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This page sucks balls! What of all the middle school geometry students who stop by to learn about the line? We should introduce things a little more gently, in the context of ordinary Euclidean geometry in the plane, and then in 3D, before getting into the fancier abstract concepts. I'll make these changes if there's no objection and no one else does first. Deco 04:09, 2 June 2005 (UTC)
Here Line and Line segment are treated as part of the same article. In Polish Wikipedia - they are separated. How can it refer to both articles in Polish now? — Preceding unsigned comment added by Xyzzer ( talk • contribs) 16:16, 31 January 2006 (UTC)
This is because the content of this article is in two bg articles. Nplus 17:22, 4 March 2006 (UTC)
The article on Interval_(mathematics) includes various notations on number sets, including the French notation I grew up with. This article doesn't include the equivalent notation I learned, which was:
I don't know if that's also used for American notation, but an equivalent reference here would be handy. (I see something vaguely similar, but less complete, in the french version of Segment 67.171.149.4 20:07, 16 March 2006 (UTC)
No where in the article does it say that a line contains infinite number of points. Is this correct, or does Quantum Physics state that this is incorrect? —The preceding unsigned comment was added by 70.59.199.11 ( talk • contribs) 14:43, 10 March 2006.
Perhaps this article could benefit from some discussion of more abstract definitions of a line? e.g. ("a straight line is a curve, any part of which is similar to the whole" from topology) Brentt 11:52, 16 August 2006 (UTC)
POLYLINE redirects to this page but is then not discussed. Can someone mention POLYLINE here, or make a distinct page?
Beau Wilkinson 18:34, 27 September 2006 (UTC)
The article said "In three dimensions, a line must be described by parametric equations". This is wrong: a line in any dimension can be described by a linear equation. I changed "may" to "must" and added a couple of linear equations for a 3D line. This leaves the definitions section a bit rambly, in my opinion --- why describe a 2D line in slope-intercent, versus a 3D line in parametric and standard form? --- but I felt it was a step in the right direction since it is at least correct. It does have the advantage of getting the link to linear equation earlier in the page. Owsteele 14:39, 14 December 2006 (UTC)
The result of the proposal was No Move.-- Hús ö nd 02:30, 31 July 2007 (UTC)
Line (mathematics) to
Line - Most basic usage of line, and the basis for all other uses ~
JohnnyMrNinja
01:01, 26 July 2007 (UTC)
These arguments (among others) also appear to apply to the requested move of Square (geometry) to Square, again displacing a disambiguation page which was recently moved to Square (disambiguation). Andrewa 04:02, 27 July 2007 (UTC)
Should the title be renamed to "Line (geometry)"? This would match "Point (geometry)" and "Square (geometry)". Jason Quinn 17:51, 4 April 2007 (UTC)
If 3 points are collinear, are the necessarily coplanar? 76.111.81.183 —Preceding signed but undated comment was added at 22:52, 22 September 2007 (UTC)
This is a cool example of a line with two origins and I was thinking of making a wikipedia article on it, but I wasn't sure if it deserves its own article or if it should be put in some other related article? LkNsngth ( talk) 20:09, 6 April 2008 (UTC)
I was surprized to find that collinear (with two l's) seems to be more widely used than colinear. Are they both correct? Colinear makes more sense to me (as in co-linear), but I'm not a native speaker. -- CyHawk ( talk) 21:45, 3 February 2008 (UTC)
Lines in a Cartesian plane can be described algebraically by linear equations and linear functions. In two dimensions, the characteristic equation is often given by the slope-intercept form:
where:
m is the slope of the line. c is the y-intercept of the line. x is the independent variable of the function y. In three dimensions, a line is described by parametric equations:
where:
x, y, and z are all functions of the independent variable t. x0, y0, and z0 are the initial values of each respective variable. a, b, and c are related to the slope of the line, such that the vector (a, b, c) is a parallel to the line.
Formal definitions This intuitive concept of a line can be formalized in various ways. If geometry is developed axiomatically (as in Euclid's Elements and later in David Hilbert's Foundations of Geometry), then lines are not defined at all, but characterized axiomatically by their properties. While Euclid did define a line as "length without breadth", he did not use this rather obscure definition in his later development.
