explanation here], in case someone thinks anything needs to be salvaged. [[Special:Contributions/172.71.146.188|172.71.146.188]] 07:09, 26 September 2024 (UTC)
:::Really, it wasn't as bad as others have been. Anyway, I tweaked it myself. In particular, changing the 'constant' would seem to just be a matter of recalibrating what kelvin are. i.e. what was 273.15K is no longer that value, even if 0K still is ({{w|Zero-point energy|maybe}}? ...or only in the most theoretical of ideal gas?) the same. The AI seemingly cannot explain how you might feel hotter or colder with a same temperature, or understand why that's a nonsensical description (in leiu of windchill or fever effects).
:::And describing the dial range (and setting) in terms of temperature is problematic if we're not changing the heat energy (of whatever value) in any way and only adjusting what the temperature callibration may be against that. But whatever. [[Special:Contributions/172.69.43.171|172.69.43.171]] 13:32, 26 September 2024 (UTC)

: On the use of "/" for "or", unless the article has changed significantly since that complaint, it's clearly using the slash in a way that is intended to be a "higher and more localised grammatical priority" than the more wordy conjunction. You might typically say "this slash that" as a monatomic grammatical element which doesn't confuse with wider-scope conjunctions.
: "I prefer fish-and-chips to either no fish-and-chips or just fish/chips on their own...", to give an example (with a British slant, sorry; batter-fried fish and (home)fries in a more US context). You have to distinguish the "and"s as not part of the wider grammar (using "&" would work, also) and the slashness of the choice of thing to be on its own needs to not be confused with the either-or "or". (Maybe "...or just fish or chips on their own" might work, but could be read as "just fish" or "''individual'' chips", depending upon how you interpret the plural tense, especially given that "fish" can be "a fish" or "some fish(es)". Whilst "...or just fish, or chips, on their own" hits a problem that "...or just fish, or chips, on its own" might solve, but at the expense of straining grammar in a different way.)
: In the article, though, it is "heating/cooling" as an atomic choice of effect (that cross-relate directly with the following "below/above" similar alternates). The implication is that reading the sentence with either option is valid, without writing the whole sentence out twice, each time dedicated to the different singular option involved. This is different from lazily using "/" for a 'full' "or" (and "&" for "and").
: Above all, does it aid greater understanding? Yes, I think it does. The mental parsing is less ambiguous. Just like <code>variable1 and variable2||variable3</code> may be different from <code>variable1&&variable2 or variable3</code> for {False,<either>,True} input sets, in a coding language that supports both forms of logical operators at different levels of precedence – and avoiding/alleviating thebkind of bracket-soup that it might already be floating within as part of a larger logical assessment.
: It's probably intrinsically way more understandable than my above analysis and its analogies! But it's how I read it and my best way of explaining my understanding. [[Special:Contributions/172.70.91.253|172.70.91.253]] 06:58, 25 September 2024 (UTC)
::Is this supposed to be a joke?!? If it were more understandable, why isn't it a common abbreviation? [[Special:Contributions/172.71.147.146|172.71.147.146]] 06:10, 26 September 2024 (UTC)
:::¿Qué? [[Special:Contributions/172.69.194.65|172.69.194.65]] 12:43, 26 September 2024 (UTC)
Maybe worth noting: this is almost the joke of https://www.xkcd.com/2292/. {{unsigned ip|172.68.210.15|08:52, 25 September 2024}}
:1) Sign, please; 2) May be more handy to internally use something like <code><nowiki>[[2292: Thermometer]]</nowiki></code>, here, i.e. [[2292: Thermometer]]; 3) Similar basis, but weird/obscure/awkward ''actual'' measure, with the necessary constant (for conversion purposes) still being constant.
:Randall's clearly well versed in the use, or at least the generalities, of ''k''. I think these are two different jokes(/punchlines) taken from the same root physics. I'm sure there's a good way of noting the link, even if it's not as close as you suggest. [[Special:Contributions/172.70.160.182|172.70.160.182]] 10:14, 25 September 2024 (UTC)

Isn't this like SCP-536? [[User:ISaveXKCDpapers|ISaveXKCDpapers]] ([[User talk:ISaveXKCDpapers|talk]]) 00:36, 27 September 2024 (UTC)

Talk:2966: Exam Numbers

2024-08-03T13:59:11Z

ISaveXKCDpapers:

pre-algebra: 4, calculus: pi^2 / 4 (about 2.467), physics: cosmological constant: depends on how you measure it [[Special:Contributions/162.158.167.48|162.158.167.48]] 18:11, 31 July 2024 (UTC)

Game theory: -5x10⁶ (maybe helpful, maybe not... just be thankful I didn't include an ''i'' factor in there somewhere...) [[Special:Contributions/172.70.162.185|172.70.162.185]] 18:20, 31 July 2024 (UTC)
:Interesting; I went with ∞+10. So, between our answers, that makes the average... [[User:ProphetZarquon|ProphetZarquon]] ([[User talk:ProphetZarquon|talk]]) 05:21, 1 August 2024 (UTC)

Could somebody reformat all the math here in whatever LaTeX plugin this wiki uses? --[[Special:Contributions/162.158.222.102|162.158.222.102]] 18:35, 31 July 2024 (UTC)
:Probably not, because the MathML here is broken. But, also, nothing I see requires anything particularly complicated, it can all stay in fairly straightforward (standardly formatted) text. [[Special:Contributions/141.101.98.224|141.101.98.224]] 18:44, 31 July 2024 (UTC)
I had to look up "TREE(3)." Seriousness aside, I think the largest number would be the astrological sign 1 that has its end_points_ as galaxy clusters. [[Special:Contributions/172.68.245.184|172.68.245.184]] 19:26, 31 July 2024 (UTC)
:Which astrological sign? Search engines aren't helping. [[User:Onestay|Onestay]] ([[User talk:Onestay|talk]]) 20:41, 31 July 2024 (UTC)
::The nonexistent one I just made up that looks like a "1." 😃 [[Special:Contributions/172.71.222.6|172.71.222.6]] 21:06, 31 July 2024 (UTC)
:'OAK'? 'ELM'? 'ASH?' 'BOX'? 'YEW'? [[Special:Contributions/141.101.98.165|141.101.98.165]] 08:52, 1 August 2024 (UTC)

If infinity _is_ a number, it might be a possible solution to the game theory question. The average of any set of numbers that includes infinity is infinity, and infinity + 10 is still infinity. I probably wouldn't try that in most classes, but a game theory professor might approve "gaming" the system, as it were. {{unsigned ip|172.70.39.44}}
:If I would prefer no-one (else) to win, I might submit -∞ as my answer. [[Special:Contributions/172.70.90.74|172.70.90.74]] 20:13, 31 July 2024 (UTC)
::If I really wanted to mess with them, I would submit i. [[Special:Contributions/172.70.160.248|172.70.160.248]] 08:54, 1 August 2024 (UTC)

I did a bit of a deep dive into wikipedia and the googology wiki and the answer to the last question depends on a few things (along with assuming ZFC). If transfinite ordinals count as numbers, then those at the end of {{w|List of large cardinal properties}} take the cake (if i'm reading it right). Otherwise, something based off [https://googology.fandom.com/wiki/Rayo%27s_number Rayo's number] is the best googologists have come up with so far. [[Special:Contributions/172.69.246.149|172.69.246.149]] 20:18, 31 July 2024 (UTC)Bumpf

:How about "On, in the context of MK set theory"? MK is a standard way to extend ZFC by allowing classes as mathematical objects, so On (the class of all set-size ordinals) is a class-sized "ordinal". But MK doesn't allow proper classes to be contained in any object, so "On+1" doesn't exist except as a definable hyperclass. Thus, On is the biggest "number" in a model of MK set theory.{{unsigned ip|172.68.205.151}}

::I personally would not call a class a number, but that is very very subjective! And why would we take a standard extension of ZFC in the first place and not just keep ZFC? yadda yadda yadda you get the idea :) [[Special:Contributions/172.69.246.142|172.69.246.142]] 21:43, 2 August 2024 (UTC)Bumpf

Isn’t the joke in the pre-algebra that it would require algebra in order ro calculate? [[Special:Contributions/172.68.70.135|172.68.70.135]] 20:36, 31 July 2024 (UTC)
:Yes. I agree that it would be worth adding wording along the lines that “the joke here is that you need algebra to solve the equation”. [[User:Dúthomhas|Dúthomhas]] ([[User talk:Dúthomhas|talk]]) 20:56, 31 July 2024 (UTC)
: I interpreted the 'pre-' bit as being more like 'proto-' - i.e. it's not fully proper algebra, but it's the kind of work you would do in preparation for tackling proper algebra.[[Special:Contributions/172.68.186.156|172.68.186.156]] 08:58, 1 August 2024 (UTC)
::That is actually exactly correct, at least in the US. Pre-algebra teaches the basics of algebra, and any seventh-grade student _should_ actually be able to solve the given problem. IDK if Randall gave this thought when formulating the joke, though... [[User:Dúthomhas|Dúthomhas]] ([[User talk:Dúthomhas|talk]]) 05:33, 2 August 2024 (UTC)

You know, formatting math on this wiki would be a lot easier if the Math extension were correctly installed, but evidently it's not: <math>\int_0^\pi x \sin^2 x \;dx</math> [[User:Zmatt|Zmatt]] ([[User talk:Zmatt|talk]]) 22:22, 31 July 2024 (UTC)

Is that integral really correct? I asked Wolfram Alpha and it gave me
: integral x sin^2(x) dx = 1/8 (2 x (x - sin(2 x)) - cos(2 x)) + constant
which does not seem to be the same as
: −2x sin(2x)+cos(2x)−2x)/28 + C.
But maybe there's something with half-angle formulas that makes them the same? … but I don't think so, they don't evaluate the same for x=0. [[User:JohnHawkinson|JohnHawkinson]] ([[User talk:JohnHawkinson|talk]]) 02:56, 1 August 2024 (UTC)

:Yup, looks like it was supposed to be
:: -(2x sin(2x)+cos(2x)-2x^2)/8
:but they messed up the places of the negation and square.
:Though the important part here isn't what it is at any f(x), but what it is for any f(x)-f(y). In this particular case, f(pi)-f(0). [[Special:Contributions/162.158.41.121|162.158.41.121]] 04:49, 1 August 2024 (UTC)

:The answer
:: (-2x sin(2x)+cos(2x)-2x)/28 + C
:is indeed incorrect. However, your reasoning for thinking this is wrong. Two functions can have the same derivative while also having different outputs for the same input. For example, sin^2(x) and -cos^2(x) have the same derivative, but sin^2(0) = 0, whereas -cos^2(0) = -1. [[User:ISaveXKCDpapers|ISaveXKCDpapers]] ([[User talk:ISaveXKCDpapers|talk]]) 13:59, 3 August 2024 (UTC)

As to '''biggest numbers:''' I thought most people would say the answers revolved around "nine-stuffing." For a kindergartener, stuff in as many bare <code>9</code>s as possible. For a postgrad, mix in exponentiation and write your numbers even smaller than a kindergartener can. <code>9^9^9^9...</code> or perhaps <code>99^99^99...</code> or…I'm not sure what's optimal. Of course, I'm not math postdoc ;) Or maybe some integrals or big-∏ notation. [[User:JohnHawkinson|JohnHawkinson]] ([[User talk:JohnHawkinson|talk]]) 12:41, 1 August 2024 (UTC)
:Mastered division in preschool, learned exponents and logarithms in kindergarten. When I got asked this very question, my answer was (10^(10^9-1)-1), which is 999999999 "9"s. When told off for a "wrong answer format", I asked the teacher exactly how long she expected me to spend writing out literally nearly a billion digits to answer "properly" and whether she could afford that much stationery. She have me an A+. I knew the lower numbers don't actually matter so much, but it still took me until first grade to properly get into the programmatic mindset, and now the biggest finite integer I can properly consider is (2^(2^48)), which...
::10^9-1 = 999999999
::2^48 = 281474976710656
:And those are the higher numbers, so even though the lower ones are 10 versus 2, it's pretty clear which number is bigger, no? [[Special:Contributions/172.71.150.197|172.71.150.197]] 23:22, 1 August 2024 (UTC)
::If you answer "the biggest number you can think of" with "<some number> minus one", then I think you would rightfully have done yourself out of a mark... [[Special:Contributions/141.101.98.128|141.101.98.128]] 23:59, 1 August 2024 (UTC)
:::What even is thinking? Does math necessarily count? What are numbers? Whole numbers only? No living human can yet prove whether or not pi^(pi^(pi^pi)) is an integer, even though it's less than 2^(2^61). To me, "nine nines of nines" is entirely reasonable for a kindergartener who knows about exponents to answer, and while I may be slightly surprised at knowing proper expression, b^p-1 (seen both as 10^9-1 for "nine nines" and 10^(...)-1 for "of nines") appears in a lot of math things, e.g. the biggest signed 32 bit integer, 2147483647, is 2^31-1. Just because a normal internet connected computer can only count octets from 0 to 255, does that mean 255 being 2^8-1 is a more wrong answer than 2^8, which it "thinks" is 0? Does that make 0 automatically the largest number anybody can think of? [[Special:Contributions/172.68.23.200|172.68.23.200]] 00:29, 2 August 2024 (UTC)
::::I learned about tetration in college, and always thought it was the coolest thing. It is to exponentiation that exponentiation is to multiplication. The playground rules of infinity + 1 basically applies here. Whatever largest number you can write down, Just say 999^^999 for example, which being 999 to the 999th power 999 times. a googolplex is 10^^4, so this is an extremely fastly growing number, but not being 100% math nerd, don't know if this counts in any way. Rayo^^rayo, does it even make sense, who knows.--[[User:Youj ying|Youj ying]] ([[User talk:Youj ying|talk]]) 04:36, 2 August 2024 (UTC)
::::: googolplex = 10^googol = 10^10^100 = 10^10^10^2. Whereas 10^^4 = 10^10^10^10 = googolplex^(10^10^10-10^10^2), a number so much larger that the number of times you could write out googolplex using the same number of digits as 10^^4 has is itself a number so large as to have about 9999999900 digits. Even 4^^4 is bigger: ln(a^b)=ln(a)*b so ln(ln(10^10^100))=ln(ln(10)*10^100)=231.09254... and ln(ln(4^^4))=ln(ln(4)*4^4^4)=355.21799... which can be double-checked on many calculators. [[Special:Contributions/162.158.41.180|162.158.41.180]] 09:32, 2 August 2024 (UTC)

The PhD Cosmology question is easy, actually. Just write down H0 (imagine that 0 is subscript, I don't know how (if) I can format this comment). It doesn't ask you to write down the ''value'' of the constant, just the constant itself.[[Special:Contributions/172.71.103.118|172.71.103.118]] 14:56, 1 August 2024 (UTC)
:Well if you're going to get smartass about it, the last question is easy too - you just write out "THE BIGGEST NUMBER YOU CAN THINK OF"[[Special:Contributions/172.68.186.56|172.68.186.56]] 15:17, 1 August 2024 (UTC)

If you asked a psychologist what is the biggest number you can think of, they'd probably say "about 5". Anything more than that, and you're not really thinking of the number - you're just thinking of the name of the number.[[Special:Contributions/172.68.186.128|172.68.186.128]] 15:44, 1 August 2024 (UTC)
:I can absolutely visualize a grid of 9, 16, or 25. [[Special:Contributions/162.158.41.22|162.158.41.22]] 04:23, 2 August 2024 (UTC)
::Exactly, though - a grid. You're really thinking about 5 and 5 and an arrangement pattern.[[Special:Contributions/172.70.85.19|172.70.85.19]] 08:13, 2 August 2024 (UTC)

Given that postgraduate math should be real math, according to [[899: Number Line]] the largest number should be 8. My first thought for that question was card(R), as it is not really possible to prove that the number you thought about is larger than that. (Granted, the powerset of the reals is larger, if you fix an interpretation of R). --[[Special:Contributions/172.68.253.131|172.68.253.131]] 17:25, 1 August 2024 (UTC)

'''Prime notation in integrand?''': Here we go again, I guess. In [https://www.explainxkcd.com/wiki/index.php?title=2966:_Exam_Numbers&diff=next&oldid=347876 this edit], 172.68.2.126 changed the working of the integral to include expressions with a prime inside the integrand, like <code> ¼ ∫0π x (sin 2x)’ dx</code>.
I…have no idea what the prime is supposed to be indicating here…differentiation?
Even if this notation is meaningful (as it surely must be?), I don't think we should be using it here. Sure, many lay readers won't understand calculus at all, but for those that do, keeping it at a level understandable to a high-school calculus student seems wise. But since I don't understand it, I wanted to post before changing it back. What does it mean? [[User:JohnHawkinson|JohnHawkinson]] ([[User talk:JohnHawkinson|talk]]) 19:58, 1 August 2024 (UTC)

== What is a number? ==

Infinity is _not_ a number. [[User:Dúthomhas|Dúthomhas]] ([[User talk:Dúthomhas|talk]]) 19:39, 31 July 2024 (UTC)

Infinity is absolutely not a number, and is the one answer I would mark as unambiguously wrong for the last one. Just say TREE(G_64) or something. [[Special:Contributions/162.158.154.31|162.158.154.31]] 20:15, 31 July 2024 (UTC)
:This is correct. No one in post-grad math would write “infinity” and expect that answer to work. Infinity is NOT a number except for seven-year-olds. Yet the explanation above continues to posit it as a possible correct answer. [[User:Dúthomhas|Dúthomhas]] ([[User talk:Dúthomhas|talk]]) 20:49, 31 July 2024 (UTC)
::I qualify as a "post-grad math", and yet, I think infinity would have been a perfectly valid answer. Let me explain. The term "number" without further context is a bit vague, because there are several possible generalizations of natural numbers (something that presumably everyone agrees to call a "number"), and they are not compatible, ie. there is not a single generalization that generalizes them all. So we have to choose which generalization makes sense in the current context. Since the question is about thinking how big a number is, I naturally thought that the adequate generalization would be one that focuses on the order on natural numbers, ie. ordinals. In that case, my answer to this question would be "the class of numbers I can think of is not bounded, therefore there is no such thing such as a 'biggest number I can think of'". But if I had to write down a big number, I would write ε_{ε_{ε_{...}}} up until I filled the page, because that's the most efficient way I know to write a big, *big* infinity. Which is a number. (and I'm not seven, just to be clear) [[User:Jthulhu|Jthulhu]] ([[User talk:Jthulhu|talk]]) 08:35, 1 August 2024 (UTC)
:In IEEE floating point math, Infinity is ''not'' Not A Number. The latter is an indication of error (in a context where errors can't be signalled immediately) and an entirely separate concept to infinity. But both are not Normal Numbers. Or even Denormalized Numbers. Floating point math is a whole lot trickier than it appears to be at first glance, and only extremely tangentially related to mathematical reals. --[[Special:Contributions/172.68.205.54|172.68.205.54]] 00:48, 1 August 2024 (UTC)
::I would have written this, but I saw that your comment already explained the two points I would have made, so, well, well done! [[User:Jthulhu|Jthulhu]] ([[User talk:Jthulhu|talk]]) 08:35, 1 August 2024 (UTC)
:If I write a song titled "Infinity" that was part of an opera, then it would be a [https://en.wikipedia.org/wiki/Musical_number number]. [[Special:Contributions/162.158.175.141|162.158.175.141]] 13:26, 1 August 2024 (UTC)
A number, by definition, is a construct used to classify and/or compare values. How rigorous this needs be for one limits the extent to which they accept things as being a number. Even things like "apple" could be interpreted as (dimensioned) numbers, with a possible value being "1 fruit"; In that regard, one may consider things like apple=orange<grapes.

Just "infinity" is nearly useless in this regard, as it's "no end thing". Usually interpreted (when necessary) as the countable infinite cardinal x=aleph_null, this prevents most useful comparisons, including dimensional analysis since x^n=x for all counting (aka. finite positive integer) n. Spacetime may or may not be boundless, but we can't tell how many edges may or may not loop. Is it infinity? Yes. Is it infinite? God only knows. Can you *count to it*? God can. Does that make it a number? Depends. Is "infinity plus one" a sane concept? No, it can't be finite, ordinal, and/or real in a way addition is defined; It's without end, and if you could add to it, that would indicate an end.

In contrast, classification has its roots in trade, and barter, and tipping. How much of a thing is enough, but not too much. Somebody may accept between 1/2 and 2/3 of a pie you're splitting, because less wouldn't be fair and more may give them a stomach ache; Is 3<=6x<=4 a number? It's similar in uselessness to "infinity", but whether something is less or more can at least still be established within its range. In the limit, Surreal numbers are the principal example of classification, taking the arithmetic mean of the maximum and minimum of their lower and upper bounds, or the predecessor or successor, or zero. For example, y={y|1} is the biggest number less than one, with z<=y<1 for all z<1. It's less than one, but not any "smaller" than one, with an immeasurably infinitesimal difference 0<1-y.

Choice of axioms is very important for all this, since its full extent can render everything except finite non-negative integers "not a number" (by Presburger Arithmetic), or allow everything up to and including unique antichain cardinalities (by Martin's Maximum).

The sixth power of the smallest ordinal with the cardinality of the continuum in the constructed universe (w_1^6 where beth_n=C(w_n)) is the biggest number I can personally conceptualize, although I can consistently work with w_2 in this system as well. Does the fact that this is infinite make it any less useful as a number than 2.5? No. It says I can think accurately about all the standard ways of comparing things in up to 6 infinitely divisible dimensions. Just because one cannot necessarily picture something others can't doesn't mean it doesn't exist. If a one-eyed person can only see a 2 spatial + 1 temporal dimensional image, that doesn't mean depth doesn't exist, it just means it's "hidden" from that perspective. 3+1+2 has two "hidden" dimensions compared to normal 3+1 spacetime, and beth_1 is infinitely divisible unlike the quantum (at most beth_0) nature of our known universe, but I can still work with 3+1+1, and 3+1+2 in the same way people can think about a (possibly looping) universe where everything can be bigger or smaller, and spatial geometry itself may be some degree of spherical, and people have been working with fractions since antiquity, so why should I limit myself to what other people can grasp?

