Talk:Mathematical fallacy

2964:'consistency check'.) Yes, both share a 90 degree angle, but clearly the angles RBO and QCO are NOT equal! (nor, of course are angles ROB and QOC.) I guess the "hidden fallacy" is assuming that BOQ and COR are (straight) lines (line segments). They're not. (Depending on how the fallacy is set up, you could also propose/pretend that the perpendicular line from AC to B (and also from AB to C) intersects at O, which it doesn't (generally).) I find it hard to believe that this has been allowed to exist for 12 (plus?) years! I keep going over the thing, and ...nope, Line #7 does NOT follow from any preceding line, nor is it correct. Segments BR and CQ aren't (necessarily) equal, nor are angles BOR and COQ. Segments BO and CO are equal, since O is on the perp. bisector. So we have two triangles with one (90°) angle and one (opposite) side equal. So what? What follows line 7 is utter rubbish, by the way. This needs a rewrite, and this time I politely suggest it be coherently written by someone who understands middle-school geometry. 367:, and judging by what I just archived, alot of the stuff in this article is exactly that. Surely, some of the fallacies are notable in their commonalities (dividing or multiplying by zero, using multivalued complex functions incorrectly), but these things should be discussed briefly in an encyclopedic fashion. They shouldn't be accumulation points for a giant collection of invalid deductions. This is on my mental to-do list, and I may start cutting redundant or unnecessary things from the article, so if anyone wants to help or contribute their thoughts, I'd encourage you to do so. I'd be more than happy if someone else did some of the work here or contributed their ideas about what should happen to this article. -- 2530:. However, the derivative of a function of integer argument is not defined (in the naive calculus sense, at least) because the integers are not interior points (of the set of integers). So we don't even get to the point where it is possible to apply the differentiation rule for sums. As PST points out, there are generalizations of the derivative that do make sense. An alternative analysis might run something like this: "equality of two functions evaluated at integer points does not imply equality of the derivatives." But I think that this is less clear, since the function is a priori only defined for integer arguments: there is no canonical way to "interpolate" it. 2455:

subtle errors, and how subtle alterations may be necessary (and in this case, the proof cannot be fixed). If you defend your revert here, it is certainly possible to restore your edit. However, conforming to WP guidelines, once an editor has challenged an edit (by reverting), the person behind the challenged edit must defend the edit (and reach consensus) before adding it back. Generally in Knowledge, we add rather than delete content and I think that this should be at the forefront of your mind when editing. You may certainly rewrite the article, but please be careful in terms of deleting content which was maintained for so many years by hard-working editors. --

237: 2914: 227: 206: 178: 316: 4434: 5015:... and the divide by zero is invoked, so to speak, by having a disguised zero = zero in line 3, i.e. false or irrelevant logic e.g. a fallacy, this is then on-sold by factoring ((needlessly in reality) before dividing. Surely getting a bad expression past the reader (the student, fallacy target etc. not wiki reader here) is the actual invoking of the fallacy condition. Happy to learn, hence re-visiting, cheers 991:

proof that 1=0 as an exercise. In my experience these sorts of things do make it as "interesting" exercises in basic texts on grade school algebra and introductory calculus. Of course, I agree with the previous comments that the current article is heavily redundant, and more or less not encyclopedic in its current form. I think a few good examples can be lifted and put into a more general article.

22: 4499:

has to do is find a book or paper or newspaper article or something that says something and reference that if other people disagree with something being included. Verifiability does not mean everything needs a citation in the article - but it does mean that if anyone disagrees and thinks something needs a citation they can remove the stuff if a citation can't be found in a reasonable time see

2473:

another. Moreover, the templates at the top of the page indicate that (1) much of the content is not encyclopedic (and so should logically be removed), and (2) that the article requires cleanup. I assume that you disagree with these, but you seem to be in the minority. My purpose is to completely rewrite the article from sources: in particular, this will establish appropriate context for

363:

exact same thing. That's, by my estimation, half or a third of the article. I believe with a little effort, this article can be substantially tidied. I would, however, raise my concerns now that this article isn't encyclopedic as it stands. Nothing in this article is verifiable, it's simply an anecdotal collection of all those impossible "proofs" that kids hear in math class. Like they say,

2569:, the phrase "proof is false" is at best confusing. I suggest something along the lines of "proof is faulty", "demonstration fails" (to convince), "argument is invalid", i. e. concentrate on showing that there is a flaw in the argument, rather than pointing to the conclusion's truth value. Proof is generally understood to imply validity, an invalid argument loses its proofness. 1630:

be deduced that one third of one is not 3.3 recurring, but a very slightly different number, or that three thirds of one isnt one, a paradox. This cannot be applied to numbers divisible by three i.e. 9. One third of nine is three. Three times three is nine. However one third of 100 is in fact 33.3 recurring, and 33.3 recurring times three 99.9 recurrring. Phew! Maths.

1513: 2994: 80: 53: 5124: 4797: 4471:

repeatedly reverted and removed with reasons as "the original version was correct", and then "it was in a wrong section" after explanation was given why the original version was not appropriate, and then "there was no reference" after the modification was moved a separate section, and then "it was not a reference" after a reference was provided.

4727: 90: 2580:, nothing prevents us from defining a fractional summation compatible with the traditional definition, in which the bounds may range over the reals, a simple case being rounding of the bounds. This concept can be generalized to functions, leading to the possiblity of applying a function π times to an argument. A instructive case is the 4359:

The fallacy as illustrated by the original example is commonly discussed with extraneous solutions. I agree this article should not be an endless list of fallacies, but rather it should address the notion, and types of common fallacies. Extraneous solution is such an important type that should not be

3226:

I just want to say that I agree that this is not a kind of article I'm proud of having in wikipedia. On the other hand, I don't see what can be done, either. Reading the thread above, this seems like a long-time problem, and no one figured out how to solve it. Maybe someone does! (I would concentrate

3124:

This article primarily consists of (unreferenced) examples of false proofs (for which there is no hope of ever supplying references). There is no shortage of published accounts of individual mathematical fallacies from which to draw, if necessary. But the article simply cannot be an endless account

990:

WolfKeeper mentioned that he "strongly suspects it is quite referenceable to maths textbooks". I think this is likely to be somewhat true, especially if the article is merged to a mathematical fallacy article. For instance, I believe Stewart's calculus contains the "+C" indefinite/definite integral

740:

At the very least, if we keep this we should get rid of the "Proof that 2+2=5" style section headings. They convey no useful information about what happens in the sections; once one has "derived" a falsehood, it is trivial and arbitrary how one cleans up that falsehood to arrive at a more spectacular

5037:

Yes, there is a "disguised" 0=0 in lines 3 and 4, but there is no fallacy because indeed 0=0. The action of factoring out (a-b) is not irrelevant. On the contrary, it is necessary as a preparation for the actual fallacy of dividing by zero, which happens in line 5. And of course, also note that what

4498:

I have removed the addition again. It referenced another Knowledge article - one without any references. That other problems exist is no reason to start sticking in more when other people disagree. This article really has had too much uncited stuff stuck into it and needed to be cleaned up. What one

4482:

As I'm leaving, I challenge the readers of this article to find a "proper" reference for the original example of "3=0" and have this example included back into this page, because the Knowledge community is better served with this example on this page than without. I'd be happy to see this example be

4441:

Construct a rectangle ABCD. Now identify a point E such that CD = CE and the angle DCE is a non-zero angle. Take the perpendicular bisector of AD, crossing at F, and the perpendicular bisector of AE, crossing at G. Label where the two perpendicular bisectors intersect as H and join this point to A,

3177:

There's no shortage of cool fallacies out there, many of them in published books. That's kind of my point. This article isn't a book on the subject of mathematical fallacies, and our purpose isn't to emulate them. Instead, it's an encyclopedia article about mathematical fallacies. So I think the

2805:

The page has been moved, and I have rewritten the lead section. I have also cut six of the examples for various reasons. I'm not too worried about "original research" in the article, since most of the ideas involved are at least a century old, and the generation of a fresh example from a traditional

2213:

Both are right; you get to pick your poison. If you work with a square root function defined only on the non-negative reals the error is where Pokipsy points at. But it is not uncommon either to work with a square root that is defined for all complex numbers but discontinuous along the negative real

750:

Ha! "Unverifiable material may be challenged and removed." I love it. OK, let's verify the invalid proofs. Seriously though, mathematical proofs (including, as here, proofs that certain proofs are invalid) should stand by themselves; it's useful to know who wrote about them first, or before, but the

444:

What, like a complex analysis textbook will explain how the square root is really "multi-valued"? Or an algebra book will explain that the 0 of a ring has no inverse? These joke proofs circulate on message boards and people show each other, but they have little value and I doubt they are laid out in

4470:

I came across this article while searching for a "sophisticated proof" of 1=0, and found the original section "x cubed =1" perfect in that regard. Having realized that the original statement didn't convey what the fallacy intends to convey, I contributed a modification. However, my contribution was

3206:

I think you're taking this article too seriously. We should have a category 'recreational mathematics' on Knowledge where stuff like this can go perhaps. What you are talking about sounds more like maths pedagogy and correcting students' misconceptions which is nothing at all like what this article

3100:

The first sentence: "Functions that are multivalued have poorly defined inverse functions over their entire range." is false. For example a vertical line is a multivalued function because for the given value of x there are infinitely many value of y. The inverse of a vertical line is a horizontal

1629:

Could the fact that three mathematical thirds of one isnt in fact one but 9.9 recurring be included in the article, since it is very simplistic. 9.9 recurring is infinitely close, but as infinity is a destination and not a fixed point, it is not equal to one. If this logic applied correctly, it can

908:

article). Then, rather than laying out as a series of "proofs" followed by "the fallacy here is that ..." explanations, it could be categorised by types of fallacy, with an explanation of each followed by an illustrative example "proof". That way it might seem more like an encylopedia article. Matt

663:

I think you misread the meaning of facts in this context. WP writes about facts as in well known existing terms/notions/phenomenons/concepts/etc and not as in writing only about things, which are scientifically proven to exist. Those invalid proofs exists as philosophical concepts, religious terms,

362:

I hate to say it, but this article is a complete mess. It contains a substantial amount of redundant content. Square roots produce 1=-1 because they can be multivalued. So do fourth roots, or 3/2 powers, or 3/2 powers of things that are equal to -1 but obfuscated. We don't need five examples of the

4958:

In line 3 of section Division by zero, show me any number where the expression does not come back to 0 = 0, there is none, zero then populates line 4 to factor zero and line 5 to reduce/divide by zero, if we are going to use this example it needs to be correct and shown so that non-math people can

1126:

from MAA Online for a discussion of how Euler himself was misled into believing the result for the real numbers since mathematics at the time lacked a definition of convergence. Since the result is false for the standard metric, but true for the 2-adic metric, this example may be too subtle for a

3104:

In the explanation of the "proof" that 2\pi = 0 the text states that the arcsin is a infinitely mulitvalued function. This is also false. The arcsin function is a single valued function. Given any value of x, arcsin(x) is a number between -\pi/2 and \pi/2. This means that arcsin(sin(x)) only

2963:

WOW! 12 years later and this section is in gross error! Specifically: Line 7 in the "proof" claims that the two triangles BOR and COQ are congruent! LMFAO!! Simple inspection of the accompanying diagram shows they are not. (Not that that means much, but still your first clue ought to be a visual

2472:

If your attitude is simply to revert any good faith effort to reform the article, then I see no way that progress can possibly be made. There is a clear and obvious consensus on this very discussion page that there are too many examples, and that many of the examples are almost exactly like one

2454:

I am sure that you are editing with good faith, but your two most recent edits were not appropriate. One of the (false) proofs your deleted was an attempt to correct another similar proof, and should be included in the article. This example is a particularly basic instance of how proofs can have

2412:

I have put citation needed on all the proofs without any citation. I think asking for a citation is a good way of cutting down the proofs to those which people though were worth printing and might help to remove errors. I will have a look at providing citations for some which I know I've seen in

1117:

There isn't anything particularly twisted about the validity of the convergence of this series to -1 under the 2-adic metric. In fact it is a rather pedestrian result and a standard preliminary example in texts on the p-adic numbers (see Gouvea, "p-adic Numbers", 1997). It is fallacious for the

1054:

Yes, there is a fundamental tension between a "clean" invalid proof (where the fallacy is laid bare, like dividing by zero) and an invalid proof where it's not obvious what's gone wrong. (For example here, you could even use "S = 1+1+1+...".) But anyway, I concur with what you're saying -- the

941:

Another vote for renaming this to 'mathematical fallacy'. In fact, I came across this article because I was looking for a reference on logic, not (non)algebra. I also concur that the section headings should be re-named, and perhaps each fallacy could be done (only) twice: once in an 'obscured'

916:

I agree: the encyclopedic content is the list of common fallacies, ideally with sources. What we can prove with them is interesting but secondary material to illustrate their use. Of course, statements like 2=1, 3=0 etc. are equivalent because given any one of them as an axiom, the rest can be

4478:

Now this begs the question why Sławomir kept removing such a common example of fallacy from this article, given there has been other section in the article without a reference. I don't have the answer, but I hope it is not to try to avoid the embarrassment of, as a regular of this article, not

4474:

I understand the need to police the page. However, by looking at the history of the article, it is not the case that the article had a history of being vandalized previously. After all, this is a small article compared with most Knowledge articles. Even if the section is not perfect, would the

4452:

The error in the proof comes in the diagram and the final point. An accurate diagram would show that the triangle ECH is a reflection of the triangle DCH in the line CH rather than being on the same side, and so while the angles DCH and ECH are equal in magnitude, there is no justification for

3063:

I think it would be nice if somebody could clean up the inappropriate "Q.E.D."s that have been used all over the place. For example by making them appropriate by stating up front what is going "to be demonstrated". It would make things a lot more readable and as well as not look so illiterate.

