Talk:Projectively extended real line

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Created new article[edit]

I have started the article. It is nowhere near complete, but I must take a break now (will continue working on it later today). In the meantime, I will gladly hear any comments, and invite everyone to improve the format. -- Meni Rosenfeld (talk) 10:55, 24 January 2006 (UTC)

On the format, the use of self-links is against basic good practice.
On the content, I'm concerned that this has little of the content I'd expect of a projective line explanation; such as homogeneous coordinates, the transitivity of Möbius transformations, the interpretation as one-dimensional subspaces of a two-dimensional space. Charles Matthews 21:51, 24 January 2006 (UTC)
Agree with Charles. The emphasis of this article should be on projective geometry—of which there is barely a mention—not on arthimetic operations on this structure. I would hardly say that "the most interesting feature of this structure is that it allows division by zero". -- Fropuff 23:09, 24 January 2006 (UTC)

About format, what did you mean by self-links?

Agree with the above. I'm afraid it does not have the most important information about Real Projective Line, which is some geometric motivation and construction; relation to Projective Geometry. — Preceding unsigned comment added by 79.179.208.66 (talk) 00:08, 6 May 2013 (UTC)

About "little of the content...", like I said, this article is nowhere near complete. I will need the help of all of you to finish it...

About what is interesting, that is of course subjective. I have written mostly about things that I find interesting and that I understand. Admittedly, I don't understand very well all the topological and geometrical implications, and I find them less interesting than the arithmetic and analytical ones. Also I wrote more about what is special to this structure, rather than generic things which hold for every 1-point compactification (which can be found in the projective line article). These things should of course be here as well, and I will continue to add things that I know, but I will need your help in those points that I'm not proficient at. For now, I'll rewrite the "most interesting feature" part to be more NPOV.

And also, if anyone is skilled enough to create an image of the circle representing this structure, that would be great. -- Meni Rosenfeld (talk) 08:02, 25 January 2006 (UTC)

BTW did you mean the projectively extended real numbers thing? I did that mostly for emphasis, I will change that if you think it's wrong. -- Meni Rosenfeld (talk) 08:04, 25 January 2006 (UTC)

Yes, that's a self-link because the redirect comes back to the page. Charles Matthews 08:17, 25 January 2006 (UTC)
fixed :) -- Meni Rosenfeld (talk) 08:37, 25 January 2006 (UTC)

a / 0 = ∞[edit]

Regarding this edit, I think that the definitions of the artithmetic operations should be given separately for real numbers and ∞. That is, a will always stand for a real number, and ∞ will be explicitly called ∞. I think it will be clearer this way how these definitions extend the operations on real numbers. Any ideas? -- Meni Rosenfeld (talk) 15:27, 25 August 2006 (UTC)

Intrval arithmetic[edit]

From the article, we have for a,b\in\widehat{\mathbb{R}}

x \in [a, b] \iff \frac{1}{x} \in \left [\frac{1}{b}, \frac{1}{a}\right ]

But what about [-1, 1]? In 'ordinary' interval arithmetic this would be either undefined or the negation of (-1, 1) -- it can be infinite, but not strictly smaller than 1 in absolute value. But the formula above gives [1,-1]=\emptyset, which seems to contadict the article's claim that the result is always an interval. Or rather, if we accept this as an interval, it no longer follows that the interval contains the results of calculations with points inside the original interval (since 1/0.8 is defined and not in [1, -1]), which would make interval arithmetic useless. CRGreathouse (t | c) 23:24, 25 September 2006 (UTC)

I'm not sure I understand. If x \in [-1, 1] then, by the formula above, \frac{1}{x} \in [1, -1] = [1, \infty) \cup \{\infty\} \cup (\infty, -1]. What's the problem? Are you sure you've read the "Definitions for intervals" subsection just before the "Interval arithmetic" section? -- Meni Rosenfeld (talk) 08:47, 26 September 2006 (UTC)
Ah, sorry, I must have missed that. CRGreathouse (t | c) 06:32, 27 September 2006 (UTC)

In English?[edit]

I would love to understand this article. Especially since I put the image in my signature. Any work that can be done to better explain it is awesome, and if you want to try to break it down for me too, it's much appreciated!-- Patrick {oѺ} 06:37, 24 July 2009 (UTC)

==Function of numbers==

Apart from aesthetic writing purposes, the function of numbers may be said to be the means of quantifying the relative size of things. And the number line and the related rules of notation are the means of achieving a universal agreement on the quantity value of each individual number. And it works fine for simple addition and subtraction operations, and for counting quantities of sets of things. And after the advent of rules of geometry and trigonometry its use has been expanded into exponential and angular quantification values. But I can't see any reason to curve the number line since that distorts your concept of what you are doing as you move along it.WFPM (talk) 14:43, 18 August 2009 (UTC)