In summary: "number" is too vague for claiming most things "aren't" to be reasonable. Infinite values (that aren't just "infinity", that's vague enough by itself to be almost as unreasonable) are just one one example of a valid answer most people seem to be up in arms about. [[Special:Contributions/162.158.41.181|162.158.41.181]] 01:06, 1 August 2024 (UTC)

:All right, all right. I yield. That’s some... _impressive_ reasoning. If we are going to redefine words to meaninglessness then there is no hope of engaging in useful discussion. I’m sure Randall will at least get a good laugh out of the idea that post-grad math students would submit “infinity” as the largest number they could think of. I still think it a disservice to readers to posit infinity as a _valid_ answer, though. [[User:Dúthomhas|Dúthomhas]] ([[User talk:Dúthomhas|talk]]) 05:05, 1 August 2024 (UTC)
::This isn't redefining words to meaninglessness. Do you know how many branches of math there are that generalize the concept of "number"? 2500 years ago the Greeks didn't believe irrational numbers were numbers (and Hippasus supposedly drowned for discovering them), and it was a long time before 0 was accepted as a number, but now we have entire hierarchies of numbers. 100 years ago Georg Cantor created another upheaval in mathematics when he invented Set theory along with the infinite ordinals and cardinals, which are usually referred to as "numbers". Complex analysis defines the "extended complex numbers", modeled by the [https://en.wikipedia.org/wiki/Riemann_sphere Riemann sphere], which includes a "point at infinity" (and of course you can [https://en.wikipedia.org/wiki/Projectively_extended_real_line extend the real numbers]] similarly). Non-standard analysis defines not only infinite but infinitesimal numbers as [https://en.wikipedia.org/wiki/Hyperreal_number Hyperreal number]s. Granted, I'll agree that using "infitity" without any qualification or context is not really precise enough to do anything useful with. [[Special:Contributions/172.69.23.49|172.69.23.49]] 03:12, 3 August 2024 (UTC)
:::Um, hi... Nothing really to do with what you wrote, but how you wrote it. You use URL-links for wikipedia. You have <code><nowiki>[https://en.wikipedia.org/wiki/Riemann_sphere Riemann sphere]</nowiki></code> ("[https://en.wikipedia.org/wiki/Riemann_sphere Riemann sphere]") instead of the more nice templated <code><nowiki>{{w|Riemann sphere}}</nowiki></code> ("{{w|Riemann sphere}}").
:::But you also seem aware of the 'plural modifier' suffix without doing it either 'normal' way. You have given <code><nowiki>[https://en.wikipedia.org/wiki/Hyperreal_number Hyperreal number]s</nowiki></code> ("[https://en.wikipedia.org/wiki/Hyperreal_number Hyperreal number]s") instead of either <code><nowiki>[https://en.wikipedia.org/wiki/Hyperreal_number Hyperreal numbers]</nowiki></code> ("[https://en.wikipedia.org/wiki/Hyperreal_number Hyperreal numbers]"), with no point in adding the pluralisation outside the link at all, or <code><nowiki>{{w|Hyperreal number}}s</nowiki></code> ("{{w|Hyperreal number}}s") which does ''most'' of the work for you...
:::Wouldn't have commented on this at all (would have rationalised any case in an Explanation, but wouldn't change a person's Discussion contribution, even purely as a background aesthetic), and there have been a lot of 'lazy' []-linking by IPs recently where the {{template|w}}-template could have been used, but that last one seems to show an intent to do it right/nice. Just crucially missing the probable intent. So... FYI. General informative meta-comment that I hope helps you/others a bit, if you care to pass by this way again and appreciate it. [[Special:Contributions/172.69.195.175|172.69.195.175]] 08:48, 3 August 2024 (UTC)

Y'all, the answer is clearly 1. Sincerely, someone who has studied probability.[[Special:Contributions/162.158.137.155|162.158.137.155]] 14:04, 1 August 2024 (UTC)
: I came to the same conclusion by different means: A number that fills 100% of the answer area is the biggest valid answer, and is very clearly a one turned on its side. [[Special:Contributions/108.162.238.68|108.162.238.68]] 18:59, 2 August 2024 (UTC)

No actual cosmologist denotes the Hubble constant in Hz. It's about 70 (km/s)/Mpc. [[Special:Contributions/162.158.41.22|162.158.41.22]] 04:23, 2 August 2024 (UTC)
:The ''true'' Cosmic cosmologist surely knows that H0 is 1 (exactly and always), redefining all subsequently derived units in terms of that and the other 'true constants' of the universe... ;) [[Special:Contributions/141.101.99.172|141.101.99.172]] 10:43, 2 August 2024 (UTC)

Infinity comes up in kindergarten more often than most people think. Kids learn to keep asking "why?" in sequence around four, and they don't stop until they get an answer which makes them face some terrifying reality, usually concerning mortality. Infinity is small potatoes and involved in the answer to a suprising number of simple math questions. [[Special:Contributions/172.70.214.124|172.70.214.124]] 05:50, 3 August 2024 (UTC)

528: Windows 7

2024-06-18T19:48:01Z

ISaveXKCDpapers: spelling

{{comic
| number = 528
| date = January 9, 2009
| title = Windows 7
| image = windows_7.png
| titletext = Disclaimer: I have not actually tried the beta yet. I hear it's quite pleasant and hardly Hitler-y at all.
}}

==Explanation==
This comic came out 2 weeks after the beta version of the at that time not yet released {{w|Windows 7}} got leaked on the internet, and 2 days after a trojan-infected version got leaked as well. [[Megan]] is observing [[Cueball]] use a laptop on which he has installed the Windows 7 beta. However, the alleged Windows 7 beta is showing nothing but a picture of {{w|Adolf Hitler}} and Cueball is unable to do anything. This could hint at the version containing malware. Cueball then presses {{w|Control-Alt-Delete}} (the well-known Windows Secure Attention Sequence which opens Task Manager or displays a list of options which includes 'Shut Down' and 'Restart') as suggested by Megan, but {{tvtropes|AdolfHitlarious|only manages to make the picture's eyes flash}}.

The fourth panel shows Megan commenting that this Windows 7 beta is better than {{w|Windows Vista}}, to which Cueball agrees. The joke is that Megan deems a mostly non-functional and vaguely sinister OS, which is likely malware, better than Windows Vista, which was generally perceived as one of the worst Windows OSes.

The title text is a disclaimer stating that [[Randall]] has not tried the beta at the time this comic was written, but what he has heard about it he regards as at least mildly positive. However, he also {{tvtropes|DamnedByFaintPraise|damns it by faint praise}}, referring to it as hardly Hitler-y at all, a statement that could raise warning flags, as one would not generally expect an operating system to be able to be described as Hitler-y to any degree at all. "Hitler-y" is implied to mean pertaining to or having qualities similar to Adolf Hitler, the late German Nazi Party leader and perhaps the most notorious instigator of mass murder in history.

==Transcript==
:[Megan is standing behind Cueball sitting at a desk using his laptop.]
:Megan: What are you doing?
:Cueball: Trying the Windows 7 beta.
:Megan: Why is it showing a picture of Hitler?
:[The laptop's screen is shown with Adolf Hitler's face on it, drawn in regular ''xkcd'' style with no facial features other than his hairstyle and mustache.]
:Cueball: I don't know. I can't get it to do anything else.
:Megan: There's no UI?
:Cueball: No, just Hitler.
:[Return to the original scene, except Megan is now scratching her head in confusion.]
:Megan: Did you try Ctrl-Alt-Delete?
:Cueball: It just makes Hitler's eyes flash.
:Megan: Huh.
:[Scene remains basically the same, except Megan is no longer scratching her head and Cueball is no longer typing on the laptop.]
:Megan: Well, it's better than Vista.
:Cueball: True.

{{comic discussion}}

[[Category:Comics featuring Cueball]]
[[Category:Comics featuring Megan]]
[[Category:Comics featuring Adolf Hitler]]
[[Category:Computers]]
[[Category:Cueball Computer Problems]]
[[Category:Comics featuring politicians]]

2117: Differentiation and Integration

2024-06-11T14:46:47Z

ISaveXKCDpapers:

{{comic
| number = 2117
| date = February 27, 2019
| title = Differentiation and Integration
| image = differentiation_and_integration.png
| titletext = "Symbolic integration" is when you theatrically go through the motions of finding integrals, but the actual result you get doesn't matter because it's purely symbolic.
}}

==Explanation==
This comic illustrates the old saying [https://mathoverflow.net/q/66377 "Differentiation is mechanics, integration is art."] It does so by providing a {{w|flowchart}} purporting to show the process of differentiation, and another for integration.

{{w|Derivative|Differentiation}} and {{w|Antiderivative|Integration}} are two major components of {{w|calculus}}. As many Calculus 2 students are painfully aware, integration is much more complicated than the differentiation it undoes.

However, Randall dramatically overstates this point here. After the first step of integration, Randall assumes that any integration can not be solved so simply, and then dives into a step named "????", suggesting that it is unknowable how to proceed. The rest of the flowchart is (we can assume deliberately) even harder to follow, and does not reach a conclusion. This is in contrast to the simple, straightforward flowchart for differentiation. The fact that the arrows in the bottom of the integration part leads to nowhere indicates that "Phone calls to mathematicians", "Oh no" and "Burn the evidence" are not final steps in the difficult journey. The flowchart could be extended by Randall to God-knows-where extents.

It should be noted that Randall slightly undermines his point by providing four different methods, and an "etc", and a "No"-branch for attempting differentiation with no guidelines for selecting between them.

===Differentiation===
'''{{w|Chain rule}}'''

For any <math> \frac{d}{dx}f(x)=f'(x)</math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx}(f(g(x)))=f'(g(x))\cdot g'(x)</math>.

'''{{w|Power Rule}}'''

For any <math> f(x)=g(x)^a </math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx}f(x)=a\cdot g(x)^{a-1}\cdot g'(x) </math>.

'''{{w|Quotient rule}}'''

For any <math> \frac{d}{dx}f(x)=f'(x)</math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx} \frac{f(x)}{g(x)}=\frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{(g(x))^2}</math> if <math>g(x)\ne 0</math>.

'''{{w|Product rule}}'''

For any <math> \frac{d}{dx}f(x)=f'(x)</math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx}(f(x)\cdot g(x))=f'(x)\cdot g(x)+f(x)\cdot g'(x)</math>.

===Integration===
'''{{w|Integration by parts}}'''

The "product rule" run backwards. Since <math>(uv)' = uv' + u'v</math>, it follows that by integrating both sides you get <math> uv = \int u dv + \int v du</math>, which is more commonly written as <math>\int u dv = uv - \int v du</math>. By finding appropriate values for functions <math>u, v</math> such that your problem is in the form <math>\int u dv</math>, your problem ''may'' be simplified. The catch is, there exists no algorithm for determining what functions they might possibly be, so this approach quickly devolves into a guessing game - this has been the topic of an earlier comic, [[1201: Integration by Parts]].

'''{{w|Integration by substitution|Substitution}}'''

The "chain rule" run backwards. Since <math> d(f(u)) = f'(u) \frac{du}{dx} dx</math>, it follows that <math>f(u) = \int f'(u) \frac{du}{dx} dx</math>. By finding appropriate values for functions <math>f, u</math> such that your problem is in the form <math>\int f'(u) \frac{du}{dx} dx</math> your problem ''may'' be simplified.

'''{{w|Cauchy's integral formula|Cauchy's Formula}}'''

Cauchy's Integral formula is a result in complex analysis that relates the value of a contour integral in the complex plane to properties of the singularities in the interior of the contour. <math> \frac{d^n}{da^n}f(a) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(z)}{\left(z-a\right)^{n+1}}\, dz.</math> It is often used to compute integrals on the real line by extending the path of the integral from the real line into the complex plane to apply the formula, then proving that the integral from the parts of the contour not on the real line has value zero.

'''{{w|Partial_fraction_decomposition#Application_to_symbolic_integration|Partial Fractions}}'''

Partial fractions is a technique for breaking up a function that comprises one polynomial divided by another into a sum of functions comprising constants over the factors of the original denominator, which can easily be integrated into logarithms.

'''Install {{w|Mathematica}}'''

Mathematica is a modern technical computing system spanning most areas. One of its features is to compute mathematical functions. This step in the flowchart is to install and use Mathematica to do the integration for you. Here is a description about the [https://web.archive.org/web/20180727184709/http://reference.wolfram.com/language/tutorial/IntegralsThatCanAndCannotBeDone.html intricacies of integration and how Mathematica handles those]. (It would be quicker to try [https://www.wolframalpha.com Wolfram Alpha] instead of installing Mathematica, which uses the same backend for mathematical calculations.)

'''{{w|Riemann integral|Riemann Integration}}'''

The Riemann integral is a definition of definite integration. Elementary textbooks on calculus sometimes present finding a definite integral as a process of approximating an area by strips of equal width, as in <math>\sum_{i=0}^{n-1} f(t_i) \left(x_{i+1}-x_i\right).</math>, and then taking the limit as the strips become narrower. Riemann integration removes the requirement that the strips have equal width, and so is a more flexible definition. However there are still many functions for which the Riemann integral doesn't converge, and consideration of these functions leads to the {{w|Lebesgue integration|Lebesgue integral}}. Riemann integration is not a method of calculus appropriate for finding the anti-derivative of an elementary function.

'''{{w|Stokes' Theorem}}'''

Stokes' theorem is a statement about the integration of differential forms on manifolds. <math>\int_{\partial \Omega}\omega=\int_\Omega d\omega\,.</math> It is invoked in science and engineering during control volume analysis (that is, to track the rate of change of a quantity within a control volume, it suffices to track the fluxes in and out of the control volume boundary), but is rarely used directly (and even when it is used directly, the functions that are most frequently used in science and engineering are well-behaved, like sinusoids and polynomials).

'''{{w|Risch Algorithm}}'''

The Risch algorithm is a notoriously complex procedure that, given a certain class of symbolic integrand, either finds a symbolic integral or proves that no elementary integral exists. (Technically it is only a semi-algorithm, and cannot produce an answer unless it can determine if a certain symbolic expression is {{w|Constant problem|equal to 0}} or not.) Many computer algebra systems have chosen to implement only the simpler Risch-Norman algorithm, which does not come with the same guarantee. A series of extensions to the Risch algorithm extend the class of allowable functions to include (at least) the error function and the logarithmic integral. A human would have to be pretty desperate to attempt this (presumably) by hand.

'''{{w|Bessel function}}'''

Bessel functions are the solution to the differential equation <math> x^2 \frac{d^2y}{dx^2}+x \frac{dy}{dx}+(x^2-n^2)*y=0</math>, where n is the order of Bessel function. Though they do show up in some engineering, physics, and abstract mathematics, in lower levels of calculus they are often a sign that the integration was not set up properly before someone put them into a symbolic algebra solver.

'''Phone calls to mathematicians'''

This step would indicate that the flowchart user, desperate from failed attempts to solve the problem, contacts some more skilled mathematicians by phone, and presumably asks them for help. The connected steps of "Oh no", "What the heck is a Bessel function?" and "Burn the evidence" may suggest the possibility that this interaction might not play out very well and could even get the caller in trouble.
Specialists and renowned experts being bothered - not to their amusement - by strangers, often at highly inconvenient times or locations, is a common comedic trope, also previously utilized by xkcd (for example in [[163: Donald Knuth]]).

'''Burn the evidence'''

This phrase parodies a common trope in detective fiction, where characters burn notes, receipts, passports, etc. to maintain secrecy. This may refer to the burning of one's work to avoid the shame of being associated with such a badly failed attempt to solve the given integration problem. Moreover, such a poor attempt at integration could be viewed as a 'crime against mathematics', with the working of the problem being criminal 'evidence' that must be destroyed to avoid exposure as the culprit. Alternatively, it could be an ironic hint to the fact that in order to find the integral, it may even be necessary to break the law or upset higher powers, so the negative consequences of a prosecution can only be avoided by destroying the evidence.

'''{{w|Symbolic integration}}'''

Symbolic integration is the basic process of finding an antiderivative function (defined with symbols), as opposed to numerically integrating a function. The title text is a pun that defines the term not as integration that works with symbols, but rather as integration as a symbolic act, as if it were a component of a ritual. A symbolic act in a ritual is an act meant to evoke something else, such as burning a wooden figurine of a person to represent one’s hatred of that person. Alternatively, the reference could be seen as a joke that integration might as well be a symbol, like in a novel, because Randall can't get any meaningful results from his analysis.

==Transcript==
:[Two flow charts are shown.]

:[The first flow chart has four steps in simple order, one with multiple recommendations.]
:DIFFERENTIATION
:Start
:Try applying
::Chain Rule
::Power Rule
::Quotient Rule
::Product Rule
::Etc.
:Done?
::No [Arrow returns to "Try applying" step.]
::Yes
:Done!

[The second flow chart begins like the first, then descends into chaos.]
:INTEGRATION
:Start
:Try applying
::Integration by Parts
::Substitution
:Done?
:Haha, Nope!

:[Chaos, Roughly from left to right, top to bottom, direction arrows not included.]
::Cauchy's Formula
::????
::???!?
::???
::???
::?
::Partial Fractions
::??
::?
::Install Mathematica
::?
::Riemann Integration
::Stokes' Theorem
::???
::?
::Risch Algorithm
::???
::[Sad face.]
::?????
::???
::What the heck is a Bessel Function??
::Phone calls to mathematicians
::Oh No
::Burn the Evidence
::[More arrows pointing out of the image to suggest more steps.]

{{comic discussion}}
[[Category:Analysis]]
[[Category:Flowcharts]]
[[Category:Math]]

2117: Differentiation and Integration

2024-06-11T14:44:40Z

ISaveXKCDpapers: integrated sum expression into text in Riemann integral section

{{comic
| number = 2117
| date = February 27, 2019
| title = Differentiation and Integration
| image = differentiation_and_integration.png
| titletext = "Symbolic integration" is when you theatrically go through the motions of finding integrals, but the actual result you get doesn't matter because it's purely symbolic.
}}

==Explanation==
This comic illustrates the old saying [https://mathoverflow.net/q/66377 "Differentiation is mechanics, integration is art."] It does so by providing a {{w|flowchart}} purporting to show the process of differentiation, and another for integration.

{{w|Derivative|Differentiation}} and {{w|Antiderivative|Integration}} are two major components of {{w|calculus}}. As many Calculus 2 students are painfully aware, integration is much more complicated than the differentiation it undoes.

However, Randall dramatically overstates this point here. After the first step of integration, Randall assumes that any integration can not be solved so simply, and then dives into a step named "????", suggesting that it is unknowable how to proceed. The rest of the flowchart is (we can assume deliberately) even harder to follow, and does not reach a conclusion. This is in contrast to the simple, straightforward flowchart for differentiation. The fact that the arrows in the bottom of the integration part leads to nowhere indicates that "Phone calls to mathematicians", "Oh no" and "Burn the evidence" are not final steps in the difficult journey. The flowchart could be extended by Randall to God-knows-where extents.

It should be noted that Randall slightly undermines his point by providing four different methods, and an "etc", and a "No"-branch for attempting differentiation with no guidelines for selecting between them.

===Differentiation===
'''{{w|Chain rule}}'''

For any <math> \frac{d}{dx}f(x)=f'(x)</math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx}(f(g(x)))=f'(g(x))\cdot g'(x)</math>.

'''{{w|Power Rule}}'''

For any <math> f(x)=g(x)^a </math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx}f(x)=a\cdot g(x)^{a-1}\cdot g'(x) </math>.

'''{{w|Quotient rule}}'''

For any <math> \frac{d}{dx}f(x)=f'(x)</math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx} \frac{f(x)}{g(x)}=\frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{(g(x))^2}</math> if <math>g(x)\ne 0</math>.

'''{{w|Product rule}}'''

For any <math> \frac{d}{dx}f(x)=f'(x)</math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx}(f(x)\cdot g(x))=f'(x)\cdot g(x)+f(x)\cdot g'(x)</math>.

===Integration===
'''{{w|Integration by parts}}'''

The "product rule" run backwards. Since <math>(uv)' = uv' + u'v</math>, it follows that by integrating both sides you get <math> uv = \int u dv + \int v du</math>, which is more commonly written as <math>\int u dv = uv - \int v du</math>. By finding appropriate values for functions <math>u, v</math> such that your problem is in the form <math>\int u dv</math>, your problem ''may'' be simplified. The catch is, there exists no algorithm for determining what functions they might possibly be, so this approach quickly devolves into a guessing game - this has been the topic of an earlier comic, [[1201: Integration by Parts]].

'''{{w|Integration by substitution|Substitution}}'''

The "chain rule" run backwards. Since <math> d(f(u)) = f'(u) \frac{du}{dx} dx</math>, it follows that <math>f(u) = \int f'(u) \frac{du}{dx} dx</math>. By finding appropriate values for functions <math>f, u</math> such that your problem is in the form <math>\int f'(u) \frac{du}{dx} dx</math> your problem ''may'' be simplified.

'''{{w|Cauchy's integral formula|Cauchy's Formula}}'''

Cauchy's Integral formula is a result in complex analysis that relates the value of a contour integral in the complex plane to properties of the singularities in the interior of the contour. <math> \frac{d^n}{da^n}f(a) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(z)}{\left(z-a\right)^{n+1}}\, dz.</math> It is often used to compute integrals on the real line by extending the path of the integral from the real line into the complex plane to apply the formula, then proving that the integral from the parts of the contour not on the real line has value zero.