2821:

Excellent job on the lead. The title of the article should probably be "mathematical fallacy" in the singular, rather than the plural of "mathematical fallacies", per Knowledge's typical naming conventions, but otherwise I strongly support the new title. I agree with most (if not all) of the

2382:

There are too many examples of essentially the same thing. I propose that some effort should be made to give sources for things, and ultimately when a source is found to be lacking, content should be removed. (A better place for a lot of the article is Wikibooks. Maybe someone could move it

4986:

in line 5", suggests that the fallacy started before line 5. It didn't. I think that non-math people deserve precision, so I reverted your edit because of this sloppiness, and because the original wording was correct and sufficient. There were no errors or fallacies in lines 3 and 4, and your

2742:

While we are certainly free to redefine things in any way that suits us, as long as we stick to conventional mathematics, the meaning of a finite sum is fairly unambiguous. In fact, most professional mathematicians are probably scarcely aware of the existence of the fractional calculus. So,

2730:

point is one that could be made of pretty much the entire article, by the way. So far I think I am the only editor who has made any systematic effort to source the claims of the article. At any rate, the OR issue in this particular case is not serious, since the fallacy itself appeared in a

4591:. We can quibble over the semantics of the best way to lay out the fallacy then, but I still firmly believe that the original version was better than your proposed one. Your proposed version makes no mention whatsoever of complex numbers. Nowhere does it say that the two non-real roots of 1075:

Judging by the main page on this series this actually has some twisted mathematical merit to it. At any rate the current objection to the proof is "you don't use a convergence formula on something that doesn't converge", but the s=1+2s manipulation yields this without assuming anything about

3079:

I have just made a number of minor edits to deal with this. In doing so I found that the entries with inappropriate QEDs were invariably lacking a citation. For this and other reasons there is more scope for cleaning up for somebody who has the time. It is a nice collection of examples.

1205: 2681:

here is what is meant by "derivative with respect to x", further implying the question of what x's domain is intended to be. I daresay that the majority of readers will imply "naive" calculus and real numbers, respectively, but without some serious evaluation, that is just an educated

4911:

People write books with 'mathematical fallacy' in the title. They don't write ones about 'invalid proof'. The better common name for the topic is mathematical fallacy. As described in the lead there's a lot more to being a mathematical fallacy than it simply being an invalid proof.

963:. However i do consider the content (posssibly modified) of encyclopaedic value in the sense of listing typical (and well known) mathematical fallacies. And as mentioned above you can find at least some (or most?) examples in literature as well. One example would be Heuser's 3012:

I think that's backward. The text does indeed explain that BOR and COQ are congruent by hypotenuse-leg equality; but it asserts without proof that the hypotenuses are equal, and they aren't. This is actually quite clear from the diagram. The real error is trusting the

2431:

I seem to have mislaid the old book I was going to look in "Riddles in Mathematiics' by Eugene P Northop. If itr is the one I was thinking of it has a number of interesting fallacies in it plus puzzles like the one about the probable length of a chord of a circle.

2689:": Sorry, but first and foremost, they are expressions. You may use these expressions to define functions, of course, but doing so constitutes a choice not implicit in the text of the argument. Of course, you could avail yourself to the argument directly above. ;) 382:

You are right; none of these things appear to have encyclopedic value in themselves. A few of them might make sense as examples in an article about mathematical fallacies in general, but they would need to be subordinate to a general discussion of such fallacies.

4445:

Now, AH=DH because FH is a perpendicular bisector; similarly BH = CH. AH=EH because GH is a perpendicular bisector, so DH = EH. And by construction BA = CD = CE. So the triangles ABH, DCH and ECH are congruent, and so the angles ABH, DCH and ECH are equal.

2509:

in a case where the bounds of summation are not independent of the differentiation variable. I do not know how the current section meshes with the texts it cites, and whether these sources are really relevant here, but this needs to be addressed. Regards,

1149:

I'm going to tag all the sections in this article. If they're not referenced within 3 months, then they're probably never going to be, and so they will be removed. I'm also not going to allow any new sections to be added unless they're referenced.-

613:(unindent) My biggest concern is that this is essentially one big piece of original research, and we'd do best to simply cite a couple of well-known and commonly mistaken "proofs", rather than do the infinite number of ways of "proving" 1=-1. 2692:"don't even get to the point where it is possible to apply the differentiation rule for sums": I see what you're getting at, but as I think I've shown above, that would only be true under assumptions not implicit in the text. Would "in a case 3158:

There's been a number of books on mathematical fallacies and I think it should be straightforward enough to provide citations for most of the article. The references at the end aren't just published false proofs, they are books on fallacies.

4466:

After a few days of trying to contribute and make this article a little better (it is already great), I'm leaving this article and will stop modifying it again simply because I don't have enough time to deal with the ongoing events.

4382:, and left the stub section header in, in the hopes that it would encourage someone with sources to add those. We don't need more uncited stuff in this article. What we need are more reliable sources on mathematical fallacies. 2525:

I have fixed the issue you point out. The cited source isn't clear on where the error lies. I disagree with your proposed analysis of the fallacy, however. Each side of the equation is a well-defined function of the integer

1576:

square(-a) = square(a) x square(-1) square(-a) = a^2 x 1 root(a^2) = a squareroot on a number give a possitive awnser. The logic written in the "paradox" is used in equations because in equations you are "reverse-engeneering"

483:

instructional value? The whole point of an encyclopedia is to be instructional, and the wikipedia, being the biggest encyclopedia ever, has plenty of scope to include information that some will not find in any way important.-

1897: 1129:=1, and hence the series fails the convergence criterion. The validity of the result under the p-adic metric should be relegated to an afterthought. I'll see if I can find a reference which has free online access. pAddict-- 2754:. The derivative (in what amounts to a "naive" calculus interpretation) is defined for functions, not expressions. Is the line between expression and function really so sharp that anyone is likely to be confused by this? 2000: 1508:{\displaystyle {\begin{aligned}1&=1+0+0+0+0+\cdots \\&=0+1+0+0+0+\cdots \\&=0+0+1+0+0+\cdots \\&=0+0+0+1+0+\cdots \\&=0+0+0+0+1+\cdots \\&\vdots \\&=0+0+0+0+0+\cdots \\&=0\end{aligned}}} 2098: 1842: 3178:

question we need to address is: what do we put in it? It's clear (to me at least) that it shouldn't just be a laundry list of fallacies sorted by topic. Otherwise we would look like the already excellent

4899:– As stated in the lead section, mathematical fallacy is a special case of invalid proof. And because they are both covered by this article, I suggest to use the broader title, which was used until 2009. 3034:

The hypotenuses of BOR and COQ are resp. BO and OC, and by design, of course they are equal. That is immediately inferred by looking at the triangles BOD and DOC of which they are the hypotenuses too. -

2747:

point would probably be doubly an issue for using an unconventional notion of "summation", as opposed to the usual notion of "differentiation". It is, at any rate, an utterly unsupported interpretation.

3207:

is about. There is a popular topic of amusing fallacies which is exploited by Cut the Knot along with some books. The article just summarizes a notable topic just like most other articles in Knowledge.

2663: 3137:

in their own right, which is the way things currently stand. I suggest stubbing the article, and trying to compile some sources on this talk page. Perhaps some help can be conscripted from users at

2167: 3021: 3020:

be where the drawing shows it. The error is not in assuming that O is inside the triangle. If you limit the claim to "a triangle that contains its circumcenter is isoceles" then it's still wrong.

1038:

Given that the point of this example of an invalid proof is to show that you can't apply the usual rules to divergent series, an even simpler example of the perils is this proof of 0 is infinite:

4978:. Your wording was sloppy: lines don't have values, lines don't factor values. You could have said that "both terms in line 3 equal zero, a factor zero is factored out in line 4, and the fallacy 1127:

page on mathematical fallacies, and the best course of action may be to remove it. At the very least, this section should state that this is a fallacy under the standard metric since lim_{n-: -->

4479:

spotting the issue with the example previously, and subsequently insisting that the original version was correct after it was pointed out that the original example was not appropriately stated.

1210: 1778: 2986:

The reason why BOR and COQ are congruent is neatly explained in the text, and supported by relevant sources. The point is that O cannot be where it is the drawing, so you better don't do a "

2268: 1937: 293: 2670:

The pertinent question is whether the statement about line 3 is in the literature. If it is, an inline citation is essential, as this is a central point in the analyis of the argument.

2481:

of almost identical ones. Knowledge is not an indiscriminate collection of information and, as I suggest above, a better place for the article (as it currently stands) is WikiBooks.

3246:

I originally removed the following as failing verification. It did not appear in the originally cited source. It has since been restored without a reference, which makes the entry

5132: 3590: 2039: 394:

I disagree, I find this encyclopaedic, albeit currently unreferenced. I strongly suspect it is quite referenceable to maths textbooks though, and I would hate to see it deleted.- (

3101:

line which is perfectly well defined. The sentence should read: Functions that are not injective have poorly defined inverse functions (multivalued functions) over their range.

3649: 3519: 5066:

Just been reading an book by Eugene P Northrop which covers practicaly all the ground of this article and a lot more in detail. It dates from 1944 but Dover have reprinted it

3811: 3359: 4299: 3409: 4041: 3693: 3460: 2397:

I have a feeling that a good way to approach cleanup overall is to attempt to rewrite the article from sources. There is no shortage of books on mathematical fallacies.