In Euclidean space Rn (and analogously in all other vector spaces), we define a line L as a subset of the form
where a and b are given vectors in Rn with b non-zero. The vector b describes the direction of the line, and a is a point on the line. Different choices of a and b can yield the same line.
Properties
In a two-dimensional space, such as the plane, two different lines must either be parallel lines or must intersect at one point. In higher-dimensional spaces however, two lines may do neither, and two such lines are called skew lines.
In R2, every line L is described by a linear equation of the form
with fixed real coefficients a, b and c such that a and b are not both zero (see Linear equation for other forms). Important properties of these lines are their slope, x-intercept and y-intercept. The eccentricity of a straight line is infinity.
More abstractly, one usually thinks of the real line as the prototype of a line, and assumes that the points on a line stand in a one-to-one correspondence with the real numbers. However, one could also use the hyperreal numbers for this purpose, or even the long line of topology.
The "straightness" of a line, interpreted as the property that it minimizes distances between its points, can be generalized and leads to the concept of geodesics on differentiable manifolds.
Ray
In Euclidean geometry, a ray, or half-line, given two distinct points A (the origin) and B on the ray, is the set of points C on the line containing points A and B such that A is not strictly between C and B. In geometry, a ray starts at one point, then goes on forever in one direction. —Preceding
unsigned comment added by
Gon56 (
talk •
contribs)
07:26, 14 June 2008 (UTC)
Hi Tango. Glad you were happy to compromise without disturbing the "straight curve" lead. The fact that lines are fundamental might distinguish them from less fundamental objects, but otherwise makes little headway towards saying what a line actually is. A lead needs to be concise, and therefore should not be wasting words on such peripheral issues as the degree of fundamentality of "line" (which is highly debatable in any case, see below) when it should be trying to get to the essence of the concept as soon as possible.
Leisurely introductions are fine pedagogically, but the length necessary to get anywhere with them makes them better suited to the main body. As an example of this sort of thing see my attempt at exposing the motivation for, and underlying machinery of, toposes, admittedly a dry read but there is no way a mathematically capable reader new to toposes can extract that information from the preceding material in the article oneself---but no way should that go in the lead! Then compare it with John Baez's attempt at a similar thing, which is great fun but throws the baby out with the bathwater by failing to adequately convey what's going on under the hood (bonnet). The automotive analogy is a good one: John tries to convey what it feels like to drive one in various terrains while I try to explain the principle of the engine inside. Both help, but engineers and mathematicians are more likely to want the latter.
That said, your objection to my long list of characterizing properties in the lead was well taken, as was your suggestion that this more detailed material should be in the main body. After sleeping on what I'd written I'd come to the same conclusion myself and was going to move it to the body and write simply "straight line" in the lead when I noticed that you'd beaten me to it and had simply reverted my edits, which was fine by me by that time.
In the meantime I'd come to the realization that my wordy characterization had arisen from too hasty an attempt to replace "fundamental object" with something more specific, at a time when I didn't really have a suitable replacement ready and so just threw the kitchen sink at it. I now think that most of the other characteristics belong elsewhere than in the line article, namely in the more general classes of which lines are a particular subclass. This is what inheritance is all about in object-oriented programming, and the inheritance concept seems to provide an equally good organizing principle for encyclopedias.
Whether lines actually are more or less fundamental than curves is a nice question. A line as a subspace of the Cartesian plane can be defined without loss of generality as the set of zeros of a two-variable affine form, namely the solutions of ax + by + c = 0. Curves in the same setting cannot be defined in that way without significant loss of generality as one only obtains algebraic curves with that approach, no sine waves, space filling curves, etc. etc. But if you do limit yourself to algebraic curves then a line as the linear case of an algebraic curve is less fundamental by virtue of being an instance of a more general and therefore more fundamental notion.
Moving beyond Euclidean space, intensionally defined curves, those structured with suitable data appropriate to curves expressed without reference to a higher dimensional embedding space, are arguably a fundamental concept in their own right. A particularly simple example is a curve as a structure endowed with two metrics for respectively arc and chord length, the sort of entity one might run across in a CAD system like Autocad. This is a self-contained yet simple concept of "curve" admitting an equally simple notion of "straight," namely that the two metrics agree! Curves of such a kind are fundamental in the same sense that rings, lattices, etc. are fundamental, making their special cases slightly less fundamental (but only slightly less when the definition is as simple as mere coincidence of the two metrics).
