https://www.explainxkcd.com/wiki/api.php?action=feedcontributions&feedformat=atom&user=162.158.133.138explain xkcd - User contributions [en]2026-05-25T14:58:25ZUser contributionsMediaWiki 1.30.0https://www.explainxkcd.com/wiki/index.php?title=1757:_November_2016&diff=1304491757: November 20162016-11-09T14:17:41Z<p>162.158.133.138: 2016-41=1975</p>
<hr />
<div>{{comic<br />
| number = 1757<br />
| date = November 9, 2016<br />
| title = November 2016<br />
| image = november_2016.png<br />
| titletext = Once you've done this, make a note of how old they were. Then, when their age reaches double that, show them this chart again.<br />
}}<br />
<br />
==Explanation==<br />
{{incomplete|Need individual explanations.}}<br />
This is yet another comic designed to [[Feel old|make you feel old]]. It lists ages between 16 and 41, and a major event that happened a little over half their life ago. So, for an age of 16 (people born in 2000), it lists the release of GTA IV, which happened in 2008. Thus, GTA IV has been around for the majority of their life. The joke at the end is that people over 41 don't need anything to make them feel old, because they are in fact old and thus already feel so.<br />
<br />
The titletext points out that the same chart can be used for the same person much later in their life. However, the major event shifts earlier and earlier into their life; when their age has doubled, the event in the chart has happend in the year of their birth.<br />
year of event(age) = 2016 - age/2<br />
year of event(2*age) = 2016 - (2* age)/2 = 2016-age = year of birth<br />
<br />
===Table===<br />
{| class = "wikitable"<br />
! Age<br />
! Birthyear<br />
! Release date<br />
! Thing<br />
! Explanation<br />
|-<br />
| 16<br />
| 2000<br />
| April 29, 2008<br />
| Grand Theft Auto IV<br />
|<br />
|-<br />
| 17<br />
| 1999<br />
| May 2007<br />
| Rickrolling<br />
|<br />
|-<br />
| 18<br />
| 1998<br />
|<br />
| Aqua Teen Hunger Force, Colon Movie Film for Theaters<br />
|<br />
|-<br />
| 19<br />
| 1997<br />
| November 19, 2006<br />
| The Nintendo Wii<br />
|<br />
|-<br />
| 20<br />
| 1996<br />
| March 2006<br />
| Twitter<br />
|<br />
|-<br />
| 21<br />
| 1995<br />
| November 22, 2005 (Xbox), <br />
| The Xbox 360, xkcd<br />
|<br />
|-<br />
| 22<br />
| 1994<br />
|<br />
| Chuck Norris Facts<br />
|<br />
|-<br />
| 23<br />
| 1993<br />
| January 25, 2004<br />
| Opportunity's Mars Exploration<br />
|<br />
|-<br />
| 24<br />
| 1992<br />
| February 4, 2004<br />
| Facebook<br />
|<br />
|-<br />
| 25<br />
| 1991<br />
| April 1, 2004 (Gmail), July 9, 2003 (Pirates of the Caribbean)<br />
| Gmail, Pirates of the Caribbean<br />
| <br />
|-<br />
| 26<br />
| 1990<br />
|<br />
| In da Club<br />
|<br />
|-<br />
| 27<br />
| 1989<br />
| September 20, 2002<br />
| Firefly<br />
|<br />
|-<br />
| 28<br />
| 1988<br />
|<br />
| The War in Afghanistan<br />
|<br />
|-<br />
| 29<br />
| 1987<br />
| 2001<br />
| The iPod<br />
|<br />
|-<br />
| 30<br />
| 1986<br />
|<br />
| Shrek, Wikipedia<br />
|<br />
|-<br />
| 31<br />
| 1985<br />
|<br />
| Those X-Men movies<br />
|<br />
|-<br />
| 32<br />
| 1984<br />
|<br />
| The Sims<br />
|<br />
|-<br />
| 33<br />
| 1983<br />
|<br />
| Autotuned hit songs<br />
|<br />
|-<br />
| 34<br />
| 1982<br />
|<br />
| The Star Wars Prequels<br />
|<br />
|-<br />
| 35<br />
| 1981<br />
|<br />
| The Matrix<br />
|<br />
|-<br />
| 36<br />
| 1980<br />
| 1998 (outside Japan)<br />
| Pok&eacute;mon Red & Blue<br />
|<br />
|-<br />
| 37<br />
| 1979<br />
|<br />
| Netflix, Harry Potter, Google<br />
|<br />
|-<br />
| 38<br />
| 1978<br />
|<br />
| Deep Blue's Victory<br />
|<br />
|-<br />
| 39<br />
| 1977<br />
|<br />
| Tupac's Death<br />
|<br />
|-<br />
| 40<br />
| 1976<br />
|<br />
| The last Calvin and Hobbes strip<br />
|<br />
|-<br />
| 41<br />
| 1975<br />
|<br />
| Toy Story<br />
|<br />
|-<br />
| >41<br />
| Before 1975<br />
| n/a<br />
| [Don't worry, they've got this covered]<br />
|<br />
|}<br />
<br />
==Transcript==<br />
<br />
The November 2016<br />
Guide to making people<br />
feel old<br />
<br />
[In rectangle]<br />
If they're [age], you say:<br />
"Did you know <u> [thing] </u> has been around for the majority of your life?"<br />
<br />
{| class = "wikitable"<br />
! <u>Age</u><br />
! <u>Thing</u><br />
|-<br />
| 16<br />
| Grand Theft Auto IV<br />
|-<br />
| 17<br />
| Rickrolling<br />
|-<br />
| 18<br />
| <i>Aqua Teen Hunger Force <br> Colon Movie Film for Theaters</i><br />
|-<br />
| 19<br />
| The Nintendo Wii<br />
|-<br />
| 20<br />
| Twitter<br />
|-<br />
| 21<br />
| The Xbox 360, xkcd<br />
|-<br />
| 22<br />
| Chuck Norris Facts<br />
|-<br />
| 23<br />
| Opportunity's Mars Exploration<br />
|-<br />
| 24<br />
| Facebook<br />
|-<br />
| 25<br />
| Gmail, <i>Pirates of the Caribbean</i><br />
|-<br />
| 26<br />
| In da Club<br />
|-<br />
| 27<br />
| <i>Firefly</i><br />
|-<br />
| 28<br />
| The War in Afghanistan<br />
|-<br />
| 29<br />
| The iPod<br />
|-<br />
| 30<br />
| <i>Shrek</i>, Wikipedia<br />
|-<br />
| 31<br />
| Those X-Men movies<br />
|-<br />
| 32<br />
| <i>The Sims</i><br />
|-<br />
| 33<br />
| Autotuned hit songs<br />
|-<br />
| 34<br />
| The <i>Star Wars</i> Prequels<br />
|-<br />
| 35<br />
| <i>The Matrix</i><br />
|-<br />
| 36<br />
| <i>Pok&eacute;mon Red & Blue</i><br />
|-<br />
| 37<br />
| Netflix, <i>Harry Potter</i>, Google<br />
|-<br />
| 38<br />
| Deep Blue's Victory<br />
|-<br />
| 39<br />
| Tupac's Death<br />
|-<br />
| 40<br />
| The last <i>Calvin and Hobbes</i> strip<br />
|-<br />
| 41<br />
| <i>Toy Story</i><br />
|-<br />
| >41<br />
| [Don't worry, they've got this covered]<br />
|}<br />
{{comic discussion}}<br />
<br />
[[Category:Comics to make one feel old]]</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=Talk:1724:_Proofs&diff=125891Talk:1724: Proofs2016-08-29T14:35:07Z<p>162.158.133.138: </p>
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<div>Judging from my experience when I first encountered proofs in math classes (or my general experience from math classes), the teacher is going to write down a "proof" which makes absolutely no sense to students and is also never explained in a way that actually makes them understand. Instead, they are just going to use "dark magic" and write what seems to be completely senseless to students.<br />
[[Special:Contributions/141.101.91.223|141.101.91.223]] 04:24, 24 August 2016 (UTC)<br />
:: 'Dark magic' might also refer to the supernatural, so when the teacher said that an answer 'will be written' in a specific location, Cueball took this to mean that a spirit would be summoned to write it, like a ouija chalk board. [[Special:Contributions/141.101.70.67|141.101.70.67]] 09:27, 25 August 2016 (UTC)<br />
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Transcript generated by the BOT was murdering me, had to change it. Proposing miss Lenhart is party 1. [[User:EppOch|EppOch]] ([[User talk:EppOch|talk]]) 04:45, 24 August 2016 (UTC)<br />
:: I support that. [[Special:Contributions/141.101.91.223|141.101.91.223]] 06:13, 24 August 2016 (UTC)<br />
:: Me to, but I am on mobile, so editing is a pain [[Special:Contributions/162.158.86.71|162.158.86.71]] 06:51, 24 August 2016 (UTC)<br />
:: Done [[User:Elektrizikekswerk|Elektrizikekswerk]] ([[User talk:Elektrizikekswerk|talk]]) 08:26, 24 August 2016 (UTC)<br />
:::Note that the BOT doesn't create any text - [http://www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&oldid=125654 see here]. The transcript was made by several people. Agree completely that this is Miss Lenhart, but even if it was not "[http://www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&direction=next&oldid=125660 party 1 and party 2]" is not the way to describe a woman with long blonde hair and Cueball ;-) There is at the moment [[explain_xkcd:Community_portal/Proposals#New_character_category_for_blonde_woman_news_reporter_.28from_1699.29|a discussion]] what to call other women looking like this (i.e. those that are not clearly Miss Lenhart, [[Mrs. Roberts]] or her daughter [[Elaine Roberts]]). Chip in there if you have any opinions on that regard... --[[User:Kynde|Kynde]] ([[User talk:Kynde|talk]]) 11:01, 24 August 2016 (UTC)<br />
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<br />
Irrationality proof isn't really a proof by contradiction (it doesn't use double negation elimination). You're showing (exists a,b. ...) -> False by assuming (exists a, b. ...) and showing False, which is implication introduction --[[Special:Contributions/162.158.85.105|162.158.85.105]] 07:33, 24 August 2016 (UTC)<br />
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I'm thinking she's doing one of those proof that write down a formula or function out of nowhere, and proceeds to proof everything with it. [[Special:Contributions/108.162.222.125|108.162.222.125]] 08:43, 24 August 2016 (UTC)<br />
<br />
This comic reminds me of "divination" rituals, where a magical spirit is summoned to write out an answer. Usually not something as complex as here, but hey, XKCD! --[[User:Henke37|Henke37]] ([[User talk:Henke37|talk]]) 10:04, 24 August 2016 (UTC)<br />
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Man, Reductio ad absurdum never made any logic. If we could assume any thing, why use logic?<br />
Oh wait, it has already been covered in XKCD {{unsigned ip|162.158.49.12}}<br />
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"Dark magic" proofs are centered around properties of functions, and abstract concepts, rather than manipulating the functions themselves?? [[Special:Contributions/108.162.246.113|108.162.246.113]] 11:26, 24 August 2016 (UTC)<br />
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My assumptions is that the "Dark Magic" being referred to here is more "A technique that works, though nobody really understands why." [see http://catb.org/jargon/html/B/black-magic.html] In this case, the teacher is setting up a proof in an manner which will lead to the desired goal, but to the student it is exceedingly unobvious as to why one would do it this way, other than "it works" [[Special:Contributions/108.162.219.52|108.162.219.52]] 15:30, 24 August 2016 (UTC)<br />
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I was thinking that a "dark magic proof" referred to those ridiculous "party trick" proofs like 'proving' that 1 = 0 via some confusing train of logic, and mathematical sleight of hand. {{unsigned ip|108.162.237.213}}<br />
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Maybe he meant "dark patterns"? {{unsigned ip|162.158.126.139}}<br />
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It seems pretty obvious to me that by "weird, dark magic proofs", the student is talking about proofs that drag in far-flung reaches of mathematics so distant that they no longer appear to be mathematics, especially ones that involve meta-reasoning. Gödel's proof of the incompleteness of Peano arithmetic is the archetypical example, but others include Lob's theorem and any proof by contradiction involving the halting problem. Ms Lenhart's proof starts out by setting up a proof-by-contradiction, already a warning sign, and she then escalates it at the end by implying that this proof will somehow involve the actual physics of where the solution can and cannot be written. [[Special:Contributions/108.162.241.123|108.162.241.123]] 17:27, 24 August 2016 (UTC)<br />
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:: Agreed, although I think starting out with a proof by contradiction setup is by itself not that much of a warning sign. However it heads straight into meta-space by making the assumption of the existence of a function that produces a solution of something. [[User:Zmatt|Zmatt]] ([[User talk:Zmatt|talk]]) 18:52, 24 August 2016 (UTC)<br />
<br />
:: The fact that the proof mentions the actual blackboard on which it is written is of course problematic in numerous ways, as is predicating on whether something "will eventually" happen. This is well outside the scope of the [https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory usual mathematical foundations]. Since careless use of meta-recursion is a [https://en.wikipedia.org/wiki/Curry's_paradox trap], such a proof would have to very very carefully consider foundational issues and cannot handwave over them. [[User:Zmatt|Zmatt]] ([[User talk:Zmatt|talk]]) 19:13, 24 August 2016 (UTC)<br />
<br />
----<br />
"''In the title text the decision of whether to take the axiom of choice is made by a deterministic process. The axiom of determinacy is incompatible with the axiom of choice...''" The axiom of determinacy is not really relevant to deterministic processes - it is about (certain types of two-players-) games and says that any such game is determined (that is, some player has a winning strategy). So this axiom is not relevant to the title text --[[Special:Contributions/162.158.83.66|162.158.83.66]] 17:39, 24 August 2016 (UTC)<br />
:I agree. I read the title text in almost exactly the opposite way - that the proof relies on the existence of a deterministic process for selecting objects, and therefore the invocation of the axiom of choice as a part of the process is superfluous (but not a contradiction). Anyhow, the axiom of determinacy isn't ever mentioned, so it probably shouldn't be shoehorned in here. [[Special:Contributions/162.158.74.53|162.158.74.53]] 20:36, 24 August 2016 (UTC)<br />
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I feel like it is a stretch to assert Lenhart is setting up a proof by contradiction. It sounded to me more like an prior knowledge proof (not sure it's technical name). For example, "calculate the space between two concentric circles of differing diameter when the longest straight line you can draw is length d." If you assume there is a function F(r1, r2) which has been previously proven to calculate this space, then it is easy to show that the space is in fact .5*pi*(.5*d)^2 (as you have a degenerative case where r1=0, and you have an ordinary circle). I also think this type of proof is more "dark magic"-feeling than a simple proof by contradiction. {{unsigned ip|108.162.216.87}}<br />