4661: 3729: 4151: 4080: 4622: 4589: 4556: 4222: 3955: 3884: 1719: 1652:

Although we could include an invalid proof based on a failure to understand that 0.999... = 1, I'm not sure this is a good idea. We already have an in-depth article on

2548:

A few comments, but let's take of care business first: Thanks for the quick reaction, this gives me the warm and fuzzy feeling that keeps me hooked to my keyboard. ;)

3840: 3758: 111:

on Knowledge. If you would like to support the project, please visit the project page, where you can get more details on how you can help, and where you can join the

3916: 4339:

Quite right I think remove the stuff. This article in particular needs citations to stop people's brainwaves over breakfast being stuck in. I also stuck a query at

4328: 4251: 4183: 4112: 3987: 1198: 5168: 5094: 3309: 2576:

on each side is an integer": (as do lines 2&4, BTW) It is true that this notation presupposes integer x in the case of "naive" calculus . But, analogous to

449:

a textbook, and unless these are meaningful beyond instructional value (or to confuse/trick people on an internet forum) I don't see how it's encyclopedic. --

2675:

As long as an argument is not fully formal, you're always faced with the task of interpreting the text, which practically always introduces ambiguities. The

1076:

convergence, or at least not explicitly. This seems rather out of place in this page along with all the ha-ha-gotcha-we-divided-by-zero stuff and whatnot. --

682:

Spelling errors exist, but an article on the topic should not consist of an indiscriminate collection of made-up examples of spelling errors, but instead of

2863:∞(0.999......) ^ k Therefore, 1 = 0 Can someone explain here what was the wrong step? If this proof were right can we conclude that 1 <: --> 5195: 148: 138: 5220: 3129:

of a mathematical fallacy (as it has been discussed in the peer-reviewed literature: e.g., mathematics, philosophy, psychology). This discussion can be

2214:

axis (namely, for a nonzero argument choose the root that has positive real part or is pure imaginary with positive imaginary part). In this latter case

283: 5215: 4663:. I don't see how you can claim to offer a correct resolution of the fallacy without saying these these things that lie at the heart of the matter. 1035:

2S = 2 + 4 + 6 ... S = 1 + 2 + 3 + 4 + 5 + 6 + 7 ... ------------------------------------ - S = - 1 - 3 - 5 - 7 ...

1544:

This is simply trying to show that mathematical proofs must contain finite steps by showing that a proof with infinite steps lead to absurd things.

5190: 1848: 2920:"Furthermore, it can be further shown that, if AB is shorter than AC, then E will lie outside of AB, while F will lie within AC (and vice versa)." 1200:

that only uses convergent infinite sums, and only the law allowing to interchange two consecutive terms in such a sum, which is definitely valid:

4453:

subtracting one from the other; to find the angle DCE you need to subtract the angles DCH and ECH from the angle of a full circle (2π or 360°).

1945: 5205: 259: 2874: 1610: 113: 4403: 2923:

This doesn't make any sense, because there is no mention of points "E" and "F" on the text, neither in the picture accompanying the text.

2823: 2398: 1584: 669: 2722:

I'll make some adjustments to the text to attempt to address some of these points. Let me respond to the ones that I disagree with here:

2057: 4862:

Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a

5200: 5016: 4960: 4525:. I'm certainly not embarrassed about anything. I think the fallacy is a better illustration of the fallacy of extracting the root of 3106: 2990:". You have been tricked into trying to reason by diagram inspection, which is the actual point of this fallacy puzzle. You've been had 2965: 2930: 2759: 1637: 1551: 1130: 664:

fanous mistakes or misnomers do and as such WP can of course write about them as long as it correctly (and factually) describes them as

333: 3025: 5210: 5116: 5078: 2531: 2482: 2352: 4792:

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

2393:

in general, rather than on some specific litany of non-notable examples. Knowledge is not an indiscriminate collection of factoids.

1793: 1118:

standard metric on real numbers, since the formula sum_{k=0}^n a^n = (1-a^{n+1})/(1-a) only converges to 1/(1-a) for |a|<1 (type

752: 250: 211: 4782: 103: 58: 4503:. In particular verifiability is not provided by someone working something out for themselves and being sure it is right, see 5010:

The division-by-zero fallacy has many variants. The following example uses a disguised division by zero to "prove" that 2 = 1

3105:

equals x if x is between -\pi/2 and \pi/2. This is the fallacy in the given proof since 2\pi is not in the requisite range.

2597: 2461: 1570:

square(root(-a))=-a root(-a) = i root(a) square(i) = -1 square(root(a)) = a With me so far? you have to consider every step.

4670: 4433: 4389: 3257: 3189: 3148: 766:

Back in 2004 I observed that there was already a lot of redundancy, and I added this paragraph near the top of the article:

564:

There is a big difference between something that is instructional (as in educational) and a how-to. This is not a how-to.- (

4475:

Knowledge community be better served with this particular example or without? I'd definitely say the answer is the former.

2126: 3182:. It has to do something other than this. Our pillars (NOR, V, NOT) might help to produce a more encyclopedic article. 1536: 1123: 1107: 711: 695:

WRT 01:50, 20 January 2008 (UTC), Fact is anything that can be proven - these are factual, all right. Just not correct.

5172: 4834: 624: 33: 2386:

The article should have some prose indicating why the fallacies are notable, and in some sense what they are all about.

5144: 185: 63: 426:

I'm a lot less certain. And I'm 100% certain that there are books with these kinds of things in that can be cited.- (

1733: 2811: 1041:

S = 1 + 2 + 3 + 4 + 5 + ... S = 1 + 2 + 3 + 4 + ... ---------------------------- - 0 = 1 + 1 + 1 + 1 + 1 + ...

4360:

excluded. Furthermore, the whole section on Multivalued functions of this article doesn't have a reference either.

2217: 1905: 4931: 4853: 2456: 2275: 909:

20:10, 20 February 2008 (UTC) (Actually, I've just noticed that "Mathematical fallacy" already redirects here...)

4664: 4383: 3251: 3183: 3142: 2505:

integer is an interior point of the real line. The mistake lies in applying the differentiation rule for finite

1091:

It is refreshing to see an example that isn't a simple division by zero. Raekuul 12:00, 17 December 2008 (UTC)

585:

mentions "instruction manuals" and "textbooks" in separate bulleted items. Knowledge is supposed to be neither.

342: 2878: 2827: 2806:

idea for a fallacy is not in a serious sense research at all. I felt justified in taking down two of the tags.

1614: 4407: 2402: 1588: 673: 21: 2934: 2763: 2379:. The section titles "Proof that..." can then be renamed to the more correct and descriptive "Fallacy of..." 1555: 1134: 5020: 4964: 4825: 4774: 3850:

The fallacy is in the last part of the "proof" before "Substituting...", which is to conclude that because

3250:, and thus forbidden by policy. I have moved it here in case there is need to discuss the matter further. 3236: 3152: 3110: 2969: 2356: 2345:

Not sure how such an obvious error was missed. If I've missed something out, or am horribly wrong, do say.

1641: 1520:

I think that the fallacy in this "proof" also needs to be explained. Raekuul 12:12, 17 December 2008 (UTC)

1060: 947: 756: 5140: 4770: 3232: 2535: 2486: 845:. Similarly, the argument below that purports to demonstrate 5=4 makes this same error with the function 771: 3525: 2011: 4941: 4863: 4809:

If you have discovered URLs which were erroneously considered dead by the bot, you can report them with

4706: 4696: 2807: 2736: 996: 39: 4773:. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit 2913: 2861:

1 = 1 / 3 + 1 / 3 + 1 / 3 =(0.333...) +(0.333...) + 0.333...) =(.0999......) Lmt k --: -->

236: 1603:

How if we change the title to "Invalid Proofs". There are more than 1 invalid proof(s) on this page.

5135:

until a consensus is reached, and readers of this page are welcome to contribute to the discussion.

4890: 4766: 4188:

is also a solution to the original equation. No where in the "proof" it is shown that a solution to

3596: 3085: 3069: 2926: 2870: 2847: 2788: 2710: 2515: 2372: 2348: 2271: 1633: 1606: 1580: 1547: 1524: 1159: 1095: 742: 699: 683: 590: 573: 493: 435: 403: 384: 4699:, △ROB ≅ △QOC (RO = QO since △RAO ≅ △QAO; BO = CO since △ODB ≅ △ODC; ∠ORB and ∠OQC are right angles) 3471: 4705:

I must say, I have no idea "what the heck" HL might be either. I went to to the wikilinked article

4449:

But if the angles DCH and ECH are equal then the angle DCE must be zero, which is a contradiction.

2577: 2188: 4959:

understand it. Please revert your revert and check the math first and show me a number, any number

4521:

I don't really appreciate having ulterior motives suggested for my actions. This suggests that I

3769: 3317: 2111:

are positive real numbers, which is not the case in the proof above. Thus, the proof is invalid.

258:

on Knowledge. If you would like to participate, please visit the project page, where you can join

5105: 5088: 4880: 4340: 4259: 3370: 3138: 2566: 2203: 1056: 943: 917:"proved" easily. Debatably, most of the fallacies are also equivalent, but that is bordering on 640: 531: 454: 417: 372: 348: 242: 4783:

https://web.archive.org/web/20071101040235/http://www.jokes-funblog.com/categories/49-Math-Jokes

4001: 3655: 2270:

is a valid conclusion, but then you're not allowed to distribute the roots over the divisions. –

226: 205: 4810: 4484: 4361: 3420: 5075: 4627: 3995:

The first part of proof is correct, which is to show that a solution to the original equation

3699: 3228: 3141:. This is, after all, something of a "fun" area that might be a nice collaborative project. 2051:

The proof is invalid because it applies the following principle for square roots incorrectly:

1661: 1532: 1103: 1081: 1045: 1015: 890: 707: 687: 95: 4123: 4052: 4594: 4561: 4528: 4194: 3927: 3856: 1690: 992: 976: 926: 814: 621: 364: 344: 315: 4817: 2198:

The complex number i is obtained by applying square root over -1. How is this a problem? --

4901: 4343:

about the citation that was there which didn't actually pan out - that is a bit worrying.

3817: 3735: 3081: 3065: 2843: 2784: 2706: 2676: 2551:

If the source isn't clear on where the error lies, providing our own analysis constitutes

2511: 1725: 1151: 565: 485: 427: 395: 5128: 3895: 770:

Most of these proofs depend on some variation of the same error. The error is to take a

4786: 4307: 4230: 4162: 4091: 3966: 3125:

of published false proofs. What, then? It seems to me that the article should discuss

2938: 1177: 1119: 5047: 4992: 4917: 4747: 4714: 4512: 4488: 4365: 4348: 3212: 3164: 3040: 3003: 2950: 2898: 2890: 2585: 2581: 2437: 2418: 2184: 4987:

explanation sort of drew the attention away from the real fallacy. Hope this helps. -

4816:

If you found an error with any archives or the URLs themselves, you can fix them with

3294: 177: 5184: 5101: 4876: 4522: 3133:, but it does not seem like a worthy subject of an encyclopedia article for it to be 2909:

Fallacy of the isosceles triangle - Someone clean the segment up, doesn't make sense.

2893:. And please put in a terminating pre like I stuck in in above if you put it there. 2697: 2376: 2199: 636: 582: 527: 523: 450: 413: 368: 5004:

Thanks for the reply, however I still cannot see the distinction, the lead reads...

4504: 3247: 2744: 2732: 2727: 2562:"original proof": Is there another proof discussed in the section I'm not aware of? 2556: 2552: 1657: 1528: 1099: 1077: 1011: 918: 886: 703: 2750:

The distinction between "expression" and "function" in your post is somewhat of a

1032:

Let S = 1 + 2 + 3 + 4 + ..., so 2S = 2 + 4 + 6 + 8 + .... Now subtract S from 2S:

4974:

Yes, the equation (not the "expression") comes back to 0 = 0. But you wrote that

4709:

and couldn't find any clue as to what HL might stand for. Can someone clarify? -

3179: 968: 4976:"Line 3 equals zero, line 4 factors zero and the fallacy is completed in line 5" 4500: 2751: 2413:

books but otherwise I think the remainder should be removed after a week or so.

1784: 1122:

to verify that this condition on convergence is explicit for real numbers). See

972: 922: 615: 346: 255: 5176: 5150: 5109: 5051: 5024: 4996: 4968: 4921: 4905: 4884: 4839: 4751: 4735: 4718: 4674: 4516: 4492: 4411: 4393: 4378:

Indeed, it doesn't. There was a bunch of stuff there that lacked references.