If anything makes lines fundamental it would surely be that they are conceptually simple, being the path referred to in Newton's first law of motion (Newton assumed space was always flat), and encountered early on as one of the simplest instances of a geometric object, only points being simpler (unless a point is defined as the intersection of two nonparallel lines!). But this brings us to the difficult question of what it even means to be "fundamental." I guess this is a big part of what bothers me when I see it in the first sentence of an article, the other equally big part being that even if we all agreed on what it meant it still says very little: few if any concepts list "fundamental" among their defining characteristics. -- Vaughan Pratt ( talk) 23:40, 7 December 2008 (UTC)
I changed it because that is how the formula actually is written, according to the math courses I have taken. —The preceding unsigned comment was added by 75.4.13.98 ( talk) 23:46, 14 May 2007 (UTC).
c is usually used in calculus for the constant. It really could be anything though, there is no rules for choosing the particular letter you use, just guidlines and some loose standards that are always changing accoring to the math course your in or the time your taking it or personal preference. It makes no difference to the equation. Brentt ( talk) 21:41, 9 December 2008 (UTC)
The ray section makes it sound like a ray is half as long as a line, when they are in fact the same length (half of infinity is infinity). M00npirate ( talk) 01:44, 27 January 2009 (UTC)
Tango has argued for retention of the second sentence of the lead, "It is a fundamental object in geometry." I am just as strongly against it. However I don't want to get into an edit war with Tango because these often turn out badly. What do others feel about what this sentence contributes to the lead?
Articles on circles, angles, etc. content themselves with characterizing the concept and its applications without trying to position them in the hierarchy of fundamentality. My feeling is that lines should be described in the same spirit, and that those responsible for the article on them should take a neutral point of view on whether lines deserve to be singled out from other concepts as "fundamental." Otherwise we're going to get into interminable arguments as to whether circles, angles, etc. should also be accorded this special status of "fundamental." I much prefer the terminable kind. -- Vaughan Pratt ( talk) 06:35, 8 December 2008 (UTC)
Hmm. I see Vaughan's point, but I can't say I feel very strongly about it one way or the other. Certainly no one is going to defend the assertion "you can learn geometry just fine without ever bothering about lines", so in that sense they're fundamental. But I don't see the need to say so just here. Are there likely to be readers who are confused on this point? -- Trovatore ( talk) 09:16, 8 December 2008 (UTC)
I'll have to admit that I find this particular controversey rather humorous, since my OR has involved geometries where lines are indeed the fundamental notion and which are rather pointless in the sense that the automorphism group of the geometry does not even preserve the points, so that points are not even definable. -- Ramsey2006 ( talk) 17:13, 8 December 2008 (UTC)
Please see my commentary in Talk:Lineaments about the word lineaments which currently points only to this article. Reply there please. 66.102.204.49 ( talk) 00:49, 8 October 2009 (UTC)
The declaration that the line is the shortest distance between two points is limited to Euclidean Space and further the sitation (number 3) states that it was "sort-of" proved by Euclid and "assumed" by Pytheagoras. Neither of these are proofs and neither should they be stated as such. —Preceding unsigned comment added by 82.68.215.206 ( talk) 11:50, 19 November 2008 (UTC)
Furthermore , such a definition should be explained to have no tangent with the real, physical world,
being a construct that only works on paper or monitor screens.
A straight line might be possible as an entry vector for particles entering a black hole , if those exist, anywhere else in the Universe a line would be a curve , since space is bent and folds with each and any gravity source around.
