Talk:Pythagorean theorem

I think a visual of the triangles involved in this proof would be very helpful- even if they are really crude like mine.

Something like this:

Jacob Bronowski demonstrated this proof on The Ascent of Man. The four copies of the original triangle plus the small square in the middle form a square with an area equal to the square of the hypotenuse. Moving the triangle at top left to bottom right and the triangle at top right to bottom left without changing their orientation, so that their hypotenuses lie along those of the other two triangles, results in a shape formed from the squares of the other two sides, which can be proved by extending the right-hand side of the square downward to form a dividing line. QED. This is a lot simpler and more elegant than most of the proofs that are given in textbooks. -- Lee M 16:21, 17 Aug 2003 (UTC)

Just an interesting side note, it is believed that Pythagoras stumbled onto this proof as he was climbing the stairs to his office and he looked down at the courtyard and in the mosaic tiles, he saw the pattern of three circles and a right angle triangle.

Poser: (3,4,5) is a Pythagorean triplet since 3^2 + 4^2 = 5^2. Which positive integers are not part of a Pythagorean triplet?

perhaps add a mention of the fact that in the UK it's known as "Pythagoras' Theorem" ?

(6,8,10) is a Pythagorean triple, but is not listed in the table. All the triples listed have no common divisor (unlike (6,8,10)). They are called primitive Pythagorean triples.

Why is Pythagorean theorem correct and Pythagorean Theorem incorrect? .... (snip)

this issue has cropped up in 3 separate places this last week. I'm creating Wikipedia talk:Naming conventions (theorems) & moving everything there. -- Tarquin

How big were Gauss's triangles? Did he measure distances on the earth or through the air (or earth)? -phma

I have now changed the 20th-century revisionist account of the Pythagorean theorem, which appeared in this article, and which speaks of squares of numbers that are the lengths of the sides, to the more traditional and more geometric version that speaks of areas of squares. The modern version is what I would expect of people whose only acquaintance with the Pythagorean theorem comes from high-school or undergraduate courses, and not from anyone who's read Euclid. Euclid and his fellow Greek geometers did not have the concept of real number, and so had to use Eudoxos' theory of proportions instead. That necessarily means their statement of this theorem had to be different from one that speaks of squaring of numbers. -- Mike Hardy

PS: An illustration on the article page should show those three squares! Can someone provide one?

OK, what's wrong with pointing out the lack of rigour of the sample proof, and why it matters, while acknowledging its usefulness for purposes of illustration? If no persuasive reply by tomorrow, I shall try a variant edit to bring this out in a different way. PML.

The proof is rigorous. It uses the fact that the angle sum in a triangle is 180 degrees, the angle sum in a square is 360 degrees, and that the area of a square is the side squared. These are not true in non-Euclidean geometries. AxelBoldt 06:34 Jan 9, 2003 (UTC)

The proof is not rigorous; it relies on those facts without being sufficiently explicit about them. The particular fact you mention it using in one particular way is not a sufficient use. One could start out doing all the steps of this proof with a particular triangle suitably placed on the surface of a cone, and rather than being ruled out of order one would come up against a practical difficulty. Though, yes, I was too quick off the mark in suggesting a sphere as a counterexample. PML.

Ok, why not add a bit to the sentence I wrote about the proof not working in spherical geometry? AxelBoldt 21:34 Jan 10, 2003 (UTC)

There's a typo in the caption to the figure - it says 'the *are* of the square on the hypotenuse'. I'd fix it, but it's difficult with an image. Can someone regenerate it please?

Done. Michael Hardy 22:29 Mar 30, 2003 (UTC)

>>I think a visual of the triangles involved in this proof would be very helpful-<<

I did it! I programmed a vector display computer terminal called the Vectrex Arcade to do an interactive proof of the Pythagorean theorem according to the method demonstrated by Jacob Bronowski in the Ascent of Man. I dedicated the program to his memory and released it as a cartridge for the Vectrex system in fall 2002. BTW: The program speaks through a special voice circuit as the user works through the proof and displays the four triangles that are user adjustable and moveable.

See here: http://vgdb.vectrex.com/vec.pl?vgdbcode=VGDB077

Rob Mitchell, Atlanta, GA

Here's another version of the picture proof that doesn't require any math:

Is this image too complicated, and does anyone think it ought to be included in the main page?

(This is another version of the proof that appears as Proof 9 in the link, but I drew the picture myself.)

I prefer this picture to the one in the article. "Doesn't require any math"? What does that mean? This sort of thing is what math is. Michael Hardy 21:52, 15 Feb 2004 (UTC)

...and now I've edited the proof and included this new picture. Perhaps the old proof and an accompanying picture should also be there, but I think it may need rephrasing in light of the new illustration. Michael Hardy 00:29, 16 Feb 2004 (UTC)

...maybe the a's and b's can be exchanged in the right picture, so the correspond with the a's and b's in the left picture...