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:While technically the same pattern, I would assume something more like NP-complete proofs: Assume we have function F which solves this problem in polynomial time ... then we can solve that problem in polynomial time as well. Just, instead of "polynomial time", the existence of function is the question here, so it will likely be something around {{w|Recursively enumerable set|recursively enumerable}}/{{w|countable set|countable}} stuff. -- [[User:Hkmaly|Hkmaly]] ([[User talk:Hkmaly|talk]]) 13:02, 25 August 2016 (UTC)<br />
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I don't like how this explanation uses the word "standard". Non-standard mathematical objects are subjects of non-standard analysis, not metamathematics. --[[Special:Contributions/108.162.218.185|108.162.218.185]] 02:25, 27 August 2016 (UTC)<br />
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Simplest explanation would be Cueball suspect Ms Lenhart already made-up an answer for a made-up function (hence ''magic''), which is confirmed at the last panel. Laymen like myself wouldn't grasp any of those methamathematical stuff explanation. :) [[Special:Contributions/162.158.167.35|162.158.167.35]] 07:20, 29 August 2016 (UTC)<br />
: There is no such thing, like "answer for a function", so you can't be right. And this interpretation is completely ignoring the mathematical similarities, yet it was [http://www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=125861&oldid=125829 introduced] as a summary of the mathematical explanation. If you don't grasp the idea, don't try to summarize it, please. [[Special:Contributions/162.158.133.138|162.158.133.138]] 14:35, 29 August 2016 (UTC)</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=1258891724: Proofs2016-08-29T14:29:24Z<p>162.158.133.138: no, it's wrong summary</p>
<hr />
<div>{{comic<br />
| number = 1724<br />
| date = August 24, 2016<br />
| title = Proofs<br />
| image = proofs.png<br />
| titletext = Next, let's assume the decision of whether to take the Axiom of Choice is made by a deterministic process ...<br />
}}<br />
<br />
==Explanation==<br />
[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those dark-magic (unclear, incomprehensible) proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.<br />
<br />
The proof she starts setting up resembles a {{w|proof by contradiction}}. This kind of proof assumes that a particular condition is true, and shows that this assumption leads to a contradiction, which disproves the initial assumption. For example assumption that √2 is a {{w|rational number}} means that, for some natural ''a'' and ''b'', √2=''a/b'', where ''a/b'' is an {{w|irreducible fraction}}. Yet, multiplying this equation by itself, we get 2=''a²/b²'' which in turn rearranges to 2''b²''=''a²''. Therefore ''a²'' is even (as any integer multiplied by 2 is even), which means that ''a'' is an even number, as an even number squared is always even and an odd number squared is always odd. This means, that ''a=2k'' and ''2b²=(2k)²=4k²'', meaning ''b²''=2''k²'', so ''b'' must be even too. But if both ''a'' and ''b'' are even, ''a/b'' cannot be irreducible. Contradiction means that the initial assumption is false, and √2 cannot be a rational number.<br />
<br />
Alternatively, instead of a proof by contradiction the setup could be for a one way function. For example, it is relatively easy to test that a solution to a differential equation is valid but choosing the correct solution to test can seem like black magic to students.<br />
<br />
The way that Ms Lenhart's proof refers to the act of doing math itself, is characteristic of metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if in the end they must be a "perfectly sensible chain of reasoning" like the rest of good mathematics. While typical mathematical theorems and their proofs deal with such mathematical objects as numbers, functions, points or lines, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but can be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken as a part of the axiomatic system.<br />
<br />
Using a position on the blackboard as a part of the proof is a joke, but it bears a resemblance to {{w|Cantor's diagonal argument}} where a position in a sequence of digits of a real number was a tool in a proof that not all infinite sets have the same {{w|cardinality}} (rough equivalent of the number of elements). This "diagonal method" is also often used in metamathematical proofs.<br />
<br />
The axiom of choice itself states that for every collection of nonempty sets, you can have a function that draws one element from each set of the collection. This axiom, once considered controversial, was added relatively late to the axiomatic set theory, and even contemporary mathematicians still study which theorems really require its inclusion. In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. {{w|Determinacy}} of infinite games is used as a tool in the set theory, however the deterministic process is rather a term of the {{w|stochastic process|stochastic processes theory}}, and the {{w|dynamical systems theory}}, branches of mathematics far from the abstract set theory, which makes the proof even more exotic. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].<br />
<br />
Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in her prior class.<br />
<br />
==Transcript==<br />
:[Miss Lenhart is standing facing left in front of a whiteboard writing on it. Eleven left aligned lines of writing is shown as unreadable scribbles. A voice interrupts her from off-panel right.]<br />
:Miss Lenhart: ... Let's assume there exists some function ''F''(''a,b,c''...) which produces the correct answer-<br />
:Cueball (off-panel): Hang on.<br />
<br />
:[In a frame-less panel Cueball is sitting on a chair at a desk with a pen in his hand taking notes.]<br />
:Cueball: This is going to be one of those weird, dark magic proofs, isn't it? I can tell.<br />
<br />
:[Miss Lenhart has turned right towards Cueball, who is again speaking off-panel. The white board is also off-panel.]<br />
:Miss Lenhart: What? No, no, it's a perfectly sensible chain of reasoning.<br />
:Cueball (off-panel): All right...<br />
<br />
:[Miss Lenhart is facing the whiteboard again writing more scribbles behind some of the lines from before (the first line has disappeared). The lines that have more text added are now number three and five (four and six before). Cueball again speaks off-panel.]<br />
:Miss Lenhart: Now, let's assume that the correct answer will eventually be written on the board at the coordinates (''x, y''). If we—<br />
:Cueball (off-panel): I ''knew'' it!<br />
<br />
{{comic discussion}}<br />
<br />
[[Category:Comics featuring Miss Lenhart]]<br />
[[Category:Comics featuring Cueball]]<br />
[[Category:Math]]</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=1725:_Linear_Regression&diff=1258881725: Linear Regression2016-08-29T13:57:55Z<p>162.158.133.138: </p>
<hr />
<div>{{comic<br />
| number = 1725<br />
| date = August 26, 2016<br />
| title = Linear Regression<br />
| image = linear_regression.png<br />
| titletext = The 95% confidence interval suggests Rexthor's dog could also be a cat, or possibly a teapot.<br />
}}<br />
<br />
==Explanation==<br />
{{w|Linear regression}} is a method for modeling the relationship between multiple variables. In the simplest case, it can be used for two variables wherein the model determines a "{{w|least squares|best-fit}}" line through a {{w|scatter plot}} of the datasets, together with a {{w|coefficient of determination}}, usually denoted ''r''<sup>2</sup> or ''R''<sup>2</sup>. When only two variables are included in the regression, ''R''<sup>2</sup> is merely the square of the correlation between the two variables. ''R''<sup>2</sup> is a number between 0 and 1 that indicates how well one variable can be used to predict the value of another. A value of 1 means perfect correlation, while a value close to 0 indicates a weak relationship between the variables.<br />
<br />
{{w|Asterism_(astronomy)|Asterism}}s are patterns created by linking the apparent positions of stars as seen in the sky from Earth. Strictly, "Rexthor" is an asterism, as a {{w|constellation}} is the region of sky containing the asterism, although "constellation" is used informally in place of "asterism" by even seasoned astronomers. Different civilizations have recognized different constellations (the modern IAU, for example, lists 88 "official" constellations), and one could create their own constellations by connecting assorted points.<br />
<br />
In this comic, a set of data has had linear regression and some form of statistical analysis applied to it, indicating that there is low correlation between the two. The data points are so widely scattered that (as noted in the comic) it is easier to connect the data points in a constellation-like pattern than it is to determine whether the correlation is negative or positive (without looking at the trendline, of course). Because of this, [[Randall]] suggests we should be suspicious of any conclusions drawn from this data.<br />
<br />
"Rexthor the Dog bearer" seems to be a spoof on Thor, a Norse god who wields a hammer. <br />
By replacing his hammer with a dog and adding "Rex" (an archetypal dog name), Randall creates a comical, dog-bearing version of Thor. <br />
<br />
The 95% {{w|confidence interval}} in statistics is such a range of an estimate, that the probability of the real value (the estimated population parameter) to lie inside the range is at least 95%. The confidence interval is a standard method to provide evaluation of the estimation error in statistics. On the right panel the resulting estimate seems to be a drawing, so the 95% confidence interval would be a set of all drawings derived from the sample such that the probability of the right drawing to be among them is at least 95%. According to the title text among these drawings you can find a cat and a teapot as well, so we can't be 95% confident that a dog exists in the data.<br />
<br />
The teapot may be a reference to {{w|Russell's_teapot|Russell's teapot}}, or possibly to the "teapot" asterism in the constellation Sagittarius.<br />
<br />
==Transcript==<br />
:[Two square panels show identical sets of scattered black dots, with only the red additions being different.]<br />
<br />
:[The left panel shows a slightly rising red line drawn through the middle of the panel, passing near a few dots but not obviously related to most of them. A red text is below the dots:]<br />
:<span style="color:red">R<sup>2</sup>=0.06</span><br />
<br />
:[The right panel shows many of the dots connected by red lines to form a stick figure of a man resembling the constellation Orion, with the hand on the reader's right raised and holding an object. A red text is below the dots:]<br />
:<span style="color:red">Rexthor, the Dog-Bearer</span><br />
<br />
:[A caption is below and spanning both panels:] <br />
:I don't trust linear regressions when it's harder to guess the direction of the correlation from the scatter plot than to find new constellations on it.<br />
<br />
{{comic discussion}}<br />
<br />
[[Category:Comics with color]]<br />
[[Category:Scatter plots]]<br />
[[Category:Astronomy]]<br />
[[Category:Animals]]</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=1725:_Linear_Regression&diff=1258871725: Linear Regression2016-08-29T13:56:40Z<p>162.158.133.138: /* Explanation */</p>
<hr />
<div>{{comic<br />
| number = 1725<br />
| date = August 26, 2016<br />
| title = Linear Regression<br />
| image = linear_regression.png<br />
| titletext = The 95% confidence interval suggests Rexthor's dog could also be a cat, or possibly a teapot.<br />
}}<br />
<br />
==Explanation==<br />
{{w|Linear regression}} is a method for modeling the relationship between multiple variables. In the simplest case, it can be used for two variables wherein the model determines a "{{w|least squares|best-fit}}" line through a {{w|scatter plot}} of the datasets, together with a {{w|coefficient of determination}}, usually denoted ''r''<sup>2</sup> or ''R''<sup>2</sup>. When only two variables are included in the regression, ''R''<sup>2</sup> is merely the square of the correlation between the two variables. ''R''<sup>2</sup> is a number between 0 and 1 that indicates how well one variable can be used to predict the value of another. A value of 1 means perfect correlation, while a value close to 0 indicates a weak relationship between the variables.<br />
<br />
{{w|Asterism_(astronomy)|Asterism}}s are patterns created by linking the apparent positions of stars as seen in the sky from Earth. Strictly, "Rexthor" is an asterism, as a {{w|constellation}} is the region of sky containing the asterism, although "constellation" is used informally in place of "asterism" by even seasoned astronomers. Different civilizations have recognized different constellations (the modern IAU, for example, lists 88 "official" constellations), and one could create their own constellations by connecting assorted points.<br />
<br />
In this comic, a set of data has had linear regression and some form of statistical analysis applied to it, indicating that there is low correlation between the two. The data points are so widely scattered that (as noted in the comic) it is easier to connect the data points in a constellation-like pattern than it is to determine whether the correlation is negative or positive (without looking at the trendline, of course). Because of this, [[Randall]] suggests we should be suspicious of any conclusions drawn from this data.<br />
<br />
"Rexthor the Dog bearer" seems to be a spoof on Thor, a Norse god who wields a hammer. <br />
By replacing his hammer with a dog and adding "Rex" (an archetypal dog name), Randall creates a comical, dog-bearing version of Thor. <br />
<br />