'''{{w|Partial_fraction_decomposition#Application_to_symbolic_integration|Partial Fractions}}'''

Partial fractions is a technique for breaking up a function that comprises one polynomial divided by another into a sum of functions comprising constants over the factors of the original denominator, which can easily be integrated into logarithms.

'''Install {{w|Mathematica}}'''

Mathematica is a modern technical computing system spanning most areas. One of its features is to compute mathematical functions. This step in the flowchart is to install and use Mathematica to do the integration for you. Here is a description about the [https://web.archive.org/web/20180727184709/http://reference.wolfram.com/language/tutorial/IntegralsThatCanAndCannotBeDone.html intricacies of integration and how Mathematica handles those]. (It would be quicker to try [https://www.wolframalpha.com Wolfram Alpha] instead of installing Mathematica, which uses the same backend for mathematical calculations.)

'''{{w|Riemann integral|Riemann Integration}}'''

The Riemann integral is a definition of definite integration. Elementary textbooks on calculus sometimes present finding a definite integral as a process of approximating an area by strips of equal width, as in <math>\sum_{i=0}^{n-1} f(t_i) \left(x_{i+1}-x_i\right),</math> and then taking the limit as the strips become narrower. Riemann integration removes the requirement that the strips have equal width, and so is a more flexible definition. However there are still many functions for which the Riemann integral doesn't converge, and consideration of these functions leads to the {{w|Lebesgue integration|Lebesgue integral}}. Riemann integration is not a method of calculus appropriate for finding the anti-derivative of an elementary function.

'''{{w|Stokes' Theorem}}'''

Stokes' theorem is a statement about the integration of differential forms on manifolds. <math>\int_{\partial \Omega}\omega=\int_\Omega d\omega\,.</math> It is invoked in science and engineering during control volume analysis (that is, to track the rate of change of a quantity within a control volume, it suffices to track the fluxes in and out of the control volume boundary), but is rarely used directly (and even when it is used directly, the functions that are most frequently used in science and engineering are well-behaved, like sinusoids and polynomials).

'''{{w|Risch Algorithm}}'''

The Risch algorithm is a notoriously complex procedure that, given a certain class of symbolic integrand, either finds a symbolic integral or proves that no elementary integral exists. (Technically it is only a semi-algorithm, and cannot produce an answer unless it can determine if a certain symbolic expression is {{w|Constant problem|equal to 0}} or not.) Many computer algebra systems have chosen to implement only the simpler Risch-Norman algorithm, which does not come with the same guarantee. A series of extensions to the Risch algorithm extend the class of allowable functions to include (at least) the error function and the logarithmic integral. A human would have to be pretty desperate to attempt this (presumably) by hand.

'''{{w|Bessel function}}'''

Bessel functions are the solution to the differential equation <math> x^2 \frac{d^2y}{dx^2}+x \frac{dy}{dx}+(x^2-n^2)*y=0</math>, where n is the order of Bessel function. Though they do show up in some engineering, physics, and abstract mathematics, in lower levels of calculus they are often a sign that the integration was not set up properly before someone put them into a symbolic algebra solver.

'''Phone calls to mathematicians'''

This step would indicate that the flowchart user, desperate from failed attempts to solve the problem, contacts some more skilled mathematicians by phone, and presumably asks them for help. The connected steps of "Oh no", "What the heck is a Bessel function?" and "Burn the evidence" may suggest the possibility that this interaction might not play out very well and could even get the caller in trouble.
Specialists and renowned experts being bothered - not to their amusement - by strangers, often at highly inconvenient times or locations, is a common comedic trope, also previously utilized by xkcd (for example in [[163: Donald Knuth]]).

'''Burn the evidence'''

This phrase parodies a common trope in detective fiction, where characters burn notes, receipts, passports, etc. to maintain secrecy. This may refer to the burning of one's work to avoid the shame of being associated with such a badly failed attempt to solve the given integration problem. Moreover, such a poor attempt at integration could be viewed as a 'crime against mathematics', with the working of the problem being criminal 'evidence' that must be destroyed to avoid exposure as the culprit. Alternatively, it could be an ironic hint to the fact that in order to find the integral, it may even be necessary to break the law or upset higher powers, so the negative consequences of a prosecution can only be avoided by destroying the evidence.

'''{{w|Symbolic integration}}'''

Symbolic integration is the basic process of finding an antiderivative function (defined with symbols), as opposed to numerically integrating a function. The title text is a pun that defines the term not as integration that works with symbols, but rather as integration as a symbolic act, as if it were a component of a ritual. A symbolic act in a ritual is an act meant to evoke something else, such as burning a wooden figurine of a person to represent one’s hatred of that person. Alternatively, the reference could be seen as a joke that integration might as well be a symbol, like in a novel, because Randall can't get any meaningful results from his analysis.

==Transcript==
:[Two flow charts are shown.]

:[The first flow chart has four steps in simple order, one with multiple recommendations.]
:DIFFERENTIATION
:Start
:Try applying
::Chain Rule
::Power Rule
::Quotient Rule
::Product Rule
::Etc.
:Done?
::No [Arrow returns to "Try applying" step.]
::Yes
:Done!

[The second flow chart begins like the first, then descends into chaos.]
:INTEGRATION
:Start
:Try applying
::Integration by Parts
::Substitution
:Done?
:Haha, Nope!

:[Chaos, Roughly from left to right, top to bottom, direction arrows not included.]
::Cauchy's Formula
::????
::???!?
::???
::???
::?
::Partial Fractions
::??
::?
::Install Mathematica
::?
::Riemann Integration
::Stokes' Theorem
::???
::?
::Risch Algorithm
::???
::[Sad face.]
::?????
::???
::What the heck is a Bessel Function??
::Phone calls to mathematicians
::Oh No
::Burn the Evidence
::[More arrows pointing out of the image to suggest more steps.]

{{comic discussion}}
[[Category:Analysis]]
[[Category:Flowcharts]]
[[Category:Math]]

Talk:2117: Differentiation and Integration

2024-06-11T14:39:45Z

ISaveXKCDpapers: /* Change to explanation of substitution */

Darn, I have no idea what this comic is about. Randal has eluded my yet again. [[User:Linker|Linker]] ([[User talk:Linker|talk]]) 17:43, 27 February 2019 (UTC)
:Calculus. [[Special:Contributions/162.158.79.143|162.158.79.143]] 18:16, 27 February 2019 (UTC)
::Basically, differentiation is easy to do by hand, but integration, even of things that look simple on paper, can be very difficult, as well as easy to mess up or get lost in. [[User:Glassvein|Glassvein]] ([[User talk:Glassvein|talk]]) 01:52, 6 March 2019 (UTC)

And Calc 2 is why I stopped being a Computer Science major and moved (eventually) to majoring in English. Consistent 4.0s in math through Trig and Calc I ... 1.6 in Calc II, retook and got a 1.8. Without the Calc, couldn't do the physics; without the physics, couldn't get my 2-yr degree and move on from community college to a full university. I don't know what all the integration stuff in the flowchart is (since I didn't do well in Calc and it was a long time ago), but there's so very many things that become [https://en.wikipedia.org/wiki/Nonelementary_integral nonelementary integrals] that all sorts of special tricks have to be employed for things that look like they should be easy. It's like having a problem that's very easy to do division on, but requires special advanced mathematical tricks to use multiplication upon.[[Special:Contributions/108.162.216.208|108.162.216.208]] 19:07, 27 February 2019 (UTC)

Going to start learning integration in 3 weeks... Wish me luck. [[Special:Contributions/162.158.62.96|162.158.62.96]] 12:53, 3 November 2021 (UTC)

Basic ideas:
Integration by parts is the reverse of the Product Rule.
Substitution is the reverse of the Chain Rule.
Cauchy's Formula gives the result of a contour integration in the complex plane, using "singularities" of the integrand.
Partial fractions is just splitting up one complex fraction into a sum of simple fractions, which is relevant because they are easier to integrate.
Stokes theorem is the relationship between an integral over an area, and an integral over the boundary of said area.
Riemann integration was the first rigorous definition of integration. This has been superseded by Lesbesgue integration.
Bessel functions are like 2d versions of sin and cos, and turn up sometimes when doing integration.[[Special:Contributions/162.158.89.61|162.158.89.61]] 20:14, 27 February 2019 (UTC)
:"Lesbesgue integration." Best. Freudian. Slip. Ever. SCNR :P [[Special:Contributions/162.158.91.59|162.158.91.59]] 08:28, 28 February 2019 (UTC)
I know what you mean ;). After all, Gen 8 Pokemon was announced the other day, so you read it as "Pokemon League Integration". Completely understandable. [[Special:Contributions/172.68.78.28|172.68.78.28]] 14:40, 28 February 2019 (UTC)
Shouldn't Wolfram Alpha be somewhere in that flowchart? [[Special:Contributions/162.158.255.142|162.158.255.142]] 20:54, 27 February 2019 (UTC)

Glad to see I'm not the only one who is too dumb to integrate [[Special:Contributions/162.158.90.36|162.158.90.36]] 21:02, 27 February 2019 (UTC)
:Symbolic differentiation is just going through algorithm ; there are few functions which don't have it but they tend to be constructed in complicated way, and if function have differentiation it's usually easy to find it. Symbolic integration requires lot of thinking and trial and error ; even very easy function may lack primitive function and even if they don't, you may be unable to find it except randomly. If it's exercise in book, the ones for differentiation are done by thinking about some interesting function and putting it there. The ones for integration are done by thinking about some interesting function and putting it's differentiation there. -- [[User:Hkmaly|Hkmaly]] ([[User talk:Hkmaly|talk]]) 23:38, 27 February 2019 (UTC)

Oddly enough it mentions Riemann integration, but that is the integral most people know how to use. Turns out there are a lot more (e.g. lebesgue and generalized riemann integrals). I'm halfway through a second semester of real analysis and was floored by how involved integration can be. [[Special:Contributions/172.68.34.106|172.68.34.106]] 21:36, 27 February 2019 (UTC)

One of my professors once said: "Never try to integrate a function. Almost all (in a strict mathematical sense) functions are impossible to integrate, so there is no reason why you should even try." --[[Special:Contributions/162.158.88.128|162.158.88.128]] 07:52, 28 February 2019 (UTC)

How is there no "+ C" joke in there [[User:Blagae|Blagae]] ([[User talk:Blagae|talk]]) 13:16, 28 February 2019
(UTC)

Probably because he put a +C joke in 1201:_Integration_by_Parts. [[Special:Contributions/108.162.219.160|108.162.219.160]] 13:48, 2 March 2019 (UTC)

Why is all the maths broken [[User:GcGYSF(asterisk)P(vertical line)e|GcGYSF(asterisk)P(vertical line)e]] ([[User talk:GcGYSF(asterisk)P(vertical line)e|talk]]) 22:24, 4 May 2022 (UTC)
== Risch algorithm ==

I thought I could contribute to the article with a better explanation of the Risch algorithm, since I have a bit of expertise here -- I've read all the original papers, plus the Cherry papers that add the extra features like Li and erf. I pulled out some of the old papers to review my knowledge of symbolic differential algebra (it's been a while!) then typed up a careful explanation which corrected some errors in the original description and fleshed out many more details... possibly excessively, but hey, that's kind of our calling here.

Then I saw that Glassvein completely removed my version for what appears to be the original without so much as a mention in the edit description. What gives? I

[[User:CRGreathouse|CRGreathouse]] ([[User talk:CRGreathouse|talk]]) 04:59, 28 February 2019 (UTC)
: Probably due to simultaneous editing. I've restored your definition. [[Special:Contributions/162.158.88.128|162.158.88.128]] 16:52, 28 February 2019 (UTC)

:: OK, wasn't sure if it was intentional (if somehow it was worse). Thanks! [[User:CRGreathouse|CRGreathouse]] ([[User talk:CRGreathouse|talk]]) 01:34, 1 March 2019 (UTC)
That was indeed an accident due to simultaneous editing. My bad!
[[User:Glassvein|Glassvein]] ([[User talk:Glassvein|talk]]) 02:47, 5 March 2019 (UTC)

== Numerical Integration ==
Better still...plot the graph - cut along the line - weigh the part under the line. :-) [[User:SteveBaker|SteveBaker]] ([[User talk:SteveBaker|talk]]) 20:46, 28 February 2019 (UTC)

[Anonymous: I understand mathematically that integration is much more difficult than differentiation, but is there a possibility that Randall is making the comment that the same is true for Society? Integration has proved very difficult, and has led to riots, but experience shows that dividing our society up into small subgroups (that then argue with each other, but don't spend enough time together for riots) is relatively easy.]{{unsigned ip|108.162.219.112}}

:Good luck plotting {{w|Weierstrass function}}. -- [[User:Hkmaly|Hkmaly]] ([[User talk:Hkmaly|talk]]) 00:05, 9 March 2019 (UTC)

== overstates the case? ==

Current summary says that the comic overstates the case of how difficult integration can be. I'm not sure that's true. Sure, you can use numerical integration to get a specific area under the curve, but that's not what the comic is referring to. Unless some mathematician can show here how integration can be done by repeatedly following a set of fixed rules, Ithis comic is actually completely accurate. And that's why it's funny. :-)

== Purify the Power rule? ==
The derivative power rule shown is combined with the chain rule. I think it should be stated to be a pure power rule, without the chain rule components. When I tried making that adjustment, I got error messages that I could not resolve, so could someone who knows how the MATH feature works remove the chain rule from the power rule? [[User:Nutster|Nutster]] ([[User talk:Nutster|talk]]) 18:44, 26 December 2021 (UTC)

== Change to explanation of substitution ==

So the LaTeX on this wiki is apparently broken, meaning we can't change the math things without stuff screwing up. Unfortunately, there are a lot of alterations I'd like to make. Most are minor, like simplifying the power rule explanation, but more importantly, the explanation for integration by substitution is straight-up wrong. I'm going ahead with the change anyway, because hey, a broken explanation can't be any worse than a wrong explanation, right? I'm just gonna wait and hope that this gets fixed in the future. [[User:ISaveXKCDpapers|ISaveXKCDpapers]] ([[User talk:ISaveXKCDpapers|talk]]) 14:39, 11 June 2024 (UTC)

Talk:2117: Differentiation and Integration

2024-06-11T14:39:32Z

ISaveXKCDpapers: /* Change to explanation of substitution */ new section

2117: Differentiation and Integration

2024-06-11T14:32:30Z

ISaveXKCDpapers: /* Integration */

{{comic
| number = 2117
| date = February 27, 2019
| title = Differentiation and Integration
| image = differentiation_and_integration.png
| titletext = "Symbolic integration" is when you theatrically go through the motions of finding integrals, but the actual result you get doesn't matter because it's purely symbolic.
}}

==Explanation==
This comic illustrates the old saying [https://mathoverflow.net/q/66377 "Differentiation is mechanics, integration is art."] It does so by providing a {{w|flowchart}} purporting to show the process of differentiation, and another for integration.

{{w|Derivative|Differentiation}} and {{w|Antiderivative|Integration}} are two major components of {{w|calculus}}. As many Calculus 2 students are painfully aware, integration is much more complicated than the differentiation it undoes.

However, Randall dramatically overstates this point here. After the first step of integration, Randall assumes that any integration can not be solved so simply, and then dives into a step named "????", suggesting that it is unknowable how to proceed. The rest of the flowchart is (we can assume deliberately) even harder to follow, and does not reach a conclusion. This is in contrast to the simple, straightforward flowchart for differentiation. The fact that the arrows in the bottom of the integration part leads to nowhere indicates that "Phone calls to mathematicians", "Oh no" and "Burn the evidence" are not final steps in the difficult journey. The flowchart could be extended by Randall to God-knows-where extents.

It should be noted that Randall slightly undermines his point by providing four different methods, and an "etc", and a "No"-branch for attempting differentiation with no guidelines for selecting between them.

===Differentiation===
'''{{w|Chain rule}}'''

For any <math> \frac{d}{dx}f(x)=f'(x)</math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx}(f(g(x)))=f'(g(x))\cdot g'(x)</math>.

'''{{w|Power Rule}}'''

For any <math> f(x)=g(x)^a </math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx}f(x)=a\cdot g(x)^{a-1}\cdot g'(x) </math>.

'''{{w|Quotient rule}}'''

For any <math> \frac{d}{dx}f(x)=f'(x)</math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx} \frac{f(x)}{g(x)}=\frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{(g(x))^2}</math> if <math>g(x)\ne 0</math>.

'''{{w|Product rule}}'''

For any <math> \frac{d}{dx}f(x)=f'(x)</math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx}(f(x)\cdot g(x))=f'(x)\cdot g(x)+f(x)\cdot g'(x)</math>.

===Integration===
'''{{w|Integration by parts}}'''

The "product rule" run backwards. Since <math>(uv)' = uv' + u'v</math>, it follows that by integrating both sides you get <math> uv = \int u dv + \int v du</math>, which is more commonly written as <math>\int u dv = uv - \int v du</math>. By finding appropriate values for functions <math>u, v</math> such that your problem is in the form <math>\int u dv</math>, your problem ''may'' be simplified. The catch is, there exists no algorithm for determining what functions they might possibly be, so this approach quickly devolves into a guessing game - this has been the topic of an earlier comic, [[1201: Integration by Parts]].

'''{{w|Integration by substitution|Substitution}}'''

The "chain rule" run backwards. Since <math> d(f(u)) = f'(u) \frac{du}{dx} dx</math>, it follows that <math>f(u) = \int f'(u) \frac{du}{dx} dx</math>. By finding appropriate values for functions <math>f, u</math> such that your problem is in the form <math>\int f'(u) \frac{du}{dx} dx</math> your problem ''may'' be simplified.

'''{{w|Cauchy's integral formula|Cauchy's Formula}}'''

Cauchy's Integral formula is a result in complex analysis that relates the value of a contour integral in the complex plane to properties of the singularities in the interior of the contour. <math> \frac{d^n}{da^n}f(a) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(z)}{\left(z-a\right)^{n+1}}\, dz.</math> It is often used to compute integrals on the real line by extending the path of the integral from the real line into the complex plane to apply the formula, then proving that the integral from the parts of the contour not on the real line has value zero.

'''{{w|Partial_fraction_decomposition#Application_to_symbolic_integration|Partial Fractions}}'''

Partial fractions is a technique for breaking up a function that comprises one polynomial divided by another into a sum of functions comprising constants over the factors of the original denominator, which can easily be integrated into logarithms.

'''Install {{w|Mathematica}}'''

Mathematica is a modern technical computing system spanning most areas. One of its features is to compute mathematical functions. This step in the flowchart is to install and use Mathematica to do the integration for you. Here is a description about the [https://web.archive.org/web/20180727184709/http://reference.wolfram.com/language/tutorial/IntegralsThatCanAndCannotBeDone.html intricacies of integration and how Mathematica handles those]. (It would be quicker to try [https://www.wolframalpha.com Wolfram Alpha] instead of installing Mathematica, which uses the same backend for mathematical calculations.)

'''{{w|Riemann integral|Riemann Integration}}'''

The Riemann integral is a definition of definite integration. <math>\sum_{i=0}^{n-1} f(t_i) \left(x_{i+1}-x_i\right).</math> Elementary textbooks on calculus sometimes present finding a definite integral as a process of approximating an area by strips of equal width and then taking the limit as the strips become narrower. Riemann integration removes the requirement that the strips have equal width, and so is a more flexible definition. However there are still many functions for which the Riemann integral doesn't converge, and consideration of these functions leads to the {{w|Lebesgue integration|Lebesgue integral}}. Riemann integration is not a method of calculus appropriate for finding the anti-derivative of an elementary function.

'''{{w|Stokes' Theorem}}'''

Stokes' theorem is a statement about the integration of differential forms on manifolds. <math>\int_{\partial \Omega}\omega=\int_\Omega d\omega\,.</math> It is invoked in science and engineering during control volume analysis (that is, to track the rate of change of a quantity within a control volume, it suffices to track the fluxes in and out of the control volume boundary), but is rarely used directly (and even when it is used directly, the functions that are most frequently used in science and engineering are well-behaved, like sinusoids and polynomials).

'''{{w|Risch Algorithm}}'''

The Risch algorithm is a notoriously complex procedure that, given a certain class of symbolic integrand, either finds a symbolic integral or proves that no elementary integral exists. (Technically it is only a semi-algorithm, and cannot produce an answer unless it can determine if a certain symbolic expression is {{w|Constant problem|equal to 0}} or not.) Many computer algebra systems have chosen to implement only the simpler Risch-Norman algorithm, which does not come with the same guarantee. A series of extensions to the Risch algorithm extend the class of allowable functions to include (at least) the error function and the logarithmic integral. A human would have to be pretty desperate to attempt this (presumably) by hand.

'''{{w|Bessel function}}'''

Bessel functions are the solution to the differential equation <math> x^2 \frac{d^2y}{dx^2}+x \frac{dy}{dx}+(x^2-n^2)*y=0</math>, where n is the order of Bessel function. Though they do show up in some engineering, physics, and abstract mathematics, in lower levels of calculus they are often a sign that the integration was not set up properly before someone put them into a symbolic algebra solver.

'''Phone calls to mathematicians'''

This step would indicate that the flowchart user, desperate from failed attempts to solve the problem, contacts some more skilled mathematicians by phone, and presumably asks them for help. The connected steps of "Oh no", "What the heck is a Bessel function?" and "Burn the evidence" may suggest the possibility that this interaction might not play out very well and could even get the caller in trouble.
Specialists and renowned experts being bothered - not to their amusement - by strangers, often at highly inconvenient times or locations, is a common comedic trope, also previously utilized by xkcd (for example in [[163: Donald Knuth]]).