Contents

1 Relationship to non-Euclidean geometry and physical space

2 ONCE MORE on Relationship to non-Euclidean geometry and physical space

3 Relationship to non-Euclidean geometry and physical space 3 (Answer to Wile E. Heresiarch)

4 NB

5 Not an elementary proof

6 A Numerical proof

7 copyright status of the images

8 Application

9 Historical note on the theorem not related to the theorem?

10 Euclid's definition of area

Relationship to non-Euclidean geometry and physical space

I think this section should be removed it says nothing. Tosha 19:41, 17 Mar 2004 (UTC)

I'm not sure what you mean, "it says nothing", it's clear what it says. The part about non-Euclidean geometry could easily be integrated into the additions you made giving the theorem in hyperbolic and spherical plane, maybe a new section on the theorem's form in these contexts. The discussion about physical space is certainly relevant; although it's not strictly a mathematical issue, we are writing a general encyclopedia, not a math textbook, and articles on math topics shouldn't restrict themselves to just math. The Pythagorean theorem is relevant to the curvature of the universe, because in principle it could be used as a test to check this curvature and see if the universe is Euclidean or not -- Gauss (a mathematician, also) actually tried this. Revolver 03:51, 18 Mar 2004 (UTC)

Re: Gauss, there is apparently some disagreement about whether Gauss actually carried out the experiment or if it's a myth, or if he did but for some other reason. I don't think it makes a mention of the possibility less relevant (even though the question of whether such a "test" would answer the question requires some physics background.) Revolver 04:07, 18 Mar 2004 (UTC)

See [1] (http://www.mathpages.com/rr/s8-06/8-06.htm) and [2] (http://mathforum.org/epigone/math-history-list/merphexproi/199807211543.LAA17351@pfaff.mit.edu) for discussions. Revolver 04:11, 18 Mar 2004 (UTC)

I just want to say that it says nothing usefull or interesting for a reader,

That's pretty subjective. Lots of mathematicians, physicists, and philosophers have debated the issue ad nauseum. They must have thought it was "interesting". Revolver 02:40, 19 Mar 2004 (UTC)

clearly it says something. Moreover it is not exactly relevent, it should go to non-Euclidean geometryTosha 06:24, 18 Mar 2004 (UTC)

If a paragraph mentioning the lack of rigor of the visual proof is relevant, I hardly see how this section isn't. Certainly more people will appreciate it and understand it. The failure of the (Euclidean) Pythagorean theorem on the sphere is easy for people to visualise and see, and this may give many people some appreciation for why the postulate of parallel lines is essential in the conclusion. And if this section belongs only in an article on non-Euclidean geometry, why on earth did you include the non-Euclidean versions of the Pythagorean theorem here??? By the same reasoning, shouldn't THOSE belong only in the articles on non-Euclidean geometry, but not here?? Revolver 02:40, 19 Mar 2004 (UTC)

It is hard to talk to you, you should not use this kind of arguments, we are tolking about this subsection and it is not directly connected with area-problem. Once more this subsection is not relevent, if you want to know about non-Euclidean geometry go to non-Euclidean geometry, it is not right idea to include something about non-Euclidean geometry in every Euclidean theorem. (Hope you agree)Tosha 04:26, 19 Mar 2004 (UTC)

Both the "area problem" and "non-Euclidean discussion" are similar in that they venture outward from the central topic of the article, and so it's okay to ask how much and where this should happen. As for the rest of what you say, you're contradicting yourself. You say "if you want to know about non-Euclidean geometry go to non-Euclidean geometry, it is not right idea to include something about non-Euclidean geometry in every Euclidean theorem", yet YOU were the person who edited in the hyperbolic and spherical versions of the Pythagorean theorem in the article! Don't you see what's contradictory about that??????? Revolver 06:30, 19 Mar 2004 (UTC)

There's stuff about normed vector spaces, simplexes, etc. here...these are a far cry from the original theorem...why should they stay here?? Revolver 06:31, 19 Mar 2004 (UTC)

That was a rhetorical question. Revolver 19:08, 19 Mar 2004 (UTC)

BTW the proof number 6 on the link seems to be ok, it is indeed elementary.Tosha 19:54, 17 Mar 2004 (UTC)

In reference to the remark above -- "I think this section should be removed it says nothing." -- That is quite mistaken; the section is very informative. The breakthrough of Gauss etc was to think of the geometry of the universe as an empirical question -- that is, to be decided by experiment instead of deduction. The Pythogorean theorem is tied to the experimental part: if one measures some triangles and finds that the Pythagorean theorem is not satisfied, that is experimental evidence for a non-Euclidean geometry. Happy editing, Wile E. Heresiarch 00:50, 20 Mar 2004 (UTC)

ONCE MORE on Relationship to non-Euclidean geometry and physical space

It does not belong here, the experiment is not at all related to the theorem, it is related to the fith postulat,

I cannot see how this is true. Playfair's axiom is known to be equivalent to the Pythagorean theorem, so at least in the context of comparing hyperbolic and parabolic geometry, the issue is the same. And if by the 5th postulate, you mean "Playfair's axiom", then the issue is always the same. Playfair = Pythagorean theorem (http://www.cut-the-knot.org/triangle/pythpar/PTimpliesPP.shtml#I)

one should avoid to talk about common places. This subsection tels you that math is just a model for real world not real world itself (the first surprise) plus it says that this theorem as well as the most theorems in Euclidean geometry is not true in non-Euclidean (yet an other surprise).

It is difficult to infer your meaning. If you mean to be sarcastic, you are overreaching. For many people, it may be a good surprise to find out that Euclidean geometry is only one possible model for the world, or that there are geometries in which Euclidean theorems no longer hold. It was a shock to mathematicians 150 years ago; I assume it is a shock to the general public today. I know it was a shock for me when I first learned it.