The 95% {{w|confidence interval}} in statistics is such a range of an estimate, that the probability of the real value (the estimated population parameter) to lie inside the range is at least 95%. The confidence interval is a standard method to provide evaluation of the estimation error in statistics. On the right panel the resulting estimate seems to be a drawing, so the 95% confidence interval would be a set of all drawings derived from the sample such that the probability of the right drawing to be among them is at least 95%. According to the title text among these drawings you can find a cat and a teapot as well, so we can't be 95% confident that a cat exists in the data.<br />
<br />
The teapot may be a reference to {{w|Russell's_teapot|Russell's teapot}}, or possibly to the "teapot" asterism in the constellation Sagittarius.<br />
<br />
==Transcript==<br />
:[Two square panels show identical sets of scattered black dots, with only the red additions being different.]<br />
<br />
:[The left panel shows a slightly rising red line drawn through the middle of the panel, passing near a few dots but not obviously related to most of them. A red text is below the dots:]<br />
:<span style="color:red">R<sup>2</sup>=0.06</span><br />
<br />
:[The right panel shows many of the dots connected by red lines to form a stick figure of a man resembling the constellation Orion, with the hand on the reader's right raised and holding an object. A red text is below the dots:]<br />
:<span style="color:red">Rexthor, the Dog-Bearer</span><br />
<br />
:[A caption is below and spanning both panels:] <br />
:I don't trust linear regressions when it's harder to guess the direction of the correlation from the scatter plot than to find new constellations on it.<br />
<br />
{{comic discussion}}<br />
<br />
[[Category:Comics with color]]<br />
[[Category:Scatter plots]]<br />
[[Category:Astronomy]]<br />
[[Category:Animals]]</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=1725:_Linear_Regression&diff=1258851725: Linear Regression2016-08-29T13:52:17Z<p>162.158.133.138: about 95% confidence interval</p>
<hr />
<div>{{comic<br />
| number = 1725<br />
| date = August 26, 2016<br />
| title = Linear Regression<br />
| image = linear_regression.png<br />
| titletext = The 95% confidence interval suggests Rexthor's dog could also be a cat, or possibly a teapot.<br />
}}<br />
<br />
==Explanation==<br />
{{w|Linear regression}} is a method for modeling the relationship between multiple variables. In the simplest case, it can be used for two variables wherein the model determines a "{{w|least squares|best-fit}}" line through a {{w|scatter plot}} of the datasets, together with a {{w|coefficient of determination}}, usually denoted ''r''<sup>2</sup> or ''R''<sup>2</sup>. When only two variables are included in the regression, ''R''<sup>2</sup> is merely the square of the correlation between the two variables. ''R''<sup>2</sup> is a number between 0 and 1 that indicates how well one variable can be used to predict the value of another. A value of 1 means perfect correlation, while a value close to 0 indicates a weak relationship between the variables.<br />
<br />
{{w|Asterism_(astronomy)|Asterism}}s are patterns created by linking the apparent positions of stars as seen in the sky from Earth. Strictly, "Rexthor" is an asterism, as a {{w|constellation}} is the region of sky containing the asterism, although "constellation" is used informally in place of "asterism" by even seasoned astronomers. Different civilizations have recognized different constellations (the modern IAU, for example, lists 88 "official" constellations), and one could create their own constellations by connecting assorted points.<br />
<br />
In this comic, a set of data has had linear regression and some form of statistical analysis applied to it, indicating that there is low correlation between the two. The data points are so widely scattered that (as noted in the comic) it is easier to connect the data points in a constellation-like pattern than it is to determine whether the correlation is negative or positive (without looking at the trendline, of course). Because of this, [[Randall]] suggests we should be suspicious of any conclusions drawn from this data.<br />
<br />
"Rexthor the Dog bearer" seems to be a spoof on Thor, a Norse god who wields a hammer. <br />
By replacing his hammer with a dog and adding "Rex" (an archetypal dog name), Randall creates a comical, dog-bearing version of Thor. <br />
<br />
The 95% {{w|confidence interval}} in statistics is such a range of an estimate, that the probability of the real value (the estimated population parameter) to lie inside the range is at least 95%. The confidence interval is a standard method to provide evaluation of the estimation error in statistics. On the right panel the resulting estimate seems to be a drawing, so the 95% confidence interval would be a set of all drawings connecting points from the sample such that the probability of the right drawing to be among them is at least 95%. According to the title text among these drawings you can find a cat and a teapot as well, so we can't be 95% confident that a cat exists in the data.<br />
<br />
The teapot may be a reference to {{w|Russell's_teapot|Russell's teapot}}, or possibly to the "teapot" asterism in the constellation Sagittarius.<br />
<br />
==Transcript==<br />
:[Two square panels show identical sets of scattered black dots, with only the red additions being different.]<br />
<br />
:[The left panel shows a slightly rising red line drawn through the middle of the panel, passing near a few dots but not obviously related to most of them. A red text is below the dots:]<br />
:<span style="color:red">R<sup>2</sup>=0.06</span><br />
<br />
:[The right panel shows many of the dots connected by red lines to form a stick figure of a man resembling the constellation Orion, with the hand on the reader's right raised and holding an object. A red text is below the dots:]<br />
:<span style="color:red">Rexthor, the Dog-Bearer</span><br />
<br />
:[A caption is below and spanning both panels:] <br />
:I don't trust linear regressions when it's harder to guess the direction of the correlation from the scatter plot than to find new constellations on it.<br />
<br />
{{comic discussion}}<br />
<br />
[[Category:Comics with color]]<br />
[[Category:Scatter plots]]<br />
[[Category:Astronomy]]<br />
[[Category:Animals]]</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=1257811724: Proofs2016-08-26T12:27:36Z<p>162.158.133.138: </p>
<hr />
<div>{{comic<br />
| number = 1724<br />
| date = August 24, 2016<br />
| title = Proofs<br />
| image = proofs.png<br />
| titletext = Next, let's assume the decision of whether to take the Axiom of Choice is made by a deterministic process ...<br />
}}<br />
<br />
==Explanation==<br />
[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those dark-magic (unclear, incomprehensible) proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.<br />
<br />
The proof she starts setting up resembles a {{w|proof by contradiction}}. This kind of proof assumes that a particular theorem is true, and shows that this assumption leads to a contradiction, which disproves the initial assumption. For example assumption that √2 is a {{w|rational number}} means that, for some natural ''a'' and ''b'', √2=''a/b'', where ''a/b'' is an {{w|irreducible fraction}}. Yet, multiplying this equation by itself, we get 2=''a²/b²'' which in turn rearranges to 2''b²''=''a²'', therefor ''a²'' is even (as any integer multiplied by 2 is even), which means that ''a'' is an even number, as an even number squared is always even and an odd number squared is always odd. This means, that ''a=2k'' and ''2b²=(2k)²=4k²'', meaning ''b²''=2''k²'', so ''b'' must be even too. But if both ''a'' and ''b'' are even, ''a/b'' cannot be irreducible. Contradiction means that the initial assumption is false, and √2 cannot be a rational number.<br />
<br />
Alternatively, instead of a proof by contradiction the setup could be for a one way function. For example, it is relatively easy to test that a solution to a differential equation is valid but choosing the correct solution to test can seem like black magic to students.<br />
<br />
The way, Ms Lenhart's proof refers to the act of doing math itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if in the end they must be a "perfectly sensible chain of reasoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions, points or lines, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but can be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken as a part of the axiomatic system.<br />
<br />
Using a position on the blackboard as a part of the proof is a joke, but it bears a resemblance to the {{w|Cantor's diagonal argument}} where a position in a sequence of digits of a real number was a tool in a proof that not all infinite sets have the same {{w|cardinality}} (rough equivalent of the number of elements). This "diagonal method" is also often used in metamathematical proofs.<br />
<br />
The axiom of choice itself states that for every collection of nonempty sets, you can have a function that draws one element from each set of the collection. This axiom, once considered controversial, was added relatively late to the axiomatic set theory, and even contemporary mathematicians still study which theorems really require its inclusion. In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. {{w|Determinacy}} of infinite games is used as a tool in the set theory, however the deterministic process is rather a term of the {{w|stochastic process|stochastic processes theory}}, and the {{w|dynamical systems theory}}, branches of mathematics far from the abstract set theory, which makes the proof even more exotic. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].<br />
<br />
Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in her prior class.<br />
<br />
==Transcript==<br />
:[Miss Lenhart is standing facing left in front of a whiteboard writing on it. Eleven left aligned lines of writing is shown as unreadable scribbles. A voice interrupts her from off-panel right.]<br />
:Miss Lenhart: ... Let's assume there exists some function ''F''(''a,b,c''...) which produces the correct answer-<br />
:Cueball (off-panel): Hang on.<br />
<br />
:[In a frame-less panel Cueball is sitting on a chair at a desk with a pen in his hand taking notes.]<br />
:Cueball: This is going to be one of those weird, dark magic proofs, isn't it? I can tell.<br />
<br />
:[Miss Lenhart has turned right towards Cueball, who is again speaking off-panel. The white board is also off-panel.]<br />
:Miss Lenhart: What? No, no, it's a perfectly sensible chain of reasoning.<br />
:Cueball (off-panel): All right...<br />
<br />
:[Miss Lenhart is facing the whiteboard again writing more scribbles behind some of the lines from before (the first line has disappeared). The lines that have more text added are now number three and five (four and six before). Cueball again speaks off-panel.]<br />
:Miss Lenhart: Now, let's assume that the correct answer will eventually be written on the board at the coordinates (''x, y''). If we—<br />
:Cueball (off-panel): I ''knew'' it!<br />
<br />
{{comic discussion}}<br />
<br />
[[Category:Comics featuring Miss Lenhart]]<br />
[[Category:Comics featuring Cueball]]<br />
[[Category:Math]]</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=1257801724: Proofs2016-08-26T12:07:42Z<p>162.158.133.138: /* Explanation */</p>
<hr />
<div>{{comic<br />
| number = 1724<br />
| date = August 24, 2016<br />
| title = Proofs<br />
| image = proofs.png<br />
| titletext = Next, let's assume the decision of whether to take the Axiom of Choice is made by a deterministic process ...<br />
}}<br />
<br />
==Explanation==<br />
[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those dark-magic (unclear, incomprehensible) proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.<br />
<br />
The proof she starts setting up resembles a {{w|proof by contradiction}}. This kind of proof assumes that a particular theorem is true, and shows that this assumption leads to a contradiction, which disproves the initial assumption. For example assumption that √2 is a {{w|rational number}} means that, for some natural ''a'' and ''b'', √2=''a/b'', where ''a/b'' is an {{w|irreducible fraction}}. Yet, multiplying this equation by itself, we get 2=''a²/b²'' which in turn rearranges to 2''b²''=''a²'', therefor ''a²'' is even (as any integer multiplied by 2 is even), which means that ''a'' is an even number, as an even number squared is always even and an odd number squared is always odd. This means, that ''a=2k'' and ''2b²=(2k)²=4k²'', meaning ''b²''=2''k²'', so ''b'' must be even too. But if both ''a'' and ''b'' are even, ''a/b'' cannot be irreducible. Contradiction means that the initial assumption is false, and √2 cannot be a rational number.<br />
<br />
Alternatively, instead of a proof by contradiction the setup could be for a one way function. For example, it is relatively easy to test that a solution to a differential equation is valid but choosing the correct solution to test can seem like black magic to students.<br />
<br />
The way, Ms Lenhart's proof refers to the act of doing math itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if in the end they must be a "perfectly sensible chain of reasoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions, points or lines, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but can be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken as a part of the axiomatic system.<br />
<br />
Using a position on the blackboard as a part of the proof is a joke, but it bears a resemblance to the {{w|Cantor's diagonal argument}} where a position in a sequence of digits of a real number was a tool in a proof that not all infinite sets have the same {{w|cardinality}} (rough equivalent of the number of elements). This "diagonal method" is also often used in metamathematical proofs.<br />
<br />