"real projective line": the standard name?[edit]

Yeah, I was kinda testing the waters with that edit as to whether it would be necessary to discuss this, and it would appear that it is. This looks to me, as a relative outsider, like a fairly clear case of a term (real projective line) being adopted and used from a more general concept, and ultimately usurped, by a specialist field in mathematics (namely analysis in this instance). Since the term almost certainly has not lost its original related but conflicting meaning in other areas of maths such as geometry and group theory, in an encyclopaedic context this kind of insular use of an adopted term essentially exclusively the way one subdiscipline uses it seems inappropriate. This article is about a set with a specific identification of points on such a structure, and is thus about the additional structure and not the concept in the title. I would like to see others' opinion on the standardness of this term in this particular use across mathematics at large, with a view to finding the most suitable title for this article. Please note that there are several articles that form a family that use the term in a different way: Projective geometry, Projective line, Projective space, Real projective plane, Real projective space, Complex projective plane, Complex projective space, and no doubt many others. —Quondum 17:27, 14 December 2014 (UTC)

Completely standard and there is no difference between the object described here and the specialization to the reals of the object described in all those projective geometry articles. It's a matter of viewpoint. Different disciplines will describe or define an object in a way that is most conducive to what that discipline is interested in examining. In analysis, the simplest way to get to the object is by adjoining a new point to the real (number) line, but the drawback of that approach is that the new point "looks different" from the others when in actuality the space is homogeneous and all points "look alike". Other disciplines will describe the real projective line in a clearly homogeneous manner but will then have to "pick" a point to play this special role. It's six of one or half a dozen of the other as far as how it should be presented. I know that you are now going to say that we should present both views, and I agree, but only to a limited extent. In the present article I think it would suffice to put into the lead a statement to the effect that an alternative view of this object can be obtained from the Projective line article. This will keep the article honest without overburdening it in precision. Bill Cherowitzo (talk) 18:59, 14 December 2014 (UTC)
User:Quondum wrote: there are several articles that form a family that use the term in a different way: Projective geometry, Projective line, Projective space, Real projective plane, Real projective space, Complex projective plane, Complex projective space. What does this mean? In what way is their usage of this term different? Tkuvho (talk) 09:24, 15 December 2014 (UTC)
I will ignore for the moment that geometers sometimes use the adjective "projective" to apply to any space that preserves lines, noting only that this leads to a bigger class of objects in 1 and 2 dimensions; this further point will only confuse the argument, and I am thus for my argument here excluding this interpretation.
Consider the class of projective planes for which the homographies are the transformations. I could write an article about the real elliptic plane, and name it "real projective plane". I could then argue that it is the same object as anyone means by "real projective plane", since it is undoubtedly a real projective plane. The problem is that I would be discussion only a subclass of the real projective plane, and I'd be talking about additional structure such as a metric that does not apply to the general real projective plane. I'd fail to mention many of the interesting homographies, and focus on a subgroup thereof. As a result, a reader would get a completely different picture of what the definition of a real projective plane is. The problem is that I've added structure in the definition that I'm using; it is like defining a ring and calling it a group. I'm not saying that the projectively extended real numbers are not a real projective line; I'm saying that the class of real projective lines is not the same class as the projectively extended real numbers. This article has a deficient section Real projective line#Geometry that hardly dispels this; Tkuvho has only now added "in geometry" to the lead. And the section Real projective line#Hyperbolic involutions is simply wrong: there are more than two hyperbolic involutions on the real projective line, unless one has an incredibly restrictive definition of "elementary arithmetic involution". —Quondum 15:07, 15 December 2014 (UTC)
Even though you wish to exclude this from consideration, I think something must be addressed because it bears on your main concern. No geometer (or anyone else really) would use the expression "... space that preserves lines". Preserving lines is not a property of a space, rather it is a property of the transformation group of a space. Even under the strictest interpretation of the Kleinian viewpoint, a space and its transformation group are not identified. The Kleinian view, succinctly put, is that the geometry of a space is the study of the invariants of the transformation group of that space. Notice that the concept of the space is not being defined by its transformation group, only the geometry of that space. Now, on to your main points. To exclude the problem areas we can restrict the discussion to the projective spaces PG(2,F), the projective planes defined over a field F (and here I must apologize for the notational choices of my fellow geometers, the "PG" in this notation stands for Projective Geometry but it should really be "PS" for Projective Space.) The transformation group of PG(2,ℝ), the real projective plane, is PΓL(3,ℝ) = PGL(3,ℝ) only because ℝ has no non-trivial automorphisms. Note that I am not picking the transformation group to consist only of homographies, it turns out this way because of a field property. Thus, in more generality, your class of projective planes for which the homographies are the transformations, is the class of projective planes defined over fields which admit no non-trival automorphisms. In a similar vein, the metric that is used in your discussion of the real elliptic plane, is not intrinsic to the projective space, it is coming from ℝ. There is only one real projective plane, talking about subclasses makes no sense to me, however, one can look at this object through various filtering lenses. Ignoring all properties of ℝ except for its field properties will give you the picture as a projective geometer sees it, while also letting in the the metric properties of ℝ will give you the topological viewpoint. Articles written exclusively from either of these viewpoints will not look the same (so I am in agreement with your concerns, but for different reasons); different topics will be emphasized and others ignored. This does not mean that the subjects are different. As you know, there are difficulties with the one dimensional projective spaces that are dealt with by altering the definition. However, the modification ensures that the real projective line and the extended real line are the same. What you are seeing as different objects is the same thing viewed from different perspectives. Bill Cherowitzo (talk) 20:49, 15 December 2014 (UTC)
I have a feeling Sławomir may have an opinion on this. As far as I'm concerned, I have clearly failed to communicate what I wish to communicate. I abused the term "space" as you say, but this is a common style of abuse; it is precisely this kind of terminological abuse and consequent misunderstandings in mathematics that I find most frustrating. —Quondum 22:36, 15 December 2014 (UTC)