4369: 4352: 3261: 3216: 3193: 3168: 3114: 3089: 3073: 3044: 3029: 3007: 2973: 2954: 2902: 2882: 2851: 2831: 2815: 2792: 2767: 2714: 2539: 2519: 2490: 2466: 2441: 2422: 2406: 2360: 2279: 2207: 2192: 1892:{\displaystyle {\frac {\sqrt {1}}{\sqrt {-1}}}={\frac {\sqrt {-1}}{\sqrt {1}}}} 1645: 1618: 1592: 1559: 1163: 1138: 1085: 1064: 1048: 1019: 1000: 980: 959:

I'm fine with renaming (and extending) the lemma to a more general article on

951: 930: 894: 760: 745: 690: 677: 644: 629: 593: 577: 535: 497: 458: 439: 421: 407: 387: 376: 4940:

Subsequent comments should be made in a new section on this talk page or in a

778: 232: 108: 85: 4558:, which makes it belong in the "Power and root" section under the heading of 3992:

is just one of them (the other two are solutions to the original equation).

2889:

This page is for improving the article. Questions like this should be put on

2501:"no integer is an interior point of the real line" That is just plain false, 1656:, so perhaps we should add a link to that article in the see-also section. -- 5043: 5001: 4988: 4955: 4913: 4743: 4710: 4508: 4344: 3208: 3160: 3036: 2999: 2946: 2894: 2506: 2433: 2414: 587:"The purpose of Knowledge is to present facts, not to teach subject matter." 1995:{\displaystyle {\sqrt {1}}\cdot {\sqrt {1}}={\sqrt {-1}}\cdot {\sqrt {-1}}} 857:. The erroneous step starts with the correct assertion that for certain 3465:

Substituting the last expression for x in the original equation we get:

2180:

which are complex numbers and this is not correct (not even meaningful).

1653: 1055:

version on the page should be replaced with the final version you give.

5161: 5123: 2375:(which currently redirects here) is a better title for the article than 2588:, permitting definition of a special case of fractional multiplication: 905: 107:, a collaborative effort to improve the coverage of content related to 4804:

An editor has reviewed this edit and fixed any errors that were found.

2857:

Please someone explain to me what is the falasy in the following proof

2685:"Each side of the equation is a well-defined function of the integer 2093:{\displaystyle {\sqrt {\frac {x}{y}}}={\frac {\sqrt {x}}{\sqrt {y}}}} 2045: 522:

An encyclopedia is a reference text, not an instructional text. See

79: 52: 1128:

infinity} |(1-2^{n+1})/(1-2)| diverges to infinity because |2|: -->

5166:

https://en.wikipedia.org/Imaginary_unit#i_raised_to_the_power_of_i

3288:

The following illustrates a subtle misuse of extraneous solution.

2912: 4483:

included in this article, even if it is in the original version.

686:

facts (or ascribable notable opinions) about spelling errors. --

5165: 2780: 904:(assuming there's too much detail to shoehorn into the existing 412:

This stuff does not appear in textbooks, I am fairly certain. --

5162:

https://en.wikipedia.org/Mathematical_fallacy#Complex_exponents

2120:

I think instead that the mistake is one step before: the rule

2005:

Any number's square root squared gives the original number, so

1837:{\displaystyle {\sqrt {\frac {1}{-1}}}={\sqrt {\frac {-1}{1}}}} 445:

such terms - perhaps as a "caution, do not do this" but we are

2559:

does not apply here, as evidenced by our current disagreement.

971:

which lists a few invalid proofs with literature references.--

349: 309: 15: 2315:

the a² becomes √a², but the x² doesn't become √x². Edited to:

4432: 635:

Countably infinite, so at least we can put them in order. --

589:

False proofs are not facts, if anything they are un-facts. –

176: 967:

on page 51 (German). Additional literature can be found at

5133:

Knowledge:Redirects for discussion/Log/2021 December 4#X=y

4777:

for additional information. I made the following changes:

751:

proof will still be valid if that information is missing

2658:{\displaystyle \Gamma (\pi +1)=\pi !=\Pi _{i=1}^{\pi }i} 4975: 4895: 4739: 4686: 4379: 2776: 1025:

Proof that the sum of all positive integers is negative

4227:

is also a solution to the original equation. In fact,

2162:{\displaystyle a=b\rightarrow {\sqrt {a}}={\sqrt {b}}} 1174:

To illustrate this, consider the following "proof" of

5127:

A discussion is taking place to address the redirect

4930:

The above discussion is preserved as an archive of a

4787:

http://www.jokes-funblog.com/categories/49-Math-Jokes

4630: 4597: 4564: 4531: 4310: 4262: 4233: 4197: 4165: 4126: 4094: 4055: 4004: 3969: 3930: 3898: 3859: 3820: 3772: 3738: 3702: 3658: 3599: 3528: 3474: 3423: 3373: 3320: 3297: 2600: 2220: 2129: 2060: 2015: 2014: 1948: 1908: 1851: 1796: 1736: 1693: 1208: 1180: 365:

Knowledge is not for things made up in school one day

254:, a collaborative effort to improve the coverage of 4866:. No further edits should be made to this section. 3763:Substituting x=1 in the original equation yields: 2842:The title of this page is "Mathematical fallaciy". 159: 4944:. No further edits should be made to this section. 4655: 4616: 4583: 4550: 4402:One removed was pretty and I couldn't remember it 4322: 4293: 4245: 4216: 4177: 4145: 4106: 4074: 4035: 3981: 3949: 3910: 3878: 3834: 3805: 3752: 3723: 3687: 3643: 3584: 3513: 3454: 3403: 3353: 3303: 2657: 2262: 2161: 2092: 2033: 1994: 1931: 1891: 1836: 1772: 1713: 1507: 1192: 4875:the page at this time, per the discussion below. 2565:Presuming that "proof" in this article refers to 1773:{\displaystyle {\frac {1}{-1}}={\frac {-1}{1}}} 1682: 805:, and to (erroneously) conclude that therefore 969:http://www.cut-the-knot.org/proofs/index.shtml 873:, and then makes the incorrect deduction that 2988:simple inspection of the accompanying diagram 2822:remaining cleanup. "Taking in hand" indeed! 2263:{\displaystyle {\sqrt {1/-1}}={\sqrt {-1/1}}} 1932:{\displaystyle {\sqrt {1}}\cdot {\sqrt {-1}}} 1010:My edit doesn't always show up. Why is this? 829:× 0, and the erroneous step is to start with 8: 5072:Riddles in mathematics : a book of paradoxes 4685:ClueBot NG rightfully reverted the question 2497:Error in Proof that 2 =1 in section Calculus 1724:Convert both sides of the equation into the 942:form, and once in an obviously-false form. 5160:I consider that there is a need to clarify 4689:added by IP 192.16.204.218 below the line: 885:It was removed in spring 2007, though. -- 19: 5093:: CS1 maint: location missing publisher ( 4852:The following is a closed discussion of a 4765:I have just modified one external link on 4416: 3266: 3119: 200: 156: 47: 4635: 4629: 4602: 4596: 4569: 4563: 4536: 4530: 4309: 4304:Therefore, it is incorrect to substitute 4267: 4261: 4232: 4202: 4196: 4164: 4131: 4125: 4093: 4060: 4054: 4009: 4003: 3968: 3935: 3929: 3897: 3864: 3858: 3819: 3777: 3771: 3737: 3707: 3701: 3675: 3663: 3657: 3622: 3604: 3598: 3557: 3533: 3527: 3479: 3473: 3442: 3422: 3378: 3372: 3325: 3319: 3296: 2646: 2635: 2599: 2250: 2242: 2226: 2221: 2219: 2152: 2142: 2128: 2076: 2061: 2059: 2016: 2013: 1982: 1969: 1959: 1949: 1947: 1919: 1909: 1907: 1872: 1852: 1850: 1817: 1797: 1795: 1755: 1737: 1735: 1692: 1209: 1207: 1179: 900:It might be better to rename the article 4982:in line 5". Saying that "the fallacy is 4624:are precisely the roots of the equation 4437:Diagram for proof that any angle is zero 4380:I removed these, and other uncited items 4341:User talk:Tuntable#Not in citation given 2389:Focus should be placed on the notion of 817:is a special case of this; the function 4742:to make it clear for future readers. - 4419: 3830: 3801: 3748: 3719: 3683: 3639: 3580: 3509: 3450: 3399: 3349: 3269: 3022:2600:1700:975F:F010:F43D:87EE:EEA7:CFDE 1709: 202: 49: 5086: 5038:you call "irrelevant logic", is still 117:about philosophy content on Knowledge. 3096:Errors in multivalued fuction section 1120:"geometric series" into wolfram alpha 479:Why would they need to be meaningful 7: 4871:The result of the move request was: 3585:{\displaystyle x^{2}+(-1-1/x)+1=0\,} 2991: 2584:, which extends to the factorial to 2034:{\displaystyle \displaystyle {1=-1}} 248:This article is within the scope of 101:This article is within the scope of 5042:logic, and certainly no fallacy. - 4156:However, this does not imply that 2694:not applicable in standard calculus 2477:choice examples, rather than be an 837:× 0 and to conclude that therefore 38:It is of interest to the following 5196:Mid-importance Philosophy articles 3291:Assume the following equation for 2783:, too many bosses in here. Later. 2632: 2601: 14: 5221:Mid-priority mathematics articles 4769:. Please take a moment to review 3120:What's the point of this article? 268:Knowledge:WikiProject Mathematics 5216:Start-Class mathematics articles 5122: 5074:(Dover ed.). Mineola, New York. 4795: 4725: 3644:{\displaystyle x^{2}+(-1/x)=0\,} 3414:Divide by x (assume x is not 0) 2992: 2737:rigorous foundations of calculus 1145:Removal of unreferenced sections 314: 271:Template:WikiProject Mathematics 235: 225: 204: 123:Knowledge:WikiProject Philosophy 88: 78: 51: 20: 5191:Start-Class Philosophy articles 5131:. The discussion will occur at 3514:{\displaystyle x^{2}+(x)+1=0\,} 288:This article has been rated as 143:This article has been rated as 126:Template:WikiProject Philosophy 3630: 3613: 3565: 3542: 3494: 3488: 3074:10:09, 16 September 2010 (UTC) 2945:They should be Q and R, fixed 2616: 2604: 2139: 965:Lehrbuch der Analysis - Band I 1: 5206:Mid-importance logic articles 5052:11:14, 25 November 2019 (UTC) 5025:10:31, 25 November 2019 (UTC) 4997:08:20, 18 November 2019 (UTC) 4969:23:12, 17 November 2019 (UTC) 4675:12:55, 12 February 2013 (UTC) 4517:08:49, 12 February 2013 (UTC) 4493:01:25, 12 February 2013 (UTC) 4394:23:56, 11 February 2013 (UTC) 4370:22:04, 11 February 2013 (UTC) 4353:21:09, 11 February 2013 (UTC) 3806:{\displaystyle 1^{2}+1+1=0\,} 3354:{\displaystyle x^{2}+x+1=0\,} 3262:19:47, 11 February 2013 (UTC) 2862:∞ 1 ^ k = Lmt k --: --> 2193:16:33, 22 February 2009 (UTC) 1593:03:01, 26 December 2008 (UTC) 1001:07:31, 22 February 2008 (UTC) 895:14:47, 19 February 2008 (UTC) 262:and see a list of open tasks. 5151:17:56, 4 December 2021 (UTC) 5110:10:14, 1 November 2021 (UTC) 5070:Northrop, Eugene P. (2014). 4845:Requested move 17 April 2018 4840:20:31, 9 December 2017 (UTC) 4752:09:32, 21 January 2014 (UTC) 4719:08:17, 21 January 2014 (UTC) 4330:into the original equation. 4294:{\displaystyle x^{2}+x+1=0.} 3404:{\displaystyle x^{2}=-x-1\,} 3090:03:16, 17 October 2010 (UTC) 2903:20:33, 6 December 2009 (UTC) 2883:20:14, 6 December 2009 (UTC) 2852:14:13, 8 November 2009 (UTC) 2832:13:14, 8 November 2009 (UTC) 2816:10:52, 8 November 2009 (UTC) 2793:13:56, 8 November 2009 (UTC) 2768:19:41, 7 November 2009 (UTC) 2715:18:21, 7 November 2009 (UTC) 2540:14:36, 7 November 2009 (UTC) 2520:10:54, 7 November 2009 (UTC) 2491:14:08, 7 November 2009 (UTC) 2467:01:59, 7 November 2009 (UTC) 2442:23:49, 7 November 2009 (UTC) 2423:17:17, 7 November 2009 (UTC) 2407:13:04, 6 November 2009 (UTC) 2319:let the positive number be a 2289:let the positive number be a 1646:22:08, 1 February 2009 (UTC) 1619:02:51, 23 January 2009 (UTC) 1164:05:24, 7 November 2008 (UTC) 1139:18:25, 6 February 2012 (UTC) 1086:02:58, 31 October 2008 (UTC) 761:21:54, 6 February 2008 (UTC) 746:04:46, 27 January 2008 (UTC) 645:01:30, 20 January 2008 (UTC) 630:01:26, 20 January 2008 (UTC) 594:01:50, 20 January 2008 (UTC) 578:01:20, 20 January 2008 (UTC) 536:00:23, 20 January 2008 (UTC) 498:00:02, 20 January 2008 (UTC) 459:23:38, 19 January 2008 (UTC) 440:22:29, 19 January 2008 (UTC) 422:21:33, 19 January 2008 (UTC) 408:20:51, 19 January 2008 (UTC) 388:19:43, 19 January 2008 (UTC) 5177:11:00, 14 August 2022 (UTC) 4523:wasn't acting in good faith 4036:{\displaystyle x^{2}+x+1=0} 3688:{\displaystyle x^{2}=1/x\,} 3008:20:43, 31 August 2022 (UTC) 2974:17:23, 31 August 2022 (UTC) 2908: 2735:: anyone familiar with the 2361:10:31, 30 August 2009 (UTC) 377:10:00, 4 January 2008 (UTC) 5237: 5201:Start-Class logic articles 4922:16:57, 17 April 2018 (UTC) 4906:15:42, 17 April 2018 (UTC) 4885:02:45, 24 April 2018 (UTC) 4762:Hello fellow Wikipedians, 4412:23:53, 1 August 2013 (UTC) 3960:has three solutions, and 3455:{\displaystyle x=-1-1/x\,} 3237:15:28, 11 March 2012 (UTC) 3217:11:00, 11 March 2012 (UTC) 3194:02:42, 11 March 2012 (UTC) 3169:02:11, 11 March 2012 (UTC) 3153:00:28, 11 March 2012 (UTC) 1065:15:02, 16 March 2008 (UTC) 1049:10:02, 16 March 2008 (UTC) 1020:19:19, 14 March 2008 (UTC) 952:15:02, 16 March 2008 (UTC) 149:project's importance scale 5211:Logic task force articles 4656:{\displaystyle x^{2}+x+1} 4085:It is also correct that 3724:{\displaystyle x^{3}=1\,} 3115:01:11, 6 March 2012 (UTC) 2864:.09999999999999.......... 2280:21:20, 2 March 2009 (UTC) 2208:15:19, 2 March 2009 (UTC) 1029:Isn't this much simpler: 931:21:29, 8 March 2008 (UTC) 287: 220: 184: 155: 142: 73: 46: 5169:2001:6B0:E:2B18:0:0:0:71 5117:Redirects for discussion 4937:Please do not modify it. 4859:Please do not modify it. 4687:"WHAT THE HECK IS HL???" 4146:{\displaystyle x^{3}=1.} 4075:{\displaystyle x^{3}=1.