'''Burn the evidence'''

This phrase parodies a common trope in detective fiction, where characters burn notes, receipts, passports, etc. to maintain secrecy. This may refer to the burning of one's work to avoid the shame of being associated with such a badly failed attempt to solve the given integration problem. Moreover, such a poor attempt at integration could be viewed as a 'crime against mathematics', with the working of the problem being criminal 'evidence' that must be destroyed to avoid exposure as the culprit. Alternatively, it could be an ironic hint to the fact that in order to find the integral, it may even be necessary to break the law or upset higher powers, so the negative consequences of a prosecution can only be avoided by destroying the evidence.

'''{{w|Symbolic integration}}'''

Symbolic integration is the basic process of finding an antiderivative function (defined with symbols), as opposed to numerically integrating a function. The title text is a pun that defines the term not as integration that works with symbols, but rather as integration as a symbolic act, as if it were a component of a ritual. A symbolic act in a ritual is an act meant to evoke something else, such as burning a wooden figurine of a person to represent one’s hatred of that person. Alternatively, the reference could be seen as a joke that integration might as well be a symbol, like in a novel, because Randall can't get any meaningful results from his analysis.

==Transcript==
:[Two flow charts are shown.]

:[The first flow chart has four steps in simple order, one with multiple recommendations.]
:DIFFERENTIATION
:Start
:Try applying
::Chain Rule
::Power Rule
::Quotient Rule
::Product Rule
::Etc.
:Done?
::No [Arrow returns to "Try applying" step.]
::Yes
:Done!

[The second flow chart begins like the first, then descends into chaos.]
:INTEGRATION
:Start
:Try applying
::Integration by Parts
::Substitution
:Done?
:Haha, Nope!

:[Chaos, Roughly from left to right, top to bottom, direction arrows not included.]
::Cauchy's Formula
::????
::???!?
::???
::???
::?
::Partial Fractions
::??
::?
::Install Mathematica
::?
::Riemann Integration
::Stokes' Theorem
::???
::?
::Risch Algorithm
::???
::[Sad face.]
::?????
::???
::What the heck is a Bessel Function??
::Phone calls to mathematicians
::Oh No
::Burn the Evidence
::[More arrows pointing out of the image to suggest more steps.]

{{comic discussion}}
[[Category:Analysis]]
[[Category:Flowcharts]]
[[Category:Math]]

2117: Differentiation and Integration

2024-06-11T14:31:59Z

ISaveXKCDpapers: /* Explanation */

{{comic
| number = 2117
| date = February 27, 2019
| title = Differentiation and Integration
| image = differentiation_and_integration.png
| titletext = "Symbolic integration" is when you theatrically go through the motions of finding integrals, but the actual result you get doesn't matter because it's purely symbolic.
}}

==Explanation==
This comic illustrates the old saying [https://mathoverflow.net/q/66377 "Differentiation is mechanics, integration is art."] It does so by providing a {{w|flowchart}} purporting to show the process of differentiation, and another for integration.

{{w|Derivative|Differentiation}} and {{w|Antiderivative|Integration}} are two major components of {{w|calculus}}. As many Calculus 2 students are painfully aware, integration is much more complicated than the differentiation it undoes.

However, Randall dramatically overstates this point here. After the first step of integration, Randall assumes that any integration can not be solved so simply, and then dives into a step named "????", suggesting that it is unknowable how to proceed. The rest of the flowchart is (we can assume deliberately) even harder to follow, and does not reach a conclusion. This is in contrast to the simple, straightforward flowchart for differentiation. The fact that the arrows in the bottom of the integration part leads to nowhere indicates that "Phone calls to mathematicians", "Oh no" and "Burn the evidence" are not final steps in the difficult journey. The flowchart could be extended by Randall to God-knows-where extents.

It should be noted that Randall slightly undermines his point by providing four different methods, and an "etc", and a "No"-branch for attempting differentiation with no guidelines for selecting between them.

===Differentiation===
'''{{w|Chain rule}}'''

For any <math> \frac{d}{dx}f(x)=f'(x)</math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx}(f(g(x)))=f'(g(x))\cdot g'(x)</math>.

'''{{w|Power Rule}}'''

For any <math> f(x)=g(x)^a </math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx}f(x)=a\cdot g(x)^{a-1}\cdot g'(x) </math>.

'''{{w|Quotient rule}}'''

For any <math> \frac{d}{dx}f(x)=f'(x)</math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx} \frac{f(x)}{g(x)}=\frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{(g(x))^2}</math> if <math>g(x)\ne 0</math>.

'''{{w|Product rule}}'''

For any <math> \frac{d}{dx}f(x)=f'(x)</math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx}(f(x)\cdot g(x))=f'(x)\cdot g(x)+f(x)\cdot g'(x)</math>.

===Integration===
'''{{w|Integration by parts}}'''

The "product rule" run backwards. Since <math>(uv)' = uv' + u'v</math>, it follows that by integrating both sides you get <math> uv = \int u dv + \int v du</math>, which is more commonly written as <math>\int u dv = uv - \int v du</math>. By finding appropriate values for functions <math>u, v</math> such that your problem is in the form <math>\int u dv</math>, your problem ''may'' be simplified. The catch is, there exists no algorithm for determining what functions they might possibly be, so this approach quickly devolves into a guessing game - this has been the topic of an earlier comic, [[1201: Integration by Parts]].

'''{{w|Integration by substitution|Substitution}}'''

The "chain rule" run backwards. Since <math> d(f(u)) = f'(u) \frac{du}{dx} dx</math>, it follows that <math>f(u) = \int f(u) \frac{du}{dx} dx</math>. By finding appropriate values for functions <math>f, u</math> such that your problem is in the form <math>\int f(u) \frac{du}{dx} dx</math> your problem ''may'' be simplified.

'''{{w|Cauchy's integral formula|Cauchy's Formula}}'''

Cauchy's Integral formula is a result in complex analysis that relates the value of a contour integral in the complex plane to properties of the singularities in the interior of the contour. <math> \frac{d^n}{da^n}f(a) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(z)}{\left(z-a\right)^{n+1}}\, dz.</math> It is often used to compute integrals on the real line by extending the path of the integral from the real line into the complex plane to apply the formula, then proving that the integral from the parts of the contour not on the real line has value zero.

'''{{w|Partial_fraction_decomposition#Application_to_symbolic_integration|Partial Fractions}}'''

Partial fractions is a technique for breaking up a function that comprises one polynomial divided by another into a sum of functions comprising constants over the factors of the original denominator, which can easily be integrated into logarithms.

'''Install {{w|Mathematica}}'''

Mathematica is a modern technical computing system spanning most areas. One of its features is to compute mathematical functions. This step in the flowchart is to install and use Mathematica to do the integration for you. Here is a description about the [https://web.archive.org/web/20180727184709/http://reference.wolfram.com/language/tutorial/IntegralsThatCanAndCannotBeDone.html intricacies of integration and how Mathematica handles those]. (It would be quicker to try [https://www.wolframalpha.com Wolfram Alpha] instead of installing Mathematica, which uses the same backend for mathematical calculations.)

'''{{w|Riemann integral|Riemann Integration}}'''

The Riemann integral is a definition of definite integration. <math>\sum_{i=0}^{n-1} f(t_i) \left(x_{i+1}-x_i\right).</math> Elementary textbooks on calculus sometimes present finding a definite integral as a process of approximating an area by strips of equal width and then taking the limit as the strips become narrower. Riemann integration removes the requirement that the strips have equal width, and so is a more flexible definition. However there are still many functions for which the Riemann integral doesn't converge, and consideration of these functions leads to the {{w|Lebesgue integration|Lebesgue integral}}. Riemann integration is not a method of calculus appropriate for finding the anti-derivative of an elementary function.

'''{{w|Stokes' Theorem}}'''

Stokes' theorem is a statement about the integration of differential forms on manifolds. <math>\int_{\partial \Omega}\omega=\int_\Omega d\omega\,.</math> It is invoked in science and engineering during control volume analysis (that is, to track the rate of change of a quantity within a control volume, it suffices to track the fluxes in and out of the control volume boundary), but is rarely used directly (and even when it is used directly, the functions that are most frequently used in science and engineering are well-behaved, like sinusoids and polynomials).

'''{{w|Risch Algorithm}}'''

The Risch algorithm is a notoriously complex procedure that, given a certain class of symbolic integrand, either finds a symbolic integral or proves that no elementary integral exists. (Technically it is only a semi-algorithm, and cannot produce an answer unless it can determine if a certain symbolic expression is {{w|Constant problem|equal to 0}} or not.) Many computer algebra systems have chosen to implement only the simpler Risch-Norman algorithm, which does not come with the same guarantee. A series of extensions to the Risch algorithm extend the class of allowable functions to include (at least) the error function and the logarithmic integral. A human would have to be pretty desperate to attempt this (presumably) by hand.

'''{{w|Bessel function}}'''

Bessel functions are the solution to the differential equation <math> x^2 \frac{d^2y}{dx^2}+x \frac{dy}{dx}+(x^2-n^2)*y=0</math>, where n is the order of Bessel function. Though they do show up in some engineering, physics, and abstract mathematics, in lower levels of calculus they are often a sign that the integration was not set up properly before someone put them into a symbolic algebra solver.

'''Phone calls to mathematicians'''

This step would indicate that the flowchart user, desperate from failed attempts to solve the problem, contacts some more skilled mathematicians by phone, and presumably asks them for help. The connected steps of "Oh no", "What the heck is a Bessel function?" and "Burn the evidence" may suggest the possibility that this interaction might not play out very well and could even get the caller in trouble.
Specialists and renowned experts being bothered - not to their amusement - by strangers, often at highly inconvenient times or locations, is a common comedic trope, also previously utilized by xkcd (for example in [[163: Donald Knuth]]).

'''Burn the evidence'''

This phrase parodies a common trope in detective fiction, where characters burn notes, receipts, passports, etc. to maintain secrecy. This may refer to the burning of one's work to avoid the shame of being associated with such a badly failed attempt to solve the given integration problem. Moreover, such a poor attempt at integration could be viewed as a 'crime against mathematics', with the working of the problem being criminal 'evidence' that must be destroyed to avoid exposure as the culprit. Alternatively, it could be an ironic hint to the fact that in order to find the integral, it may even be necessary to break the law or upset higher powers, so the negative consequences of a prosecution can only be avoided by destroying the evidence.

'''{{w|Symbolic integration}}'''

Symbolic integration is the basic process of finding an antiderivative function (defined with symbols), as opposed to numerically integrating a function. The title text is a pun that defines the term not as integration that works with symbols, but rather as integration as a symbolic act, as if it were a component of a ritual. A symbolic act in a ritual is an act meant to evoke something else, such as burning a wooden figurine of a person to represent one’s hatred of that person. Alternatively, the reference could be seen as a joke that integration might as well be a symbol, like in a novel, because Randall can't get any meaningful results from his analysis.

==Transcript==
:[Two flow charts are shown.]

:[The first flow chart has four steps in simple order, one with multiple recommendations.]
:DIFFERENTIATION
:Start
:Try applying
::Chain Rule
::Power Rule
::Quotient Rule
::Product Rule
::Etc.
:Done?
::No [Arrow returns to "Try applying" step.]
::Yes
:Done!

[The second flow chart begins like the first, then descends into chaos.]
:INTEGRATION
:Start
:Try applying
::Integration by Parts
::Substitution
:Done?
:Haha, Nope!

:[Chaos, Roughly from left to right, top to bottom, direction arrows not included.]
::Cauchy's Formula
::????
::???!?
::???
::???
::?
::Partial Fractions
::??
::?
::Install Mathematica
::?
::Riemann Integration
::Stokes' Theorem
::???
::?
::Risch Algorithm
::???
::[Sad face.]
::?????
::???
::What the heck is a Bessel Function??
::Phone calls to mathematicians
::Oh No
::Burn the Evidence
::[More arrows pointing out of the image to suggest more steps.]

{{comic discussion}}
[[Category:Analysis]]
[[Category:Flowcharts]]
[[Category:Math]]

2117: Differentiation and Integration

2024-06-11T14:31:10Z

ISaveXKCDpapers: fixed substitution explanation (see talk)

{{comic
| number = 2117
| date = February 27, 2019
| title = Differentiation and Integration
| image = differentiation_and_integration.png
| titletext = "Symbolic integration" is when you theatrically go through the motions of finding integrals, but the actual result you get doesn't matter because it's purely symbolic.
}}

==Explanation==
This comic illustrates the old saying [https://mathoverflow.net/q/66377 "Differentiation is mechanics, integration is art."] It does so by providing a {{w|flowchart}} purporting to show the process of differentiation, and another for integration.

{{w|Derivative|Differentiation}} and {{w|Antiderivative|Integration}} are two major components of {{w|calculus}}. As many Calculus 2 students are painfully aware, integration is much more complicated than the differentiation it undoes.

However, Randall dramatically overstates this point here. After the first step of integration, Randall assumes that any integration can not be solved so simply, and then dives into a step named "????", suggesting that it is unknowable how to proceed. The rest of the flowchart is (we can assume deliberately) even harder to follow, and does not reach a conclusion. This is in contrast to the simple, straightforward flowchart for differentiation. The fact that the arrows in the bottom of the integration part leads to nowhere indicates that "Phone calls to mathematicians", "Oh no" and "Burn the evidence" are not final steps in the difficult journey. The flowchart could be extended by Randall to God-knows-where extents.

It should be noted that Randall slightly undermines his point by providing four different methods, and an "etc", and a "No"-branch for attempting differentiation with no guidelines for selecting between them.

===Differentiation===
'''{{w|Chain rule}}'''

For any <math> \frac{d}{dx}f(x)=f'(x)</math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx}(f(g(x)))=f'(g(x))\cdot g'(x)</math>.

'''{{w|Power Rule}}'''

For any <math> f(x)=g(x)^a </math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx}f(x)=a\cdot g(x)^{a-1}\cdot g'(x) </math>.

'''{{w|Quotient rule}}'''

For any <math> \frac{d}{dx}f(x)=f'(x)</math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx} \frac{f(x)}{g(x)}=\frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{(g(x))^2}</math> if <math>g(x)\ne 0</math>.

'''{{w|Product rule}}'''

For any <math> \frac{d}{dx}f(x)=f'(x)</math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx}(f(x)\cdot g(x))=f'(x)\cdot g(x)+f(x)\cdot g'(x)</math>.

===Integration===
'''{{w|Integration by parts}}'''

The "product rule" run backwards. Since <math>(uv)' = uv' + u'v</math>, it follows that by integrating both sides you get <math> uv = \int u dv + \int v du</math>, which is more commonly written as <math>\int u dv = uv - \int v du</math>. By finding appropriate values for functions <math>u, v</math> such that your problem is in the form <math>\int u dv</math>, your problem ''may'' be simplified. The catch is, there exists no algorithm for determining what functions they might possibly be, so this approach quickly devolves into a guessing game - this has been the topic of an earlier comic, [[1201: Integration by Parts]].

'''{{w|Integration by substitution|Substitution}}'''

The "chain rule" run backwards. Since <math> d(f(u)) = f'(u) \frac{du}{dx} dx</math>, it follows that <math>f(u) = \int f(u) \frac{du}{dx} dx</math>. By finding appropriate values for functions <math>f, u</math> such that your problem is in the form <math>\int df(u) du</math> your problem ''may'' be simplified.

'''{{w|Cauchy's integral formula|Cauchy's Formula}}'''

Cauchy's Integral formula is a result in complex analysis that relates the value of a contour integral in the complex plane to properties of the singularities in the interior of the contour. <math> \frac{d^n}{da^n}f(a) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(z)}{\left(z-a\right)^{n+1}}\, dz.</math> It is often used to compute integrals on the real line by extending the path of the integral from the real line into the complex plane to apply the formula, then proving that the integral from the parts of the contour not on the real line has value zero.

'''{{w|Partial_fraction_decomposition#Application_to_symbolic_integration|Partial Fractions}}'''

Partial fractions is a technique for breaking up a function that comprises one polynomial divided by another into a sum of functions comprising constants over the factors of the original denominator, which can easily be integrated into logarithms.

'''Install {{w|Mathematica}}'''

Mathematica is a modern technical computing system spanning most areas. One of its features is to compute mathematical functions. This step in the flowchart is to install and use Mathematica to do the integration for you. Here is a description about the [https://web.archive.org/web/20180727184709/http://reference.wolfram.com/language/tutorial/IntegralsThatCanAndCannotBeDone.html intricacies of integration and how Mathematica handles those]. (It would be quicker to try [https://www.wolframalpha.com Wolfram Alpha] instead of installing Mathematica, which uses the same backend for mathematical calculations.)

'''{{w|Riemann integral|Riemann Integration}}'''

The Riemann integral is a definition of definite integration. <math>\sum_{i=0}^{n-1} f(t_i) \left(x_{i+1}-x_i\right).</math> Elementary textbooks on calculus sometimes present finding a definite integral as a process of approximating an area by strips of equal width and then taking the limit as the strips become narrower. Riemann integration removes the requirement that the strips have equal width, and so is a more flexible definition. However there are still many functions for which the Riemann integral doesn't converge, and consideration of these functions leads to the {{w|Lebesgue integration|Lebesgue integral}}. Riemann integration is not a method of calculus appropriate for finding the anti-derivative of an elementary function.

'''{{w|Stokes' Theorem}}'''

Stokes' theorem is a statement about the integration of differential forms on manifolds. <math>\int_{\partial \Omega}\omega=\int_\Omega d\omega\,.</math> It is invoked in science and engineering during control volume analysis (that is, to track the rate of change of a quantity within a control volume, it suffices to track the fluxes in and out of the control volume boundary), but is rarely used directly (and even when it is used directly, the functions that are most frequently used in science and engineering are well-behaved, like sinusoids and polynomials).

'''{{w|Risch Algorithm}}'''

The Risch algorithm is a notoriously complex procedure that, given a certain class of symbolic integrand, either finds a symbolic integral or proves that no elementary integral exists. (Technically it is only a semi-algorithm, and cannot produce an answer unless it can determine if a certain symbolic expression is {{w|Constant problem|equal to 0}} or not.) Many computer algebra systems have chosen to implement only the simpler Risch-Norman algorithm, which does not come with the same guarantee. A series of extensions to the Risch algorithm extend the class of allowable functions to include (at least) the error function and the logarithmic integral. A human would have to be pretty desperate to attempt this (presumably) by hand.

'''{{w|Bessel function}}'''

Bessel functions are the solution to the differential equation <math> x^2 \frac{d^2y}{dx^2}+x \frac{dy}{dx}+(x^2-n^2)*y=0</math>, where n is the order of Bessel function. Though they do show up in some engineering, physics, and abstract mathematics, in lower levels of calculus they are often a sign that the integration was not set up properly before someone put them into a symbolic algebra solver.

'''Phone calls to mathematicians'''

This step would indicate that the flowchart user, desperate from failed attempts to solve the problem, contacts some more skilled mathematicians by phone, and presumably asks them for help. The connected steps of "Oh no", "What the heck is a Bessel function?" and "Burn the evidence" may suggest the possibility that this interaction might not play out very well and could even get the caller in trouble.
Specialists and renowned experts being bothered - not to their amusement - by strangers, often at highly inconvenient times or locations, is a common comedic trope, also previously utilized by xkcd (for example in [[163: Donald Knuth]]).

'''Burn the evidence'''

This phrase parodies a common trope in detective fiction, where characters burn notes, receipts, passports, etc. to maintain secrecy. This may refer to the burning of one's work to avoid the shame of being associated with such a badly failed attempt to solve the given integration problem. Moreover, such a poor attempt at integration could be viewed as a 'crime against mathematics', with the working of the problem being criminal 'evidence' that must be destroyed to avoid exposure as the culprit. Alternatively, it could be an ironic hint to the fact that in order to find the integral, it may even be necessary to break the law or upset higher powers, so the negative consequences of a prosecution can only be avoided by destroying the evidence.

'''{{w|Symbolic integration}}'''

Symbolic integration is the basic process of finding an antiderivative function (defined with symbols), as opposed to numerically integrating a function. The title text is a pun that defines the term not as integration that works with symbols, but rather as integration as a symbolic act, as if it were a component of a ritual. A symbolic act in a ritual is an act meant to evoke something else, such as burning a wooden figurine of a person to represent one’s hatred of that person. Alternatively, the reference could be seen as a joke that integration might as well be a symbol, like in a novel, because Randall can't get any meaningful results from his analysis.

==Transcript==
:[Two flow charts are shown.]

:[The first flow chart has four steps in simple order, one with multiple recommendations.]
:DIFFERENTIATION
:Start
:Try applying
::Chain Rule
::Power Rule
::Quotient Rule
::Product Rule
::Etc.
:Done?
::No [Arrow returns to "Try applying" step.]
::Yes
:Done!

[The second flow chart begins like the first, then descends into chaos.]
:INTEGRATION
:Start
:Try applying
::Integration by Parts
::Substitution
:Done?
:Haha, Nope!

:[Chaos, Roughly from left to right, top to bottom, direction arrows not included.]
::Cauchy's Formula
::????
::???!?
::???
::???
::?
::Partial Fractions
::??
::?
::Install Mathematica
::?
::Riemann Integration
::Stokes' Theorem
::???
::?
::Risch Algorithm
::???
::[Sad face.]
::?????
::???
::What the heck is a Bessel Function??
::Phone calls to mathematicians
::Oh No
::Burn the Evidence
::[More arrows pointing out of the image to suggest more steps.]