The axiom of choice itself states that for every collection of nonempty sets, you can have a function that draws one element from each set of the collection. This axiom, once considered controversial, was added relatively late to the axiomatic set theory, and even contemporary mathematicians still study which theorems really require its inclusion. In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. Determinacy of games is used as a tool in the set theory, however the deterministic process is rather a term of the {{w|stochastic process|stochastic processes theory}}, and the {{w|dynamical systems theory}}, branches of mathematics far from the abstract set theory, which makes the proof even more exotic. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].<br />
<br />
Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in her prior class.<br />
<br />
==Transcript==<br />
:[Miss Lenhart is standing facing left in front of a whiteboard writing on it. Eleven left aligned lines of writing is shown as unreadable scribbles. A voice interrupts her from off-panel right.]<br />
:Miss Lenhart: ... Let's assume there exists some function ''F''(''a,b,c''...) which produces the correct answer-<br />
:Cueball (off-panel): Hang on.<br />
<br />
:[In a frame-less panel Cueball is sitting on a chair at a desk with a pen in his hand taking notes.]<br />
:Cueball: This is going to be one of those weird, dark magic proofs, isn't it? I can tell.<br />
<br />
:[Miss Lenhart has turned right towards Cueball, who is again speaking off-panel. The white board is also off-panel.]<br />
:Miss Lenhart: What? No, no, it's a perfectly sensible chain of reasoning.<br />
:Cueball (off-panel): All right...<br />
<br />
:[Miss Lenhart is facing the whiteboard again writing more scribbles behind some of the lines from before (the first line has disappeared). The lines that have more text added are now number three and five (four and six before). Cueball again speaks off-panel.]<br />
:Miss Lenhart: Now, let's assume that the correct answer will eventually be written on the board at the coordinates (''x, y''). If we—<br />
:Cueball (off-panel): I ''knew'' it!<br />
<br />
{{comic discussion}}<br />
<br />
[[Category:Comics featuring Miss Lenhart]]<br />
[[Category:Comics featuring Cueball]]<br />
[[Category:Math]]</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=1257791724: Proofs2016-08-26T12:04:01Z<p>162.158.133.138: diagonal argument hint and more about deterministic processes</p>
<hr />
<div>{{comic<br />
| number = 1724<br />
| date = August 24, 2016<br />
| title = Proofs<br />
| image = proofs.png<br />
| titletext = Next, let's assume the decision of whether to take the Axiom of Choice is made by a deterministic process ...<br />
}}<br />
<br />
==Explanation==<br />
[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those dark-magic (unclear, incomprehensible) proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.<br />
<br />
The proof she starts setting up resembles a {{w|proof by contradiction}}. This kind of proof assumes that a particular theorem is true, and shows that this assumption leads to a contradiction, which disproves the initial assumption. For example assumption that √2 is a {{w|rational number}} means that, for some natural ''a'' and ''b'', √2=''a/b'', where ''a/b'' is an {{w|irreducible fraction}}. Yet, multiplying this equation by itself, we get 2=''a²/b²'' which in turn rearranges to 2''b²''=''a²'', therefor ''a²'' is even (as any integer multiplied by 2 is even), which means that ''a'' is an even number, as an even number squared is always even and an odd number squared is always odd. This means, that ''a=2k'' and ''2b²=(2k)²=4k²'', meaning ''b²''=2''k²'', so ''b'' must be even too. But if both ''a'' and ''b'' are even, ''a/b'' cannot be irreducible. Contradiction means that the initial assumption is false, and √2 cannot be a rational number.<br />
<br />
Alternatively, instead of a proof by contradiction the setup could be for a one way function. For example, it is relatively easy to test that a solution to a differential equation is valid but choosing the correct solution to test can seem like black magic to students.<br />
<br />
The way, Ms Lenhart's proof refers to the act of doing math itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if in the end they must be a "perfectly sensible chain of reasoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions, points or lines, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but can be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken as a part of the axiomatic system.<br />
<br />
Using a position on a blackboard as a part of the proof is a joke, but it bears a resemblance to the {{w|Cantor's diagonal argument}} where a position in a sequence of digits of a real number was a tool in a proof that not all infinite sets have the same {{w|cardinality}} (rough equivalent of the number of elements). This "diagonal method" is also often used in metamathematical proofs.<br />
<br />
The axiom of choice itself states that for every collection of nonempty sets, you can have a function that draws one element from each set of the collection. This axiom, once considered controversial, was added relatively late to the axiomatic set theory, and even contemporary mathematicians still study which theorems really require its inclusion. In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. Determinacy of games is used as a tool in the set theory, however deterministic process is rather a term of the {{w|stochastic process|stochastic processes theory}}, and {{w|dynamical systems theory}}, branches of mathematics far from the abstract set theory, which makes the proof even more exotic. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].<br />
<br />
Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in her prior class.<br />
<br />
==Transcript==<br />
:[Miss Lenhart is standing facing left in front of a whiteboard writing on it. Eleven left aligned lines of writing is shown as unreadable scribbles. A voice interrupts her from off-panel right.]<br />
:Miss Lenhart: ... Let's assume there exists some function ''F''(''a,b,c''...) which produces the correct answer-<br />
:Cueball (off-panel): Hang on.<br />
<br />
:[In a frame-less panel Cueball is sitting on a chair at a desk with a pen in his hand taking notes.]<br />
:Cueball: This is going to be one of those weird, dark magic proofs, isn't it? I can tell.<br />
<br />
:[Miss Lenhart has turned right towards Cueball, who is again speaking off-panel. The white board is also off-panel.]<br />
:Miss Lenhart: What? No, no, it's a perfectly sensible chain of reasoning.<br />
:Cueball (off-panel): All right...<br />
<br />
:[Miss Lenhart is facing the whiteboard again writing more scribbles behind some of the lines from before (the first line has disappeared). The lines that have more text added are now number three and five (four and six before). Cueball again speaks off-panel.]<br />
:Miss Lenhart: Now, let's assume that the correct answer will eventually be written on the board at the coordinates (''x, y''). If we—<br />
:Cueball (off-panel): I ''knew'' it!<br />
<br />
{{comic discussion}}<br />
<br />
[[Category:Comics featuring Miss Lenhart]]<br />
[[Category:Comics featuring Cueball]]<br />
[[Category:Math]]</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=1257781724: Proofs2016-08-26T11:37:52Z<p>162.158.133.138: /* Explanation */</p>
<hr />
<div>{{comic<br />
| number = 1724<br />
| date = August 24, 2016<br />
| title = Proofs<br />
| image = proofs.png<br />
| titletext = Next, let's assume the decision of whether to take the Axiom of Choice is made by a deterministic process ...<br />
}}<br />
<br />
==Explanation==<br />
[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those dark-magic (unclear, incomprehensible) proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.<br />
<br />
The proof she starts setting up resembles a {{w|proof by contradiction}}. This kind of proof assumes that a particular theorem is true, and shows that this assumption leads to a contradiction, which disproves the initial assumption. For example assumption that √2 is a {{w|rational number}} means that, for some natural ''a'' and ''b'', √2=''a/b'', where ''a/b'' is an {{w|irreducible fraction}}. Yet, multiplying this equation by itself, we get 2=''a²/b²'' which in turn rearranges to 2''b²''=''a²'', therefor ''a²'' is even (as any integer multiplied by 2 is even), which means that ''a'' is an even number, as an even number squared is always even and an odd number squared is always odd. This means, that ''a=2k'' and ''2b²=(2k)²=4k²'', meaning ''b²''=2''k²'', so ''b'' must be even too. But if both ''a'' and ''b'' are even, ''a/b'' cannot be irreducible. Contradiction means that the initial assumption is false, and √2 cannot be a rational number.<br />
<br />
Alternatively, instead of a proof by contradiction the setup could be for a one way function. For example, it is relatively easy to test that a solution to a differential equation is valid but choosing the correct solution to test can seem like black magic to students.<br />
<br />
The way, Ms Lenhart's proof refers to the act of doing math itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if in the end they must be a "perfectly sensible chain of reasoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions, points or lines, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but can be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken as a part of the axiomatic system.<br />
<br />
The axiom of choice itself states that for every collection of nonempty sets, you can have a function that draws one element from each set of the collection. This axiom, once considered controversial, was added relatively late to the axiomatic set theory, and even contemporary mathematicians still study which theorems really require its inclusion. In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].<br />
<br />
Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in her prior class.<br />
<br />
==Transcript==<br />
:[Miss Lenhart is standing facing left in front of a whiteboard writing on it. Eleven left aligned lines of writing is shown as unreadable scribbles. A voice interrupts her from off-panel right.]<br />
:Miss Lenhart: ... Let's assume there exists some function ''F''(''a,b,c''...) which produces the correct answer-<br />
:Cueball (off-panel): Hang on.<br />
<br />
:[In a frame-less panel Cueball is sitting on a chair at a desk with a pen in his hand taking notes.]<br />
:Cueball: This is going to be one of those weird, dark magic proofs, isn't it? I can tell.<br />
<br />
:[Miss Lenhart has turned right towards Cueball, who is again speaking off-panel. The white board is also off-panel.]<br />
:Miss Lenhart: What? No, no, it's a perfectly sensible chain of reasoning.<br />
:Cueball (off-panel): All right...<br />
<br />
:[Miss Lenhart is facing the whiteboard again writing more scribbles behind some of the lines from before (the first line has disappeared). The lines that have more text added are now number three and five (four and six before). Cueball again speaks off-panel.]<br />
:Miss Lenhart: Now, let's assume that the correct answer will eventually be written on the board at the coordinates (''x, y''). If we—<br />
:Cueball (off-panel): I ''knew'' it!<br />
<br />
{{comic discussion}}<br />
<br />
[[Category:Comics featuring Miss Lenhart]]<br />
[[Category:Comics featuring Cueball]]<br />
[[Category:Math]]</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=1257771724: Proofs2016-08-26T11:36:14Z<p>162.158.133.138: /* Explanation */ More about axiom of choice. BTW, it is used in constructive mathematics too - see https://en.wikipedia.org/wiki/Axiom_of_choice#In_constructive_mathematics</p>
<hr />
<div>{{comic<br />
| number = 1724<br />
| date = August 24, 2016<br />
| title = Proofs<br />
| image = proofs.png<br />
| titletext = Next, let's assume the decision of whether to take the Axiom of Choice is made by a deterministic process ...<br />
}}<br />
<br />
==Explanation==<br />
[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those dark-magic (unclear, incomprehensible) proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.<br />
<br />
The proof she starts setting up resembles a {{w|proof by contradiction}}. This kind of proof assumes that a particular theorem is true, and shows that this assumption leads to a contradiction, which disproves the initial assumption. For example assumption that √2 is a {{w|rational number}} means that, for some natural ''a'' and ''b'', √2=''a/b'', where ''a/b'' is an {{w|irreducible fraction}}. Yet, multiplying this equation by itself, we get 2=''a²/b²'' which in turn rearranges to 2''b²''=''a²'', therefor ''a²'' is even (as any integer multiplied by 2 is even), which means that ''a'' is an even number, as an even number squared is always even and an odd number squared is always odd. This means, that ''a=2k'' and ''2b²=(2k)²=4k²'', meaning ''b²''=2''k²'', so ''b'' must be even too. But if both ''a'' and ''b'' are even, ''a/b'' cannot be irreducible. Contradiction means that the initial assumption is false, and √2 cannot be a rational number.<br />
<br />
Alternatively, instead of a proof by contradiction the setup could be for a one way function. For example, it is relatively easy to test that a solution to a differential equation is valid but choosing the correct solution to test can seem like black magic to students.<br />
<br />
The way, Ms Lenhart's proof refers to the act of doing math itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if in the end they must be a "perfectly sensible chain of reasoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions, points or lines, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but can be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken as a part of the axiomatic system.<br />
<br />
