Talk:Projectively extended real line

Created new article

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 = ∞

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

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)

The result is $[1,-1]$, and by definition (few paragraphs before), this includes everything below or equal to -1, above or equal 1, and $\infty$. The trick is that the ends are reversed. It works. In 'ordinaty' interval arithmetic, the result would be $[-\infty ,+\infty ]$ or two intervals $[-\infty ,-1]\cup [1,+\infty ]$. 2A02:168:F609:0:DA58:D7FF:0:F02 (talk) 01:08, 17 December 2018 (UTC)

In English?

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?

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

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)

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

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)

Assuming "real projective line" is merely a variant of the more common usage "projective real line", one should count both. Together Google finds 8400 + 14700 = 23100. That "projectively extended real numbers" gets five thousand more hits may be the result of the Wolfram Mathworld entry for it by David Cantrell. Is there a precedent for Cantrell's use of that terminology?

All these terms are dwarfed however by "affinely extended real numbers" as the two-point compactification of the real line, with 70900 hits. This notion is well motivated by arithmetic concerns, unlike "projectively extended real numbers" which only becomes conceptually coherent when interpreted geometrically as a projective space, whose projective invariants are not real numbers at all but cross-ratios. With that interpretation the more appropriate (and less unwieldy) name would seem to be "projective real line". This suggests that this article be so moved, and I therefore so move. Vaughan Pratt (talk) 19:55, 4 June 2015 (UTC)

On the other hand the discussion at Projectively_extended_real_line#Merger_proposal seems to be making the opposite argument, namely that there exists such a thing as the projectively extended real numbers distinct from the projective line. Is this one of those things that distinguish analysts from topologists? If so I retract my motion, and would be interested to understand how it arises in analysis. Vaughan Pratt (talk) 20:36, 4 June 2015 (UTC)

Meanwhile I now realize that my assumption above that "real" commutes with "projective" when applied to "line" is wrong. A "projective line" can be defined as a one-dimensional subspace of a two-dimensional vector space. "Real projective line" in that case specifies that the vector space in question is over the reals, with the complex numbers in place of the reals giving the complex projective line, aka the Riemann sphere (confusingly not a Riemannian manifold though at least a conformal one). Interchanging "real" and "projective" yields the concept of this article. In light of the section Projectively_extended_real_line#Arithmetic_operations, collectively a kludge that works in some situations but not others, the concept fully deserves the unwieldy name "projectively extended real line". I was unable to see how the article's other justifications for the concept distinguish it from the (much cleaner) notion of the real projective line. Vaughan Pratt (talk) 00:14, 5 June 2015 (UTC)

– Nice summary. It's a one-element extension to a field, yielding a new object that has two exceptional elements, and is no longer even a ring. It's in some ways algebraically more regular than either the real numbers or the affinely extended real numbers. But I'd still argue for "numbers" in place of "line" in the title. —Quondum 00:37, 5 June 2015 (UTC)

"Projectively extended real numbers" is fine by me (now that I've had a chance to absorb the intended distinctions). It's more about arithmetic (including interval arithmetic) than geometry, and it's what David Cantrell calls it at Wolfram MathWorld. Vaughan Pratt (talk) 02:41, 5 June 2015 (UTC)

I hinted at such a renaming in my first post of the thread, but no-one took the bait. Maybe we should get separate opinion on this specific renaming; although renaming has been discussed in the past, the general consensus is really unclear to me. —Quondum 14:04, 5 June 2015 (UTC)

If in a week's time, based on the discussion to date, the "vote" remains two for and none against, let's do the move and see who complains. Vaughan Pratt (talk) 23:45, 7 June 2015 (UTC)

You might want to look at my first posts in § "real projective line": the standard name? and in this thread above. Despite evidence for its widespread use, certain people object that it is not the standard name (though I have yet to see the evidence for it). Tkuvho are you going to stay silent until we try this and then jump out of the woodwork? Wcherowi, are you still going to insist that the one-point compaction of the real numbers is suddenly classified as a geometric object? D.Lazard and Meni Rosenfeld might also wish to express an opinion. —Quondum 03:17, 8 June 2015 (UTC)

I'm Not really an expert on this, but from reading Projective line it seems that the content of this article is exactly a projective line where the underlying line is the real number line. Hence, "real projective line" seems appropriate.

I object to Vaughan's dismissive remarks. First, it looks like the arithmetic in this article is exactly they same as in Projective line - it's just written in a more detailed way. Also, it's not a "kludge" - it's exactly what you'd expect when you make a one-point compactification of the real number line, and extend addition and multiplication so that they are continuous. Even division is continuous everywhere except 0/0 and ∞/∞, better than what you have with real numbers. The structure follows naturally and intuitively from the desire to make rational functions continuous. It's also the same kind of extension as the more popular Riemann sphere.

We've also had a discussion about this recently at WP:RD/math.