{{comic discussion}}
[[Category:Analysis]]
[[Category:Flowcharts]]
[[Category:Math]]

2117: Differentiation and Integration

2024-06-11T13:24:02Z

ISaveXKCDpapers: added to category:math

{{comic
| number = 2117
| date = February 27, 2019
| title = Differentiation and Integration
| image = differentiation_and_integration.png
| titletext = "Symbolic integration" is when you theatrically go through the motions of finding integrals, but the actual result you get doesn't matter because it's purely symbolic.
}}

==Explanation==
This comic illustrates the old saying [https://mathoverflow.net/q/66377 "Differentiation is mechanics, integration is art."] It does so by providing a {{w|flowchart}} purporting to show the process of differentiation, and another for integration.

{{w|Derivative|Differentiation}} and {{w|Antiderivative|Integration}} are two major components of {{w|calculus}}. As many Calculus 2 students are painfully aware, integration is much more complicated than the differentiation it undoes.

However, Randall dramatically overstates this point here. After the first step of integration, Randall assumes that any integration can not be solved so simply, and then dives into a step named "????", suggesting that it is unknowable how to proceed. The rest of the flowchart is (we can assume deliberately) even harder to follow, and does not reach a conclusion. This is in contrast to the simple, straightforward flowchart for differentiation. The fact that the arrows in the bottom of the integration part leads to nowhere indicates that "Phone calls to mathematicians", "Oh no" and "Burn the evidence" are not final steps in the difficult journey. The flowchart could be extended by Randall to God-knows-where extents.

It should be noted that Randall slightly undermines his point by providing four different methods, and an "etc", and a "No"-branch for attempting differentiation with no guidelines for selecting between them.

===Differentiation===
'''{{w|Chain rule}}'''

For any <math> \frac{d}{dx}f(x)=f'(x)</math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx}(f(g(x)))=f'(g(x))\cdot g'(x)</math>.

'''{{w|Power Rule}}'''

For any <math> f(x)=g(x)^a </math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx}f(x)=a\cdot g(x)^{a-1}\cdot g'(x) </math>.

'''{{w|Quotient rule}}'''

For any <math> \frac{d}{dx}f(x)=f'(x)</math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx} \frac{f(x)}{g(x)}=\frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{(g(x))^2}</math> if <math>g(x)\ne 0</math>.

'''{{w|Product rule}}'''

For any <math> \frac{d}{dx}f(x)=f'(x)</math> and <math> \frac{d}{dx}g(x)=g'(x) </math>, it follows that <math> \frac{d}{dx}(f(x)\cdot g(x))=f'(x)\cdot g(x)+f(x)\cdot g'(x)</math>.

===Integration===
'''{{w|Integration by parts}}'''

The "product rule" run backwards. Since <math>(uv)' = uv' + u'v</math>, it follows that by integrating both sides you get <math> uv = \int u dv + \int v du</math>, which is more commonly written as <math>\int u dv = uv - \int v du</math>. By finding appropriate values for functions <math>u, v</math> such that your problem is in the form <math>\int u dv</math>, your problem ''may'' be simplified. The catch is, there exists no algorithm for determining what functions they might possibly be, so this approach quickly devolves into a guessing game - this has been the topic of an earlier comic, [[1201: Integration by Parts]].

'''{{w|Integration by substitution|Substitution}}'''

The "chain rule" run backwards. Since <math> d(f(u)) = (df(u))du</math>, it follows that <math>f(u) = \int df(u) du</math>. By finding appropriate values for functions <math>f, u</math> such that your problem is in the form <math>\int df(u) du</math> your problem ''may'' be simplified.

'''{{w|Cauchy's integral formula|Cauchy's Formula}}'''

Cauchy's Integral formula is a result in complex analysis that relates the value of a contour integral in the complex plane to properties of the singularities in the interior of the contour. <math> \frac{d^n}{da^n}f(a) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(z)}{\left(z-a\right)^{n+1}}\, dz.</math> It is often used to compute integrals on the real line by extending the path of the integral from the real line into the complex plane to apply the formula, then proving that the integral from the parts of the contour not on the real line has value zero.

'''{{w|Partial_fraction_decomposition#Application_to_symbolic_integration|Partial Fractions}}'''

Partial fractions is a technique for breaking up a function that comprises one polynomial divided by another into a sum of functions comprising constants over the factors of the original denominator, which can easily be integrated into logarithms.

'''Install {{w|Mathematica}}'''

Mathematica is a modern technical computing system spanning most areas. One of its features is to compute mathematical functions. This step in the flowchart is to install and use Mathematica to do the integration for you. Here is a description about the [https://web.archive.org/web/20180727184709/http://reference.wolfram.com/language/tutorial/IntegralsThatCanAndCannotBeDone.html intricacies of integration and how Mathematica handles those]. (It would be quicker to try [https://www.wolframalpha.com Wolfram Alpha] instead of installing Mathematica, which uses the same backend for mathematical calculations.)

'''{{w|Riemann integral|Riemann Integration}}'''

The Riemann integral is a definition of definite integration. <math>\sum_{i=0}^{n-1} f(t_i) \left(x_{i+1}-x_i\right).</math> Elementary textbooks on calculus sometimes present finding a definite integral as a process of approximating an area by strips of equal width and then taking the limit as the strips become narrower. Riemann integration removes the requirement that the strips have equal width, and so is a more flexible definition. However there are still many functions for which the Riemann integral doesn't converge, and consideration of these functions leads to the {{w|Lebesgue integration|Lebesgue integral}}. Riemann integration is not a method of calculus appropriate for finding the anti-derivative of an elementary function.

'''{{w|Stokes' Theorem}}'''

Stokes' theorem is a statement about the integration of differential forms on manifolds. <math>\int_{\partial \Omega}\omega=\int_\Omega d\omega\,.</math> It is invoked in science and engineering during control volume analysis (that is, to track the rate of change of a quantity within a control volume, it suffices to track the fluxes in and out of the control volume boundary), but is rarely used directly (and even when it is used directly, the functions that are most frequently used in science and engineering are well-behaved, like sinusoids and polynomials).

'''{{w|Risch Algorithm}}'''

The Risch algorithm is a notoriously complex procedure that, given a certain class of symbolic integrand, either finds a symbolic integral or proves that no elementary integral exists. (Technically it is only a semi-algorithm, and cannot produce an answer unless it can determine if a certain symbolic expression is {{w|Constant problem|equal to 0}} or not.) Many computer algebra systems have chosen to implement only the simpler Risch-Norman algorithm, which does not come with the same guarantee. A series of extensions to the Risch algorithm extend the class of allowable functions to include (at least) the error function and the logarithmic integral. A human would have to be pretty desperate to attempt this (presumably) by hand.

'''{{w|Bessel function}}'''

Bessel functions are the solution to the differential equation <math> x^2 \frac{d^2y}{dx^2}+x \frac{dy}{dx}+(x^2-n^2)*y=0</math>, where n is the order of Bessel function. Though they do show up in some engineering, physics, and abstract mathematics, in lower levels of calculus they are often a sign that the integration was not set up properly before someone put them into a symbolic algebra solver.

'''Phone calls to mathematicians'''

This step would indicate that the flowchart user, desperate from failed attempts to solve the problem, contacts some more skilled mathematicians by phone, and presumably asks them for help. The connected steps of "Oh no", "What the heck is a Bessel function?" and "Burn the evidence" may suggest the possibility that this interaction might not play out very well and could even get the caller in trouble.
Specialists and renowned experts being bothered - not to their amusement - by strangers, often at highly inconvenient times or locations, is a common comedic trope, also previously utilized by xkcd (for example in [[163: Donald Knuth]]).

'''Burn the evidence'''

This phrase parodies a common trope in detective fiction, where characters burn notes, receipts, passports, etc. to maintain secrecy. This may refer to the burning of one's work to avoid the shame of being associated with such a badly failed attempt to solve the given integration problem. Moreover, such a poor attempt at integration could be viewed as a 'crime against mathematics', with the working of the problem being criminal 'evidence' that must be destroyed to avoid exposure as the culprit. Alternatively, it could be an ironic hint to the fact that in order to find the integral, it may even be necessary to break the law or upset higher powers, so the negative consequences of a prosecution can only be avoided by destroying the evidence.

'''{{w|Symbolic integration}}'''

Symbolic integration is the basic process of finding an antiderivative function (defined with symbols), as opposed to numerically integrating a function. The title text is a pun that defines the term not as integration that works with symbols, but rather as integration as a symbolic act, as if it were a component of a ritual. A symbolic act in a ritual is an act meant to evoke something else, such as burning a wooden figurine of a person to represent one’s hatred of that person. Alternatively, the reference could be seen as a joke that integration might as well be a symbol, like in a novel, because Randall can't get any meaningful results from his analysis.

==Transcript==
:[Two flow charts are shown.]

:[The first flow chart has four steps in simple order, one with multiple recommendations.]
:DIFFERENTIATION
:Start
:Try applying
::Chain Rule
::Power Rule
::Quotient Rule
::Product Rule
::Etc.
:Done?
::No [Arrow returns to "Try applying" step.]
::Yes
:Done!

[The second flow chart begins like the first, then descends into chaos.]
:INTEGRATION
:Start
:Try applying
::Integration by Parts
::Substitution
:Done?
:Haha, Nope!

:[Chaos, Roughly from left to right, top to bottom, direction arrows not included.]
::Cauchy's Formula
::????
::???!?
::???
::???
::?
::Partial Fractions
::??
::?
::Install Mathematica
::?
::Riemann Integration
::Stokes' Theorem
::???
::?
::Risch Algorithm
::???
::[Sad face.]
::?????
::???
::What the heck is a Bessel Function??
::Phone calls to mathematicians
::Oh No
::Burn the Evidence
::[More arrows pointing out of the image to suggest more steps.]

{{comic discussion}}
[[Category:Analysis]]
[[Category:Flowcharts]]
[[Category:Math]]

1047: Approximations

2024-06-11T11:20:49Z

ISaveXKCDpapers: added explanation of why it's radians

{{comic
| number = 1047
| date = April 25, 2012
| title = Approximations
| before = [[#Explanation|↓ Skip to explanation ↓]]
| image = approximations.png
| 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.
}}

==Explanation==

This comic lists some approximations for numbers, most of them mathematical and physical constants, but some of them jokes and cultural references.

Approximations like these are sometimes used as {{w|mnemonic}}s by mathematicians and physicists, though most of Randall's approximations are too convoluted to be useful as mnemonics. Perhaps the best known mnemonic approximation (though not used here by Randall) is that "π is approximately equal to 22/7". Randall does mention (and mock) the common mnemonic among physicists that the {{w|fine structure constant}} is approximately 1/137. Although Randall gives approximations for the number of seconds in a year, he does not mention the common physicists' mnemonic that it is "π × 107", though he later added a statement to the top of the comic page addressing this point.

At the bottom of the comic are expressions involving {{w|transcendental numbers}} (namely π and e) that are tantalizingly close to being exactly true but are not (indeed, they cannot be, due to the nature of transcendental numbers). Such near-equations were previously discussed in [[217: e to the pi Minus pi]]. One of the entries, though, is a "red herring" that is exactly true.

Randall says he compiled this table through "a mix of trial-and-error, ''{{w|Mathematica}}'', and Robert Munafo's [http://mrob.com/pub/ries/ Ries] tool." "Ries" is a "{{w|Closed-form expression#Conversion from numerical forms|reverse calculator}}" that forms equations matching a given number.

The {{w|world population}} estimate for 2023 is still accurate. The estimate is 7.9 billion, and the population listed at the website census.gov is roughly the same. The current value can be found here: [https://www.census.gov/popclock/ United States Census Bureau - U.S. and World Population Clock]. Nevertheless there are other numbers listed by different sources.

The first part of the title text notes that "Jenny's constant," which is actually a telephone number referenced in Tommy Tutone's 1982 song {{w|867-5309/Jenny}}, is not only prime but 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. Twin primes were also referenced in [[1310: Goldbach Conjectures]].

The second part of the title text makes fun of the unusual mathematical operations contained in the comic. {{w|Pi|π}} is a useful number in many contexts, but it doesn't usually occur anywhere in an exponent. Even when it does, such as with complex numbers, taking the πth root is rarely helpful. A rare exception is an [http://gosper.org/4%5E1%C3%B7%CF%80.png identity] for the pi-th root of 4 discovered by Bill Gosper. Similarly, {{w|e (mathematical constant)|e}} typically appears in the base 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]).

===Equations===

{| class="wikitable"
|-
!align="center"|Thing to be approximated:
!align="center"|Formula proposed
!align="center"|Resulting approximate value
!align="center"|Correct value
!align="center"|Discussion
|-
|align="center"|One {{w|light year}} (meters)
|align="center"|998
|align="center"|9,227,446,944,279,201
|align="center"|9,460,730,472,580,800 (exact)
|align="left"|Based on 365.25 days per year (see below). 998 and 698 are [[487: Numerical Sex Positions|sexual references]].
|-
|align="center"|Earth's surface (m2)
|align="center"|698
|align="center"|513,798,374,428,641
|align="center"|5.10072 × 1014
|align="left"|998 and 698 are [[487: Numerical Sex Positions|sexual references]].
|-
|align="center"|Oceans' volume (m3)
|align="center"|919
|align="center"|1,350,851,717,672,992,089
|align="center"|1.332 × 1018
|align="left"|
|-
|align="center"|Seconds in a year
|align="center"|754
|align="center"|31,640,625
|align="center"|31,557,600 (Julian calendar), 31,556,952 (Gregorian calendar)
|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:
"Lots of emails mention the physicist favorite, 1 year = pi × 107 seconds. 754 is a hair more accurate, but it's hard to top 3,141,592's elegance." π × 107 is nearly equal to 31,415,926.536, and 754 is exactly 31,640,625. Randall's elegance belongs to the number π, but it should be multiplied by the factor of ten.

Using the traditional definitions that a second is 1/60 of a minute, a minute is 1/60 of an hour, and an hour is 1/24 of a day, a 365-day common year is exactly 31,536,000 seconds (the "''Rent'' method" approximation) and the 366-day leap year is 31,622,400 seconds. 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 seconds. 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.
|-
|align="center"|Seconds in a year (''Rent'' method)
|align="center"|525,600 × 60
|align="center"|31,536,000
|align="center"|31,557,600 (Julian calendar), 31,556,952 (Gregorian calendar)
|align="left"|"''Rent'' Method" refers to the song "{{w|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. This method for remembering how many seconds are in a year was also referenced in [https://what-if.xkcd.com/23/ What If? 23].
|-
|align="center"|Age of the universe (seconds)
|align="center"|1515
|align="center"|437,893,890,380,859,375
|align="center"|(4.354 ± 0.012) × 1017 (best estimate; exact value unknown)
|align="left"|This one will slowly get more accurate as the universe ages.
|-
|align="center"|Planck's constant
|align="center"|<math>\frac {1} {30^{\pi^e}}</math>
|align="center"|6.6849901410 × 10−34
|align="center"|6.62606957 × 10−34
|align="left"|Informally, the {{w|Planck constant}} is the smallest action possible in quantum mechanics.
|-
|align="center"|Fine structure constant
|align="center"|<math>\frac{1}{140}</math>
|align="center"|0.00714285
|align="center"|0.0072973525664 (accepted value as of 2014), close to 1/137
|align="left"|The {{w|fine structure constant}} indicates the strength of electromagnetism. It is unitless and around 0.007297, close to 1/137. The joke here is that Randall chose to write 140 as the denominator, when 137 is much closer to reality and just as many digits (although 137 is a less "round" number than 140, and Randall writes in the table that he's "had enough" of it). At one point the fine structure constant 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" Eddington}} who argued very strenuously that the fine structure constant "should" be 1/136 when that was what the best measurements suggested, and then argued just as strenuously for 1/137 a few years later as measurements improved.
|-
|align="center"|Fundamental charge
|align="center"|<math>\frac {3} {14 \pi^{\pi^\pi}}</math>
|align="center"|1.59895121062716 × 10−19
|align="center"|1.602176565 × 10−19
|align="left"|This is the charge of the proton, symbolized ''e'' for electron (whose charge is actually −''e''. You can blame Benjamin Franklin [[567|for that]].)
|-
|align="center"|Telephone number for the {{w|White House}} switchboard
|align="center"|<math>\frac {1} {e^ {\sqrt[\pi] {1 + \sqrt[e-1] 8}} }</math>
|align="center"|0.2024561414932
|align="center"|202-456-1414
|align="left"|
|-
|align="center"|Jenny's constant
|align="center"|<math>\left( 7^ {\frac{e}{1} - \frac{1}{e}} - 9 \right) \pi^2</math>
|align="center"|867.5309019
|align="center"|867-5309
|align="left"|A telephone number referenced in {{w|Tommy Tutone}}'s 1982 song {{w|867-5309/Jenny}}. As mentioned in the title text, the number is not only prime but a {{w|twin prime}} because 8675311 is also a prime.
|-
|align="center"|World population estimate (billions)
|align="center"|Equivalent to <math>6 + \frac {\frac34 y + \frac14 (y \operatorname{mod} 4) - 1499} {10}</math>
|align="center"|2005 — 6.5 
2006 — 6.6 
2007 — 6.7 
2008 — 6.7 
2009 — 6.8 
2010 — 6.9 
2011 — 7.0 
2012 — 7.0 
2013 — 7.1 
2014 — 7.2 
2015 — 7.3 
2016 — 7.3 
2017 — 7.4 
2018 — 7.5 
2019 — 7.6 
2020 — 7.6 
2021 — 7.7 
2022 — 7.8 
2023 — 7.9 
2024 — 7.9 
2025 — 8.0 
2026 — 8.1 
2027 — 8.2 
2028 — 8.2 
2029 — 8.3 
2030 — 8.4 
2031 — 8.5 
2032 — 8.5 
2033 — 8.6 
2034 — 8.7 
2035 — 8.8 
|align="center"|
|align="left"|Grows by 75 million every year on average. As of 2023, a bit too small.
|-
|align="center"|U.S. population estimate (millions)
|align="center"|Equivalent to <math>310 + 3(y - 2010)</math>
|align="center"|2000 — 280 
2001 — 283 
2002 — 286 
2003 — 289 
2004 — 292 
2005 — 295 
2006 — 298 
2007 — 301 
2008 — 304 
2009 — 307 
2010 — 310 
2011 — 313 
2012 — 316 
2013 — 319 
2014 — 322 
2015 — 325 
2016 — 328 
2017 — 331 
2018 — 334 
2019 — 337 
2020 — 340 
2021 — 343 
2022 — 346 
2023 — 349 
2024 — 352 
2025 — 355 
2026 — 358 
2027 — 361 
2028 — 364 
2029 — 367 
2030 — 370 
2031 — 373 
2032 — 376 
2033 — 379 
2034 — 382 
2035 — 385 
|align="center"|
|align="left"|Grows by 3 million each year. As of 2021 the actual number is ~13 million smaller.
|-
|align="center"|Electron rest energy (joules)
|align="center"|<math>\frac {e} {7^{16}}</math>
|align="center"|8.17948276564429 × 10−14
|align="center"|8.18710438 × 10−14
|align="left"|
|-
|align="center"|Light year (miles)
|align="center"|242.42
|align="center"|5,884,267,614,436.97
|align="center"|5,878,625,373,183.61 = 9,460,730,472,580,800 (meters in a light-year, by definition) / 1609.344 (meters in a mile)
|align="left"|{{w|42 (number)|42}} is, according to {{w|Douglas Adams}}' ''{{w|The Hitchhiker's Guide to the Galaxy}}'', the answer to the Ultimate Question of Life, the Universe, and Everything.
|-
|align="center"|<math>\sin\left(60^\circ\right) = \frac {\sqrt 3} {2}</math>
|align="center"|<math>\frac{e}{\pi}</math>
|align="center"|0.8652559794
|align="center"|0.8660254038
|align="left"|
|-
|align="center"|<math>\sqrt 3</math>
|align="center"|<math>\frac{2e}{\pi}</math>
|align="center"|1.7305119589
|align="center"|1.7320508076
|align="left"|Same as the above
|-
|align="center"|γ (Euler's gamma constant)
|align="center"|<math>\frac {1} {\sqrt 3}</math>
|align="center"|0.5773502692
|align="center"|0.5772156649
|align="left"|The {{w|Euler–Mascheroni constant}} (denoted γ) is a mysterious number describing the relationship between the {{w|Harmonic series (mathematics)|harmonic series}} and the {{w|natural logarithm}}.
|-
|align="center"|Feet in a meter
|align="center"|<math>\frac {5} {\sqrt[e]\pi}</math>
|align="center"|3.2815481951
|align="center"|3.280839895
|align="left"|Exactly 1/0.3048, as the {{w|international foot}} is defined as 0.3048 meters.
|-
|align="center"|<math>\sqrt 5</math>
|align="center"|<math>\frac{2}{e} + \frac32</math>
|align="center"|2.2357588823
|align="center"|2.2360679775
|align="left"|
|-
|align="center"|Avogadro's number
|align="center"|<math>69^{\pi^\sqrt{5}}</math>
|align="center"|6.02191201246329 × 1023
|align="center"|6.02214129 × 1023
|align="left"|Also called a mole for shorthand, {{w|Avogadro's number}} is (roughly) the number of individual atoms in 12 grams of pure carbon. Used in basically every application of chemistry. In 2019 the constant was redefined to 6.02214076 × 1023, making the Approximation slightly more correct.
|-
|align="center"|Gravitational constant ''G''
|align="center"|<math>\frac {1} {e ^ {(\pi-1)^{(\pi+1)}}}</math>
|align="center"|6.6736110685 × 10−11
|align="center"|6.67385 × 10−11
|align="left"|The universal {{w|gravitational constant}} G is equal to ''Fr''2/''Mm'', where ''F'' is the gravitational force between two objects, ''r'' is the distance between them, and ''M'' and ''m'' are their masses.
|-
|align="center"|''R'' (gas constant)
|align="center"|<math>(e + 1) \sqrt 5</math>
|align="center"|8.3143309279
|align="center"|8.3144622
|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).
|-
|align="center"|Proton–electron mass ratio
|align="center"|<math>6 \pi^5</math>
|align="center"|1836.1181087117
|align="center"|1836.15267246
|align="left"| The {{w|proton-to-electron mass ratio}} is the ratio between the rest mass of the proton divided by the rest mass of the electron.
|-
|align="center"|Liters in a {{w|gallon}}
|align="center"|<math>3 + \frac{\pi}{4}</math>
|align="center"|3.7853981634
|align="center"|3.785411784 (exact)
|align="left"|A U.S. liquid gallon is defined by law as 231 cubic inches. The British imperial gallon would be about 20% larger (but the litre is the same thing as the US liter).
|-
|align="center"|''g''0 or ''g''n
|align="center"|6 + ln(45)
|align="center"|9.8066624898
|align="center"|9.80665
|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.