Even more important ,the lack of such explanation results in confusion for young people as they can't
integrate the preconceptions induced by faulty teaching with the real world .(just think how you first reacted when encountering the reality of measuring the distance between two cities on different continents) —Preceding
unsigned comment added by
Pef333 (
talk •
contribs)
04:04, 7 January 2010 (UTC)
Shouldn't the article give some identities related to lines? I mean stuff like equations for the distance between two lines (as in skew lines) or a point and a line (might merge perpendicular distance here), the angle between two lines, determining if two lines are parallel or perpendicular, etc.? If you look at e.g. the triangle and plane articles, they have a lot more equations in them. -- Coffee2theorems ( talk) 12:19, 29 October 2009 (UTC) a line's degrees is —Preceding unsigned comment added by 67.251.80.240 ( talk) 23:22, 2 March 2010 (UTC)
The page Collinear points redirects to here, yet there was no mention of collinearity in the article. I've added a brief section that needs formatting and possibly expanding. Dbfirs 08:19, 4 April 2010 (UTC)
We seem to have a circular definition here because the definition of curve is "deviation from a straight line". Can we not find a better definition of a straight line, such as the extremely well-known "shortest path between two points" (in Euclidean space)? We could also mention the (possibly later addition by Heron or Diophantus) "definition" of a line in Euclid's Elements as breadthless length that lies equally with respect to the points on itself. How about using Wiktionary's "An infinitely extending one-dimensional figure that has no curvature; one that has length but not breadth or thickness"? Dbfirs 07:04, 29 June 2010 (UTC)
Given a point and a line in a plane, how do you determine what side of the line the point is on?
How about first figuring out where is "front" and where is "back"? And if you live in a 3-dimensional world you should start wondering where is "up" and where is "down". Not not mention the "second up" and "second down" and so forth if your mind is not bound by our common day experience of space and time. Good question, though, since it already lured an answer or two. Lapasotka ( talk) 23:36, 8 June 2011 (UTC)
When you refer to a line do you always mean a straight line? The article curve mentions that a curve can also be called a curved line, so lines aren't necessarily straight? -- 84.119.73.11 ( talk) 09:24, 8 June 2011 (UTC)
The new intro contains four definitions. In my opinion, only the first is ok, but it is informal. The others are useless. Moreover, the second is not consitent with the first.
I can't believe that this is the state of the art. In my opinion an expert editor is urgently needed, to rewrite the introduction.
Paolo.dL ( talk) 14:16, 13 December 2010 (UTC)
I do understand that you can define sets of n-tuples that you can call "straight lines", outside a vector space. These, however, are just sets of n-tuples, not sets of "points", so they are not really "lines", nor "straight lines". For practical applications, they are useless. But as soon as you make it a vector space, by choosing a basis, there's no way to define a basis without a coordinate system. Even without a fixed origin, a basis needs coordinate axes. I did not study affine spaces, but I can't imagine a metric which could build an Euclidean space unless you start from another space in which the coordinate axes have a know curvature or shape, and orientation in space. And how can you know the shape, and how can you define angles, whitout knowing what a straight line is? So, I am not convinced that this definition is non-circular. It may be so complex that you don't realize it is circular, perhaps. Of course, I am not sure, but at least I can say that your explanation does not convince me. Paolo.dL ( talk) 12:20, 14 December 2010 (UTC)
I like to cite: "If the concept of "order" of points of a line is defined, a ray, or half-line, ...." This is totally misleading. You need no concept of order of points to define the ray. :-(
I state: There are 3 types of "lines":
1) of infinite extend in 2 directions : Do you want to call this line?
2) of infinite extend in 1 direction : Do you want to call this ray?
3) of infinite extend in 0 directions : Do you want to call this line segment?
Because in one dimension you have at most 2 directions, this classification is complete.
Why were we not able to state these simple facts clearly in the article? :-(
Achim1999 ( talk) 18:32, 22 June 2012 (UTC)
The ray section contains a nonsense, but not those asserted by Achim1999. It lies only in the conditional statement of the first sentence. In fact a "betweenness" relation on three points is always defined on (Euclidean) lines: Given 3 points on a line, one is between the two others. This relation (or an equivalent one) is either among the axioms of the geometry or, in coordinate geometry, a consequence of the total ordering of the reals. Thus a correct first sentence for this section would be:
"Given two points A and B, the ray with initial point A and passing through B is the set of the points C of the line containing A and B such A is not between B and C."