I support moving to "real projective line", unless someone can convincingly argue that it's actually not the same as a real projective line. I'm also ok with anything that starts with "projectively extended" and ends with two words out of "real", "number" and "line". -- Meni Rosenfeld (talk) 13:22, 8 June 2015 (UTC)

This article is indeed related to Projective line, but that too is a misnomer, using the name "line" in the grade school sense of "axis" (or "number line", as D.Lazard points out) but that article mixes the two concepts inexcusably, as this article did, rather than the more general geometric concept from projective geometry. There is a distinction between the concept of this article and that of a line in projective geometry, which is the same as the gap between a ring and a group. Every ring is a group, but we do not use the word "group" when we are referring to the category of rings. Similarly, the category of "projective lines" (in the sense of this article) is a proper subcategory of the lines of projective geometry: there is additional structure that does not adhere to the latter.

Meni, yes, it is not a kludge: it is a beautiful and consistent concept. But "projective line" is a bad name: it seems designed to create misunderstanding, fuzzy mathematical thought and arguments for those learning about it. And finally, the term "projectively extended real numbers" is in widespread use. It think that it is time for notability arguments: we need to show what terms are actually in use. —Quondum 14:53, 8 June 2015 (UTC)

• Weak support of the title Projectively extended real numbers. This title is fine by me, and clearly better than the present one, that has been chosen by me for clarifying my opposition to a proposed merge. However, I would prefer Projectively extended number line, which has three advantages: firstly, keeping "line" better emphasizes on the geometric aspect of the extension; secondly, "number" emphasizes on the arithmetic content of the article; thirdly, even if this extension is more important on the reals, it works on any field, including the complexes, and thus, there is no need to emphasize on "real" (note that "Number line" and "Real line" refer to the same concept). D.Lazard (talk) 09:08, 8 June 2015 (UTC)

  What about generalization to other fields, as in Projective line? "Number line", if it implies reals, cannot be used there. What would you rename that article for consistency? —Quondum 14:53, 8 June 2015 (UTC)

  Because of the most common use of the concept (dealing with limits), the article must be restricted to the common case of the reals. However it could contain a section "Generalization" explaining that the same construction may apply to any field, and is called "Riemann surface" over the complexes. Also "rational line" and "complex line" seem rather strange, and "number line" and "real line" seem to be synonymous (as the equivalent of "number line" does not exist in French, I am not a good judge for that). By the way fr:Droite réelle contains a nice explanation of the concept of "number line", which would deserve to be included somewhere in English Wikipedia.D.Lazard (talk) 15:41, 8 June 2015 (UTC)

  Not a Riemann surface, rather a Riemann sphere (also the extended complex numbers and extended complex plane). While I don't object to the phrase "number line" to mean "real numbers", as you point out the term "line" does not work well in other contexts, even though it does work for the related geometric cases (e.g. "a line of a complex projective plane"). I would support either your or my suggested titles for this article, but does not deal with the bad title at Projective line. —Quondum 18:18, 8 June 2015 (UTC)

Just a couple of comments since I have been (reluctantly) dragged into this debate. First, I think that there should be an article about the real projective line (a 1 dimensional subspace of the real projective plane) as this is an important concept used in various mathematical disciplines. Quondum has raised the question, "Is this article it?" and has expressed the view that he doesn't think so since the content seems to be more about an algebraic construction/concept. I am on record as disagreeing and I continue to do so. I think that once you have identified the real numbers with the points on a line you have turned an algebraic concept into a geometric one (albeit with a strong algebraic flavor) and there is no turning back. Post-identification you have an object which can be looked at in many ways; algebraic, geometric, topologic, etc., and no one point of view can give the complete picture. I think the title of the article should reflect the object being discussed and not the point of view taken in that discussion and further, the discussion should be broad enough to encompass these different approaches. I am also disheartened by the lack of references in this article and the one link given does seem to have a strange perspective. Bill Cherowitzo (talk) 17:48, 9 June 2015 (UTC)

I agree with User:Wcherowi that the natural title for this page was and remains real projective line, and that it should be moved back to that title. Tkuvho (talk) 17:57, 9 June 2015 (UTC)

I agree with Wcherowi and Tkuvho that an article real projective line would be useful. However this cannot be this article, which is devoted to another mathematical structure, the projective completion of the real number line. This structure may be identified to a real projective line equipped with three fixed points named 0, 1 and ∞. The fact that these two structures are really different result clearly from the fact that a real projective line has infinitely many automorphisms (the homographies), while the "projectively extended number line" does not have any automorphism. Therefore, I oppose to naming back this article "real projective line". On the contrary, I propose to redirect Real projective line to Projective line until the creation of an article specific to the real case. D.Lazard (talk) 09:19, 10 June 2015 (UTC)