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.
|-
|align="center"|Proton–electron mass ratio
|align="center"|<math>\frac {e^8 - 10} {\phi}</math>
|align="center"|1836.1530151398
|align="center"|1836.15267246
|align="left"|φ is the {{w|golden ratio}}, or <math>\textstyle{ \frac{1+\sqrt 5}{2} }</math>. It has many interesting geometrical properties.
|-
|align="center"|Ruby laser wavelength (meters)
|align="center"|<math>\frac{1}{1200^2}</math>
|align="center"|6.9444 × 10−7
|align="center"|~6.943 × 10−7
|align="left"|The {{w|ruby laser}} wavelength varies because "ruby" is not clearly defined.
|-
|align="center"|Mean Earth radius (meters)
|align="center"|<math>5^8 6e</math>
|align="center"|6,370,973.035
|align="center"|6,371,008.7 (IUGG definition)
|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).
|-
|align="center"|<math>\sqrt 2</math>
|align="center"|<math>\frac35 + \frac{\pi}{7-\pi}</math>
|align="center"|1.4142200581
|align="center"|1.4142135624
|align="left"|There are recurring math jokes along the lines of, "<math>\textstyle{ \frac35 + \frac{\pi}{7-\pi} - \sqrt{2} = 0}</math>, but your calculator is probably not good enough to compute this correctly". See also [[217: e to the pi Minus pi]].
|-
|align="center"|<math>\cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7}</math>
|align="center"|<math>\frac12</math>
|align="center"|0.5
|align="center"|0.5 (exact)
|align="left"|This is the exactly correct equation referred to in the note, "Pro tip – Not all of these are wrong", as shown below and also [http://math.stackexchange.com/questions/140388/how-can-one-prove-cos-pi-7-cos3-pi-7-cos5-pi-7-1-2 here]. If you're still confused, the functions use {{w|radians}}, not {{w|degrees (angle)|degrees}}: when an angular measure does not specify units, radians are the assumed default.
|-
|align="center"|γ (Euler's gamma constant)
|align="center"|<math>\frac{e}{3^4} + \frac{e}{5}</math>
|align="center"|0.5772154006
|align="center"|0.5772156649
|align="left"|The {{w|Euler–Mascheroni constant}} (denoted γ) is a mysterious number describing the relationship between the {{w|Harmonic series (mathematics)|harmonic series}} and the {{w|natural logarithm}}.
|-
|align="center"|<math>\sqrt 5</math>
|align="center"|<math>\frac {13+4\pi} {24-4\pi}</math>
|align="center"|2.2360678094
|align="center"|2.2360679775
|align="left"|
|-
|align="center"|<math>\sum_{n=1}^{\infty} \frac{1}{n^n}</math>
|align="center"|<math>\ln(3)^e</math>
|align="center"|1.2912987577
|align="center"|1.2912859971
|align="left"|
|}

===Proof===

One of the "approximations" actually is precisely correct: <math>\textstyle{ \cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7} = \frac12 }</math>. Here is a proof:

:<math>\cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7}</math>

Multiplying by 1 (or by a nonzero number divided by itself) leaves the equation unchanged:

:<math>= \left( \cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7} \right) \frac{2 \sin\frac{\pi}{7}}{2 \sin\frac{\pi}{7}}</math>

The <math>\textstyle{ 2 \sin\frac{\pi}{7} }</math> on the top of the fraction is multiplied through the original equation:

:<math>= \frac {2 \cos \frac{\pi}{7} \sin\frac{\pi}{7} + 2 \cos \frac{3\pi}{7} \sin\frac{\pi}{7} + 2 \cos \frac{5\pi}{7} \sin\frac{\pi}{7}} {2 \sin\frac{\pi}{7}}</math>

Use the trigonometric identity <math>\textstyle{ 2 \cos A \sin B = \sin (A+B) - \sin(A-B)}</math> on the second and third terms in the numerator:

:<math>\begin{align}
&= \frac {2 \cos \frac{\pi}{7} \sin \frac{\pi}{7} + \left[\sin \left(\frac{3\pi}{7} + \frac{\pi}{7}\right) - \sin \left(\frac{3\pi}{7} - \frac{\pi}{7}\right) \right] + \left[\sin \left(\frac{5\pi}{7} + \frac{\pi}{7}\right) - \sin \left(\frac{5\pi}{7} - \frac{\pi}{7}\right) \right]} {2 \sin\frac{\pi}{7}} \\
&= \frac {2 \cos \frac{\pi}{7} \sin \frac{\pi}{7} + \left[\sin \frac{4\pi}{7} - \sin \frac{2\pi}{7} \right] + \left[\sin \frac{6\pi}{7} - \sin \frac{4\pi}{7} \right]} {2 \sin\frac{\pi}{7}}
\end{align}</math>

Use the trigonometric identity <math>\textstyle{ 2 \cos A \sin A = \sin 2A }</math> on the first term in the numerator:

:<math>\begin{align}
&= \frac {\sin \frac{2\pi}{7} + \left[\sin \frac{4\pi}{7} - \sin \frac{2\pi}{7} \right] + \left[\sin \frac{6\pi}{7} - \sin \frac{4\pi}{7} \right]} {2 \sin\frac{\pi}{7}} \\
&= \frac {\sin \frac{6\pi}{7} + \left[\sin \frac{4\pi}{7} - \sin \frac{4\pi}{7} \right] + \left[\sin \frac{2\pi}{7} - \sin \frac{2\pi}{7} \right]} {2 \sin\frac{\pi}{7}} \\
&= \frac {\sin \frac{6\pi}{7} } {2 \sin\frac{\pi}{7}}
\end{align}</math>

Noting that <math>\textstyle{\frac{6\pi}{7} + \frac{\pi}{7} = \pi}</math> and that the sines of supplementary angles (angles that sum to π) are equal:

:<math>\begin{align}
&= \frac {\sin \frac{\pi}{7} } {2 \sin\frac{\pi}{7}} \\
&= \frac12 \quad \quad \quad \text{Q.E.D.}
\end{align}</math>

To better see why the equation is true, it is better to go to the complex plane. cos(2k pi/7)  is the real part of the k-th 7-th root of unity, exp(2 k i pi/7). The seven 7-th roots of unity (for 0 <= k <= 6) sum up to zero, hence so do their real parts:


:0 = cos(0 pi/7) + cos(2 pi/7) + cos(4 pi/7) + cos(6 pi/7) + cos(8 pi/7) + cos(10 pi/7) + cos(12 pi/7)

But one of these roots is just 1, and all other root go by pairs of conjugate roots, which have the same real part (alternatively, consider that cos(x) = cos(2 pi - x)):


:0 = 1 + 2 (cos(2 pi/7) + cos(4 pi/7) + cos(6 pi/7))

Hence


:cos(2 pi/7) + cos(4 pi/7) + cos(6 pi/7) = - 1/2

which, because cos(x) = cos(pi - x), can be rewritten as


:cos(5 pi/7) + cos(3 pi/7) + cos(pi/7) = 1/2

Q.E.D.

==Transcript==
:'''A table of slightly wrong equations and identities useful for approximations and/or trolling teachers.'''
:(Found using a mix of trial-and-error, ''Mathematica'', and Robert Munafo's ''Ries'' tool.)
: All units are SI MKS unless otherwise noted.

:{| class="wikitable"
|-
|colspan="2" align="center" | Relation:
|align="center" | Accurate to within:
|-
|align="center" | One light-year(m)
|align="center" | 998
|align="center" | one part in 40
|-
|align="center" | Earth Surface(m2)
|align="center" | 698
|align="center" | one part in 130
|-
|align="center" | Oceans' volume(m3)
|align="center" | 919
|align="center" | one part in 70
|-
|align="center" | Seconds in a year
|align="center" | 754
|align="center" | one part in 400
|-
|align="center" | Seconds in a year (''Rent'' method)
|align="center" | 525,600 x 60
|align="center" | one part in 1400
|-
|align="center" | Age of the universe (seconds)
|align="center" | 1515
|align="center" | one part in 70
|-
|align="center" | Planck's constant
|align="center" | 1/(30πe)
|align="center" | one part in 110
|-
|align="center" | Fine structure constant
|align="center" | 1/140
|align="center" | [I've had enough of this 137 crap]
|-
|align="center" | Fundamental charge
|align="center" | 3/(14 * πππ)
|align="center" | one part in 500
|-
|align="center"|White House Switchboard
|colspan="2" align="center"|1 / (eπ√(1 + (e-1)√8))
|-
|align="center"|Jenny's Constant
|colspan="2" align="center"|(7(e/1 - 1/e) - 9) * π2
|-
|colspan="3" align="center"|Intermission: World Population Estimate which should stay current for a decade or two: 
Take the last two digits of the current year

Example: 20[14]

Subtract the number of leap years since hurricane Katrina

Example: 14 (minus 2008 and 2012) is 12

Add a decimal point

Example: 1.2

Add 6

Example: 6 + 1.2

7.2 = World population in billions.

Version for US population:

Example: 20[14]

Subtract 10

Example: 4

Multiply by 3

Example: 12

Add 10

Example: 3[22] million
|-
|align="center"|Electron rest energy
|align="center"|e/716 J
|align="center"|one part in 1000
|-
|align="center"|Light-year(miles)
|align="center"|2(42.42)
|align="center"|one part in 1000
|-
|colspan="2" align="center"|sin(60°) = √3/2 = e/π
|align="center"|one part in 1000
|-
|colspan="2" align="center"|√3 = 2e/π
|align="center"|one part in 1000
|-
|align="center"|γ(Euler's gamma constant)
|align="center"|1/√3
|align="center"|one part in 4000
|-
|align="center"|Feet in a meter
|align="center"|5/(e√π)
|align="center"|one part in 4000
|-
|colspan="2" align="center"|√5 = 2/e + 3/2
|align="center"|one part in 7000
|-
|align="center"|Avogadro's number
|align="center"|69π√5
|align="center"|one part in 25,000
|-
|align="center"|Gravitational constant G
|align="center"|1 / e(π - 1)(π + 1)
|align="center"|one part in 25,000
|-
|align="center"|R (gas constant)
|align="center"|(e+1) √5
|align="center"|one part in 50,000
|-
|align="center"|Proton-electron mass ratio
|align="center"|6*π5
|align="center"|one part in 50,000
|-
|align="center"|Liters in a gallon
|align="center"|3 + π/4
|align="center"|one part in 500,000
|-
|align="center"|g
|align="center"|6 + ln(45)
|align="center"|one part in 750,000
|-
|align="center"|Proton-electron mass ratio
|align="center"|(e8 - 10) / ϕ
|align="center"|one part in 5,000,000
|-
|align="center"|Ruby laser wavelength
|align="center"|1 / (12002)
|align="center"|[within actual variation]
|-
|align="center"|Mean Earth Radius
|align="center"|(58)*6e
|align="center"|[within actual variation]
|-
|colspan="3" align="center"|Protip - not all of these are wrong:
|-
|colspan="2" align="center"|√2 = 3/5 + π/(7-π)
|align="center"|cos(π/7) + cos(3π/7) + cos(5π/7) = 1/2
|-
|align="center"|γ(Euler's gamma constant) = e/34 + e/5
|align="center"|√5 = (13 + 4π) / (24 - 4π)
|align="center"|Σ 1/nn = ln(3)e
|}

{{comic discussion}}
[[Category:Charts]]
[[Category:Math]]
[[Category:Physics]]
[[Category:Protip]]

899: Number Line

2024-06-10T13:17:17Z

ISaveXKCDpapers: corrected information, grammar

{{comic
| number = 899
| date = May 16, 2011
| title = Number Line
| image = number line.png
| titletext = The Wikipedia page List of Numbers opens with "This list is incomplete; you can help by expanding it."
}}

== Explanation ==

Once again, [[Randall]] seems to be just messing around, this time with a number line.

*'''Negative numbers''' have the same magnitude as positive numbers but can only be used to represent the removal of that same magnitude (hence the term "difference" being used for subtraction). Negative numbers may be called imitator numbers in the comic because of their similarities to positive numbers.

*'''0.99'''.... is {{w|0.999...|equal to 1}} because if you subtract any number from one, however small, you will get a number that is less than 0.99. 1 − '''0.0000000372''' is 1 bit less than the {{w|IEEE_floating_point|IEEE 754 32-bit floating-point representation}} of 1.

*The '''{{w|golden ratio}}''' or '''ϕ''' (phi) is the number <math>\tfrac{1+\sqrt{5}}{2}</math>, about 1.61803. It has many interesting mathematical properties, mostly relating to geometry, and has occasional appearances in nature, such as spirals formed by the seeds in sunflowers. It is also subject to many less credible claims, such as the belief that phi appears in {{w|Parthenon}} (a well-disputed claim) or that rectangles proportioned after phi are more aesthetically pleasing. The speaker seems to drive off his listeners as soon as he brings it up; the golden ratio is infamous for being brought up by know-it-alls, which Randall has mocked in other comics.

* The approximate range from 2.1 to 2.3 is marked as '''The Forbidden Region'''. Why Randall marked this range as forbidden is really anyone's guess; it seems to be an entirely arbitrary designation.

*'''{{w|e (mathematical constant)|e}}''' (Euler's number) is 2.71828... and '''π''' (pi) is 3.14159265...

*'''2.9299372''' is probably a {{w|President's Day}} reference. It is the average of e and π just as the American Presidents' Day is always observed on the 3rd Monday of February (between {{w|George Washington}} and {{w|Abraham Lincoln}}'s birthdays). Washington and Lincoln were the 1st and 16th Presidents of the USA, respectively. Each has a celebrated place in American history.

*'''{{w|Gird}}''', '''ᛟ''' is a purely fictional number. (The glyph that Randall uses seems to resemble an older shape of the digit 4, such as seen on [http://www.bl.uk/learning/images/mappinghist/large2296.html archaic maps].). Canon and orthodox could mean "accepted as the offical story" and "most science-based followers", but they could also reference to organised religions. Gird could be a reference to any or all of:
**[http://strangehorizons.com/fiction/the-secret-number/ Bleem] - a fictional integer between 3 and 4
**iCarly's [http://icarly.wikia.com/wiki/Derf Derf] - a fictional integer between 5 and 6
**George Carlin's [http://www.urbandictionary.com/define.php?term=bleen Bleen] - a fictional integer between 6 and 7
**[https://scp-wiki.wikidot.com/scp-033 SCP-033] - a fictional "missed number" that causes mathematical systems to break down when it is introduced to them (manifesting as the physical destruction of the objects the mathematical formuli are contained in, such as paper and computers)
**Saturday Morning Breakfast Cereal's [http://www.smbc-comics.com/index.php?id=3913 Sorf] - a fictional integer between 2 and 3 

*'''Site of Battle of 4.108''' is another map joke, implying that 4.108 is an actual location, where an eponymous battle was previously fought. It may be a reference (or homage) to the {{w|Battle of Wolf 359}}, a famous military conflict in the fictional universe of Star Trek. 4.108 was also referenced in [[2861: X Value]], though with an added 3 in the ten-thousandths place.

*An '''Unexplored''' region obscures the line approximately ranging all values from 4.5 to 6.7. In the days when the Earth was still being mapped out, territories that had yet to be properly explored and charted were labelled in a similar manner. The placement of the '''Unexplored''' region on the number line indicates that all numbers in that range, including the integers 5 and 6, are completely unknown. This is, of course, patently ridiculous,{{cn}} and the humor seems to derive solely from how nonsensical and unbelievable it is. Correspondingly, the digits 5 and 6 cannot be found in the comic.

*It is often the case in the media that "It has been 7 years..." or "In the last 7 years..." etc. It is made to seem like a believable statistic but cannot always be true. Alternatively, it is intended as an absurd joke that the number 7 is just "not to be believed".

*'''8''' is not the largest even {{w|prime number}}, nor is it a prime at all. The largest (and only) even prime is 2. A joke intended for those who clearly know that the claim is false.

*The last entry seems to be a reference to certain fields of {{w|pure mathematics}}, which focus less on performing calculations with numbers and more on understanding structures that may be described using logic. It finishes off the tone of the comic that seems to be shaping the number line terms of what is commonly useful to certain areas of applied mathematics, rather than a complete, accurate version of the number line.

The title text is a literalism joke; at the time the comic was published, all Wikipedia articles with incomplete lists began with the message template "This list is incomplete; you can help by expanding it." In the case of the {{w|List of numbers}} page, one could infer the absurd notion that Wikipedia wanted to have the list include every number from negative infinity to infinity. But because all Wikipedia articles are necessarily finite, such a list would always be incomplete, no matter how much it was expanded. It may also be referencing his previous statements about Wikipedia being the home of compulsive list-makers, who make the most astonishingly complete lists imaginable.

As of 2022, Wikipedia's List of numbers page, as well as all pages including lists that cannot ever reach a state of completion, are headed by the message template "This is a dynamic list and may never be able to satisfy particular standards for completeness. You can help by adding missing items with reliable sources."

== Transcript ==
:[Number line ranging from −1 to 10.]
:[Arrow pointing left, towards negative numbers] Negative "imitator" numbers (do not use)
:[Line right before the number one] 0.99... (actually 0.0000000372 less than 1)
:[Line at the golden ratio.] Φ Parthenon; sunflowers; golden ratio; wait, come back, I have facts!
:[Line at a region between two and 2.2] forbidden region
:[Line at Euler's number.] e
:[Line a bit before 3] 2.9299372 (e and pi, observed)
:[Line at π.] π
:[Line at 3.5 with ᛟ as the numeral] Gird – accepted as canon by orthodox mathematicians
:[Line a bit after 4.] site of battle of 4.108
:[Blob between 4.5 and 6.5 labeled unexplored.]
:[Line at seven.] Number indicating a factoid is made up ("every 7 years...", "science says there are 7...", etc)
:[Line at eight.] Largest even prime
:[Line at 8.75.] If you encounter a number higher than this, you're not doing real math

== Trivia ==

* As for the "Gird" between 3 and 4, one might argue that the arithmetic square root of 11 may have some "integer" properties, because there exists an integer-to-integer{{Citation needed}} function f(x) such that f(f(x))=11x. (details needed)
* The "unexplored" area is actually famous for some numbers, such as twice π (also known as tau (τ), approximately 6.283185).

{{comic discussion}}
[[Category:Math]]
[[Category:Wikipedia]]

1462: Blind Trials

2024-06-10T05:49:29Z

ISaveXKCDpapers: corrected explanation (it has nothing to do with the sugar)

{{comic
| number = 1462
| date = December 19, 2014
| title = Blind Trials
| image = blind_trials.png
| titletext = Plus, you have to control for the fact that some people are into being blindfolded.
}}

==Explanation==
In research, a {{w|Blind Experiment|blind trial}} is an experiment where certain information about the test is concealed from the subjects and/or the testers, in order to reduce sources of bias in the results. A double-blind trial is one where neither the subject nor the testers know who has or has not received treatment (or for multiple treatments, which treatment).

A scientific approach also requires the use of {{w|control groups}} to determine the significance of observations in (clinical) trials. The members of the control group receive either no treatment or the "standard" treatment. However, to ensure "blindness" in the study, even if a control group is to receive no treatment, they must be given a {{w|placebo}}: an ineffective treatment given to ensure the doctors and/or patients are unaware whether they are being given the treatment.

For example, in clinical drug trials, when a treatment being tested is administered in the form of a pill, a visually-identical inert pill is given to the control group so no one will know if a subject has been given the treatment or a placebo. In pop culture, placebos in pill-form are often made of sugar, which has negligible medical effects.

Controls and blinding are crucial to distinguish the actual effects of the treatment from the {{w|placebo effect}}, or the psychologically-induced effects of a subject's belief that a treatment will or will not help them, which may have real physiologic effects or influence the reporting of subjective measures such as pain level or the presence of side effects. It is vital that there are no clues available to distinguish between the different groups. Even subtle cues from the body language of the testers are sufficient to trigger placebo effect, making double-blind trials necessary.

Challenges exist in designing placebo alternatives to certain physical treatments that might be tested, such as acupuncture; in this case the best quality trials have typically used either special 'joke' retractable needles that only give the illusion of proper penetration or the practitioner/researcher deliberately and safely avoids the traditional {{w|Meridian (Chinese medicine)|meridians}} on the body for the treatment concerned so that the patient remains 'blind' to their role in the trial. The practitioner must otherwise be consistent in treatment between groups and not be involved in the medical assessment phase for properly double-blinded conditions, where the most reliable results still seem to only show a significant placebo effect at work.