Theris no other information here, it is safe to give a ref to non-Euclidean geometry and kill this subsection. Tosha 01:34, 20 Mar 2004 (UTC)

The problem is that many people who aren't aware of non-Euclidean geometry may not ever think to visit articles about it or read about it, if they don't encounter it at some point. If someone becomes intrigued by non-Euclidean geometry as a result of a short subsection of this article, and learns more, that seems to justify its inclusion. We are creating a learning tool and a cultural reference, not a collection of facts organised in a minimal and terse fashion. Revolver 02:59, 20 Mar 2004 (UTC)

These surprises must be found somewhere else not in this article,

According to whom? Is this wikipedia policy?

but you can make links there. Just if something is equivalent to an axiom it does not mean that non-E.g. is directly connected (historically it is not a block of construction it is a theorem).

I'm not quite sure what that's supposed to mean. In any case, you're way too literal in your conception of the purpose of this place. You seem unable (or unwilling) to understand that wikipedia is NOT a math textbook.

Plus it should be some minimality to make it more useful.

THIS IS NOT BOURBAKI. Get off it, finally!

BTW, if you want you can formulate this statement (Playfair = Pythagorean theorem), it is some ineresting information (no sarcasm here), then you will have more rights to link this page with non-E.g.Tosha 05:11, 20 Mar 2004 (UTC)

The fact that the Pythagorean theorem is equivalent to Playfair's axiom is certainly sufficiently interesting to warrant explicit mention (not just a "link" to somewhere else) and a brief, concise explanation of the relevance of this to non-Euclidean geometry and physical space. I don't know where you get your ideas about what is and is not relevant. For the umpteenth time, let me say it again:

Wikipedia is not a mathematics textbook. We are not writing Bourbaki or Rudin. We are writing for a general audience with varying mathematical backgrounds who have different informational needs and wants. There are no paper limitations (hence, no need for "minimality") and the needs of the audience should be kept always in mind.

If you can't get used to these goals, you're just going to keep on running into fights with people here. I don't say this to be mean, you seem very earnest and conscientious. I'm just letting you know. Revolver 05:35, 20 Mar 2004 (UTC)

what do you mean no need for "minimality" does it mean you should put any unrelated material?

No, of course not. You just seem to have an extraordinarily narrow conception of the meaning of the word "related".

if there is no minimality why not include here everything?

Of course, I've included facts here about kick-boxing, Jean-Paul Sartre, the history of the do-do bird, and nose-picking habits of 11th century monks. (Pardon the sarcasm, I'm just getting tired of this.)

I'm tired of your guys. Clearly I'm trying to make it usefull for general public,

I have already given my argument several times why I think the deletion of this section would not be useful for the general public and would be a disservice. You haven't once addressed my arguments; your counterargument is basically, "it has no information and is irrelevant", which is hardly an argument.

even if you remove this subsection it will be far from being textbook.

Agreed, what I meant was, you seemed to be making your judgment about the worth of the inclusion of the section based upon its logical dependence of independence mathematically toward the rest of the material, not based on reader needs and audience concerns. I meant "textbook" in the sense of Bourbaki, as Bourbaki clearly had little regard for their audience (does anyone even read them today?)

But this subsection is badly written,

So improve it, don't delete it.

has almost no information inside and irrelevent. so I will rewrite it.Tosha 18:57, 20 Mar 2004 (UTC)

If by "rewrite", you really mean rewrite, not delete, then I think that's a great idea. I certainly agree, the section could be reworded and sorted out with other sections. I was going to do such a thing myself, but I was going to wait until our disagreements had settled down on the talk page first. As they never settled down (evidently), I haven't rewritten anything. But if by "rewrite", you really mean "blank out and delete", then if you insist on doing that, I'll raise it with an administrator or in arbitration, I really hope it doesn't have to come to that.

ONE SHOULD NOTE: I WAS NOT THE ORIGINAL AUTHOR OF THIS SECTION. I am not "defending my own words". I made a slight edit, but the section as it stands has existed in the article for more than TWO YEARS, it was originally written primarily by Axel Boldt around January of 2002, look at the page history. This is not my own pet section.

If anyone else has comments, don't be shy. Everyone's silence is deafening. Revolver 23:31, 20 Mar 2004 (UTC)

My advice -- Tosha, don't attempt to rewrite the paragraph in question until you've cooled off for a while. Please. Given your repeated, angry denunciations, I doubt that you can do the topic justice. For the record, I agree with Revolver's observation that non-Euclidean geometries are still a surprise to many people; making a connection between the Pythogorean theorem and non-Euclidean geometry will be a gentle introduction to the latter for many people. Wile E. Heresiarch 02:45, 21 Mar 2004 (UTC)

I'm not angry, I only surprised that someone can hoestly like this stupid paragraph I thought of making this better, and still belive that best is simply remove it. There is no need to repeat why, I only surprised that I'm the only one who want it. I do not belive that Revolver does not see it, most probobly he just want to win the game, but wining the game is not keeping this paragraph in the article (which was much better without it), we simply have to make this page better, and I belive we should remove this subsection.