However this sentence is too long and should be split, and the remainder of the section should be modified accordingly D.Lazard ( talk) 07:33, 25 June 2012 (UTC)
Having looked at the reader feedback comments on this article, it is clear to me that some segment of readers are not understanding the lead section. I am referring to the constant call for a "definition" of a line. What is said in the lead, and repeated in the first paragraphs of the Euclidean geometry section state clearly, at least to me, that there will not be a "definition". Perhaps this message is too subtlely delivered. I will attempt to address this problem head on with a section that I'll call Definitions versus descriptions in which some (but not all) of the discussion in the earlier section of this talk page will appear. This new section may at first appear to be too redundant of other material in the article, but I am hoping that other editors will help smooth out that problem. Bill Cherowitzo ( talk) 16:39, 19 November 2012 (UTC)
I am somewhat sorry that I missed last year's discussions in the previous two sections (NOT!). It seems to me that the reason folks were going round and round without coming to any conclusion is a lack of clarity about what a definition means. In any axiomatic system (and geometry is certainly such, Euclidean or otherwise) there must exist primitive notions, objects or relationships that have no definition. Every definition in the system can ultimately be traced back to rest on the primitive objects and the relations between them which are given by the axioms of the system. When you have different axiom systems that describe the same subject, they do not have to have the same set of primitive notions. So it is possible that a primitive notion in one system can be a definition in a second system, because the second system has a different set of primitives. This is relevant to the previous discussions in the following way: The concept of a line is a primitive notion in most axiomatic treatments of Euclidean geometry - specifically, and very emphatically, in Hilbert's treatment and Euclid's treatment (although he didn't realize it). In coordinate geometry you can define a line by means of a linear equation because you have changed the axiom system and line is no longer a primitive object. An axiom system for coordinate geometry will generally have an axiom that says that the points of a line are in one-to-one correspondence with the real numbers (for example, G.D. Birkhoff's treatment, circa 1936) a statement that you would not find in Hilbert or Euclid. In Artin's Geometric Algebra which has already been mentioned, a new set of axioms for Euclidean geometry is given, and he can, with respect to this new set, define a line ... because it is no longer a primitive notion. These are not examples where there are two (or more) definitions for the same object because the definitions are with respect to different axiom systems and in essence you have changed the groundrules and are now comparing apples and oranges.
The only reason that I have not gone in and edited the lead here is that I do not know how to do it in such a way that would remain faithful to what I have just written (without being as preachy as I have been) and yet be at the level that this article is trying to achieve. Any suggestions would be welcome. Wcherowi ( talk) 19:56, 16 September 2011 (UTC)
This may be an appropriate applied maths point of view, but a Formalist (and on Sundays I am one) sees no connection between a mathematical theory and whatever it is that you call "reality". It doesn't keep me up at night, but I would wonder what the physical referent of a Klein quadric in 5-dimensional projective space would be. I think it does matter whether something is a primitive or not. In the '50's Ma Bell (AT&T) had designed some telephone switching boxes which were models of the projective plane of order 5. In this model switches were lines. I would like to know how thinking about "breadthless widths" or any variant of that will help anyone understand the sentence I wrote before this one. I am not advocating dropping descriptions, but I am concerned about the limitations on our thinking processes that inappropriate descriptions can foster.
I fully agree with your outline of what the flow of an article should be. What I am grappling with, as an editor, is how to simplify something that I might understand from an advanced viewpoint without distorting it or providing a false impression. I believe this to be a very difficult task, but one that we need to master for good WP articles. Certainly one aspect of this task is to be very careful with the language that is used. So, when I see utter nonsense like defining something to be a primitive, which appears in the lead of this article, I tend to get upset and want to do something about it. Wcherowi ( talk) 19:07, 17 September 2011 (UTC)
I see that I am not making myself clear, so let me go ahead with the edit I had in mind and I'll respond to any comments about it. Wcherowi ( talk) 18:55, 19 September 2011 (UTC)
The article lede has a long quotation, translated from French, the first part of which applies equally well to curved line as to straight line. Indeed, the original starts speaking of fr:ligne, not of fr:droite! Similarly, Euclid's "breadthless length" would appear to apply to curved lines as well. It is only later in the quotation that it mentions fr:ligne droite, when referring to "equally extended between its points" (i.e., having zero curvature -- arc length differences equal Euclidean distances); I think this is the essential second half of Euclide's (striaght) line definition. Fgnievinski ( talk) 03:46, 30 June 2015 (UTC)
In this newly added section, to be consistent with the rest of the article, mention should be made of whether or not this is a formal definition of Lobachevsky and if it is, what are the primitives that he is using. If it is informal, and I suspect that it may be, it does not quite fit with the intro in this section. Bill Cherowitzo ( talk) 21:43, 22 August 2015 (UTC)
This subsection has been introduced in August 2015. Although the heading is plural, it contains only one example, which is a line is the locus of points in a plane that are equidistant to two distinct given points
. This is either misleading or misplaced. In fact, the enclosing section is about the definition of the concept. This example does not define the concept, as remarked by a recent edit pointing that this would lead to a circular definition. In fact, this is an example of a theorem asserting that some set of points (locus) is a line. The article could contain a section "Example of properties that define lines", but this is misleading to place such a section as a subsection of the section "Definitions versus descriptions", as such properties are not definitions nor descriptions. Therefore, I'll remove this subsection.