There are, however, certain cases where it is almost impossible to make the experience of the control group identical to that of the test group. Making a real and fake pill appear the same is a relatively trivial task, and the ignorance of participants to the details of a given established practice or procedure can allow for a certain level of blinding. However, it would be challenging (to say the least) to make the control group in the described experiment think that they are having lots of sex,{{Citation needed}} when in fact they are not. The description of the control group as taking sugar pills is a laughably poor placebo substitute, as the sensations of ingesting a pill and of engaging in sexual activity are wildly different.

Scientific research involving humans is extremely challenging to conduct because of the difficulty in finding appropriate control groups. This is one of the reasons animal experiments (for instance involving inbred strains of mice) are so common.

The title text adds another twist by taking “blind” literally, and noting that for some people, being blindfolded increases their enjoyment of sexual activity, thereby acting as a confounding variable.

Despite this, it should be noted that Cardiovascular health is typically measured in terms of objective data such as cholesterol levels, ejection fraction, and morbidity/mortality data like the frequency of myocardial infarctions, strokes, or sudden cardiac death. Even sighted, it would be difficult for either subjects or researchers to manipulate this kind of data.

==Transcript==
:[Megan is pointing at charts hanging on the wall.]
:Megan: We've designed a double-blind trial to test the effect of sexual activity on cardiovascular health.
:Both groups will ''think'' they're having lots of sex, but one group will actually be getting sugar pills.

:The limitations of blind trials

{{comic discussion}}
[[Category:Comics featuring Megan]]
[[Category:Sex]]
[[Category:Psychology]]
[[Category:Science]]
[[Category:Scientific research]]

1078: Knights

2024-04-03T23:48:20Z

ISaveXKCDpapers: add "white"

{{comic
| number = 1078
| date = July 6, 2012
| title = Knights
| image = knights.png
| titletext = 1. Nf3 ... ↘↘↘ 2. Nc3 ... ↘↘↘ 0-1
}}

==Explanation==
This comic is comparing the opening moves of the game of {{w|chess}} to the opening moves of the {{w|Battle of Agincourt}}, which was fought between the English and the French in the {{w|Hundred Years War}}. In the battle, just like in the comic, the English used their longbowmen effectively, neutralizing the French knights and infantry. The two pieces that are moved out of the white side of the board are both the pieces known as the Knights. White moves first in chess, and in the actual battle, the French knights on horseback attacked first; the English being the black pieces may also be a reference to {{w|Edward the Black Prince}}, who was a prominent figure in an earlier stage of the Hundred Years War. As you can see, all the {{w|Pawn_(chess)|pawns}} (foot soldiers) on the right side of the chess board have bows.

The word "{{w|gambit}}" means "an opening in chess, in which a minor piece or a pawn is sacrificed to gain an advantage". The usual gambit of sacrificing a pawn is subverted to be a sacrifice of a high-value piece, as an analogy of what happened at Agincourt.

The title text uses {{w|algebraic chess notation}}. Nf3 means a knight has moved to square f3. Nc3 means a knight has moved to square c3. N means knight because the king piece has the K abbreviation covered. What comes after the typical chess move is what can only be read as a hail of arrows. 0-1 at the end means that "Black Wins". This implies that White resigned, as he is not in checkmate (for non-timed chess games, the only ways to win are by checkmating your opponent or by having them resign). It seems to be an error that ellipses are included before Black's moves, as algebraic notation uses ellipses only to indicate that White's move has been omitted.

==Transcript==
:[A chessboard, The black pawns have all gained longbows and have specifically taken down the white knights as they move forward, without any black pieces needing to move from their opening positions. Caption below the panel:]
:The Agincourt Gambit

== Trivia ==
* In the title text, Nf3 was accidentally written as Ne3. At the beginning of a chess game, neither white knight can move to e3. The proper move (and the move actually made, in the picture) is Nf3. This was later corrected.

* There is an actual chess opening known as the Agincourt Defense: 1. c4 e6. However, in the case of the Agincourt Defense, White represents the English and Black the French, which is the opposite of the situation depicted in the comic. This is because the move 1. c4 by White is the English Opening, and the move 1... e6 by Black is the same as what Black plays in the French Defense, which is 1. e4 e6.

{{comic discussion}}

[[Category:Chess]]

1078: Knights

2024-04-03T23:47:05Z

ISaveXKCDpapers: capitalization (can't tell due to xkcd font, but opening names are traditionally capitalized)

{{comic
| number = 1078
| date = July 6, 2012
| title = Knights
| image = knights.png
| titletext = 1. Nf3 ... ↘↘↘ 2. Nc3 ... ↘↘↘ 0-1
}}

==Explanation==
This comic is comparing the opening moves of the game of {{w|chess}} to the opening moves of the {{w|Battle of Agincourt}}, which was fought between the English and the French in the {{w|Hundred Years War}}. In the battle, just like in the comic, the English used their longbowmen effectively, neutralizing the French knights and infantry. The two pieces that are moved out of the white side of the board are both the pieces known as the Knights. White moves first in chess, and in the actual battle, the French knights on horseback attacked first; the English being the black pieces may also be a reference to {{w|Edward the Black Prince}}, who was a prominent figure in an earlier stage of the Hundred Years War. As you can see, all the {{w|Pawn_(chess)|pawns}} (foot soldiers) on the right side of the chess board have bows.

The word "{{w|gambit}}" means "an opening in chess, in which a minor piece or a pawn is sacrificed to gain an advantage". The usual gambit of sacrificing a pawn is subverted to be a sacrifice of a high-value piece, as an analogy of what happened at Agincourt.

The title text uses {{w|algebraic chess notation}}. Nf3 means a knight has moved to square f3. Nc3 means a knight has moved to square c3. N means knight because the king piece has the K abbreviation covered. What comes after the typical chess move is what can only be read as a hail of arrows. 0-1 at the end means that "Black Wins". This implies that White resigned, as he is not in checkmate (for non-timed chess games, the only ways to win are by checkmating your opponent or by having them resign). It seems to be an error that ellipses are included before Black's moves, as algebraic notation uses ellipses only to indicate that White's move has been omitted.

==Transcript==
:[A chessboard, The black pawns have all gained longbows and have specifically taken down the white knights as they move forward, without any black pieces needing to move from their opening positions. Caption below the panel:]
:The Agincourt Gambit

== Trivia ==
* In the title text, Nf3 was accidentally written as Ne3. At the beginning of a chess game, neither knight can move to e3. The proper move (and the move actually made, in the picture) is Nf3. This was later corrected.

* There is an actual chess opening known as the Agincourt Defense: 1. c4 e6. However, in the case of the Agincourt Defense, White represents the English and Black the French, which is the opposite of the situation depicted in the comic. This is because the move 1. c4 by White is the English Opening, and the move 1... e6 by Black is the same as what Black plays in the French Defense, which is 1. e4 e6.

{{comic discussion}}

[[Category:Chess]]

1078: Knights

2024-04-03T23:46:02Z

ISaveXKCDpapers: punctuation

{{comic
| number = 1078
| date = July 6, 2012
| title = Knights
| image = knights.png
| titletext = 1. Nf3 ... ↘↘↘ 2. Nc3 ... ↘↘↘ 0-1
}}

==Explanation==
This comic is comparing the opening moves of the game of {{w|chess}} to the opening moves of the {{w|Battle of Agincourt}}, which was fought between the English and the French in the {{w|Hundred Years War}}. In the battle, just like in the comic, the English used their longbowmen effectively, neutralizing the French knights and infantry. The two pieces that are moved out of the white side of the board are both the pieces known as the Knights. White moves first in chess, and in the actual battle, the French knights on horseback attacked first; the English being the black pieces may also be a reference to {{w|Edward the Black Prince}}, who was a prominent figure in an earlier stage of the Hundred Years War. As you can see, all the {{w|Pawn_(chess)|pawns}} (foot soldiers) on the right side of the chess board have bows.

The word "{{w|gambit}}" means "an opening in chess, in which a minor piece or a pawn is sacrificed to gain an advantage". The usual gambit of sacrificing a pawn is subverted to be a sacrifice of a high-value piece, as an analogy of what happened at Agincourt.

The title text uses {{w|algebraic chess notation}}. Nf3 means a knight has moved to square f3. Nc3 means a knight has moved to square c3. N means knight because the king piece has the K abbreviation covered. What comes after the typical chess move is what can only be read as a hail of arrows. 0-1 at the end means that "Black Wins". This implies that White resigned, as he is not in checkmate (for non-timed chess games, the only ways to win are by checkmating your opponent or by having them resign). It seems to be an error that ellipses are included before Black's moves, as algebraic notation uses ellipses only to indicate that White's move has been omitted.

==Transcript==
:[A chessboard, The black pawns have all gained longbows and have specifically taken down the white knights as they move forward, without any black pieces needing to move from their opening positions. Caption below the panel:]
:The Agincourt gambit

== Trivia ==
* In the title text, Nf3 was accidentally written as Ne3. At the beginning of a chess game, neither knight can move to e3. The proper move (and the move actually made, in the picture) is Nf3. This was later corrected.

* There is an actual chess opening known as the Agincourt Defense: 1. c4 e6. However, in the case of the Agincourt Defense, White represents the English and Black the French, which is the opposite of the situation depicted in the comic. This is because the move 1. c4 by White is the English Opening, and the move 1... e6 by Black is the same as what Black plays in the French Defense, which is 1. e4 e6.

{{comic discussion}}

[[Category:Chess]]

232: Chess Enlightenment

2024-03-15T07:28:55Z

ISaveXKCDpapers: terminology, accuracy of chess explanation, grammar

{{comic
| number = 232
| date = March 7, 2007
| title = Chess Enlightenment
| image = chess enlightenment.png
| titletext = You know that 'sweep the pieces off the board and see it in your mind' thing? Doesn't work.
}}

==Explanation==
In this comic, [[Cueball]] finds his game of chess against [[Megan]] to be too difficult, and he attempts to tap his subconscious to find his next move. This is a common technique used in more physical competitions like baseball or golf, where overthinking can interfere with one's motion and thus "clearing one's mind" and relying on the subconscious is useful to overcome such mental barriers. However, chess is more a game of planning and strategy than natural movement, and the rules of chess are not ingrained into Cueball's subconscious, and so his subconscious ends up feeding him invalid moves and beginner questions concerning movement rules.

{{w|Chess}} is a board game in which two players take turns to move a variety of different pieces representing units on a battlefield to try to checkmate the other player's king. Chess has a lively tournament scene, and it takes much practice to attain a competent level of skill in the game. Different units can move and capture in different ways; pawns can only move forward by one square unless it's their first move, in which case they can move up two squares. They can only capture by moving diagonally, including when capturing ''{{w|En passant|en passant}}'', in which case they move behind an opposing pawn that had moved forward two squares on the previous turn. Other pieces have different rules.

{{w|Obi-Wan Kenobi}} is a character from the movie series ''{{w|Star Wars}}'' who played the mentor figure to the protagonist, {{w|Luke Skywalker}}. One of his pieces of advice to his mentee was to relax and listen to his subconscious in strenuous times. However, Obi-Wan gave this advice because Luke was connected to The Force, a mystical energy in the ''Star Wars'' universe that connects to the entire universe; not being a part of the ''Star Wars'' universe, Cueball is unable to tap into it. The Force does have similarities to real-life concepts used in various Eastern philosophies, but they are not typically used to play chess, for the same reasons given above.

The title text refers to a scene in the chess movie ''{{w|Searching for Bobby Fischer}}'', in which Sir Ben Kingsley's character dramatically sweeps the pieces off the board and instructs his student to see the pieces in his mind, which the child proceeds to do. Randall considers this impractical, presumably for similar reasons as the Obi-Wan example.

==Transcript==
:[Cueball and Megan are playing chess; Cueball is leaning forward over the chessboard.]
:Cueball (thinking): Why is chess so hard? Maybe the answers lie within me. Maybe I just need to let go, relax, and let my instincts and subconscious speak.

:[Cueball leans back and places his hands to his head.]
:''Meditate''

:Cueball's subconscious: Knight to G-4

:[Beat panel.]

:Cueball: That's not even a legal move.
:Cueball's subconscious: Okay, hold on. How do the pawns capture, again?
:Cueball: Man, Obi-Wan was full of crap.

{{comic discussion}}
[[Category:Comics featuring Cueball]]
[[Category:Comics featuring Megan]]
[[Category:Chess]]
[[Category:Star Wars]]

1078: Knights

2023-12-13T03:44:35Z

ISaveXKCDpapers: reordered paragraphs, added info about ellipses

{{comic
| number = 1078
| date = July 6, 2012
| title = Knights
| image = knights.png
| titletext = 1. Nf3 ... ↘↘↘ 2. Nc3 ... ↘↘↘ 0-1
}}

==Explanation==
This comic is comparing the opening moves of the game of {{w|chess}} to the opening moves of the {{w|Battle of Agincourt}}, which was fought between the English and the French in the {{w|Hundred Years War}}. In the battle, just like in the comic, the English used their longbowmen effectively, neutralizing the French knights and infantry. The two pieces that are moved out of the white side of the board are both the pieces known as the Knights. White moves first in chess, and in the actual battle, the French knights on horseback attacked first; the English being the black pieces may also be a reference to {{w|Edward the Black Prince}}, who was a prominent figure in an earlier stage of the Hundred Years War. As you can see, all the {{w|Pawn_(chess)|pawns}} (foot soldiers) on the right side of the chess board have bows.

The word "{{w|gambit}}" means "an opening in chess, in which a minor piece or a pawn is sacrificed to gain an advantage". The usual gambit of sacrificing a pawn is subverted to be a sacrifice of a high-value piece, as an analogy of what happened at Agincourt.

The title text uses {{w|algebraic chess notation}}. Nf3 means a knight has moved to square f3. Nc3 means a knight has moved to square c3. N means knight because the king piece has the K abbreviation covered. What comes after the typical chess move is what can only be read as a hail of arrows. 0-1 at the end means that "Black Wins". This implies that White resigned, as he is not in checkmate (for non-timed chess games, the only ways to win are by checkmating your opponent or by having them resign). It seems to be an error that ellipses are included before Black's moves, as algebraic notation uses ellipses only to indicate that White's move has been omitted.

==Transcript==
:[A chessboard, The black pawns have all gained longbows and have specifically taken down the white knights as they move forward, without any black pieces needing to move from their opening positions. Caption below the panel:]
:The Agincourt gambit.

== Trivia ==
* In the title text, Nf3 was accidentally written as Ne3. At the beginning of a chess game, neither knight can move to e3. The proper move (and the move actually made, in the picture) is Nf3. This was later corrected.

* There is an actual chess opening known as the Agincourt Defense: 1. c4 e6. However, in the case of the Agincourt Defense, White represents the English and Black the French, which is the opposite of the situation depicted in the comic. This is because the move 1. c4 by White is the English Opening, and the move 1... e6 by Black is the same as what Black plays in the French Defense, which is 1. e4 e6.

{{comic discussion}}

[[Category:Chess]]

1078: Knights

2023-12-13T03:39:33Z

ISaveXKCDpapers: acceptance of resignation is not required

{{comic
| number = 1078
| date = July 6, 2012
| title = Knights
| image = knights.png
| titletext = 1. Nf3 ... ↘↘↘ 2. Nc3 ... ↘↘↘ 0-1
}}

==Explanation==
This comic is comparing the opening moves of the game of {{w|chess}} to the opening moves of the {{w|Battle of Agincourt}}, which was fought between the English and the French in the {{w|Hundred Years War}}. In the battle, just like in the comic, the English used their longbowmen effectively, neutralizing the French knights and infantry. The two pieces that are moved out of the white side of the board are both the pieces known as the Knights. White moves first in chess, and in the actual battle, the French knights on horseback attacked first; the English being the black pieces may also be a reference to {{w|Edward the Black Prince}}, who was a prominent figure in an earlier stage of the Hundred Years War. As you can see, all the {{w|Pawn_(chess)|pawns}} (foot soldiers) on the right side of the chess board have bows.

The title text uses {{w|algebraic chess notation}}. Nf3 means a knight has moved to square f3. Nc3 means a knight has moved to square c3. N means knight because the king piece has the K abbreviation covered. What comes after the typical chess move is what can only be read as a hail of arrows. 0-1 at the end means that "Black Wins". This implies that White resigned, as he is not in checkmate (for non-timed chess games, the only ways to win are by checkmating your opponent or by having them resign).

The word "{{w|gambit}}" means "an opening in chess, in which a minor piece or a pawn is sacrificed to gain an advantage". The usual gambit of sacrificing a pawn is subverted to be a sacrifice of a high-value piece, as an analogy of what happened at Agincourt.

==Transcript==
:[A chessboard, The black pawns have all gained longbows and have specifically taken down the white knights as they move forward, without any black pieces needing to move from their opening positions. Caption below the panel:]
:The Agincourt gambit.

== Trivia ==
* In the title text, Nf3 was accidentally written as Ne3. At the beginning of a chess game, neither knight can move to e3. The proper move (and the move actually made, in the picture) is Nf3. This was later corrected.

* There is an actual chess opening known as the Agincourt Defense: 1. c4 e6. However, in the case of the Agincourt Defense, White represents the English and Black the French, which is the opposite of the situation depicted in the comic. This is because the move 1. c4 by White is the English Opening, and the move 1... e6 by Black is the same as what Black plays in the French Defense, which is 1. e4 e6.

{{comic discussion}}

[[Category:Chess]]

1078: Knights

2023-12-13T03:37:26Z

ISaveXKCDpapers: added information about the Agincourt Defense

{{comic
| number = 1078
| date = July 6, 2012
| title = Knights
| image = knights.png
| titletext = 1. Nf3 ... ↘↘↘ 2. Nc3 ... ↘↘↘ 0-1
}}

==Explanation==
This comic is comparing the opening moves of the game of {{w|chess}} to the opening moves of the {{w|Battle of Agincourt}}, which was fought between the English and the French in the {{w|Hundred Years War}}. In the battle, just like in the comic, the English used their longbowmen effectively, neutralizing the French knights and infantry. The two pieces that are moved out of the white side of the board are both the pieces known as the Knights. White moves first in chess, and in the actual battle, the French knights on horseback attacked first; the English being the black pieces may also be a reference to {{w|Edward the Black Prince}}, who was a prominent figure in an earlier stage of the Hundred Years War. As you can see, all the {{w|Pawn_(chess)|pawns}} (foot soldiers) on the right side of the chess board have bows.

The title text uses {{w|algebraic chess notation}}. Nf3 means a knight has moved to square f3. Nc3 means a knight has moved to square c3. N means knight because the king piece has the K abbreviation covered. What comes after the typical chess move is what can only be read as a hail of arrows. 0-1 at the end means that "Black Wins". This implies that White resigned, as he is not in checkmate (for non-timed chess games, the only ways to win are by checkmating your opponent or by accepting their resignation).

The word "{{w|gambit}}" means "an opening in chess, in which a minor piece or a pawn is sacrificed to gain an advantage". The usual gambit of sacrificing a pawn is subverted to be a sacrifice of a high-value piece, as an analogy of what happened at Agincourt.

==Transcript==
:[A chessboard, The black pawns have all gained longbows and have specifically taken down the white knights as they move forward, without any black pieces needing to move from their opening positions. Caption below the panel:]
:The Agincourt gambit.

== Trivia ==
* In the title text, Nf3 was accidentally written as Ne3. At the beginning of a chess game, neither knight can move to e3. The proper move (and the move actually made, in the picture) is Nf3. This was later corrected.

* There is an actual chess opening known as the Agincourt Defense: 1. c4 e6. However, in the case of the Agincourt Defense, White represents the English and Black the French, which is the opposite of the situation depicted in the comic. This is because the move 1. c4 by White is the English Opening, and the move 1... e6 by Black is the same as what Black plays in the French Defense, which is 1. e4 e6.

{{comic discussion}}

[[Category:Chess]]

569: Borders

2023-12-04T23:13:05Z

ISaveXKCDpapers: changed link

{{comic
| number = 569
| date = April 15, 2009
| title = Borders
| image = borders.png
| titletext = Eventually a UN is set up. And then a lone rebel runs down the line of flags in front of it, runs back to his base, and gets a kajillion points.
}}

==Explanation==

{{w|Capture the flag}} (CTF) is a common way of playing games where the objective is to capture the opponent's flag while protecting your own team's flag. This comic describes a CTF server that [[Cueball]] joins for an online war game which has the basic plot premise that after the bloodshed of the overthrow of the kingdom's old order, and its signing of peace accords with the leader's of the narrator's realm, peace has become a reality, and not just a dream unendingly deferred. Therefore, no one is trying to capture the opponent's flag, which according to the storyline flies over the kingdom's embassy in the narrator's country, as to ensure peace, therefore making the game unexciting and pointless, as Cueball says in the last panel.

The name "Liate" may be a reference to the {{w|Integration by parts#LIATE rule|LIATE rule}} for integration by parts.

The title text refers to the line of flags in front of UN buildings, which includes the flag of almost every country internationally recognized. If such an organisation and therefore a collection of flags of all the teams were established on the server, which seems the logical conclusion of the goal of achieving peace, one could get a very high score by rebelling against their nation and quickly capturing all of the opposing flags, getting a {{Wiktionary|kajillion}} points, which is slang for "an unspecified large number."

==Transcript==
:[Two Cueball-like guys stand on a hill overlooking a great city. One of them points at the city. Between them and the city stands an embassy flying a red flag. The text is not spoken by either of the guys.]
:Three years ago, the kingdom of Liate overthrew their old order and established a constitutional monarchy. Our leaders signed a treaty with their queen, and our borders were set by the Yarbis Accords.
:Many said war would be unending, that peace would always be a dream deferred. But today, our flag flies proudly over our embassy in their kingdom, and they walk our lands without fear.
:So come, traveller. Lay down your grudges and join us in brotherhood. It is time not to fight, but to live.
:[Cueball sitting at computer.]
:Cueball: This is the worst capture-the-flag server ever.