Merger proposal[edit]

I oppose to this merge. As the subject which is studied is essentially the same, the presentation is not the same. The main difference is that, here, the projective line is not considered independently from the choice of an embedding of the real line into it, while in Projective line, there is no distinguished point ∞. It follows that Real projective line is more elementary than Projective line. However, the different scopes of the two articles must be clarified. This is clear in Projective line, where the other article appear in a {{main}} template. I'll be bold and clarify this article by editing the top and moving this article to Projectively extended real line. D.Lazard (talk) 08:15, 11 April 2015 (UTC)

I also oppose the merger, essentially for the same reasons D.Lazard has given. This article makes a big deal of the arithmetic structure that is not canonical in the general projective line. This was essentially behind my comments in the thread above. I strongly support the move as done by D.Lazard. —Quondum 15:39, 11 April 2015 (UTC)
Oppose merger as per D.Lazard. Not withstanding my comments in the above thread, this specialization of the general projective line is the most common way for readers to approach the subject from different fields of mathematics. To have that buried as a section in a more general article seems like a disservice to our readers. I also support D. Lazard's move as it brings some clarity to the subject. Bill Cherowitzo (talk) 17:43, 11 April 2015 (UTC)
Face-smile.svg I think we have reached a generally agreeable result: the general reader who understands the specialization under the name "real projective line" is still pointed at this article via the redirect. And yes, I agree that this should not be buried as a section of Projective line. —Quondum 17:50, 11 April 2015 (UTC)

Obscure title[edit]

A Google check showed that in the whole internet today only five (5) links use "Projectively extended real line" whereas "Real projective line" has 8400. The move was unjustified and ought to be undone. If editors have a "good idea" they should raise them in Talk before making a Move as has occurred in this article. Further, the subject of projective geometry is sufficiently complex and historically rich that caution should be taken in all its topics. My impression is that editors dabble here when they should be reading and learning, not changing articles and their titles.Rgdboer (talk) 20:31, 11 April 2015 (UTC)

Ah, good point. The title should be "Projectively extended real numbers", which gets over 28000 hits. (PS: I think it would be incorrect to think of this as falling within the scope of projective geometry, despite the identification that is made with the projective line over real numbers.) —Quondum 21:04, 11 April 2015 (UTC)

Unstated interpretation of Limit of a sequence[edit]

The lead contains the statement

"More precisely, the point at infinity is the limit of every sequence of real numbers whose absolute values are increasing and unbounded."

While this seems to me to be absolutely correct under the correct topology (i.e. with a suitably chosen metric), it risks falling apart for the typical reader who would think in terms of limits on real numbers as they are normally defined, in which case for many such sequences (e.g. one of which the sign continues to change), the limit does not exist. Some careful rewording might be wise, perhaps mentioning the neighbourhood of or somesuch. —Quondum 21:25, 11 April 2015 (UTC)