D.Lazard (
talk)
00:00, 3 October 2015 (UTC)
The notion of ray occurs in different areas of mathematics, e.g. in topology and in the theory of Hilbert spaces. In this revision there are both notion but David Eppstein has thrown out them saying that they are out of place here. Then where is their place? — Preceding unsigned comment added by 89.135.79.17 ( talk) 06:27, 29 October 2020 (UTC)
{{u|
Mark viking}} {
Talk}
11:46, 29 October 2020 (UTC)
{{
redirect|ray (geometry)|other uses in mathematics|ray(disambiguation)#Science and mathematics}}
.
D.Lazard (
talk)
12:05, 29 October 2020 (UTC)
{{u|
Mark viking}} {
Talk}
16:37, 29 October 2020 (UTC)The presentation uses polar equations that is not frequently used or taught in high school or college courses or used by mathematicians. In particular, dependency on the point-slope form is not a good starting point to introduce the equation because vertical lines have undefined slope! The most common polar equation seen in textbooks is derived from the normal form in Cartesian coordinates. In addition other Wikipedia articles depend on the common polar equation form. The article needs to present the equation of the form and discuss the pedal distance, pedal angle and Harmonic Addition Theorem used to derive the equation using a single sinusoid. 2601:140:8980:4B20:ADA7:C5B3:B1CD:B655 ( talk) 18:47, 23 February 2021 (UTC)
Also, given the very close relationship between the normal form in Cartesian coordinates, the notation used in the "normal form section" and the "lines in polar coordinates section" need to be common. Use of for the polar angle of the line's pedal is not acceptable because is the polar angle dimension for the polar coordinate system. A contributor to the normal form section is opposed to the using a different Greek letter to represent the angle. (It seems very odd that after 25 years of work on this article we're still dealing with this.) Please chime in with your view! 2601:140:8980:4B20:86:3A7F:35DE:74DD ( talk) 00:44, 25 February 2021 (UTC)
If you actually read the edit, I replaced with . As I stated above, theta is not a good choice due to its use in polar coordinates. Any text that does this is NOT a good one because it invites confusion. (Given all the letters in the Greek alphabet, why choose theta?) My edits are mathetmatically correct, there is no need for professional mathematicians (I'm an applied mathematician myself), and I WILL insert citations for them -- that is the Wikipedia way! (The discussions are all high school level anyway). The current article omits geometric interpretation of the equation, a crucial element for the article. PLEASE read a good math text on the normal form and lines in polar coordinates: Here's a thourgh one written by an Iowa State University professor: https://orion.math.iastate.edu/alex/166H/polar_lines_tangents.pdf. The professor uses theta sub zero for the angle of the pedal rather than theta. (An ok choice. Open up a Schaum's on "Analytic Geometry" and you'll see what college students are reading.)