I do not see a single argument in this discussion for keeping it, if you want surprise, why not to talk about quantum mechanic it is even more surprising... so I do it once more Tosha 04:47, 21 Mar 2004 (UTC)

Tosha, I have not been debating this just for the saking of "winning a game". I believe it should stay and also be improved, for reasons I've explained several times and won't repeat. It is not for the sake of "surprise" (whatever that means). I would not include a discussion of quantum mechanics, because this does not seem directly related to the theorem. If you don't see how the discussion is related to the theorem, I'm not sure how to explain it any better. We seem to have reached an impasse. For the moment, I'm going to leave this for a while, emotions seem to be high, maybe return after some amount of time. Revolver 00:08, 22 Mar 2004 (UTC)

Tosha, I've restored "Relationship to non-Euclidean geometry and physical space". If you think the section can be improved, I'd like to know what changes you propose to improve it. Deleting the section entirely is acting in bad faith, frankly; it shows you do not have any intention of improving it. -- You have repeatedly stated that you see no merit in that section. You may wish to consider that several other people, apparently knowledgeable, have seen something in it. "I don't get it" is always a weak argument. Wile E. Heresiarch 21:42, 21 Mar 2004 (UTC)

Relationship to non-Euclidean geometry and physical space 3 (Answer to Wile E. Heresiarch)

I will iclude my comments after each staement, please do not cut it int pieces. I do not want to do it again, if you want to write something do it separetely Tosha 22:06, 22 Mar 2004 (UTC)

The Pythagorean theorem is derived from the axioms of Euclidean geometry, and in fact, it does not hold in non-Euclidean geometry.

That is true for almost any theorem in Euclidean geometry and therefore should not be here

Well, my point was that if someone is unaware of the existence of non-Euclidean geometries, then they will never be aware that the Pythagorean theorem actually depends on the 5th postulate or a parallel postulate. Given this fact, it seems reasonable to point out that the theorem is actually a theorem in Euclidean geometry, i.e. make the assumptions explicit. I don't see how different this is from making assumptions about area in the proof explicit, or mentioning something about this, which you seem to think is very necessary.

For example, in spherical geometry, there exists a right triangle whose three sides all have equal length, say a; this violates the Pythagoren theorem because a² + a² ≠ a².

That is ok but could be included into generalizations

Agreed, this could be part of an introduction to the non-Euclidean versions.

This does not mean that the Pythagorean theorem is false; it simply means that the Pythagorean theorem is a statement about triangles in Euclidean space, not non-Euclidean space.

That is just desined to confuse.

Well, I did write this particular sentence, and I certainly didn't "design" it to confuse. I was really just restating that the theorem does in fact depend on the assumption of the 5th postulate. Someone who isn't familiar with the concept of an "axiom" as "starting point for proving things" might think that the theorem has to be absolutely true or false, and not realise that it's just a consequence of certain assumptions. The example is an example where the statement of the theorem makes sense, but is not true, because another axiom is being used.

This does not settle the question of whether the Pythagorean theorem is true for physical space, because this depends upon whether physical space itself is Euclidean or non-Euclidean.

Again that says that math is a model for real world but not the world itself, it can go to mathematica

If physical space is non-Euclidean, then the Pythagorean theorem fails in physical space.

A lot of sense...

One of the first mathematicians to realize this possibility was Carl Friedrich Gauss, who then carefully measured out large right triangles as part of his geographical surveys in order to check the theorem. He found no counterexamples to the theorem within his measurement precision.

That is not clear what does it mean, but in both cases it is wrong: if he indeed made an experiment with triangle with vertexes on tops of mountanns then he was measured sum of angles, so it is not directly relevent but if he measured distances in Germany then he would find counterexamples, even in the time of Gauss it was clear.

After reading some more on this, I think the consensus seems to be that Gauss wasn't really testing the curvature of space, but calculating something to do with earth geodesy. Strictly speaking, yes, it is a check of the angle sum not the theorem. Of course, this is equivalent to checking whether the theorem would hold as well, but since I didn't find anything to indicate the triangles Gauss used were right triangles, it is an indirect check, yes.

The theory of general relativity holds that matter and energy cause space to be non-Euclidean, and the theorem therefore does not strictly apply in the presence of matter or energy.

It is irrelevent and too vague, but if you want to make something out of it you should say even more irrelevant things...

I think it may be inaccurate in the sense that matter and energy cause space-time to be non-Euclidean, which is not quite the same thing as what is stated.

However, the deviation from Euclidean space is very small except near strong gravitational sources such as black holes.

Again what does it mean?, nothing for those who does not know it and wrong statement for those who understands

Whether the theorem is violated over large cosmological scales is an open problem in cosmology, reflecting our ignorance of the ultimate curvature of the universe.

Same thing, irrelevent and too vague

Conclusion that is badly written subsection, with not much sense inside, the best one can do with it is remove it. It is unbelivable that to remove this stupid section I should write three times as much!!!Tosha 22:06, 22 Mar 2004 (UTC)

I think the section where you put the non-Euclidean forms of the theorem can't be hurt by moving the first part up there. Someone unfamiliar with non-Euclidean geometry may be justifiably confused what is meant by the "form of the theorem on the unit sphere or hyperbolic plane" otherwise. And as I said, the explicit mention of the dependence on a flat plane seems okay. (Again, if it's considered "fooling" people by not mentioning properties of area and dissection, certainly it's equally "fooling" people not to mention that the theorem depends on flatness.)