} 3045:18:52, 12 May 2023 (UTC) 3030:17:29, 12 May 2023 (UTC) 2955:07:05, 7 July 2010 (UTC) 2939:04:23, 7 July 2010 (UTC) 2891:The maths reference dssk 2572:"line 3 only holds when 2325:squaring both the sides: 2295:squaring both the sides: 1684:Start with the identity 981:16:14, 2 June 2008 (UTC) 691:21:49, 3 June 2008 (UTC) 678:13:37, 3 June 2008 (UTC) 294:project's priority scale 4758:External links modified 4617:{\displaystyle x^{3}=1} 4584:{\displaystyle x^{3}=1} 4551:{\displaystyle x^{3}=1} 4217:{\displaystyle x^{3}=1} 3950:{\displaystyle x^{3}=1} 3879:{\displaystyle x^{3}=1} 2103:This is only true when 1902:Multiply both sides by 1787:on both sides to yield 1714:{\displaystyle -1=-1\,} 1560:06:08, 3 May 2011 (UTC) 251:WikiProject Mathematics 160:Associated task forces: 5141:User:1234qwer1234qwer4 4657: 4618: 4585: 4552: 4438: 4324: 4295: 4247: 4218: 4179: 4147: 4108: 4076: 4037: 3983: 3951: 3912: 3880: 3836: 3807: 3754: 3725: 3689: 3645: 3586: 3515: 3456: 3405: 3355: 3305: 3227:on correct stuff.) -- 2917: 2659: 2264: 2163: 2113: 2094: 2035: 1996: 1933: 1893: 1838: 1774: 1715: 1509: 1194: 181: 104:WikiProject Philosophy 28:This article is rated 4707:solution of triangles 4658: 4619: 4586: 4553: 4436: 4325: 4296: 4253:is not a solution to 4248: 4219: 4180: 4148: 4109: 4077: 4038: 3984: 3952: 3913: 3881: 3837: 3835:{\displaystyle 3=0\,} 3808: 3755: 3753:{\displaystyle x=1\,} 3726: 3690: 3646: 3587: 3516: 3457: 3406: 3356: 3306: 2916: 2838:Time for another move 2660: 2450:Explanation of revert 2265: 2164: 2095: 2036: 1997: 1934: 1894: 1839: 1775: 1716: 1573:root(square(-a)) = a 1510: 1195: 961:mthematical fallacies 180: 5156:clarification needed 4891:Mathematical fallacy 4873:no consensus to move 4767:Mathematical fallacy 4628: 4595: 4562: 4529: 4308: 4260: 4231: 4195: 4163: 4124: 4092: 4053: 4002: 3967: 3928: 3911:{\displaystyle x=1.} 3896: 3857: 3818: 3770: 3736: 3700: 3656: 3597: 3526: 3472: 3421: 3371: 3318: 3295: 3279:Extraneous solutions 3131:informed by examples 2700:your bloodthirst? ;) 2598: 2557:Routine calculations 2391:mathematical fallacy 2373:Mathematical fallacy 2218: 2127: 2058: 2012: 1946: 1906: 1849: 1794: 1734: 1691: 1206: 1178: 902:Mathematical fallacy 274:mathematics articles 4323:{\displaystyle x=1} 4246:{\displaystyle x=1} 4178:{\displaystyle x=1} 4107:{\displaystyle x=1} 3982:{\displaystyle x=1} 2739:can spot the error. 2651: 2578:fractional calculus 1193:{\displaystyle 1=0} 1071:1+2+4+8...equals -1 986:References possible 358:Clean up (Jan 2008) 129:Philosophy articles 4828:InternetArchiveBot 4653: 4614: 4581: 4548: 4439: 4421:moved from article 4320: 4291: 4243: 4214: 4175: 4143: 4117:is a solution to 4104: 4072: 4046:is a solution to 4033: 3979: 3947: 3921:In fact, equation 3908: 3876: 3832: 3831: 3803: 3802: 3750: 3749: 3721: 3720: 3685: 3684: 3641: 3640: 3582: 3581: 3511: 3510: 3452: 3451: 3401: 3400: 3351: 3350: 3301: 3271:moved from article 2918: 2743:incidentally, the 2655: 2631: 2567:mathematical proof 2260: 2159: 2090: 2031: 2030: 1992: 1929: 1889: 1834: 1770: 1711: 1710: 1505: 1503: 1190: 781:, to observe that 243:Mathematics portal 182: 114:general discussion 34:content assessment 4883: 4458: 4457: 4429:Any angle is zero 4335: 4334: 3304:{\displaystyle x} 3248:original research 2929:comment added by 2873:comment added by 2779:my cue for going 2586:complex arguments 2351:comment added by 2258: 2237: 2157: 2147: 2119: 2116: 2114: 2088: 2087: 2082: 2071: 2070: 1990: 1977: 1964: 1954: 1927: 1914: 1887: 1886: 1881: 1867: 1866: 1858: 1832: 1831: 1812: 1811: 1768: 1750: 1679: 1675: 1636:comment added by 1609:comment added by 1583:comment added by 1550:comment added by 1541: 1527:comment added by 1112: 1098:comment added by 919:original research 716: 702:comment added by 355: 354: 308: 307: 304: 303: 300: 299: 199: 198: 195: 194: 191: 190: 96:Philosophy portal 5228: 5149: 5148: 5126: 5115:"X=y" listed at 5098: 5092: 5084: 5007:Division by zero 4939: 4904: 4898: 4879: 4861: 4838: 4829: 4802: 4799: 4798: 4733: 4729: 4728: 4667: 4662: 4660: 4659: 4654: 4640: 4639: 4623: 4621: 4620: 4615: 4607: 4606: 4590: 4588: 4587: 4582: 4574: 4573: 4557: 4555: 4554: 4549: 4541: 4540: 4442:B, C, D, and E. 4417: 4386: 4329: 4327: 4326: 4321: 4300: 4298: 4297: 4292: 4272: 4271: 4252: 4250: 4249: 4244: 4223: 4221: 4220: 4215: 4207: 4206: 4184: 4182: 4181: 4176: 4152: 4150: 4149: 4144: 4136: 4135: 4113: 4111: 4110: 4105: 4081: 4079: 4078: 4073: 4065: 4064: 4042: 4040: 4039: 4034: 4014: 4013: 3988: 3986: 3985: 3980: 3956: 3954: 3953: 3948: 3940: 3939: 3917: 3915: 3914: 3909: 3885: 3883: 3882: 3877: 3869: 3868: 3841: 3839: 3838: 3833: 3812: 3810: 3809: 3804: 3782: 3781: 3759: 3757: 3756: 3751: 3730: 3728: 3727: 3722: 3712: 3711: 3694: 3692: 3691: 3686: 3679: 3668: 3667: 3650: 3648: 3647: 3642: 3626: 3609: 3608: 3591: 3589: 3588: 3583: 3561: 3538: 3537: 3520: 3518: 3517: 3512: 3484: 3483: 3461: 3459: 3458: 3453: 3446: 3410: 3408: 3407: 3402: 3383: 3382: 3360: 3358: 3357: 3352: 3330: 3329: 3310: 3308: 3307: 3302: 3267: 3254: 3186: 3180:cut-the-not link 3145: 2997: 2996: 2995: 2941: 2885: 2808:Charles Matthews 2664: 2662: 2661: 2656: 2650: 2645: 2464: 2459: 2367:Proposed cleanup 2363: 2285:An obvious error 2269: 2267: 2266: 2261: 2259: 2254: 2243: 2238: 2230: 2222: 2168: 2166: 2165: 2160: 2158: 2153: 2148: 2143: 2118: 2115: 2099: 2097: 2096: 2091: 2089: 2083: 2078: 2077: 2072: 2063: 2062: 2040: 2038: 2037: 2032: 2029: 2001: 1999: 1998: 1993: 1991: 1983: 1978: 1970: 1965: 1960: 1955: 1950: 1938: 1936: 1935: 1930: 1928: 1920: 1915: 1910: 1898: 1896: 1895: 1890: 1888: 1882: 1874: 1873: 1868: 1859: 1854: 1853: 1843: 1841: 1840: 1835: 1833: 1827: 1819: 1818: 1813: 1810: 1799: 1798: 1779: 1777: 1776: 1771: 1769: 1764: 1756: 1751: 1749: 1738: 1726:vulgar fractions 1720: 1718: 1717: 1712: 1681: 1678: 1674: 1648: 1621: 1595: 1562: 1540: 1521: 1514: 1512: 1511: 1506: 1504: 1199: 1197: 1196: 1191: 1111: 1092: 815:Division by zero 715: 696: 628: 627: 350: 318: 310: 276: 275: 272: 269: 266: 245: 240: 239: 229: 222: 221: 216: 208: 201: 167: 157: 131: 130: 127: 124: 121: 98: 93: 92: 91: 82: 75: 74: 69: 66: 55: 48: 31: 25: 24: 16: 5236: 5235: 5231: 5230: 5229: 5227: 5226: 5225: 5181: 5180: 5158: 5139: 5136: 5120: 5085: 5081: 5069: 5064: 4953: 4948: 4935: 4900: 4894: 4857: 4847: 4832: 4827: 4800: 4796: 4775:this simple FaQ 4760: 4726: 4724: 4683: 4665: 4631: 4626: 4625: 4598: 4593: 4592: 4565: 4560: 4559: 4532: 4527: 4526: 4464: 4459: 4422: 4384: 4336: 4306: 4305: 4263: 4258: 4257: 4229: 4228: 4198: 4193: 4192: 4161: 4160: 4127: 4122: 4121: 4090: 4089: 4056: 4051: 4050: 4005: 4000: 3999: 3965: 3964: 3931: 3926: 3925: 3894: 3893: 3860: 3855: 3854: 3816: 3815: 3773: 3768: 3767: 3734: 3733: 3703: 3698: 3697: 3659: 3654: 3653: 3600: 3595: 3594: 3529: 3524: 3523: 3475: 3470: 3469: 3419: 3418: 3374: 3369: 3368: 3321: 3316: 3315: 3293: 3292: 3286: 3281: 3272: 3252: 3244: 3242:Uncited content 3184: 3143: 3122: 3098: 3061: 2993: 2924: 2911: 2875:208.120.156.114 2868: 2865: 2859: 2840: 2803: 2596: 2595: 2499: 2462: 2457: 2452: 2369: 2346: 2287: 2272:Henning Makholm 2216: 2215: 2172:is applied for 2125: 2124: 2056: 2055: 2010: 2009: 1944: 1943: 1904: 1903: 1847: 1846: 1820: 1803: 1792: 1791: 1757: 1742: 1732: 1731: 1689: 1688: 1673:The text says: 1671: 1631: 1627: 1611:125.161.199.246 1604: 1601: 1578: 1568: 1545: 1542: 1522: 1502: 1501: 1491: 1488: 1487: 1447: 1444: 1443: 1436: 1433: 1432: 1392: 1389: 1388: 1348: 1345: 1344: 1304: 1301: 1300: 1260: 1257: 1256: 1216: 1204: 1203: 1176: 1175: 1171: 1147: 1093: 1073: 1042: 1036: 1027: 1008: 988: 743:Henning Makholm 697: 620: 614: 591:Henning Makholm 385:Henning Makholm 360: 351: 345: 323: 273: 270: 267: 264: 263: 241: 234: 214: 165: 128: 125: 122: 119: 118: 94: 89: 87: 67: 61: 32:on Knowledge's 29: 12: 11: 5: 5234: 5232: 5224: 5223: 5218: 5213: 5208: 5203: 5198: 5193: 5183: 5182: 5157: 5154: 5138: 5119: 5113: 5080:978-0486780160 5079: 5063: 5060: 5059: 5058: 5057: 5056: 5055: 5054: 5030: 5029: 5028: 5027: 5013: 5012: 5011: 5008: 4952: 4949: 4947: 4946: 4932:requested move 4926: 4925: 4924: 4888: 4869: 4868: 4854:requested move 4848: 4846: 4843: 4822: 4821: 4814: 4790: 4789: 4781:Added archive 4759: 4756: 4755: 4754: 4703: 4702: 4701: 4700: 4682: 4679: 4678: 4677: 4666:Sławomir Biały 4652: 4649: 4646: 4643: 4638: 4634: 4613: 4610: 4605: 4601: 4580: 4577: 4572: 4568: 4547: 4544: 4539: 4535: 4519: 4463: 4460: 4456: 4455: 4424: 4423: 4420: 4415: 4404:109.152.222.39 4401: 4399: 4398: 4397: 4396: 4385:Sławomir Biały 4373: 4372: 4356: 4355: 4333: 4332: 4319: 4316: 4313: 4302: 4301: 4290: 4287: 4284: 4281: 4278: 4275: 4270: 4266: 4242: 4239: 4236: 4225: 4224: 4213: 4210: 4205: 4201: 4186: 4185: 4174: 4171: 4168: 4154: 4153: 4142: 4139: 4134: 4130: 4115: 4114: 4103: 4100: 4097: 4083: 4082: 4071: 4068: 4063: 4059: 4044: 4043: 4032: 4029: 4026: 4023: 4020: 4017: 4012: 4008: 3990: 3989: 3978: 3975: 3972: 3958: 3957: 3946: 3943: 3938: 3934: 3919: 3918: 3907: 3904: 3901: 3887: 3886: 3875: 3872: 3867: 3863: 3843: 3842: 3829: 3826: 3823: 3813: 3800: 3797: 3794: 3791: 3788: 3785: 3780: 3776: 3761: 3760: 3747: 3744: 3741: 3731: 3718: 3715: 3710: 3706: 3695: 3682: 3678: 3674: 3671: 3666: 3662: 3651: 3638: 3635: 3632: 3629: 3625: 3621: 3618: 3615: 3612: 3607: 3603: 3592: 3579: 3576: 3573: 3570: 3567: 3564: 3560: 3556: 3553: 3550: 3547: 3544: 3541: 3536: 3532: 3521: 3508: 3505: 3502: 3499: 3496: 3493: 3490: 3487: 3482: 3478: 3463: 3462: 3449: 3445: 3441: 3438: 3435: 3432: 3429: 3426: 3412: 3411: 3398: 3395: 3392: 3389: 3386: 3381: 3377: 3362: 3361: 3348: 3345: 3342: 3339: 3336: 3333: 3328: 3324: 3300: 3285: 3282: 3280: 3277: 3274: 3273: 3270: 3265: 3253:Sławomir Biały 3243: 3240: 3224: 3223: 3222: 3221: 3220: 3219: 3199: 3198: 3197: 3196: 3185:Sławomir Biały 3172: 3171: 3144:Sławomir Biały 3135:about examples 3121: 3118: 3097: 3094: 3093: 3092: 3060: 3057: 3056: 3055: 3054: 3053: 3052: 3051: 3050: 3049: 3048: 3047: 3014: 2979: 2978: 2977: 2976: 2958: 2957: 2910: 2907: 2906: 2905: 2860: 2858: 2855: 2839: 2836: 2835: 2834: 2824:71.182.244.158 2802: 2799: 2798: 2797: 2796: 2795: 2756: 2755: 2748: 2740: 2720: 2719: 2718: 2717: 2703: 2702: 2701: 2690: 2683: 2672: 2671: 