The axiom of choice itself states that for every collection of nonempty sets, you can have a function that draws one element from each set of the collection. This, once considered controversial, axiom was added relatively late to the axiomatic set theory, and even contemporary mathematicians still study which theorems really require its inclusion. In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].<br />
<br />
Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in her prior class.<br />
<br />
==Transcript==<br />
:[Miss Lenhart is standing facing left in front of a whiteboard writing on it. Eleven left aligned lines of writing is shown as unreadable scribbles. A voice interrupts her from off-panel right.]<br />
:Miss Lenhart: ... Let's assume there exists some function ''F''(''a,b,c''...) which produces the correct answer-<br />
:Cueball (off-panel): Hang on.<br />
<br />
:[In a frame-less panel Cueball is sitting on a chair at a desk with a pen in his hand taking notes.]<br />
:Cueball: This is going to be one of those weird, dark magic proofs, isn't it? I can tell.<br />
<br />
:[Miss Lenhart has turned right towards Cueball, who is again speaking off-panel. The white board is also off-panel.]<br />
:Miss Lenhart: What? No, no, it's a perfectly sensible chain of reasoning.<br />
:Cueball (off-panel): All right...<br />
<br />
:[Miss Lenhart is facing the whiteboard again writing more scribbles behind some of the lines from before (the first line has disappeared). The lines that have more text added are now number three and five (four and six before). Cueball again speaks off-panel.]<br />
:Miss Lenhart: Now, let's assume that the correct answer will eventually be written on the board at the coordinates (''x, y''). If we—<br />
:Cueball (off-panel): I ''knew'' it!<br />
<br />
{{comic discussion}}<br />
<br />
[[Category:Comics featuring Miss Lenhart]]<br />
[[Category:Comics featuring Cueball]]<br />
[[Category:Math]]</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=1257671724: Proofs2016-08-25T15:29:28Z<p>162.158.133.138: </p>
<hr />
<div>{{comic<br />
| number = 1724<br />
| date = August 24, 2016<br />
| title = Proofs<br />
| image = proofs.png<br />
| titletext = Next, let's assume the decision of whether to take the Axiom of Choice is made by a deterministic process ...<br />
}}<br />
<br />
==Explanation==<br />
[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those dark-magic (unclear, incomprehensible) proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.<br />
<br />
The proof she starts setting up resembles a {{w|proof by contradiction}}. This kind of proof assumes that a particular theorem is true, and shows that this assumption leads to a contradiction, which disproves the initial assumption. For example assumption that √2 is a {{w|rational number}} means that, for some natural ''a'' and ''b'', √2=''a/b'', where ''a/b'' is an {{w|irreducible fraction}}. Yet, multiplying this equation by itself, we get 2=''a²/b²'', which means that ''a'' is an even number. This means, that ''a=2k'' and ''2b²=(2k)²=4k²'', so ''b'' must be even too. But if both ''a'' and ''b'' are even, ''a/b'' cannot be irreducible. Contradiction means that the initial assumption is false, and √2 cannot be a rational number.<br />
<br />
The way, Ms Lenhart's proof refers to the act of doing math itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if in the end they must be a "perfectly sensible chain of resoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions, points or lines, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but can be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken as a part of the axiomatic system.<br />
<br />
In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. It may be an allusion to the proposed {{w|axiom of determinacy}} of the set theory. It is, however, {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with the axiom of choice, which builds another layer of the joke. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].<br />
<br />
Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in the her undergraduate class.<br />
<br />
==Transcript==<br />
:[Miss Lenhart is standing facing left in front of a whiteboard writing on it. Eleven left aligned lines of writing is shown as unreadable scribbles. A voice interrupts her from off-panel right.]<br />
:Miss Lenhart: ... Let's assume there exists some function ''F''(''a,b,c''...) which produces the correct answer-<br />
:Cueball (off-panel): Hang on.<br />
<br />
:[In a frame-less panel Cueball is sitting on a chair at a desk with a pen in his hand taking notes.]<br />
:Cueball: This is going to be one of those weird, dark magic proofs, isn't it? I can tell.<br />
<br />
:[Miss Lenhart has turned right towards Cueball, who is again speaking off-panel. The white board is also off-panel.]<br />
:Miss Lenhart: What? No, no, it's a perfectly sensible chain of reasoning.<br />
:Cueball (off-panel): All right...<br />
<br />
:[Miss Lenhart is facing the whiteboard again writing more scribbles behind some of the lines from before (the first line has disappeared). The lines that have more text added are now number three and five (four and six before). Cueball again speaks off-panel.]<br />
:Miss Lenhart: Now, let's assume that the correct answer will eventually be written on the board at the coordinates (''x, y''). If we—<br />
:Cueball (off-panel): I ''knew'' it!<br />
<br />
{{comic discussion}}<br />
<br />
[[Category:Comics featuring Miss Lenhart]]<br />
[[Category:Comics featuring Cueball]]<br />
[[Category:Math]]</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=1257661724: Proofs2016-08-25T15:27:21Z<p>162.158.133.138: </p>
<hr />
<div>{{comic<br />
| number = 1724<br />
| date = August 24, 2016<br />
| title = Proofs<br />
| image = proofs.png<br />
| titletext = Next, let's assume the decision of whether to take the Axiom of Choice is made by a deterministic process ...<br />
}}<br />
<br />
==Explanation==<br />
[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those dark-magic (unclear, incomprehensible) proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.<br />
<br />
The proof she starts setting up resembles a {{w|proof by contradiction}}. This kind of proof assumes that a particular theorem is true, and shows that this assumption leads to a contradiction, which disproves the initial assumption. For example assumption that √2 is a {{w|rational number}} means that, for some natural ''a'' and ''b'', √2=''a/b'', where ''a/b'' is an {{w|irreducible fraction}}. Yet, multiplying this equation by itself, we get 2=''a²/b²'', which means that ''a'' is an even number. This means, that ''a=2k'' and ''2b²=(2k)²=4k²'', so ''b'' must be even too. But if both ''a'' and ''b'' are even, ''a/b'' cannot be irreducible. Contradiction means that the initial assumption is false, and √2 cannot be a rational number.<br />
<br />
The way, Ms Lenhart's proof refers to the act of doing math itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if in the end they must be a "perfectly sensible chain of resoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions, points or lines, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but cannot be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken as a part of the axiomatic system.<br />
<br />
In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. It may be an allusion to the proposed {{w|axiom of determinacy}} of the set theory. It is, however, {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with the axiom of choice, which builds another layer of the joke. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].<br />
<br />
Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in the her undergraduate class.<br />
<br />
==Transcript==<br />
:[Miss Lenhart is standing facing left in front of a whiteboard writing on it. Eleven left aligned lines of writing is shown as unreadable scribbles. A voice interrupts her from off-panel right.]<br />
:Miss Lenhart: ... Let's assume there exists some function ''F''(''a,b,c''...) which produces the correct answer-<br />
:Cueball (off-panel): Hang on.<br />
<br />
:[In a frame-less panel Cueball is sitting on a chair at a desk with a pen in his hand taking notes.]<br />
:Cueball: This is going to be one of those weird, dark magic proofs, isn't it? I can tell.<br />
<br />
:[Miss Lenhart has turned right towards Cueball, who is again speaking off-panel. The white board is also off-panel.]<br />
:Miss Lenhart: What? No, no, it's a perfectly sensible chain of reasoning.<br />
:Cueball (off-panel): All right...<br />
<br />
:[Miss Lenhart is facing the whiteboard again writing more scribbles behind some of the lines from before (the first line has disappeared). The lines that have more text added are now number three and five (four and six before). Cueball again speaks off-panel.]<br />
:Miss Lenhart: Now, let's assume that the correct answer will eventually be written on the board at the coordinates (''x, y''). If we—<br />
:Cueball (off-panel): I ''knew'' it!<br />
<br />
{{comic discussion}}<br />
<br />
[[Category:Comics featuring Miss Lenhart]]<br />
[[Category:Comics featuring Cueball]]<br />
[[Category:Math]]</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=1257651724: Proofs2016-08-25T15:01:53Z<p>162.158.133.138: </p>
<hr />
<div>{{comic<br />
| number = 1724<br />
| date = August 24, 2016<br />
| title = Proofs<br />
| image = proofs.png<br />
| titletext = Next, let's assume the decision of whether to take the Axiom of Choice is made by a deterministic process ...<br />
}}<br />
<br />
==Explanation==<br />
[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those dark-magic (unclear, incomprehensible) proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.<br />
<br />
The proof she starts setting up resembles a {{w|proof by contradiction}}. This kind of proof assumes that a particular theorem is true, and shows that this assumption leads to a contradiction, which disproves the initial assumption. For example assumption that √2 is a {{w|rational number}} means that, for some natural ''a'' and ''b'', √2=''a/b'', where ''a/b'' is an {{w|irreducible fraction}}. Yet, multiplying this equation by itself, we get 2=''a²/b²'', which means that ''a'' is an even number. This means, that ''a=2k'' and ''2b²=(2k)²=4k²'', so ''b'' must be even too. But if both ''a'' and ''b'' are even, ''a/b'' cannot be irreducible. Contradiction means that the initial assumption is false, and √2 cannot be a rational number.<br />
<br />
The way, Ms Lenhart's proof refers to the act of proving itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if in the end they must be a "perfectly sensible chain of resoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions, points or lines, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but cannot be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken as a part of the axiomatic system.<br />
<br />
In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. It may be an allusion to the proposed {{w|axiom of determinacy}} of the set theory. It is, however, {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with the axiom of choice, which builds another layer of the joke. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].<br />
<br />
Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in the her undergraduate class.<br />
<br />
==Transcript==<br />
:[Miss Lenhart is standing facing left in front of a whiteboard writing on it. Eleven left aligned lines of writing is shown as unreadable scribbles. A voice interrupts her from off-panel right.]<br />
:Miss Lenhart: ... Let's assume there exists some function ''F''(''a,b,c''...) which produces the correct answer-<br />
:Cueball (off-panel): Hang on.<br />
<br />
:[In a frame-less panel Cueball is sitting on a chair at a desk with a pen in his hand taking notes.]<br />
:Cueball: This is going to be one of those weird, dark magic proofs, isn't it? I can tell.<br />
<br />
:[Miss Lenhart has turned right towards Cueball, who is again speaking off-panel. The white board is also off-panel.]<br />
:Miss Lenhart: What? No, no, it's a perfectly sensible chain of reasoning.<br />
:Cueball (off-panel): All right...<br />
<br />
:[Miss Lenhart is facing the whiteboard again writing more scribbles behind some of the lines from before (the first line has disappeared). The lines that have more text added are now number three and five (four and six before). Cueball again speaks off-panel.]<br />
:Miss Lenhart: Now, let's assume that the correct answer will eventually be written on the board at the coordinates (''x, y''). If we—<br />
:Cueball (off-panel): I ''knew'' it!<br />
<br />
{{comic discussion}}<br />
<br />
[[Category:Comics featuring Miss Lenhart]]<br />
[[Category:Comics featuring Cueball]]<br />
[[Category:Math]]</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=1257641724: Proofs2016-08-25T15:00:03Z<p>162.158.133.138: /* Explanation */</p>
<hr />
<div>{{comic<br />
| number = 1724<br />
| date = August 24, 2016<br />
| title = Proofs<br />
| image = proofs.png<br />
| titletext = Next, let's assume the decision of whether to take the Axiom of Choice is made by a deterministic process ...<br />
}}<br />
<br />
==Explanation==<br />
[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those dark-magic (unclear, incomprehensible) proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.<br />
<br />
The proof she starts setting up resembles a {{w|proof by contradiction}}. This kind of proof assumes that a particular theorem is true, and shows that this assumption leads to a contradiction, which disproves the initial assumption. For example assumption that √2 is a {{w|rational number}} means that, for some natural ''a'' and ''b'', √2=''a/b'', where ''a/b'' is an irreducible fraction. But multiplying this equation by itself we get 2=''a²/b²'', which means that ''a'' is an even number. This means, that ''a=2k'' and ''2b²=(2k)²=4k²'', so ''b'' must be even too. But if both ''a'' and ''b'' are even, ''a/b'' cannot be irreducible. Contradiction means that the initial assumption is false, and √2 cannot be a rational number.<br />
<br />