Perhaps the anecdote about Gauss is more confusing than necessary, esp. given the historical questions surrounding the incident. With regard to physical space, I'll take your comments into account that the way it was written was unclear, but I'm going to rewrite that part mentioning that the reader may have wondered if checking the theorem could tell if space was flat or not, but then indicate that this doesn't directly work and physicists consider other ways of answering this question. Revolver 03:34, 25 Mar 2004 (UTC)

good, piece, it looks much better now, I think the relation to phisical world still can be removed, but not by me. Tosha 06:30, 25 Mar 2004 (UTC)

Thanks for the comment. You make some good points; I think one of the reasons you encountered such resistance was you didn't bring up these specific criticisms originally. Removing an entire subsection which has been present for 2 years or more is generally a big change and usually has explanation. I was not aware of some of the problems with it until looking into it more, so I took the section (written by others) at face value and didn't see a reason for any change. So, I guess I'm saying it's fine to want to delete subsections, but just keep in mind that most people will want some justification (as you eventually gave). It's not that anyone's judgment is being questioned; it's just a general policy that with major changes or deletions, these are usually discussed and reasons given first, so everyone understands. Revolver 07:45, 25 Mar 2004 (UTC)

NB: This proof is often considered very simple, one but it has a hidden gap. The properties of area used here are not as elementary as one might think; in fact, proving the necessary properies is harder than the Pythagorean theorem itself. That is the reason why this proof is not used in good introductions to Euclidean geometry.

I agree with you that there is a hidden gap in this proof, based upon the properties of areas, but I disagree that it's significantly harder than proving the Pythagorean theorem. The "elementary" theorem you quote as #6 being valid suffers the same blemishes as the visual proof given, because it relies on properties of similarity of figures, which is almost as much work to justify (in terms of isometries of the plane and magnifications) as properties of area. I don't know why it isn't used in introductions to Euclidean geometry, but the reason isn't because of this logical blemish. After all, if we were to be completely precise and rigorous, almost all of Euclid's Elements is garbage, because the axioms are incomplete. Should we given a proof that refers to Hilbert's 21 axioms?? I don't know exactly how it would proceed, but I'm almost positive the Pythagorean theorem cannot be proved from Euclid's axioms, without invoking one of Hilbert's axioms to cover up logical flaws. Since this is a general introduction for readers, and since 99.9% of them aren't going to be concerned with the technical family matters of foundations and axioms and complete rigor, I don't see anything wrong with the proof, or any need to mention that it's "deficient". I actually find the visual proof far more powerful and convincing than either Euclid's proof or proof #6, (both are based on the same idea); Euclid's proof I have to write out vertices of triangles and check the order and cross-multiply, follow all steps, when the visual proof makes it obvious. Revolver 03:26, 18 Mar 2004 (UTC)

Shure, it uses similarity, but it can be easely derived from any right set of axioms (it is some work but it is elemetary, just look in a reasonable book),

I have not idea how you use the word "elementary" or "reasonable", those are subjective, to say the least (esp. among a wide readership). Yes, you're absolutely right, the Euclidean Pythagorean theorem is strictly speaking, a theorem of Euclidean similarity geometry (it can be proved using the geometry determined by the group of similarities of R^2), but 99.9% of the people reading the article aren't even going to know what that statement means, let alone that's it's true. As far as general readers are concerned, each method of RIGOROUS proof is equally incomprehensible, intangible, and confusing. Most people reading this article won't even know what a group is, let alone the interpretation of geometry as properties invariant under a group, so if this is something worth pointing out (and perhaps it is...) it should be as a separate section at the end, since it's really speaking to the "family". Putting it right after the proof makes the invited dinner guests feel left out of the current conversation.

regarding area, to introduce it you need to be strong, it is just a bit easier than Lebegue measure (infat I do not know any elemetary book in Eclidean geometry which introduces area on a correct way, most comon gap is to assume that there is an additive area-function).

Good grief, in essence you're saying we can't use formulas for the areas of triangles and rectangles in Wikipedia, without making disclaimers that we haven't yet rigorously derived the existence, uniqueness and properties of Lebesgue measure in Euclidean space. You seem to be forgetting the audience. This is Wikipedia, not "Bourbakipedia". 90% of the people visiting this article will have little or no exposure to any kind of real deductive mathematical reasoning at all. The purpose of this article is to convince them it's true, not to give a bleached proof using the easiest set of axioms. Revolver 02:40, 19 Mar 2004 (UTC)

But I do not want (and never wanted) to remove this proof from here it is a nice proof in a way.Tosha 06:24, 18 Mar 2004 (UTC)

Well, good. I still make my objection that the comment about the "deficiency" in the proof is unwarranted, at least at that spot, and could go in a separate section near the end. Revolver 02:40, 19 Mar 2004 (UTC)

I think the remark stay at the right place, I do not know more than 99% of people but I'm sure they do not want to be fooled, maybe I'm wrong but then I only care about remaining 1%. Tosha 04:26, 19 Mar 2004 (UTC)