2668: 2667: 2666: 2654: 2649: 2644: 2641: 2638: 2634: 2630: 2627: 2624: 2621: 2618: 2615: 2612: 2609: 2606: 2603: 2590: 2589: 2582:Gamma function 2570: 2563: 2560: 2543: 2542: 2498: 2495: 2494: 2493: 2479:endless litany 2451: 2448: 2447: 2446: 2445: 2444: 2426: 2425: 2399:71.182.236.206 2395: 2394: 2387: 2384: 2380: 2368: 2365: 2342: 2340: 2336: 2334: 2328: 2326: 2324: 2320: 2316: 2312: 2310: 2306: 2304: 2298: 2296: 2294: 2290: 2286: 2283: 2257: 2253: 2249: 2246: 2241: 2236: 2233: 2229: 2225: 2211: 2210: 2170: 2169: 2156: 2151: 2146: 2141: 2138: 2135: 2132: 2101: 2100: 2086: 2081: 2075: 2069: 2066: 2042: 2041: 2028: 2025: 2022: 2019: 2003: 2002: 1989: 1986: 1981: 1976: 1973: 1968: 1963: 1958: 1953: 1926: 1923: 1918: 1913: 1900: 1899: 1885: 1880: 1877: 1871: 1865: 1862: 1857: 1844: 1830: 1826: 1823: 1816: 1809: 1806: 1802: 1781: 1780: 1767: 1763: 1760: 1754: 1748: 1745: 1741: 1722: 1721: 1708: 1705: 1702: 1699: 1696: 1670: 1669:About radicals 1667: 1666: 1665: 1626: 1623: 1600: 1597: 1585:83.176.244.236 1567: 1564: 1519: 1518: 1517: 1516: 1515: 1500: 1497: 1494: 1492: 1490: 1489: 1486: 1483: 1480: 1477: 1474: 1471: 1468: 1465: 1462: 1459: 1456: 1453: 1450: 1448: 1446: 1445: 1442: 1439: 1437: 1435: 1434: 1431: 1428: 1425: 1422: 1419: 1416: 1413: 1410: 1407: 1404: 1401: 1398: 1395: 1393: 1391: 1390: 1387: 1384: 1381: 1378: 1375: 1372: 1369: 1366: 1363: 1360: 1357: 1354: 1351: 1349: 1347: 1346: 1343: 1340: 1337: 1334: 1331: 1328: 1325: 1322: 1319: 1316: 1313: 1310: 1307: 1305: 1303: 1302: 1299: 1296: 1293: 1290: 1287: 1284: 1281: 1278: 1275: 1272: 1269: 1266: 1263: 1261: 1259: 1258: 1255: 1252: 1249: 1246: 1243: 1240: 1237: 1234: 1231: 1228: 1225: 1222: 1219: 1217: 1215: 1212: 1211: 1189: 1186: 1183: 1170: 1167: 1146: 1143: 1142: 1141: 1114: 1113: 1072: 1069: 1068: 1067: 1040: 1034: 1026: 1023: 1007: 1004: 987: 984: 957: 956: 955: 954: 936: 935: 934: 933: 911: 910: 883: 882: 738: 737: 736: 735: 734: 733: 732: 731: 730: 729: 728: 727: 726: 725: 724: 723: 722: 721: 720: 719: 718: 717: 670:84.174.254.199 611: 610: 609: 608: 607: 606: 605: 604: 603: 602: 601: 600: 599: 598: 597: 596: 549: 548: 547: 546: 545: 544: 543: 542: 541: 540: 539: 538: 509: 508: 507: 506: 505: 504: 503: 502: 501: 500: 468: 467: 466: 465: 464: 463: 462: 461: 391: 390: 359: 356: 353: 352: 347: 343: 341: 338: 337: 329: 328: 325: 324: 319: 313: 306: 305: 302: 301: 298: 297: 286: 280: 279: 277: 260:the discussion 247: 246: 230: 218: 217: 209: 197: 196: 193: 192: 189: 188: 183: 173: 172: 170: 168: 162: 161: 153: 152: 145:Mid-importance 141: 135: 134: 132: 100: 99: 83: 71: 70: 68:Mid‑importance 56: 44: 43: 37: 26: 13: 10: 9: 6: 4: 3: 2: 5233: 5222: 5219: 5217: 5214: 5212: 5209: 5207: 5204: 5202: 5199: 5197: 5194: 5192: 5189: 5188: 5186: 5179: 5178: 5174: 5170: 5167: 5163: 5155: 5153: 5152: 5146: 5142: 5134: 5130: 5125: 5118: 5114: 5112: 5111: 5107: 5103: 5099: 5096: 5090: 5082: 5077: 5073: 5067: 5061: 5053: 5049: 5045: 5041: 5036: 5035: 5034: 5033: 5032: 5031: 5026: 5022: 5018: 5017:121.99.108.78 5014: 5009: 5006: 5005: 5003: 5000: 4999: 4998: 4994: 4990: 4985: 4981: 4977: 4973: 4972: 4971: 4970: 4966: 4962: 4961:121.99.108.78 4957: 4951:Line 3 = zero 4950: 4945: 4943: 4938: 4933: 4928: 4927: 4923: 4919: 4915: 4910: 4909: 4908: 4907: 4903: 4897: 4896:Invalid proof 4892: 4887: 4886: 4882: 4878: 4874: 4867: 4865: 4860: 4855: 4850: 4849: 4844: 4842: 4841: 4836: 4831: 4830: 4819: 4815: 4812: 4808: 4807: 4806: 4805: 4793: 4788: 4784: 4780: 4779: 4778: 4776: 4772: 4768: 4763: 4757: 4753: 4749: 4745: 4741: 4737: 4732: 4723: 4722: 4721: 4720: 4716: 4712: 4708: 4698: 4694: 4693: 4692: 4691: 4690: 4688: 4680: 4676: 4672: 4668: 4650: 4647: 4644: 4641: 4636: 4632: 4611: 4608: 4603: 4599: 4578: 4575: 4570: 4566: 4545: 4542: 4537: 4533: 4524: 4520: 4518: 4514: 4510: 4506: 4502: 4497: 4496: 4495: 4494: 4490: 4486: 4480: 4476: 4472: 4468: 4462:Parting words 4461: 4454: 4450: 4447: 4443: 4435: 4431: 4430: 4426: 4425: 4418: 4414: 4413: 4409: 4405: 4395: 4391: 4387: 4381: 4377: 4376: 4375: 4374: 4371: 4367: 4363: 4358: 4357: 4354: 4350: 4346: 4342: 4338: 4337: 4331: 4317: 4314: 4311: 4288: 4285: 4282: 4279: 4276: 4273: 4268: 4264: 4256: 4255: 4254: 4240: 4237: 4234: 4211: 4208: 4203: 4199: 4191: 4190: 4189: 4172: 4169: 4166: 4159: 4158: 4157: 4140: 4137: 4132: 4128: 4120: 4119: 4118: 4101: 4098: 4095: 4088: 4087: 4086: 4069: 4066: 4061: 4057: 4049: 4048: 4047: 4030: 4027: 4024: 4021: 4018: 4015: 4010: 4006: 3998: 3997: 3996: 3993: 3976: 3973: 3970: 3963: 3962: 3961: 3944: 3941: 3936: 3932: 3924: 3923: 3922: 3905: 3902: 3899: 3892: 3891: 3890: 3873: 3870: 3865: 3861: 3853: 3852: 3851: 3848: 3847: 3827: 3824: 3821: 3814: 3798: 3795: 3792: 3789: 3786: 3783: 3778: 3774: 3766: 3765: 3764: 3745: 3742: 3739: 3732: 3716: 3713: 3708: 3704: 3696: 3680: 3676: 3672: 3669: 3664: 3660: 3652: 3636: 3633: 3627: 3623: 3619: 3616: 3610: 3605: 3601: 3593: 3577: 3574: 3571: 3568: 3562: 3558: 3554: 3551: 3548: 3545: 3539: 3534: 3530: 3522: 3506: 3503: 3500: 3497: 3491: 3485: 3480: 3476: 3468: 3467: 3466: 3447: 3443: 3439: 3436: 3433: 3430: 3427: 3424: 3417: 3416: 3415: 3396: 3393: 3390: 3387: 3384: 3379: 3375: 3367: 3366: 3365: 3346: 3343: 3340: 3337: 3334: 3331: 3326: 3322: 3314: 3313: 3312: 3298: 3289: 3283: 3278: 3276: 3275: 3268: 3264: 3263: 3259: 3255: 3249: 3241: 3239: 3238: 3234: 3230: 3218: 3214: 3210: 3205: 3204: 3203: 3202: 3201: 3200: 3195: 3191: 3187: 3181: 3176: 3175: 3174: 3173: 3170: 3166: 3162: 3157: 3156: 3155: 3154: 3150: 3146: 3140: 3136: 3132: 3128: 3117: 3116: 3112: 3108: 3107:67.142.166.22 3102: 3095: 3091: 3087: 3083: 3078: 3077: 3076: 3075: 3071: 3067: 3058: 3046: 3042: 3038: 3033: 3032: 3031: 3027: 3023: 3019: 3015: 3011: 3010: 3009: 3005: 3001: 2989: 2985: 2984: 2983: 2982: 2981: 2980: 2975: 2971: 2967: 2966:98.21.219.152 2962: 2961: 2960: 2959: 2956: 2952: 2948: 2944: 2943: 2942: 2940: 2936: 2932: 2931:87.94.140.223 2928: 2921: 2915: 2904: 2900: 2896: 2892: 2888: 2887: 2886: 2884: 2880: 2876: 2872: 2856: 2854: 2853: 2849: 2845: 2837: 2833: 2829: 2825: 2820: 2819: 2818: 2817: 2813: 2809: 2800: 2794: 2790: 2786: 2782: 2778: 2774: 2773: 2772: 2771: 2770: 2769: 2765: 2761: 2760:71.182.216.82 2753: 2749: 2746: 2741: 2738: 2734: 2729: 2725: 2724: 2723: 2716: 2712: 2708: 2704: 2699: 2695: 2691: 2688: 2684: 2680: 2679: 2678:Gretchenfrage 2674: 2673: 2669: 2652: 2647: 2642: 2639: 2636: 2628: 2625: 2622: 2619: 2613: 2610: 2607: 2594: 2593: 2592: 2591: 2587: 2583: 2579: 2575: 2571: 2568: 2564: 2561: 2558: 2554: 2550: 2549: 2547: 2546: 2545: 2544: 2541: 2537: 2533: 2529: 2524: 2523: 2522: 2521: 2517: 2513: 2508: 2504: 2496: 2492: 2488: 2484: 2480: 2476: 2471: 2470: 2469: 2468: 2465: 2460: 2449: 2443: 2439: 2435: 2430: 2429: 2428: 2427: 2424: 2420: 2416: 2411: 2410: 2409: 2408: 2404: 2400: 2392: 2388: 2385: 2381: 2378: 2377:invalid proof 2374: 2371: 2370: 2366: 2364: 2362: 2358: 2354: 2350: 2343: 2339: 2332: 2321: 2317: 2313: 2309: 2302: 2291: 2284: 2282: 2281: 2277: 2273: 2255: 2251: 2247: 2244: 2239: 2234: 2231: 2227: 2223: 2209: 2205: 2201: 2197: 2196: 2195: 2194: 2190: 2186: 2183:Am I right?-- 2181: 2179: 2175: 2154: 2149: 2144: 2136: 2133: 2130: 2123: 2122: 2121: 2112: 2110: 2106: 2084: 2079: 2073: 2067: 2064: 2054: 2053: 2052: 2049: 2048: 2047: 2026: 2023: 2020: 2017: 2008: 2007: 2006: 1987: 1984: 1979: 1974: 1971: 1966: 1961: 1956: 1951: 1942: 1941: 1940: 1924: 1921: 1916: 1911: 1883: 1878: 1875: 1869: 1863: 1860: 1855: 1845: 1828: 1824: 1821: 1814: 1807: 1804: 1800: 1790: 1789: 1788: 1786: 1765: 1761: 1758: 1752: 1746: 1743: 1739: 1730: 1729: 1728: 1727: 1706: 1703: 1700: 1697: 1694: 1687: 1686: 1685: 1680: 1676: 1668: 1663: 1659: 1655: 1651: 1650: 1649: 1647: 1643: 1639: 1638:82.39.116.172 1635: 1624: 1622: 1620: 1616: 1612: 1608: 1598: 1596: 1594: 1590: 1586: 1582: 1574: 1571: 1565: 1563: 1561: 1557: 1553: 1552:99.38.249.252 1549: 1538: 1534: 1530: 1526: 1498: 1495: 1493: 1484: 1481: 1478: 1475: 1472: 1469: 1466: 1463: 1460: 1457: 1454: 1451: 1449: 1440: 1438: 1429: 1426: 1423: 1420: 1417: 1414: 1411: 1408: 1405: 1402: 1399: 1396: 1394: 1385: 1382: 1379: 1376: 1373: 1370: 1367: 1364: 1361: 1358: 1355: 1352: 1350: 1341: 1338: 1335: 1332: 1329: 1326: 1323: 1320: 1317: 1314: 1311: 1308: 1306: 1297: 1294: 1291: 1288: 1285: 1282: 1279: 1276: 1273: 1270: 1267: 1264: 1262: 1253: 1250: 1247: 1244: 1241: 1238: 1235: 1232: 1229: 1226: 1223: 1220: 1218: 1213: 1202: 1201: 1187: 1184: 1181: 1173: 1172: 1168: 1166: 1165: 1161: 1157: 1153: 1144: 1140: 1136: 1132: 1131:164.76.72.206 1125: 1121: 1116: 1115: 1109: 1105: 1101: 1097: 1090: 1089: 1088: 1087: 1083: 1079: 1070: 1066: 1062: 1058: 1057:not-just-yeti 1053: 1052: 1051: 1050: 1047: 1039: 1033: 1030: 1024: 1022: 1021: 1017: 1013: 1005: 1003: 1002: 998: 994: 985: 983: 982: 978: 974: 970: 966: 962: 953: 949: 945: 944:not-just-yeti 940: 939: 938: 937: 932: 928: 924: 920: 915: 914: 913: 912: 907: 903: 899: 898: 897: 896: 892: 888: 880: 876: 872: 868: 864: 860: 856: 852: 848: 844: 840: 836: 832: 828: 824: 820: 816: 812: 808: 804: 800: 796: 792: 788: 784: 780: 776: 773: 769: 768: 767: 764: 762: 758: 754: 748: 747: 744: 713: 709: 705: 701: 694: 693: 692: 689: 685: 681: 680: 679: 675: 671: 667: 662: 661: 660: 659: 658: 657: 656: 655: 654: 653: 652: 651: 650: 649: 648: 647: 646: 642: 638: 634: 633: 632: 631: 626: 