The way, Ms Lenhart's proof refers to the act of proving itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if in the end they must be a "perfectly sensible chain of resoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions, points or lines, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but cannot be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken as a part of the axiomatic system.<br />
<br />
In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. It may be an allusion to the proposed {{w|axiom of determinacy}} of the set theory. It is, however, {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with the axiom of choice, which builds another layer of the joke. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].<br />
<br />
Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in the her undergraduate class.<br />
<br />
==Transcript==<br />
:[Miss Lenhart is standing facing left in front of a whiteboard writing on it. Eleven left aligned lines of writing is shown as unreadable scribbles. A voice interrupts her from off-panel right.]<br />
:Miss Lenhart: ... Let's assume there exists some function ''F''(''a,b,c''...) which produces the correct answer-<br />
:Cueball (off-panel): Hang on.<br />
<br />
:[In a frame-less panel Cueball is sitting on a chair at a desk with a pen in his hand taking notes.]<br />
:Cueball: This is going to be one of those weird, dark magic proofs, isn't it? I can tell.<br />
<br />
:[Miss Lenhart has turned right towards Cueball, who is again speaking off-panel. The white board is also off-panel.]<br />
:Miss Lenhart: What? No, no, it's a perfectly sensible chain of reasoning.<br />
:Cueball (off-panel): All right...<br />
<br />
:[Miss Lenhart is facing the whiteboard again writing more scribbles behind some of the lines from before (the first line has disappeared). The lines that have more text added are now number three and five (four and six before). Cueball again speaks off-panel.]<br />
:Miss Lenhart: Now, let's assume that the correct answer will eventually be written on the board at the coordinates (''x, y''). If we—<br />
:Cueball (off-panel): I ''knew'' it!<br />
<br />
{{comic discussion}}<br />
<br />
[[Category:Comics featuring Miss Lenhart]]<br />
[[Category:Comics featuring Cueball]]<br />
[[Category:Math]]</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=1257631724: Proofs2016-08-25T14:54:28Z<p>162.158.133.138: </p>
<hr />
<div>{{comic<br />
| number = 1724<br />
| date = August 24, 2016<br />
| title = Proofs<br />
| image = proofs.png<br />
| titletext = Next, let's assume the decision of whether to take the Axiom of Choice is made by a deterministic process ...<br />
}}<br />
<br />
==Explanation==<br />
[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those dark-magic (unclear, incomprehensible) proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.<br />
<br />
The proof she starts setting up resembles a {{w|proof by contradiction}}. This kind of proof assumes that a particular theorem is true, and shows that this assumption leads to a contradiction, which disproves the initial assumption. For example assumption that √2 is a {{w|rational number}} means that, for some natural ''a'' and ''b'', √2=''a/b'', where ''a/b'' is an irreducible fraction. But multiplying this equation by itself we get 2=''a²/b²'', which means that ''a'' is an even number. This means, that ''a=2k'' and ''2b²=(2k)²=4k²'', so ''b'' must be even too. But if both ''a'' and ''b'' are even, ''a/b'' cannot be irreducible. Contradiction means that the initial assumption is false, and √2 cannot be a rational number.<br />
<br />
The way, Ms Lenhart's proof refers to the act of proving itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if in the end they must be a "perfectly sensible chain of resoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions, points or lines, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but cannot be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken as a part of the axiomatic system.<br />
<br />
In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. The {{w|axiom of determinacy}} is {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with the {{w|axiom of choice}}, which is the continuation of the joke of these dark magic proofs. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].<br />
<br />
Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in the her undergraduate class.<br />
<br />
==Transcript==<br />
:[Miss Lenhart is standing facing left in front of a whiteboard writing on it. Eleven left aligned lines of writing is shown as unreadable scribbles. A voice interrupts her from off-panel right.]<br />
:Miss Lenhart: ... Let's assume there exists some function ''F''(''a,b,c''...) which produces the correct answer-<br />
:Cueball (off-panel): Hang on.<br />
<br />
:[In a frame-less panel Cueball is sitting on a chair at a desk with a pen in his hand taking notes.]<br />
:Cueball: This is going to be one of those weird, dark magic proofs, isn't it? I can tell.<br />
<br />
:[Miss Lenhart has turned right towards Cueball, who is again speaking off-panel. The white board is also off-panel.]<br />
:Miss Lenhart: What? No, no, it's a perfectly sensible chain of reasoning.<br />
:Cueball (off-panel): All right...<br />
<br />
:[Miss Lenhart is facing the whiteboard again writing more scribbles behind some of the lines from before (the first line has disappeared). The lines that have more text added are now number three and five (four and six before). Cueball again speaks off-panel.]<br />
:Miss Lenhart: Now, let's assume that the correct answer will eventually be written on the board at the coordinates (''x, y''). If we—<br />
:Cueball (off-panel): I ''knew'' it!<br />
<br />
{{comic discussion}}<br />
<br />
[[Category:Comics featuring Miss Lenhart]]<br />
[[Category:Comics featuring Cueball]]<br />
[[Category:Math]]</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=1257621724: Proofs2016-08-25T14:50:28Z<p>162.158.133.138: more about proof by contradiction</p>
<hr />
<div>{{comic<br />
| number = 1724<br />
| date = August 24, 2016<br />
| title = Proofs<br />
| image = proofs.png<br />
| titletext = Next, let's assume the decision of whether to take the Axiom of Choice is made by a deterministic process ...<br />
}}<br />
<br />
==Explanation==<br />
[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those dark-magic (unclear, incomprehensible) proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.<br />
<br />
The proof she starts setting up resembles a {{w|proof by contradiction}}. This kind of proof assumes that a particular theorem is true, and shows that this assumption leads to a contradiction, which disproves the initial assumption. For example assumption that √2 is a {{w|rational number}} means that, for some natural ''a'' and ''b'', √2=''a/b'', where ''a/b'' is an irreducible fraction. But multiplying this equation by itself we get 2=''a²/b²'', which means that ''a'' is an even number. This means, that ''a=2k'' and ''2b²=(2k)²=4k²'', so ''b'' must be even too. But if both ''a'' and ''b'' are even, ''a/b'' cannot be irreducible. Contradiction means that the initial assumption is false, and √2 cannot be a rational number.<br />
<br />
The way, Ms Lenhart's proof refers to the act of proving itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if finally they must be a "perfectly sensible chain of resoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions or curves, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but cannot be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken.<br />
<br />
In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. The {{w|axiom of determinacy}} is {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with the {{w|axiom of choice}}, which is the continuation of the joke of these dark magic proofs. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].<br />
<br />
Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in the her undergraduate class.<br />
<br />
==Transcript==<br />
:[Miss Lenhart is standing facing left in front of a whiteboard writing on it. Eleven left aligned lines of writing is shown as unreadable scribbles. A voice interrupts her from off-panel right.]<br />
:Miss Lenhart: ... Let's assume there exists some function ''F''(''a,b,c''...) which produces the correct answer-<br />
:Cueball (off-panel): Hang on.<br />
<br />
:[In a frame-less panel Cueball is sitting on a chair at a desk with a pen in his hand taking notes.]<br />
:Cueball: This is going to be one of those weird, dark magic proofs, isn't it? I can tell.<br />
<br />
:[Miss Lenhart has turned right towards Cueball, who is again speaking off-panel. The white board is also off-panel.]<br />
:Miss Lenhart: What? No, no, it's a perfectly sensible chain of reasoning.<br />
:Cueball (off-panel): All right...<br />
<br />
:[Miss Lenhart is facing the whiteboard again writing more scribbles behind some of the lines from before (the first line has disappeared). The lines that have more text added are now number three and five (four and six before). Cueball again speaks off-panel.]<br />
:Miss Lenhart: Now, let's assume that the correct answer will eventually be written on the board at the coordinates (''x, y''). If we—<br />
:Cueball (off-panel): I ''knew'' it!<br />
<br />
{{comic discussion}}<br />
<br />
[[Category:Comics featuring Miss Lenhart]]<br />
[[Category:Comics featuring Cueball]]<br />
[[Category:Math]]</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=1257611724: Proofs2016-08-25T14:34:55Z<p>162.158.133.138: why?</p>
<hr />
<div>{{comic<br />
| number = 1724<br />
| date = August 24, 2016<br />
| title = Proofs<br />
| image = proofs.png<br />
| titletext = Next, let's assume the decision of whether to take the Axiom of Choice is made by a deterministic process ...<br />
}}<br />
<br />
==Explanation==<br />
[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those dark-magic (unclear, incomprehensible) proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.<br />
<br />
The proof she starts setting up resembles a {{w|proof by contradiction}}. These often involve making an assumption that there exists some formula or figure that fulfills the requirements given and plucking that answer out of abstract mathematics, much like summoning of demons is associated with black magic. This is usually done by relying on knowledge of the constraints of the form (for example, having the square root of 2 be ''a/b'' where ''a'' and ''b'' are both integers and have no common factors when proving that the square root of 2 is irrational).<br />
<br />
The way, Ms Lenhart's proof refers to the act of proving itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if finally they must be a "perfectly sensible chain of resoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions or curves, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but cannot be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken.<br />
<br />
In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. The {{w|axiom of determinacy}} is {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with the {{w|axiom of choice}}, which is the continuation of the joke of these dark magic proofs. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].<br />
<br />
Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in the her undergraduate class.<br />
<br />
==Transcript==<br />
:[Miss Lenhart is standing facing left in front of a whiteboard writing on it. Eleven left aligned lines of writing is shown as unreadable scribbles. A voice interrupts her from off-panel right.]<br />
:Miss Lenhart: ... Let's assume there exists some function ''F''(''a,b,c''...) which produces the correct answer-<br />
:Cueball (off-panel): Hang on.<br />
<br />
:[In a frame-less panel Cueball is sitting on a chair at a desk with a pen in his hand taking notes.]<br />
:Cueball: This is going to be one of those weird, dark magic proofs, isn't it? I can tell.<br />
<br />
:[Miss Lenhart has turned right towards Cueball, who is again speaking off-panel. The white board is also off-panel.]<br />
:Miss Lenhart: What? No, no, it's a perfectly sensible chain of reasoning.<br />
:Cueball (off-panel): All right...<br />
<br />
:[Miss Lenhart is facing the whiteboard again writing more scribbles behind some of the lines from before (the first line has disappeared). The lines that have more text added are now number three and five (four and six before). Cueball again speaks off-panel.]<br />
:Miss Lenhart: Now, let's assume that the correct answer will eventually be written on the board at the coordinates (''x, y''). If we—<br />
:Cueball (off-panel): I ''knew'' it!<br />
<br />
{{comic discussion}}<br />
<br />
[[Category:Comics featuring Miss Lenhart]]<br />
[[Category:Comics featuring Cueball]]<br />
[[Category:Math]]</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=1257601724: Proofs2016-08-25T14:32:13Z<p>162.158.133.138: </p>
<hr />
<div>{{comic<br />
| number = 1724<br />
| date = August 24, 2016<br />
| title = Proofs<br />
| image = proofs.png<br />
| titletext = Next, let's assume the decision of whether to take the Axiom of Choice is made by a deterministic process ...<br />
}}<br />
<br />
==Explanation==<br />
[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those dark-magic (unclear, incomprehensible) proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.<br />
<br />
The proof she starts setting up resembles a {{w|proof by contradiction}}. These often involve making an assumption that there exists some formula or figure that fulfills the requirements given and plucking that answer out of abstract mathematics, much like summoning of demons is associated with black magic. This is usually done by relying on knowledge of the constraints of the form (for example, having the square root of 2 be ''a/b'' where ''a'' and ''b'' are both integers and have no common factors when proving that the square root of 2 is irrational). This common usage is then shown to be not the case in the comic as the proof then goes to claim that the answer will be written in a specific place (though this could be taken as indicating that the result is finite or has a simple algorithm for continuing it).<br />