I think that about says it all. If you think that moving around triangles and rectangles in a general article without making some immediate disclaimer about the translation-invariance of Lebesgue measure is "fooling" (!) people, then you have a serious misunderstanding of the purpose of wikipedia. This place is written for the 99%, not the 1%. If that bothers you, write a graduate math textbook. Revolver 06:30, 19 Mar 2004 (UTC)

Regarding the visual proof, this is a minor detail, but the colours don't match up, e.g. the blue triangle doesn't go to the blue triangle, etc.... Revolver 03:40, 18 Mar 2004 (UTC)

I apologise, it's not the colours that don't match up, it's that a and b need to be switched in the right hand figure, they lengths don't match. Revolver 03:43, 18 Mar 2004 (UTC)

You're right. Can the person who created this picture, or someone else familiar with whatever software it takes to edit the thing, correct the problem? Michael Hardy 22:19, 18 Mar 2004 (UTC)

Seems to be User:Jellyvista - not contributed since November, so I'd suggest emailing him/her. --Trainspotter 12:41, 19 Mar 2004 (UTC)

Old image

I agree that the new image is better than the one I previously contributed (once the mis-labelling discussed above is fixed), but this is just to paste the old one here on the talk page just so that there's still a record of it, in case it's still useful to anyone.

--Trainspotter 12:41, 19 Mar 2004 (UTC)

NB

I revert it, if you change it make sure you understand it first, the problem is not congruence, the problem is that you can introduce notion of area such that area of equi-decomposed figures is equal. It was nice worning and after your changes became useless. Tosha 22:13, 25 Mar 2004 (UTC)

Okay, well then maybe it can mention exactly what is the problem. As it stands, all it says is "this relies on properties of area, which is difficult". I found the original wording unhelpful, for this reason. It says, "there is a problem, but I won't tell you what it is". It seems to me if the problem is severe enough to acknowledge, it can be explained. The very fact that I misunderstood the nature of the problem involved only indicates that it's unclear what is going on.

By "area such that equi-decomposed figures is equal", are you referring to finite additivity? Saying this isn't helpful unless the reader knows exactly what equi-decomposed means (I have an idea what you mean, but I'm not completely certain.) It seems congruence has to be involved somewhere, triangles are be moved around and assumed their area remains unchanged.

All I am saying is, if foundational or logical technicality problems seem important enough to mention to the reader, then they should at least be precisely stated and explained. I am a graduate student in math and obviously I didn't understand the precise problem involved according to you, so certainly the general reader won't understand. (This is why I favor moving this remark to a separate section devoted to foundational questions of various proofs, or not having it at all.) Equivalently, if some foundational question is not precisely explained, I don't see why it is mentioned. The whole point of foundational questions is that they deal with issues of precise logical foundations. It is the one area where precision is demanded.

I also think the comment that "good Euclidean geometry texts don't include this proof" is POV. I venture to say that up until the late 1800s, most mathematicians throughout history would have accepted the proof as perfectly okay. Of course, this is 2004, but my point is that not all geometry texts strive for perfect rigor, so this comment makes an assumption. Almost every introduction to Pythagorean theorem I have seen for general readers includes this as one of the most easily understood. It is only unacceptable in good texts striving for modern mathematical rigor.

Certainly the statement that PT is a statement of similarity geometry is okay. This is true, right? I welcome your suggestions, but I become a little frustrated with how often you make a blanket denunciation that something is completely "useless" as is. Almost everything (not always) currently written, even if bad flawed, has something worthwhile and useful in it. Revolver 18:47, 26 Mar 2004 (UTC)

Everything is understandable, my point is that you have too much energy but often do not think before making changes. I changed NB a little using your sugestions. Tosha 19:49, 26 Mar 2004 (UTC)

What is your definition of "area"? There seem to me at least 3 approaches I can think of,

Lebesgue measure approach, as outer measure generated by open rectangles
Riemannian manifold approach, area as determined by some Riemannian metric (I am not very familiar with this approach)
Synthetic approach, area defined using some axioms of a synethic geometry

I'm not sure which definition you mean. In any case, any proof that area of rectangles is finitely additive would depend on which approach you take. And in any case, I don't think it's necessary to justify to people that the area of the union of 2 rectangles is the sum of their areas. This is an extremely technical point that really is only of interest to professionals. Saying it here only confuses people. No one will understand what you are talking about.

The fact that area is invariant under isometries and with rectangles is finitely additive makes the proof rigorous; I don't think either of these needs to be mentioned. What is really important is that we assume Euclidean at all (in non-Euclidean, there are no rectangles, so you can't even "draw" the figure on the left or right. Revolver 18:40, 28 Mar 2004 (UTC)

I mean the synethic one, others require integration, and that means that they are not elementary. More over if you take a model fo E.plane then PT almost in the definition of distance, therefore no need to prove it.You are right it is enough to prove that area is finitely additive, but you need also triangles here. It is not an easy statement if you use synethic area, try to do it your-self you will see (and do it with out using PT). Tosha 20:22, 28 Mar 2004 (UTC)

So, what is the definition of area in Euclidean geometry? (this isn't meant to be sarcastic; in the U.S., synthetic geometry is no longer taught at any level of math instruction, it's possible to get a Ph.D. in math here without knowing anything about synthetic geometry. (In fact, in secondary school, proofs are rarely taught, most geometry classes have no proofs in them.) This may explain why I seem to almost nothing about it myself. I cannot verify that area is finitely additive for triangles, because I honestly don't know the definition of "area".