By reading the Wikipedia article, a high school student should be able to understand and write the equation for the normal line to the line is, identity the normal's angle with the x-axis or polar axis, the distance the line is from the origin (or pole) sometimes called the pedal distance, and convert back-and-forth between the normal and polar forms using their understanding of the geometry of the line. In polar coordinates, the student should be able to to identify the coordinates of the point of intersection of the pedal with the line by inspection; and a 30 second scribble to give the coordinates in Cartesian coordinates using the normal form. I think this should take a few hours of writing rather than a couple of decades. We shouldn't let the readers down. If you'd like to initiate with Professor Alexander's text or Schaum's please -- the kids are waiting. 2601:140:8980:4B20:3124:B672:E0D2:96B1 ( talk) 16:15, 25 February 2021 (UTC)
Not to duplicate the article on the real line, the whole notion about the geometry of needs to be addressed in some way. Some essential topics for encyclopedic converage would include postulates related to lines, distance, the absolute value function, ordering, the triangle inequalities, dense sets, and continuity. 69.138.197.204 ( talk) 09:35, 3 March 2021 (UTC)
![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | Archive 2 | Archive 3 |
This page sucks balls! What of all the middle school geometry students who stop by to learn about the line? We should introduce things a little more gently, in the context of ordinary Euclidean geometry in the plane, and then in 3D, before getting into the fancier abstract concepts. I'll make these changes if there's no objection and no one else does first. Deco 04:09, 2 June 2005 (UTC)
Here Line and Line segment are treated as part of the same article. In Polish Wikipedia - they are separated. How can it refer to both articles in Polish now? — Preceding unsigned comment added by Xyzzer ( talk • contribs) 16:16, 31 January 2006 (UTC)
This is because the content of this article is in two bg articles. Nplus 17:22, 4 March 2006 (UTC)
The article on Interval_(mathematics) includes various notations on number sets, including the French notation I grew up with. This article doesn't include the equivalent notation I learned, which was:
I don't know if that's also used for American notation, but an equivalent reference here would be handy. (I see something vaguely similar, but less complete, in the french version of Segment 67.171.149.4 20:07, 16 March 2006 (UTC)
No where in the article does it say that a line contains infinite number of points. Is this correct, or does Quantum Physics state that this is incorrect? —The preceding unsigned comment was added by 70.59.199.11 ( talk • contribs) 14:43, 10 March 2006.
Perhaps this article could benefit from some discussion of more abstract definitions of a line? e.g. ("a straight line is a curve, any part of which is similar to the whole" from topology) Brentt 11:52, 16 August 2006 (UTC)
POLYLINE redirects to this page but is then not discussed. Can someone mention POLYLINE here, or make a distinct page?
Beau Wilkinson 18:34, 27 September 2006 (UTC)
The article said "In three dimensions, a line must be described by parametric equations". This is wrong: a line in any dimension can be described by a linear equation. I changed "may" to "must" and added a couple of linear equations for a 3D line. This leaves the definitions section a bit rambly, in my opinion --- why describe a 2D line in slope-intercent, versus a 3D line in parametric and standard form? --- but I felt it was a step in the right direction since it is at least correct. It does have the advantage of getting the link to linear equation earlier in the page. Owsteele 14:39, 14 December 2006 (UTC)
The result of the proposal was No Move.-- Hús ö nd 02:30, 31 July 2007 (UTC)
Line (mathematics) to
Line - Most basic usage of line, and the basis for all other uses ~
JohnnyMrNinja
01:01, 26 July 2007 (UTC)
These arguments (among others) also appear to apply to the requested move of Square (geometry) to Square, again displacing a disambiguation page which was recently moved to Square (disambiguation). Andrewa 04:02, 27 July 2007 (UTC)
Should the title be renamed to "Line (geometry)"? This would match "Point (geometry)" and "Square (geometry)". Jason Quinn 17:51, 4 April 2007 (UTC)
If 3 points are collinear, are the necessarily coplanar? 76.111.81.183 —Preceding signed but undated comment was added at 22:52, 22 September 2007 (UTC)
This is a cool example of a line with two origins and I was thinking of making a wikipedia article on it, but I wasn't sure if it deserves its own article or if it should be put in some other related article? LkNsngth ( talk) 20:09, 6 April 2008 (UTC)
I was surprized to find that collinear (with two l's) seems to be more widely used than colinear. Are they both correct? Colinear makes more sense to me (as in co-linear), but I'm not a native speaker. -- CyHawk ( talk) 21:45, 3 February 2008 (UTC)
Lines in a Cartesian plane can be described algebraically by linear equations and linear functions. In two dimensions, the characteristic equation is often given by the slope-intercept form:
where:
m is the slope of the line. c is the y-intercept of the line. x is the independent variable of the function y. In three dimensions, a line is described by parametric equations:
where:
x, y, and z are all functions of the independent variable t. x0, y0, and z0 are the initial values of each respective variable. a, b, and c are related to the slope of the line, such that the vector (a, b, c) is a parallel to the line.