<br />
The way, Ms Lenhart's proof refers to the act of proving itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if finally they must be a "perfectly sensible chain of resoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions or curves, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but cannot be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken.<br />
<br />
In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. The {{w|axiom of determinacy}} is {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with the {{w|axiom of choice}}, which is the continuation of the joke of these dark magic proofs. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].<br />
<br />
Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in the her undergraduate class.<br />
<br />
==Transcript==<br />
:[Miss Lenhart is standing facing left in front of a whiteboard writing on it. Eleven left aligned lines of writing is shown as unreadable scribbles. A voice interrupts her from off-panel right.]<br />
:Miss Lenhart: ... Let's assume there exists some function ''F''(''a,b,c''...) which produces the correct answer-<br />
:Cueball (off-panel): Hang on.<br />
<br />
:[In a frame-less panel Cueball is sitting on a chair at a desk with a pen in his hand taking notes.]<br />
:Cueball: This is going to be one of those weird, dark magic proofs, isn't it? I can tell.<br />
<br />
:[Miss Lenhart has turned right towards Cueball, who is again speaking off-panel. The white board is also off-panel.]<br />
:Miss Lenhart: What? No, no, it's a perfectly sensible chain of reasoning.<br />
:Cueball (off-panel): All right...<br />
<br />
:[Miss Lenhart is facing the whiteboard again writing more scribbles behind some of the lines from before (the first line has disappeared). The lines that have more text added are now number three and five (four and six before). Cueball again speaks off-panel.]<br />
:Miss Lenhart: Now, let's assume that the correct answer will eventually be written on the board at the coordinates (''x, y''). If we—<br />
:Cueball (off-panel): I ''knew'' it!<br />
<br />
{{comic discussion}}<br />
<br />
[[Category:Comics featuring Miss Lenhart]]<br />
[[Category:Comics featuring Cueball]]<br />
[[Category:Math]]</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=1257591724: Proofs2016-08-25T14:31:19Z<p>162.158.133.138: </p>
<hr />
<div>{{comic<br />
| number = 1724<br />
| date = August 24, 2016<br />
| title = Proofs<br />
| image = proofs.png<br />
| titletext = Next, let's assume the decision of whether to take the Axiom of Choice is made by a deterministic process ...<br />
}}<br />
<br />
==Explanation==<br />
[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those "Black Magic" (unclear, incomprehensible) proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.<br />
<br />
The proof she starts setting up resembles a {{w|proof by contradiction}}. These often involve making an assumption that there exists some formula or figure that fulfills the requirements given and plucking that answer out of abstract mathematics, much like summoning of demons is associated with black magic. This is usually done by relying on knowledge of the constraints of the form (for example, having the square root of 2 be ''a/b'' where ''a'' and ''b'' are both integers and have no common factors when proving that the square root of 2 is irrational). This common usage is then shown to be not the case in the comic as the proof then goes to claim that the answer will be written in a specific place (though this could be taken as indicating that the result is finite or has a simple algorithm for continuing it).<br />
<br />
The way, Ms Lenhart's proof refers to the act of proving itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if finally they must be a "perfectly sensible chain of resoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions or curves, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but cannot be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken.<br />
<br />
In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. The {{w|axiom of determinacy}} is {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with the {{w|axiom of choice}}, which is the continuation of the joke of these dark magic proofs. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].<br />
<br />
Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in the her undergraduate class.<br />
<br />
==Transcript==<br />
:[Miss Lenhart is standing facing left in front of a whiteboard writing on it. Eleven left aligned lines of writing is shown as unreadable scribbles. A voice interrupts her from off-panel right.]<br />
:Miss Lenhart: ... Let's assume there exists some function ''F''(''a,b,c''...) which produces the correct answer-<br />
:Cueball (off-panel): Hang on.<br />
<br />
:[In a frame-less panel Cueball is sitting on a chair at a desk with a pen in his hand taking notes.]<br />
:Cueball: This is going to be one of those weird, dark magic proofs, isn't it? I can tell.<br />
<br />
:[Miss Lenhart has turned right towards Cueball, who is again speaking off-panel. The white board is also off-panel.]<br />
:Miss Lenhart: What? No, no, it's a perfectly sensible chain of reasoning.<br />
:Cueball (off-panel): All right...<br />
<br />
:[Miss Lenhart is facing the whiteboard again writing more scribbles behind some of the lines from before (the first line has disappeared). The lines that have more text added are now number three and five (four and six before). Cueball again speaks off-panel.]<br />
:Miss Lenhart: Now, let's assume that the correct answer will eventually be written on the board at the coordinates (''x, y''). If we—<br />
:Cueball (off-panel): I ''knew'' it!<br />
<br />
{{comic discussion}}<br />
<br />
[[Category:Comics featuring Miss Lenhart]]<br />
[[Category:Comics featuring Cueball]]<br />
[[Category:Math]]</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=1257581724: Proofs2016-08-25T14:26:35Z<p>162.158.133.138: /* Explanation */</p>
<hr />
<div>{{comic<br />
| number = 1724<br />
| date = August 24, 2016<br />
| title = Proofs<br />
| image = proofs.png<br />
| titletext = Next, let's assume the decision of whether to take the Axiom of Choice is made by a deterministic process ...<br />
}}<br />
<br />
==Explanation==<br />
{{incomplete|More on the match, especially the title text.}}<br />
[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those {{w|Magic_(programming)#Variants|Dark Magic}} proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.<br />
<br />
If this actually refers to the proof being magical, or just to the fact that many students often feel like the resulting proof just appeared without any reason, i.e. either the teacher did not do it clearly, or the student is not up to the task of understanding proofs of that complexity, is not clear.<br />
<br />
The proof she starts setting up resembles a {{w|proof by contradiction}}. These often involve making an assumption that there exists some formula or figure that fulfills the requirements given and plucking that answer out of abstract mathematics, much like summoning of demons is associated with black magic. This is usually done by relying on knowledge of the constraints of the form (for example, having the square root of 2 be ''a/b'' where ''a'' and ''b'' are both integers and have no common factors when proving that the square root of 2 is irrational). This common usage is then shown to be not the case in the comic as the proof then goes to claim that the answer will be written in a specific place (though this could be taken as indicating that the result is finite or has a simple algorithm for continuing it).<br />
<br />
The way, Ms Lenhart's proof refers to the act of proving itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if finally they must be a "perfectly sensible chain of resoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions or curves, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but cannot be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements become impossible to prove and disprove if the {{w|axiom of choice}} is not taken.<br />
<br />
In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. The {{w|axiom of determinacy}} is {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with the {{w|axiom of choice}}, which is the continuation of the joke of these dark magic proofs. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].<br />
<br />
Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in the her undergraduate class.<br />
<br />
==Transcript==<br />
:[Miss Lenhart is standing facing left in front of a whiteboard writing on it. Eleven left aligned lines of writing is shown as unreadable scribbles. A voice interrupts her from off-panel right.]<br />
:Miss Lenhart: ... Let's assume there exists some function ''F''(''a,b,c''...) which produces the correct answer-<br />
:Cueball (off-panel): Hang on.<br />
<br />
:[In a frame-less panel Cueball is sitting on a chair at a desk with a pen in his hand taking notes.]<br />
:Cueball: This is going to be one of those weird, dark magic proofs, isn't it? I can tell.<br />
<br />
:[Miss Lenhart has turned right towards Cueball, who is again speaking off-panel. The white board is also off-panel.]<br />
:Miss Lenhart: What? No, no, it's a perfectly sensible chain of reasoning.<br />
:Cueball (off-panel): All right...<br />
<br />
:[Miss Lenhart is facing the whiteboard again writing more scribbles behind some of the lines from before (the first line has disappeared). The lines that have more text added are now number three and five (four and six before). Cueball again speaks off-panel.]<br />
:Miss Lenhart: Now, let's assume that the correct answer will eventually be written on the board at the coordinates (''x, y''). If we—<br />
:Cueball (off-panel): I ''knew'' it!<br />
<br />
{{comic discussion}}<br />
<br />
[[Category:Comics featuring Miss Lenhart]]<br />
[[Category:Comics featuring Cueball]]<br />
[[Category:Math]]</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=1257571724: Proofs2016-08-25T14:23:13Z<p>162.158.133.138: </p>
<hr />
<div>{{comic<br />
| number = 1724<br />
| date = August 24, 2016<br />
| title = Proofs<br />
| image = proofs.png<br />
| titletext = Next, let's assume the decision of whether to take the Axiom of Choice is made by a deterministic process ...<br />
}}<br />
<br />
==Explanation==<br />
{{incomplete|More on the match, especially the title text.}}<br />
[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those {{w|Magic_(programming)#Variants|Dark Magic}} proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.<br />
<br />
If this actually refers to the proof being magical, or just to the fact that many students often feel like the resulting proof just appeared without any reason, i.e. either the teacher did not do it clearly, or the student is not up to the task of understanding proofs of that complexity, is not clear.<br />
<br />
The proof she starts setting up resembles a {{w|proof by contradiction}}. These often involve making an assumption that there exists some formula or figure that fulfills the requirements given and plucking that answer out of abstract mathematics, much like summoning of demons is associated with black magic. This is usually done by relying on knowledge of the constraints of the form (for example, having the square root of 2 be ''a/b'' where ''a'' and ''b'' are both integers and have no common factors when proving that the square root of 2 is irrational). This common usage is then shown to be not the case in the comic as the proof then goes to claim that the answer will be written in a specific place (though this could be taken as indicating that the result is finite or has a simple algorithm for continuing it).<br />
<br />
The way, Ms Lenhart's proof refers to the act of proving itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if finally they must be a "perfectly sensible chain of resoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions or curves, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but cannot be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements becomes impossible to prove and disprove if the {{w|axiom of choice}} is not taken.<br />
<br />
In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. The {{w|axiom of determinacy}} is {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with the {{w|axiom of choice}}, which is the continuation of the joke of these dark magic proofs. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].<br />
<br />
Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in the her undergraduate class.<br />
<br />
==Transcript==<br />
:[Miss Lenhart is standing facing left in front of a whiteboard writing on it. Eleven left aligned lines of writing is shown as unreadable scribbles. A voice interrupts her from off-panel right.]<br />
:Miss Lenhart: ... Let's assume there exists some function ''F''(''a,b,c''...) which produces the correct answer-<br />
:Cueball (off-panel): Hang on.<br />
<br />
:[In a frame-less panel Cueball is sitting on a chair at a desk with a pen in his hand taking notes.]<br />
:Cueball: This is going to be one of those weird, dark magic proofs, isn't it? I can tell.<br />
<br />
:[Miss Lenhart has turned right towards Cueball, who is again speaking off-panel. The white board is also off-panel.]<br />
:Miss Lenhart: What? No, no, it's a perfectly sensible chain of reasoning.<br />
:Cueball (off-panel): All right...<br />
<br />
:[Miss Lenhart is facing the whiteboard again writing more scribbles behind some of the lines from before (the first line has disappeared). The lines that have more text added are now number three and five (four and six before). Cueball again speaks off-panel.]<br />
:Miss Lenhart: Now, let's assume that the correct answer will eventually be written on the board at the coordinates (''x, y''). If we—<br />
:Cueball (off-panel): I ''knew'' it!<br />
<br />