I don't see how the PT is a part of the definition of 2-dim Lebesgue measure, and the PT can be interpreted in terms of 2-dim Lebesgue measure, as a statement about 2-dim areas of certain squares. 2-dim measure is just product measure of 1-dim measure, which has distance defined, but only in trivial way. It also seems that you can model Euclidean similarity geometry by using R^2 with the usual inner product (and simply forget or pretend you "know" the distance formula) and then again derive the PT as statement about areas of squares (of course, I don't know what area means, so...)

I still maintain that the proof isn't supposed to elementary, but explanatory. This particular proof happens to be one of the oldest proofs known (about 2000-2500 years old), so this is how humans first discovered it (or justified it). Most people have no clue what synthetic geometry is, and any involved discussion will confuse them. This is not misleading people -- there are lots of informal and unrigorous proofs on articles here. There is probably enough information about proofs of the theorem, axiomatic discussions, and lots of details to warrant a separate article, say Proof of the Pythagorean theorem. If people are interested they can be prompted to go there. Again, it's not that I think it's irrelevant; but the PT is one of the most widely known theorems in the world -- billions of people who know very little about math are familiar with it, to some extent. These are the readers who will see the "visual proof", and almost all of them will be confused by a lengthy discussion. Maybe a short statement saying something like, "This proof appears simple, but uses unstated assumptions -- for more details of this and discussion of other proofs, go to..." Revolver 09:06, 30 Mar 2004 (UTC)

I fixed the visual proof image (switched "a" and "b" on the right side, using Gimp). --Mihai 02:08, 7 Apr 2004 (UTC)

Not an elementary proof

Also, I must say I also don't like the NB below the visual proof, at least in the current form. The way it's written now, the reaction of many Wikipedia readers is: "What do you mean the area of a square is not the sum of the areas of its pieces?! This text must be some sort of vandalism." I suggest two solutions:

A much shorter NB that says: "This proof makes use of assumptions only valid in Euclidian geometry."
A NB similar to the current one, but rewritten to include a link to more discussion and rewritten to sounds less confusing to people who only know simple math.

In conclusion, in this little debate, I'm on Revolver's side. By the way, I'm a physics PhD student. --Mihai 02:24, 7 Apr 2004 (UTC)

I think you misunderstand the note. It's pointing out that the visual proof relies on an unstated lemma that you can dissect a figure and reassemble the pieces to get a new figure with the same area. This lemma is not true in general (see Banach-Tarski paradox) but it is true for a suitably restrained notion of dissection (in which the pieces are measurable). So turning this simple and convincing visual proof into a fully rigorous proof is going to take some pretty complex mathematics. Other proofs are more elementary because they need concepts only from geometry and not also measure theory.

The note probably needs a little rewriting to make it clear to non-mathematicians. Gdr 12:17, 2004 Jul 26 (UTC)

Note that since the beggining of discussion the NB was reworded (nicely) on 3 Jun 2004 by User:Stevenj. Tosha 19:14, 26 Jul 2004 (UTC)

The way the NB is currently worded, is about the best it's possible to be worded. I don't see how it can be more clear without sacrificing what it's saying. Revolver 19:54, 26 Jul 2004 (UTC)

A Numerical proof

I would remove it, I think it does not add anything to the article, but do not want to do that before I will get somebody on my side. Tosha 11:17, 8 Aug 2004 (UTC)

copyright status of the images

what is it? -- Ævar Arnfjörð Bjarmason (https://academickids.com:443/encyclopedia/index.php?title=User_talk:%C6var_Arnfj%F6r%F0_Bjarmason&action=edit&section=new) 17:54, 2004 Sep 4 (UTC)

I suspect they were created for the express purpose of putting them in this Wikipedia article. Michael Hardy 22:14, 4 Sep 2004 (UTC)

Application

Shouldn't there be a section regarding the application of the pyth-theorem in this text. For instance i wondered that sqrt(cos(φ)^2 + sin(φ)^2) = 1 (because of the sinus definition in the the unit circle) wasn't mentioned anywhere, although it would fit in here quite well.--Slicky 18:54, Sep 18, 2004 (UTC)

Historical note on the theorem not related to the theorem?

I thorougly do not understand Tosha's reasons here for flat-out deleting this note about Pythagoras not proving the theorem (Euclid did). See this version (http://en.wikipedia.org/w/wiki.phtml?title=Pythagorean_theorem&oldid=8344208) for what I'm talking about. I think this historical fact is worth noting since I'm sure only a few people know/realize that Pythagoras didn't prove the theorem true that was named after him.

Though, I see this isn't Tosha's first time in deleting stuff from this article.