The Riemann sphere CP^1 similarly has a variety of structures depending on which field of mathematics you are coming from. This does not mean there should be a separate article for CP^1 in algebraic geometry, CP^1 in complex analysis, and CP^1 in projective geometry. I find the argument from additional structures very dubious. Tkuvho (talk) 09:22, 10 June 2015 (UTC)

I also don't find this argument convincing. Since the extended object is not a field, in what sense are you referring to its automorphisms? What structure is being preserved? It seems to me that if the answer is to be that there are none you must have already made several undeclared assumptions about this object; one of them being that it is not the real projective line. Bill Cherowitzo (talk) 15:44, 10 June 2015 (UTC)

The pair formed by a Euclidean vector space and its projective completion is an algebraic structure whose automorphisms are the isometries of the vector space, or, preferably, the direct isometries (preserving the orientation). Here, we are in the particular case of dimension 1, where the only direct isometry is the identity. Clearly, such a pedantic description is not worth to be inserted in the article. However, everything (except the last section) in the present version of the article deals with the properties of this structure, which may thus be considered as the subject of this article. On the other hand, an article on the "projective real line" would require a detailed description of the homographies (including their applications to optic, a lens inducing a homography of its axis), which is lacking here. Clearly, the projectively extended real line must appear as an example in the "projective real line article", and this article must say that the extended line is an example of projective line. But, I do not see how to merge these two subjects into a useful article. To summarize and to answer to Wcherowi, the embedding of the reals into a projective line is not a projective line, it is an embedding. D.Lazard (talk) 16:57, 10 June 2015 (UTC)

I anticipated these widely diverging opinions from previous discussions. Bill, surely you'll concede that the field of real numbers has a unique automorphism, but that the affine real line has an infinity of these? Similarly, there should be acknowledgement that we a dealing with two distinct objects in this case: D.Lazard is correct (except that I presume he meant "does not have any nontrivial automorphism"); it should be clear that automorphisms are transformations that preserve the defined structure, even if we are not clear on a name for that structure. —Quondum 17:45, 10 June 2015 (UTC)

Thank you Daniel, you've convinced me. I've never seen this except as an embedded object, but perhaps that is just a bias of my geometric background. This still leaves us with the problem that there is no article about the real projective line and I think there should be one (maybe combined with the complex projective line but definitely not a section of the general projective line article). Bill Cherowitzo (talk) 05:10, 11 June 2015 (UTC)

I agree that we need a separate article that will do the geometric object justice, which this article cannot do. It should not be not combined with Riemann sphere, which is what Complex projective line redirects to. —Quondum 12:44, 11 June 2015 (UTC)

Difficulties are stemming from one editor’s interest in non-Desarguesian planes where there is no structure of projective harmonic conjugates. When the real projective line is viewed in the real projective plane, or in the Riemann sphere, then the structure of harmonic conjugates is inherited, but not if the line is in a non-Desarguesian plane. Resistance to harmonic conjugates, and to the tradition of homographies revealing structure on the real projective line, has produced this ugly article. The editor is perpetuating a debased view of projective geometry put forth by Reinhold Baer that does not include harmonic conjugates as essential to projective geometry.Rgdboer (talk) 23:08, 10 June 2015 (UTC)

Unstated interpretation of Limit of a sequence

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)

Isn't the existing wording defining +∞, as distinct from either −∞ (decreasing sequences) or ∞ (as the identification of +∞ and −∞)? Vaughan Pratt (talk) 20:46, 4 June 2015 (UTC)

No, since it talks about absolute values. So, according to this sentence, the sequence 1, -2, 3, -4, 5, ... converges to infinity, so it is clearly unsigned infinity. -- Meni Rosenfeld (talk) 14:19, 8 June 2015 (UTC)

Well, by definition this article talks about the new structure, with its own topology etc. In "real numbers as they are normally defined", there's not even such a thing as ∞ (it's used as a piece of notation, but not as a first-class object). If the reader wanted to know how it is with real numbers, he'd read Real number...

Another merger proposal

This article and Real projective line seem to be talking about the same thing. Do you think both articles should be merged? Llightex (talk) 18:18, 3 August 2015 (UTC)

No. The projectively extended real line is the real line with a point added at infinity. The real projective line is the space of one-dimensional linear subspaces of a two-dimensional real vector space. These are rather different, and can only be identified when some extra structure is specified. Sławomir
Biały 18:24, 3 August 2015 (UTC)

Although the previous proposal was to merge with Projective line, most of the length discussion, above, was about the new merge proposal. This may be unclear as, as that time, the article real projective line did not yet exist. Nevertheless, a consensus has been reached for the need of two different articles Projectively extended real line and Real projective line. Thus, I oppose to this merge, for the same reasons as those that are above developed. IMO this discussion does not need to be continued, unless if new arguments are provided, that could change the consensus. D.Lazard (talk) 20:02, 3 August 2015 (UTC)