{{comic discussion}}<br />
<br />
[[Category:Comics featuring Miss Lenhart]]<br />
[[Category:Comics featuring Cueball]]<br />
[[Category:Math]]</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=1257561724: Proofs2016-08-25T14:19:17Z<p>162.158.133.138: </p>
<hr />
<div>{{comic<br />
| number = 1724<br />
| date = August 24, 2016<br />
| title = Proofs<br />
| image = proofs.png<br />
| titletext = Next, let's assume the decision of whether to take the Axiom of Choice is made by a deterministic process ...<br />
}}<br />
<br />
==Explanation==<br />
{{incomplete|More on the match, especially the title text.}}<br />
[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those {{w|Magic_(programming)#Variants|Dark Magic}} proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.<br />
<br />
If this actually refers to the proof being magical, or just to the fact that many students often feel like the resulting proof just appeared without any reason, i.e. either the teacher did not do it clearly, or the student is not up to the task of understanding proofs of that complexity, is not clear.<br />
<br />
The proof she starts setting up resembles a {{w|proof by contradiction}}. These often involve making an assumption that there exists some formula or figure that fulfills the requirements given and plucking that answer out of abstract mathematics, much like summoning of demons is associated with black magic. This is usually done by relying on knowledge of the constraints of the form (for example, having the square root of 2 be ''a/b'' where ''a'' and ''b'' are both integers and have no common factors when proving that the square root of 2 is irrational). This common usage is then shown to be not the case in the comic as the proof then goes to claim that the answer will be written in a specific place (though this could be taken as indicating that the result is finite or has a simple algorithm for continuing it). This may also be a reference to proof by induction, which can be thought of as a proof of the existence of an infinite number of more specific proofs.<br />
<br />
The way, Ms Lenhart's proof refers to the act of proving itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if finally they must be a "perfectly sensible chain of resoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions or curves, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but cannot be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements becomes impossible to prove and disprove if the {{w|axiom of choice}} is not taken.<br />
<br />
In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. The {{w|axiom of determinacy}} is {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with the {{w|axiom of choice}}, which is the continuation of the joke of these dark magic proofs. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].<br />
<br />
Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in the her undergraduate class.<br />
<br />
==Transcript==<br />
:[Miss Lenhart is standing facing left in front of a whiteboard writing on it. Eleven left aligned lines of writing is shown as unreadable scribbles. A voice interrupts her from off-panel right.]<br />
:Miss Lenhart: ... Let's assume there exists some function ''F''(''a,b,c''...) which produces the correct answer-<br />
:Cueball (off-panel): Hang on.<br />
<br />
:[In a frame-less panel Cueball is sitting on a chair at a desk with a pen in his hand taking notes.]<br />
:Cueball: This is going to be one of those weird, dark magic proofs, isn't it? I can tell.<br />
<br />
:[Miss Lenhart has turned right towards Cueball, who is again speaking off-panel. The white board is also off-panel.]<br />
:Miss Lenhart: What? No, no, it's a perfectly sensible chain of reasoning.<br />
:Cueball (off-panel): All right...<br />
<br />
:[Miss Lenhart is facing the whiteboard again writing more scribbles behind some of the lines from before (the first line has disappeared). The lines that have more text added are now number three and five (four and six before). Cueball again speaks off-panel.]<br />
:Miss Lenhart: Now, let's assume that the correct answer will eventually be written on the board at the coordinates (''x, y''). If we—<br />
:Cueball (off-panel): I ''knew'' it!<br />
<br />
{{comic discussion}}<br />
<br />
[[Category:Comics featuring Miss Lenhart]]<br />
[[Category:Comics featuring Cueball]]<br />
[[Category:Math]]</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=1724:_Proofs&diff=1257551724: Proofs2016-08-25T14:17:54Z<p>162.158.133.138: more about mathematics, especially unnoticed reference to Gödel</p>
<hr />
<div>{{comic<br />
| number = 1724<br />
| date = August 24, 2016<br />
| title = Proofs<br />
| image = proofs.png<br />
| titletext = Next, let's assume the decision of whether to take the Axiom of Choice is made by a deterministic process ...<br />
}}<br />
<br />
==Explanation==<br />
{{incomplete|More on the match, especially the title text.}}<br />
[[Miss Lenhart]] is back teaching a math class. She begins a proof when one of her students ([[Cueball]]) interrupts her asking if this is one of those {{w|Magic_(programming)#Variants|Dark Magic}} proofs. She says no, but it soon turns out that it is; Cueball exclaims that he just knew it would be.<br />
<br />
If this actually refers to the proof being magical, or just to the fact that many students often feel like the resulting proof just appeared without any reason, i.e. either the teacher did not do it clearly, or the student is not up to the task of understanding proofs of that complexity, is not clear.<br />
<br />
The proof she starts setting up resembles a {{w|proof by contradiction}}. These often involve making an assumption that there exists some formula or figure that fulfills the requirements given and plucking that answer out of abstract mathematics, much like summoning of demons is associated with black magic. This is usually done by relying on knowledge of the constraints of the form (for example, having the square root of 2 be ''a/b'' where ''a'' and ''b'' are both integers and have no common factors when proving that the square root of 2 is irrational). This common usage is then shown to be not the case in the comic as the proof then goes to claim that the answer will be written in a specific place (though this could be taken as indicating that the result is finite or has a simple algorithm for continuing it). This may also be a reference to proof by induction, which can be thought of as a proof of the existence of an infinite number of more specific proofs.<br />
<br />
The way, Ms Lenhart's proof refers to the act of proving itself, is characteristic to metamathematical proofs, for example {{w|Gödel's incompleteness theorems}}, which, at first sight, may indeed look like black magic, even if finally they must be a "completely reasonable chain of resoning" like the rest of good mathematics. While standard mathematical theorems and their proofs deal with standard mathematical objects, like numbers, functions or curves, the metamathematical theorems treat other theorems as objects of interest. In this way you can propose and prove theorems about possibility of proving other theorems. For example, in 1931 {{w|Kurt Gödel}} was able to prove that any mathematical system based on arithmetics (that is using numbers) has statements that are true, but cannot be neither proved nor disproved. This kind of metamathematical reasoning is especially useful in the {{w|set theory}}, where many statements becomes impossible to prove and disprove if the {{w|axiom of choice}} is not taken.<br />
<br />
In the title text the decision of whether to take the axiom of choice is made by a deterministic process, that is a process which future states can be developed with no randomness involved. The {{w|axiom of determinacy}} is {{w|Axiom_of_determinacy#Incompatibility_of_the_axiom_of_determinacy_with_the_axiom_of_choice|incompatible}} with the {{w|axiom of choice}}, which is the continuation of the joke of these dark magic proofs. The axiom of choice was mentioned earlier in [[804: Pumpkin Carving]].<br />
<br />
Although Miss Lenhart did retire a year ago after [[1519: Venus]], she seems to have returned here for a math course at university level, but continues the trend she finished with in the her undergraduate class.<br />
<br />
==Transcript==<br />
:[Miss Lenhart is standing facing left in front of a whiteboard writing on it. Eleven left aligned lines of writing is shown as unreadable scribbles. A voice interrupts her from off-panel right.]<br />
:Miss Lenhart: ... Let's assume there exists some function ''F''(''a,b,c''...) which produces the correct answer-<br />
:Cueball (off-panel): Hang on.<br />
<br />
:[In a frame-less panel Cueball is sitting on a chair at a desk with a pen in his hand taking notes.]<br />
:Cueball: This is going to be one of those weird, dark magic proofs, isn't it? I can tell.<br />
<br />
:[Miss Lenhart has turned right towards Cueball, who is again speaking off-panel. The white board is also off-panel.]<br />
:Miss Lenhart: What? No, no, it's a perfectly sensible chain of reasoning.<br />
:Cueball (off-panel): All right...<br />
<br />
:[Miss Lenhart is facing the whiteboard again writing more scribbles behind some of the lines from before (the first line has disappeared). The lines that have more text added are now number three and five (four and six before). Cueball again speaks off-panel.]<br />
:Miss Lenhart: Now, let's assume that the correct answer will eventually be written on the board at the coordinates (''x, y''). If we—<br />
:Cueball (off-panel): I ''knew'' it!<br />
<br />
{{comic discussion}}<br />
<br />
[[Category:Comics featuring Miss Lenhart]]<br />
[[Category:Comics featuring Cueball]]<br />
[[Category:Math]]</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=Talk:1666:_Brain_Upload&diff=117411Talk:1666: Brain Upload2016-04-11T16:59:00Z<p>162.158.133.138: </p>
<hr />
<div><!-- Please sign your posts with a ~~~~ --><br />
I don't think you can assume Cueball is doing it to live longer. I figured he did it just to allow the researcher to experiment with the system. If I uploaded my consciousness into a computer, *it* might 'live' longer than me, but *I* will not live any longer for having done it. demiller9 [[Special:Contributions/162.158.68.47|162.158.68.47]] 16:15, 11 April 2016 (UTC)<br />
<br />
Are we completely certain that its Cueball? Initially from reading it i would be more inclined to believe that its Beret Guy<br />
[[Special:Contributions/162.158.133.138|162.158.133.138]] 16:59, 11 April 2016 (UTC)</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=Danish&diff=114939Danish2016-03-15T14:32:20Z<p>162.158.133.138: </p>
<hr />
<div>{{Infobox character<br />
| image = Danish.png<br />
| caption = Danish, as seen in [[1027: Pickup Artist]]<br />
| first_appearance = [[377: Journal 2]]<br />
}}<br />
<br />
'''Danish''' is a [[stick figure]] character in [[xkcd]]. She is very similar in appearance to [[Megan]], but has heavier hair. Personality-wise, she is to [[Black Hat]] as [[Megan]] is to [[Cueball]].<br />
<br />
Like most xkcd characters, her real name is unknown. The name "Danish" is picked from when [[Black Hat]] called her "my dearest darling danish", as a term of endearment, in [[515: No One Must Know]]. As she replies to him with "my lovely cutie pie", Black Hat's name should thus have been Pie, but he already had a better name. Obviously this is not her name but just part of the joke that they call each other cake names, i.e. {{w|danish pastry}} and {{w|Pie}}. But it is as good a name as any.<br />
<br />
A girl that physically looks like Danish appeared in [[139: I Have Owned Two Electric Skateboards]], long before she was introduced in the [[:Category:Journal|Journal series]].<br />
In [[177: Alice and Bob]] a similar looking character with propensity to destruction is called Eve – a reference to cryptographic schemes involving communication between Alice and Bob with Eve playing role of an attacker.<br />
<br />
She has also appeared in [[515: No One Must Know]] and [[542: Cover-Up]], in which she appears to be in a continued romantic relationship with [[Black Hat]].<br />
<br />
==Characteristics==<br />
Danish is characterized by her cynicism and devilish tricks; and physically by her hair, which is somewhat longer than [[Megan]]'s.<br />
<br />
{{navbox-characters}}<br />
<br />
[[Category:Characters]]</div>162.158.133.138https://www.explainxkcd.com/wiki/index.php?title=Talk:606:_Cutting_Edge&diff=108894Talk:606: Cutting Edge2016-01-08T16:46:14Z<p>162.158.133.138: /* update on Command & Conquer servers */ new section</p>
<hr />
<div>Today is early 2013. This comic is right. [[User:Greyson|Greyson]] ([[User talk:Greyson|talk]]) 16:52, 5 March 2013 (UTC)<br />
I tested the hypothesis in the last panel by announcing "FREE CAKE!", setting down an empty cake box with "The cake is a lie" written inside it, and running. The results were highly conclusive: memes do not magically revive after five years. [[Special:Contributions/107.204.46.198|107.204.46.198]] 00:43, 2 May 2013 (UTC)<br />
<br />
"The title text also points to another flaw in this strategy: multi-player gaming requires other players, so if you play a game five years after everybody else, there's nobody else to play with. It's even worse with online gaming, as the company hosting the online server may have shut it down a long time ago." Unless the game happens to be team fortress 2.[[Special:Contributions/162.158.6.246|162.158.6.246]] 23:06, 21 September 2015 (UTC)<br />
<br />
== update on Command & Conquer servers ==<br />
<br />
The servers have been shut down in 2014, about five years after this comic was launched, but game enthusiasts managed to run their own servers.<br />
So it is still possible to play this game now in 2016. :-) <br />
Source: http://www.eurogamer.net/articles/2014-07-07-command-and-conquer-multiplayer-saved-following-gamespy-shutdown<br />
[[Special:Contributions/162.158.133.138|162.158.133.138]] 16:46, 8 January 2016 (UTC)</div>162.158.133.138