623: 619: 618: 595: 592: 588: 584: 581: 580: 579: 575: 571: 567: 563: 562: 561: 560: 559: 558: 557: 556: 555: 554: 553: 552: 551: 550: 537: 533: 529: 525: 521: 520: 519: 518: 517: 516: 515: 514: 513: 512: 511: 510: 499: 495: 491: 487: 482: 478: 477: 476: 475: 474: 473: 472: 471: 470: 469: 460: 456: 452: 448: 443: 442: 441: 437: 433: 429: 425: 424: 423: 419: 415: 411: 410: 409: 405: 401: 397: 393: 392: 389: 386: 381: 380: 379: 378: 374: 370: 366: 357: 340: 339: 336: 335: 331: 330: 327: 326: 322: 317: 312: 311: 295: 291: 285: 282: 281: 278: 261: 257: 253: 252: 244: 238: 233: 231: 228: 224: 223: 219: 213: 210: 207: 203: 187: 179: 175: 174: 171: 169: 164: 163: 158: 154: 150: 146: 140: 137: 136: 133: 116: 115: 110: 106: 105: 97: 86: 84: 81: 77: 76: 72: 65: 60: 57: 54: 50: 45: 41: 35: 27: 23: 18: 17: 5164:in light of 5159: 5121: 5100: 5071: 5068: 5065: 5039: 4983: 4979: 4954: 4936: 4929: 4889: 4872: 4870: 4858: 4851: 4826: 4823: 4803: 4794: 4791: 4764: 4761: 4730: 4704: 4684: 4507:about that. 4481: 4477: 4473: 4469: 4465: 4451: 4448: 4444: 4440: 4428: 4427: 4400: 4303: 4226: 4187: 4155: 4116: 4084: 4045: 3994: 3991: 3959: 3920: 3888: 3849: 3845: 3844: 3762: 3464: 3413: 3363: 3290: 3287: 3245: 3225: 3134: 3130: 3126: 3123: 3103: 3099: 3062: 3017: 2987: 2922: 2919: 2866: 2841: 2804: 2757: 2721: 2693: 2686: 2677: 2573: 2532:74.98.44.216 2527: 2502: 2500: 2483:74.98.44.216 2478: 2474: 2453: 2396: 2390: 2353:88.107.76.20 2344: 2337: 2330: 2322: 2318: 2314: 2307: 2300: 2292: 2288: 2212: 2182: 2177: 2173: 2171: 2117: 2108: 2104: 2102: 2050: 2044: 2043: 2004: 1901: 1785:square roots 1782: 1723: 1683: 1677: 1672: 1628: 1602: 1599:Title Change 1575: 1572: 1569: 1543: 1155: 1148: 1124:this article 1074: 1043: 1037: 1031: 1028: 1009: 989: 964: 960: 958: 901: 884: 878: 874: 870: 866: 862: 858: 854: 850: 846: 842: 838: 834: 830: 826: 822: 818: 810: 806: 802: 798: 794: 790: 786: 782: 777:that is not 774: 765: 749: 741:absurdity. – 739: 665: 616: 612: 586: 569: 489: 480: 446: 431: 399: 361: 332: 320: 290:Mid-priority 289: 249: 215:Mid‑priority 144: 112: 102: 40:WikiProjects 4942:move review 4864:move review 3016:O actually 2925:—Preceding 2869:—Preceding 2752:red herring 2347:—Preceding 2323:let x = -a 2293:let x = -a 1632:—Preceding 1605:—Preceding 1579:—Preceding 1546:—Preceding 1523:—Preceding 1169:Wait, What? 1094:—Preceding 993:JackSchmidt 797:) for some 753:76.67.98.29 698:—Preceding 583:WP:NOT#TEXT 524:WP:NOT#TEXT 265:Mathematics 256:mathematics 212:Mathematics 30:Start-class 5185:Categories 4956:user:DVdm 4902:Petr Matas 4835:Report bug 4731:OK, got it 3127:the notion 3082:AlexFekken 3066:AlexFekken 2844:Paradoctor 2785:Paradoctor 2707:Paradoctor 2512:Paradoctor 2327:x² = (-a)² 2297:x² = (-a)² 1939:to obtain 1625:Suggestion 1156:Wolfkeeper 779:one-to-one 684:verifiable 570:WolfKeeper 490:WolfKeeper 432:WolfKeeper 400:WolfKeeper 120:Philosophy 109:philosophy 59:Philosophy 5089:cite book 4984:completed 4818:this tool 4811:this tool 4738:. Made a 4734:, thanks 2705:Regards, 2338:√x² = √a² 2185:pokipsy76 668:proofs.-- 334:Archive 1 5102:NadVolum 5062:Northrop 4877:Dekimasu 4824:Cheers.— 2927:unsigned 2871:unsigned 2867:Thanks 2775:I think 2349:unsigned 2308:x² = √a² 2200:Secretss 1654:0.999... 1634:unsigned 1607:unsigned 1581:unsigned 1548:unsigned 1537:contribs 1525:unsigned 1108:contribs 1096:unsigned 772:function 712:contribs 700:unsigned 637:Cheeser1 528:Cheeser1 451:Cheeser1 414:Cheeser1 369:Cheeser1 321:Archives 4771:my edit 3889:then 3059:Q.E.D.? 2801:Cleanup 2383:there?) 2335:x² = a² 2329:x² = -a 2305:x² = a² 2299:x² = -a 1658:Zundark 1566:1 != -1 1529:Raekuul 1100:Raekuul 1078:AceMyth 1046:Lambiam 1012:Oboeboy 906:fallacy 887:Dominus 704:Raekuul 688:Lambiam 666:invalid 292:on the 147:on the 4980:occurs 4740:change 4736:Google 4695:9. By 3846:Q.E.D. 3364:Then: 3139:WT:WPM 2777:that's 2682:guess. 2341:x = a 2311:x = a 2046:Q.E.D. 1783:Apply 973:Kmhkmh 923:Certes 833:× 0 = 617:x42bn6 481:beyond 36:scale. 5040:valid 4681:By HL 4505:WP:OR 3284:3 = 0 3013:text. 2745:WP:OR 2733:WP:RS 2728:WP:OR 2698:slake 2503:every 2475:a few 1006:Error 186:Logic 64:Logic 5173:talk 5145:talk 5137:~~~~ 5106:talk 5095:link 5076:ISBN 5048:talk 5044:DVdm 5021:talk 5002:DVdm 4993:talk 4989:DVdm 4965:talk 4918:talk 4914:Dmcq 4748:talk 4744:DVdm 4715:talk 4711:DVdm 4671:talk 4513:talk 4509:Dmcq 4501:WP:V 4489:talk 4485:Hguy 4408:talk 4390:talk 4366:talk 4362:Hguy 4349:talk 4345:Dmcq 3258:talk 3233:talk 3229:Taku 3213:talk 3209:Dmcq 3190:talk 3165:talk 3161:Dmcq 3149:talk 3111:talk 3086:talk 3070:talk 3041:talk 3037:DVdm 3026:talk 3004:talk 3000:DVdm 2970:talk 2951:talk 2947:Dmcq 2935:talk 2899:talk 2895:Dmcq 2879:talk 2848:talk 2828:talk 2812:talk 2789:talk 2781:AWOL 2764:talk 2726:The 2711:talk 2536:talk 2516:talk 2507:sums 2487:talk 2438:talk 2434:Dmcq 2419:talk 2415:Dmcq 2403:talk 2357:talk 2276:talk 2204:talk 2189:talk 2176:and 2107:and 1662:talk 1642:talk 1615:talk 1589:talk 1556:talk 1533:talk 1160:Talk 1152:User 1135:talk 1104:talk 1082:talk 1061:talk 1016:talk 997:talk 977:talk 948:talk 927:talk 891:talk 861:and 853:) = 801:and 789:) = 757:talk 708:talk 674:talk 641:talk 625:Mess 622:Talk 574:Talk 566:User 532:talk 526:. -- 494:Talk 486:User 455:talk 436:Talk 428:User 418:talk 404:Talk 396:User 373:talk 5129:X=y 4934:. 4785:to 3018:can 821:is 813:. 447:not 284:Mid 139:Mid 5187:: 5175:) 5108:) 5091:}} 5087:{{ 5050:) 5023:) 4995:) 4967:) 4920:) 4893:→ 4881:よ! 4856:. 4750:) 4717:) 4697:HL 4673:) 4515:) 4491:) 4410:) 4392:) 4368:) 4351:) 4289:0. 4141:1. 4070:1. 3906:1. 3617:− 3552:− 3546:− 3437:− 3431:− 3394:− 3388:− 3311:: 3260:) 3235:) 3215:) 3192:) 3167:) 3151:) 3113:) 3088:) 3072:) 3043:) 3028:) 3006:) 2998:- 2972:) 2953:) 2937:) 2901:) 2881:) 2850:) 2830:) 2814:) 2791:) 2766:) 2758:-- 2713:) 2696:" 2648:π 2633:Π 2623:π 2608:π 2602:Γ 2555:. 2553:OR 2538:) 2518:) 2489:) 2458:PS 2440:) 2421:) 2405:) 2359:) 2333:-a 2303:-a 2278:) 2245:− 2232:− 2206:) 2191:) 2140:→ 2024:− 1985:− 1980:⋅ 1972:− 1957:⋅ 1922:− 1917:⋅ 1876:− 1861:− 1822:− 1805:− 1759:− 1744:− 1704:− 1695:− 1644:) 1617:) 1591:) 1558:) 1539:) 1535:• 1485:⋯ 1441:⋮ 1430:⋯ 1386:⋯ 1342:⋯ 1298:⋯ 1254:⋯ 1162:) 1154:) 1137:) 1110:) 1106:• 1084:) 1063:) 1044:-- 1018:) 999:) 979:) 950:) 929:) 921:. 893:) 877:= 869:= 865:, 841:= 825:→ 809:= 763:. 759:) 714:) 710:• 676:) 643:) 576:) 568:) 534:) 496:) 488:) 457:) 438:) 430:) 420:) 406:) 398:) 375:) 166:/ 62:: 5171:( 5147:) 5143:( 5104:( 5097:) 5083:. 5046:( 5019:( 4991:( 4963:( 4916:( 4837:) 4833:( 4820:. 4813:. 4801:Y 4746:( 4713:( 4669:( 4651:1 4648:+ 4645:x 4642:+ 4637:2 4633:x 4612:1 4609:= 4604:3 4600:x 4579:1 4576:= 4571:3 4567:x 4546:1 4543:= 4538:3 4534:x 4511:( 4487:( 4406:( 4388:( 4364:( 4347:( 4318:1 4315:= 4312:x 4286:= 4283:1 4280:+ 4277:x 4274:+ 4269:2 4265:x 4241:1 4238:= 4235:x 4212:1 4209:= 4204:3 4200:x 4173:1 4170:= 4167:x 4138:= 4133:3 4129:x 4102:1 4099:= 4096:x 4067:= 4062:3 4058:x 4031:0 4028:= 4025:1 4022:+ 4019:x 4016:+ 4011:2 4007:x 3977:1 3974:= 3971:x 3945:1 3942:= 3937:3 3933:x 3903:= 3900:x 3874:1 3871:= 3866:3 3862:x 3828:0 3825:= 3822:3 3799:0 3796:= 3793:1 3790:+ 3787:1 3784:+ 3779:2 3775:1 3746:1 3743:= 3740:x 3717:1 3714:= 3709:3 3705:x 3681:x 3677:/ 3673:1 3670:= 3665:2 3661:x 3637:0 3634:= 3631:) 3628:x 3624:/ 3620:1 3614:( 3611:+ 3606:2 3602:x 3578:0 3575:= 3572:1 3569:+ 3566:) 3563:x 3559:/ 3555:1 3549:1 3543:( 3540:+ 3535:2 3531:x 3507:0 3504:= 3501:1 3498:+ 3495:) 3492:x 3489:( 3486:+ 3481:2 3477:x 3448:x 3444:/ 3440:1 3434:1 3428:= 3425:x 3397:1 3391:x 3385:= 3380:2 3376:x 3347:0 3344:= 3341:1 3338:+ 3335:x 3332:+ 3327:2 3323:x 3299:x 3256:( 3231:( 3211:( 3188:( 3163:( 3147:( 3109:( 3084:( 3068:( 3039:( 3024:( 3002:( 2968:( 2949:( 2933:( 2897:( 2877:( 2846:( 2826:( 2810:( 2787:( 2762:( 2709:( 2687:x 2665:. 2653:i 2643:1 2640:= 2637:i 2629:= 2626:! 2620:= 2617:) 2614:1 2611:+ 2605:( 2574:x 2534:( 2528:x 2514:( 2485:( 2463:T 2436:( 2417:( 2401:( 2355:( 2331:. 2301:. 2274:( 2256:1 2252:/ 2248:1 2240:= 2235:1 2228:/ 2224:1 2202:( 2187:( 2178:b 2174:a 2155:b 2150:= 2145:a 2137:b 2134:= 2131:a 2109:y 2105:x 2085:y 2080:x 2074:= 2068:y 2065:x 2027:1 2021:= 2018:1 1988:1 1975:1 1967:= 1962:1 1952:1 1925:1 1912:1 1884:1 1879:1 1870:= 1864:1 1856:1 1829:1 1825:1 1815:= 1808:1 1801:1 1766:1 1762:1 1753:= 1747:1 1740:1 1707:1 1701:= 1698:1 1664:) 1660:( 1640:( 1613:( 1587:( 1554:( 1531:( 1499:0 1496:= 1482:+ 1479:0 1476:+ 1473:0 1470:+ 1467:0 1464:+ 1461:0 1458:+ 1455:0 1452:= 1427:+ 1424:1 1421:+ 1418:0 1415:+ 1412:0 1409:+ 1406:0 1403:+ 1400:0 1397:= 1383:+ 1380:0 1377:+ 1374:1 1371:+ 1368:0 1365:+ 1362:0 1359:+ 1356:0 1353:= 1339:+ 1336:0 1333:+ 1330:0 1327:+ 1324:1 1321:+ 1318:0 1315:+ 1312:0 1309:= 1295:+ 1292:0 1289:+ 1286:0 1283:+ 1280:0 1277:+ 1274:1 1271:+ 1268:0 1265:= 1251:+ 1248:0 1245:+ 1242:0 1239:+ 1236:0 1233:+ 1230:0 1227:+ 1224:1 1221:= 1214:1 1188:0 1185:= 1182:1 1158:( 1150:( 1133:( 1102:( 1080:( 1059:( 1014:( 995:( 975:( 946:( 925:( 889:( 881:. 879:y 875:x 871:y 867:x 863:y 859:x 855:x 851:x 849:( 847:f 843:y 839:x 835:y 831:x 827:x 823:x 819:f 811:y 807:x 803:y 799:x 795:y 793:( 791:f 787:x 785:( 783:f 775:f 755:( 706:( 672:( 639:( 572:( 530:( 492:( 484:( 453:( 434:( 416:( 402:( 383:– 371:( 296:. 151:. 42::

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

Index