{{comic discussion}}

[[Category:Comics with color]]
[[Category:Comics featuring Cueball]]
[[Category:Multiple Cueballs]]
[[Category:Video games]]

569: Borders

2023-12-04T23:11:19Z

ISaveXKCDpapers: added sentence about LIATE rule

{{comic
| number = 569
| date = April 15, 2009
| title = Borders
| image = borders.png
| titletext = Eventually a UN is set up. And then a lone rebel runs down the line of flags in front of it, runs back to his base, and gets a kajillion points.
}}

==Explanation==

{{w|Capture the flag}} (CTF) is a common way of playing games where the objective is to capture the opponent's flag while protecting your own team's flag. This comic describes a CTF server that [[Cueball]] joins for an online war game which has the basic plot premise that after the bloodshed of the overthrow of the kingdom's old order, and its signing of peace accords with the leader's of the narrator's realm, peace has become a reality, and not just a dream unendingly deferred. Therefore, no one is trying to capture the opponent's flag, which according to the storyline flies over the kingdom's embassy in the narrator's country, as to ensure peace, therefore making the game unexciting and pointless, as Cueball says in the last panel.

The name "Liate" may be a reference to the LIATE rule for {{w|integration by parts}}.

The title text refers to the line of flags in front of UN buildings, which includes the flag of almost every country internationally recognized. If such an organisation and therefore a collection of flags of all the teams were established on the server, which seems the logical conclusion of the goal of achieving peace, one could get a very high score by rebelling against their nation and quickly capturing all of the opposing flags, getting a {{Wiktionary|kajillion}} points, which is slang for "an unspecified large number."

==Transcript==
:[Two Cueball-like guys stand on a hill overlooking a great city. One of them points at the city. Between them and the city stands an embassy flying a red flag. The text is not spoken by either of the guys.]
:Three years ago, the kingdom of Liate overthrew their old order and established a constitutional monarchy. Our leaders signed a treaty with their queen, and our borders were set by the Yarbis Accords.
:Many said war would be unending, that peace would always be a dream deferred. But today, our flag flies proudly over our embassy in their kingdom, and they walk our lands without fear.
:So come, traveller. Lay down your grudges and join us in brotherhood. It is time not to fight, but to live.
:[Cueball sitting at computer.]
:Cueball: This is the worst capture-the-flag server ever.

{{comic discussion}}

[[Category:Comics with color]]
[[Category:Comics featuring Cueball]]
[[Category:Multiple Cueballs]]
[[Category:Video games]]

1112: Think Logically

2023-08-23T07:38:13Z

ISaveXKCDpapers: shorten, changed "dozens and dozens" (an understatement) to "lots"

{{comic
| number = 1112
| date = September 24, 2012
| title = Think Logically
| image = think logically.png
| titletext = I've developed a more logical set of rules but the people on the chess community have a bunch of stupid emotional biases and won't reply to my posts.
}}

==Explanation==
{{w|Chess}} is a centuries-old board game in which two players take turns moving one of their 16 pieces to try and checkmate the other player's king (one of the pieces). When one player is in a position to capture their opponent's king on their next move, and the opponent has no legal move available to avoid such capture, the opponent is said to be in "checkmate", and by definition immediately loses.

The game, with origins around the sixth century, and with the modern rules being essentially set in the late 15th century, has a significant amount of history. The rules and traditions are well established. The knight is a piece that can only move in an L-shaped pattern (two squares in one direction, and one square perpendicular), but has the unique ability to jump over other pieces.

The comic highlights two mistakes players often make in chess: complete fixation on the king at the cost of their other pieces, and failure to take advantage of the knight's movement patterns. At the same time this is a jab at how people sometimes oversimplify an argument when confronted with a topic they are not familiar with. Previously this was depicted in [[675: Revolutionary]] and [[793: Physicists]]. See also the {{w|Dunning–Kruger effect}}. The units in chess are widely agreed to be well-balanced, and [[Cueball|Cueball's]] criticism of the knight shows an obvious lack of knowledge of the knight's potential.

Given the long history of chess, a significant amount of writing and research has been dedicated to the game and its strategies. This is inadvertently mocked by Cueball who naively suggests it would be trivial to make a list of all situations in which a piece would move backwards (called a "retreat" in chess). Such a list — at least a partial one — certainly does exist, as do lists of numerous other chess moves and situations.

[[Knit Cap]] proceeds to demonstrate Cueball's lack of knowledge by beating him in four moves, which typically would only occur when an experienced player plays a novice. The checkmate depicted is the {{w|Scholar's mate|scholar's mate}}, being a classic early-game checkmate in chess. It is in fact extremely easy to defend against it (blocking the queen's vision of f7, without letting the queen take any of your pieces, would do), thus proving Cueball's inexperience. Careful scrutiny of the board suggests a scholar's mate, something along these lines (using chess algebraic notation): 1.e4 e5 2. Bc4 Nc6 3. Qh5 Nf6 4. Qxf7#.

Cueball, instead of admitting he underestimated the game, believes the failure is in the game itself. The title text indicates that Cueball attempted to suggest revisions to the rules of chess. Given that Cueball has no experience as a chess player, it is likely many of the changes are illogical or ridiculous. In the face of hundreds of years of history, it is not surprising that the chess community is ignoring them. The last major changes to the rules of chess occurred more than 400 years ago when, among other things, the pawn was given its two-space starting move and the queen was made into the most powerful piece (previously it was the weakest). The chess community's ties to the traditions of the game and their refusal to accept Cueball's suggestions are written off by Cueball as "{{w|emotional bias}}" suggesting his changes are logical, but that the community is letting their emotions cloud their rational decision making abilities, while in reality it is he who is being affected. However, Cueball may feel better if he learns that lots of {{w|Chess variants|chess variants}} do exist out there.

The comic may also be a jab at competitive online games whose fans call for "buffs" (power additions) and "nerfs" (power reductions) to characters they believe to be underpowered or overpowered, often with inadequate knowledge of those characters. On the other hand, some online games and multiplayer computer games in general are unbalanced since they lack centuries of history to balance themselves, unlike chess.

[[Knit Cap]] is called ''knit hat guy'' in the [http://xkcd.com/1112/info.0.json official transcript]. There are two other cases (after this comic) where a person with hair has been shown with a knit cap. The first was [[Randall|Randall's]] wife after chemotherapy in [[1141: Two Years]] and the second time it was [[1350:_Lorenz#Knit_Cap|Knit Cap]] in [[1350: Lorenz]].

==Transcript==
:[Knit Cap is sitting down at a computer touching the keyboard with one hand. Cueball is standing behind watching the screen.]
:Laptop: ''*Move*''
:Cueball: Why'd you move your knight away?

:[Knit Cap turns around and rests an arm on the chair looking at Cueball who holds out both arms.]
:Cueball: Just think ''logically''. The goal is checkmate, so you should always move pieces ''toward'' the other player's king.

:[Closeup of Cueball holding a hand to his chin.]
:Cueball: I guess occasionally you need to move backward, but it'd be trivial to make a list of those circumstances and-

:[Knit Cap is leaning back in chair facing Cueball, panel is so slim that the lap top is not included.]
:Knit Cap: Have you ever ''played'' chess?
:Cueball: Not much, but—
:Knit Cap: Wanna?
:Cueball: Uh, ok.

:[Knit Cap sitting and Cueball standing is playing chess with a board standing between them on a very small table or a four legged stool. The board extends quite far out on either side. Their moves are indicated above with four by Knit Cap and three towards Cueball. It is clear both from this and from the pieces visible on the board that Knit Cap is playing white]
:<nowiki>*Move*</nowiki>
: _____<nowiki>*Move*</nowiki>
:<nowiki>*Move*</nowiki>
: _____<nowiki>*Move*</nowiki>
:<nowiki>*Move*</nowiki>
: _____<nowiki>*Move*</nowiki>
:<nowiki>*Move*</nowiki>
:Knit Cap: Checkmate.

:[In a frameless panel Cueball is standing staring at the chess board, where there clearly are more pieces on his side of the board.]

:[Knit Cap has turned back to the laptop with both hands on the keyboard. Cueball is standing behind the chess board holding a finger up in the air.]
:Cueball: This game isn't very well-designed. For starters, knights are too weak...

{{comic discussion}}

[[Category:Comics featuring Cueball]]
[[Category:Comics featuring Knit Cap]]
[[Category:Chess]]

1112: Think Logically

2023-08-23T07:33:00Z

ISaveXKCDpapers: caveat

{{comic
| number = 1112
| date = September 24, 2012
| title = Think Logically
| image = think logically.png
| titletext = I've developed a more logical set of rules but the people on the chess community have a bunch of stupid emotional biases and won't reply to my posts.
}}

==Explanation==
{{w|Chess}} is a centuries-old board game in which two players take turns moving one of their 16 pieces to try and checkmate the other player's king (one of the pieces). When one player is in a position to capture their opponent's king on their next move, and the opponent has no legal move available to avoid such capture, the opponent is said to be in "checkmate", and by definition immediately loses.

The game, with origins around the sixth century, and with the modern rules being essentially set in the late 15th century, has a significant amount of history. The rules and traditions are well established. The knight is a piece that can only move in an L-shaped pattern (two squares in one direction, and one square perpendicular), but has the unique ability to jump over other pieces.

The comic highlights two mistakes players often make in chess: complete fixation on the king at the cost of their other pieces, and failure to take advantage of the knight's movement patterns. At the same time this is a jab at how people sometimes oversimplify an argument when confronted with a topic they are not familiar with. Previously this was depicted in [[675: Revolutionary]] and [[793: Physicists]]. See also the {{w|Dunning–Kruger effect}}. The units in chess are widely agreed to be well-balanced, and [[Cueball|Cueball's]] criticism of the knight shows an obvious lack of knowledge of the knight's potential.

Given the long history of chess, a significant amount of writing and research has been dedicated to the game and its strategies. This is inadvertently mocked by Cueball who naively suggests it would be trivial to make a list of all situations in which a piece would move backwards (called a "retreat" in chess). Such a list — at least a partial one — certainly does exist, as do lists of numerous other chess moves and situations.

[[Knit Cap]] proceeds to demonstrate Cueball's lack of knowledge by beating him in four moves, which typically would only occur when an experienced player plays a novice. The checkmate depicted is the {{w|Scholar's mate|scholar's mate}}, being a classic early-game checkmate in chess. It is in fact extremely easy to defend against it (blocking the queen's vision of f7, without letting the queen take any of your pieces, would do), thus proving Cueball's inexperience. Careful scrutiny of the board suggests a scholar's mate, something along these lines (using chess algebraic notation): 1.e4 e5 2. Bc4 Nc6 3. Qh5 Nf6 4. Qxf7#.

Cueball, instead of admitting he underestimated the game, believes the failure is in the game itself. The title text indicates that Cueball attempted to suggest revisions to the rules of chess. Given that Cueball has no experience as a chess player, it is likely many of the changes are illogical or ridiculous. In the face of hundreds of years of history, it is not surprising that the chess community is ignoring them. The last major changes to the rules of chess occurred more than 400 years ago when, among other things, the pawn was given its two-space starting move and the queen was made into the most powerful piece (previously it was the weakest). The chess community's ties to the traditions of the game and their refusal to accept Cueball's suggestions are written off by Cueball as "{{w|emotional bias}}" suggesting his changes are logical, but that the community is letting their emotions cloud their rational decision making abilities, while in reality it is he who is being affected. If that can make Cueball feel any better, it could be pointed out to him that dozens and dozens of {{w|Chess variants|chess variants}} do exist out there.

The comic may also be a jab at competitive online games whose fans call for "buffs" (power additions) and "nerfs" (power reductions) to characters they believe to be underpowered or overpowered, often with inadequate knowledge of those characters. On the other hand, some online games and multiplayer computer games in general are unbalanced since they lack centuries of history to balance themselves, unlike chess.

[[Knit Cap]] is called ''knit hat guy'' in the [http://xkcd.com/1112/info.0.json official transcript]. There are two other cases (after this comic) where a person with hair has been shown with a knit cap. The first was [[Randall|Randall's]] wife after chemotherapy in [[1141: Two Years]] and the second time it was [[1350:_Lorenz#Knit_Cap|Knit Cap]] in [[1350: Lorenz]].

==Transcript==
:[Knit Cap is sitting down at a computer touching the keyboard with one hand. Cueball is standing behind watching the screen.]
:Laptop: ''*Move*''
:Cueball: Why'd you move your knight away?

:[Knit Cap turns around and rests an arm on the chair looking at Cueball who holds out both arms.]
:Cueball: Just think ''logically''. The goal is checkmate, so you should always move pieces ''toward'' the other player's king.

:[Closeup of Cueball holding a hand to his chin.]
:Cueball: I guess occasionally you need to move backward, but it'd be trivial to make a list of those circumstances and-

:[Knit Cap is leaning back in chair facing Cueball, panel is so slim that the lap top is not included.]
:Knit Cap: Have you ever ''played'' chess?
:Cueball: Not much, but—
:Knit Cap: Wanna?
:Cueball: Uh, ok.

:[Knit Cap sitting and Cueball standing is playing chess with a board standing between them on a very small table or a four legged stool. The board extends quite far out on either side. Their moves are indicated above with four by Knit Cap and three towards Cueball. It is clear both from this and from the pieces visible on the board that Knit Cap is playing white]
:<nowiki>*Move*</nowiki>
: _____<nowiki>*Move*</nowiki>
:<nowiki>*Move*</nowiki>
: _____<nowiki>*Move*</nowiki>
:<nowiki>*Move*</nowiki>
: _____<nowiki>*Move*</nowiki>
:<nowiki>*Move*</nowiki>
:Knit Cap: Checkmate.

:[In a frameless panel Cueball is standing staring at the chess board, where there clearly are more pieces on his side of the board.]

:[Knit Cap has turned back to the laptop with both hands on the keyboard. Cueball is standing behind the chess board holding a finger up in the air.]
:Cueball: This game isn't very well-designed. For starters, knights are too weak...

{{comic discussion}}

[[Category:Comics featuring Cueball]]
[[Category:Comics featuring Knit Cap]]
[[Category:Chess]]

1112: Think Logically

2023-08-23T07:31:39Z

ISaveXKCDpapers: fixed explanation of how to stop the scholar's mate

{{comic
| number = 1112
| date = September 24, 2012
| title = Think Logically
| image = think logically.png
| titletext = I've developed a more logical set of rules but the people on the chess community have a bunch of stupid emotional biases and won't reply to my posts.
}}

==Explanation==
{{w|Chess}} is a centuries-old board game in which two players take turns moving one of their 16 pieces to try and checkmate the other player's king (one of the pieces). When one player is in a position to capture their opponent's king on their next move, and the opponent has no legal move available to avoid such capture, the opponent is said to be in "checkmate", and by definition immediately loses.

The game, with origins around the sixth century, and with the modern rules being essentially set in the late 15th century, has a significant amount of history. The rules and traditions are well established. The knight is a piece that can only move in an L-shaped pattern (two squares in one direction, and one square perpendicular), but has the unique ability to jump over other pieces.

The comic highlights two mistakes players often make in chess: complete fixation on the king at the cost of their other pieces, and failure to take advantage of the knight's movement patterns. At the same time this is a jab at how people sometimes oversimplify an argument when confronted with a topic they are not familiar with. Previously this was depicted in [[675: Revolutionary]] and [[793: Physicists]]. See also the {{w|Dunning–Kruger effect}}. The units in chess are widely agreed to be well-balanced, and [[Cueball|Cueball's]] criticism of the knight shows an obvious lack of knowledge of the knight's potential.

Given the long history of chess, a significant amount of writing and research has been dedicated to the game and its strategies. This is inadvertently mocked by Cueball who naively suggests it would be trivial to make a list of all situations in which a piece would move backwards (called a "retreat" in chess). Such a list — at least a partial one — certainly does exist, as do lists of numerous other chess moves and situations.

[[Knit Cap]] proceeds to demonstrate Cueball's lack of knowledge by beating him in four moves, which typically would only occur when an experienced player plays a novice. The checkmate depicted is the {{w|Scholar's mate|scholar's mate}}, being a classic early-game checkmate in chess. It is in fact extremely easy to defend against it (blocking the queen's vision of f7 would do), thus proving Cueball's inexperience. Careful scrutiny of the board suggests a scholar's mate, something along these lines (using chess algebraic notation): 1.e4 e5 2. Bc4 Nc6 3. Qh5 Nf6 4. Qxf7#.

Cueball, instead of admitting he underestimated the game, believes the failure is in the game itself. The title text indicates that Cueball attempted to suggest revisions to the rules of chess. Given that Cueball has no experience as a chess player, it is likely many of the changes are illogical or ridiculous. In the face of hundreds of years of history, it is not surprising that the chess community is ignoring them. The last major changes to the rules of chess occurred more than 400 years ago when, among other things, the pawn was given its two-space starting move and the queen was made into the most powerful piece (previously it was the weakest). The chess community's ties to the traditions of the game and their refusal to accept Cueball's suggestions are written off by Cueball as "{{w|emotional bias}}" suggesting his changes are logical, but that the community is letting their emotions cloud their rational decision making abilities, while in reality it is he who is being affected. If that can make Cueball feel any better, it could be pointed out to him that dozens and dozens of {{w|Chess variants|chess variants}} do exist out there.

The comic may also be a jab at competitive online games whose fans call for "buffs" (power additions) and "nerfs" (power reductions) to characters they believe to be underpowered or overpowered, often with inadequate knowledge of those characters. On the other hand, some online games and multiplayer computer games in general are unbalanced since they lack centuries of history to balance themselves, unlike chess.

[[Knit Cap]] is called ''knit hat guy'' in the [http://xkcd.com/1112/info.0.json official transcript]. There are two other cases (after this comic) where a person with hair has been shown with a knit cap. The first was [[Randall|Randall's]] wife after chemotherapy in [[1141: Two Years]] and the second time it was [[1350:_Lorenz#Knit_Cap|Knit Cap]] in [[1350: Lorenz]].

==Transcript==
:[Knit Cap is sitting down at a computer touching the keyboard with one hand. Cueball is standing behind watching the screen.]
:Laptop: ''*Move*''
:Cueball: Why'd you move your knight away?

:[Knit Cap turns around and rests an arm on the chair looking at Cueball who holds out both arms.]
:Cueball: Just think ''logically''. The goal is checkmate, so you should always move pieces ''toward'' the other player's king.

:[Closeup of Cueball holding a hand to his chin.]
:Cueball: I guess occasionally you need to move backward, but it'd be trivial to make a list of those circumstances and-

:[Knit Cap is leaning back in chair facing Cueball, panel is so slim that the lap top is not included.]
:Knit Cap: Have you ever ''played'' chess?
:Cueball: Not much, but—
:Knit Cap: Wanna?
:Cueball: Uh, ok.

:[Knit Cap sitting and Cueball standing is playing chess with a board standing between them on a very small table or a four legged stool. The board extends quite far out on either side. Their moves are indicated above with four by Knit Cap and three towards Cueball. It is clear both from this and from the pieces visible on the board that Knit Cap is playing white]
:<nowiki>*Move*</nowiki>
: _____<nowiki>*Move*</nowiki>
:<nowiki>*Move*</nowiki>
: _____<nowiki>*Move*</nowiki>
:<nowiki>*Move*</nowiki>
: _____<nowiki>*Move*</nowiki>
:<nowiki>*Move*</nowiki>
:Knit Cap: Checkmate.

:[In a frameless panel Cueball is standing staring at the chess board, where there clearly are more pieces on his side of the board.]

:[Knit Cap has turned back to the laptop with both hands on the keyboard. Cueball is standing behind the chess board holding a finger up in the air.]
:Cueball: This game isn't very well-designed. For starters, knights are too weak...

{{comic discussion}}

[[Category:Comics featuring Cueball]]
[[Category:Comics featuring Knit Cap]]
[[Category:Chess]]

2807: Bad Map Projection: ABS(Longitude)

2023-07-27T01:53:44Z

ISaveXKCDpapers: spelling

{{comic
| number = 2807
| date = July 26, 2023
| title = Bad Map Projection: ABS(Longitude)
| image = bad_map_projection_abs_longitude_2x.png
| imagesize = 740x822px
| noexpand = true
| titletext = Positive vibes/longitudes only
}}

==Explanation==
{{incomplete|Created by both eErAeShTpEsRiNm eHhE MnIrSePtHsEeRwE - Please change this comment when editing this page. Do NOT delete this tag too soon.}}

In this map, Randall has plotted the world map featuring all the landmasses from both western and eastern hemispheres. But the longitudes west of the prime meridian, normally given negative values from the range from -180° to +180°, have been made positive using the "ABS()" function that gives the absolute value by effectively stripping off the minus sign from any value. This results in the features on one side of the world being overlaid upon those of the other side, but reversed.

Thanks to the relative sparsity of western continents (essentially just the Americas, and roughly half of Antarctica), and the landless expanses of the Pacific, this is surprisingly not too dissimilar to an east-only fragment of a world map. Just with the reversed 'new world' lands added to the usual extents of continents of Africa and Eurasia and the somewhat familiar notable Antarctic Peninsula being recognisable but in an odd position (and reversed) with no sign of the Ross and Weddel Seas (as eastern-Antarctic landmass takes precedence).

To further interest the map-connoisseur, various locations are marked and dotted upon their genuine or reflected positions. Putting into close proximity various locations that have (mainly) trans-Atlantic separation in reality.

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}

{{comic discussion}}

[[Category:Bad_Map_Projections]]