Since I really don't want a revert-war with Tosha I'll pose it here. So what's others' thoughts on this? Is Euclid first proving the theorem related and/or worth noting on an article about it? Cburnett 23:53, 12 Dec 2004 (UTC)

User:Tosha's response might have been a little too brief. I think the only information worth putting in the article is the fact that the first proof of the Pythagoraen theorem can be found in Euclid's Elements. The rest of your edits do not really add to the article, instead they should go to history of greek mathematics or history of greek science. Feel free to revert my edits. MathMartin 00:24, 13 Dec 2004 (UTC)

I think this is a bit myopic. I know math people, and we tend to have the opinion that what we think is important about (mathematical) topics is all that anyone thinks is important, and this tends to exclude anything that doesn't fit into a "definition, theorem, proof, corollary" framework. This adds up to a strange education, where mathematicians are not unusually fairly ignorant about the social and historical development of their own subject. Although I would use a different quote than Cburnett used, I know the point he was trying to make. Namely, the discovery of the empirical fact of Pythagoras's theorem led to the discovery of a fact which appeared to be self-contradictory or absurd given the current understanding, and this led people to find and discover deductive proofs, not just demonstrations of the theorem. As it turned out, the discomfort over this particular result took a long time to overcome -- not really until the invention of Dedekind cuts was the issue truly addressed. While I would not include such a discussion at the top of the article, I don't see how one can say "the only information worth putting in the article is the fact..." If any result has implications to the history of math, this one does. To say that any other historical comments should be exiled to history of greek mathematics or history of greek science seems POV. Revolver 15:00, 15 Dec 2004 (UTC)

See, now that I can live with. Although I think it's lacking, I can make a compromise.

I suppose I'll never understand some people's predilecation for the delete key. The "proof" used in Pythagoras' time was akin to "see it works, therefore it is true" which is the opposite of how a real proof is performed. This is the distinction made between Pythagoras & Euclid in their proof of the theorem. While it, perhaps, belongs in an article on mathematical history or greek math history (thus the topic of Russo's book) I thought this article was really lacking in the history of the theorem. Considering the number of peoples listed in the opening paragraph, it hardly does the theorem justice. Cburnett 01:31, 13 Dec 2004 (UTC)

From my talk page:

Well, you do not know for sure (who proved it), all you know is that it is proved in Euckids book, and it is unlikely due to Euckid, anyway it is more related to history of math. Yes, Pythagoras did not have a proof in a way we understand it now, but the same is true for Euckid (although it is closer) so there is no point in your statement. The same is true for all old theorems... Tosha 01:04, 13 Dec 2004 (UTC)

Fine, insert "known" into "...provides the first proof of the Pythagorean theorem...". There's no point in stating that the first known *real* proof of Pythagorean theorem was by Euclid???? No point??? Such point has been the most interesting thing I've read directly regarding the theorem. Cburnett 02:13, 13 Dec 2004 (UTC)

I think you misunderstand Tosha's "there is no point" comment. If I read him correctly, he's not saying the matter is unimportant; he's saying that to be really precise, not even Euclid had a *real* proof of the theorem that meets with modern standards. While he is technically correct, I think this rigid application of criteria is unreasonable to follow. Taking into account that the most basic foundations of the Elements are flawed from a modern viewpoint, the entire work can be discounted as "not a real proof" or somehow unworthy. For that matter, this criticism can be applied to almost all mathematics done before the first quarter of the 20th century, so Tosha must apparently think "old" means "older than 1925" or something. For me personally, the important thing is not whether or not Euclid's proof is an "according to Hoyle miracle" and passes a modern litmus test, but rather whether "god got involved", i.e. the fact that Euclid's argument shows a clear understanding of the distinction between (attempted) proof and demonstration, and a grasp of the concept of logical argument and deduction, and these are clearly present in Euclid in a way that wasn't present in Pythagoras's understanding. Moreover, although Euclid's proof is technically "wrong", it is (as Tosha points out) "closer" and needs little adjustment to meet modern standards. It is true that Euclid almost certainly was not the first to actually discover the proof, but the Elements is the first known work to present it (today, we would say it was first "published" in the Elements). And I fail to see how some historical comments are out of place in the article; should all historical math comments be relegated to separated articles? I agree, the way it was it seemed out of place so much at the beginning, but a separate section toward the end of the article specifically discussing the history of the theorem and proof seems okay. Revolver 14:36, 15 Dec 2004 (UTC)

Euclid's definition of area

I've read through the previous discussions related to the "n.b." after the nice illustrative proof of the Pythagorean theorem. It seems to me that these discussions have missed a very important point. The focus of these discussions is on the (supposed) difficulty of finite additivity of area, and in particular, what is usually called the scissors congruence of planar regions.

But Euclid's definition of area was that two polygonal regions have the same area if they are scissors congruent! So how to make sense of the statement, "In particular, while it is easy to give a formula for area of triangles and squares, it is not as easy to prove that the area of a square is the sum of areas of its pieces."?

Are we to always take the notion of Lebesgue measure as primitive and even in straightforward contexts (such as this) "point out" that proofs using Euclidean notions are not "elementary"?

As the NB currently stands, I feel it is misleading. The given illustrative proof is in fact very simple, presupposing the basic notions of Euclidean geometry. There is no need for an NB in my opinion. --Chan-Ho Suh 06:07, Jan 2, 2005 (UTC)

If you start with definition as two polygonal regions have the same area if they are scissors congruent then you have to show that non all polygonal regions have the same area, that the same thing as finite aditivity of area... Again it is doable, but it is much harder than Pythagorean theorem.

Pythagorean theorem is a basic result and one should expect to see the proof which follows directly from axioms. on the other hand definition of area often avoided in books on Eucl.geom. So most of the proofs using area are cheating and little worning sign is good. Tosha 20:51, 5 Jan 2005 (UTC)

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