Formal definitions This intuitive concept of a line can be formalized in various ways. If geometry is developed axiomatically (as in Euclid's Elements and later in David Hilbert's Foundations of Geometry), then lines are not defined at all, but characterized axiomatically by their properties. While Euclid did define a line as "length without breadth", he did not use this rather obscure definition in his later development.
In Euclidean space Rn (and analogously in all other vector spaces), we define a line L as a subset of the form
where a and b are given vectors in Rn with b non-zero. The vector b describes the direction of the line, and a is a point on the line. Different choices of a and b can yield the same line.
Properties
In a two-dimensional space, such as the plane, two different lines must either be parallel lines or must intersect at one point. In higher-dimensional spaces however, two lines may do neither, and two such lines are called skew lines.
In R2, every line L is described by a linear equation of the form
with fixed real coefficients a, b and c such that a and b are not both zero (see Linear equation for other forms). Important properties of these lines are their slope, x-intercept and y-intercept. The eccentricity of a straight line is infinity.
More abstractly, one usually thinks of the real line as the prototype of a line, and assumes that the points on a line stand in a one-to-one correspondence with the real numbers. However, one could also use the hyperreal numbers for this purpose, or even the long line of topology.
The "straightness" of a line, interpreted as the property that it minimizes distances between its points, can be generalized and leads to the concept of geodesics on differentiable manifolds.
Ray
In Euclidean geometry, a ray, or half-line, given two distinct points A (the origin) and B on the ray, is the set of points C on the line containing points A and B such that A is not strictly between C and B. In geometry, a ray starts at one point, then goes on forever in one direction. —Preceding
unsigned comment added by
Gon56 (
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contribs)
07:26, 14 June 2008 (UTC)
Hi Tango. Glad you were happy to compromise without disturbing the "straight curve" lead. The fact that lines are fundamental might distinguish them from less fundamental objects, but otherwise makes little headway towards saying what a line actually is. A lead needs to be concise, and therefore should not be wasting words on such peripheral issues as the degree of fundamentality of "line" (which is highly debatable in any case, see below) when it should be trying to get to the essence of the concept as soon as possible.
Leisurely introductions are fine pedagogically, but the length necessary to get anywhere with them makes them better suited to the main body. As an example of this sort of thing see my attempt at exposing the motivation for, and underlying machinery of, toposes, admittedly a dry read but there is no way a mathematically capable reader new to toposes can extract that information from the preceding material in the article oneself---but no way should that go in the lead! Then compare it with John Baez's attempt at a similar thing, which is great fun but throws the baby out with the bathwater by failing to adequately convey what's going on under the hood (bonnet). The automotive analogy is a good one: John tries to convey what it feels like to drive one in various terrains while I try to explain the principle of the engine inside. Both help, but engineers and mathematicians are more likely to want the latter.
That said, your objection to my long list of characterizing properties in the lead was well taken, as was your suggestion that this more detailed material should be in the main body. After sleeping on what I'd written I'd come to the same conclusion myself and was going to move it to the body and write simply "straight line" in the lead when I noticed that you'd beaten me to it and had simply reverted my edits, which was fine by me by that time.
In the meantime I'd come to the realization that my wordy characterization had arisen from too hasty an attempt to replace "fundamental object" with something more specific, at a time when I didn't really have a suitable replacement ready and so just threw the kitchen sink at it. I now think that most of the other characteristics belong elsewhere than in the line article, namely in the more general classes of which lines are a particular subclass. This is what inheritance is all about in object-oriented programming, and the inheritance concept seems to provide an equally good organizing principle for encyclopedias.
Whether lines actually are more or less fundamental than curves is a nice question. A line as a subspace of the Cartesian plane can be defined without loss of generality as the set of zeros of a two-variable affine form, namely the solutions of ax + by + c = 0. Curves in the same setting cannot be defined in that way without significant loss of generality as one only obtains algebraic curves with that approach, no sine waves, space filling curves, etc. etc. But if you do limit yourself to algebraic curves then a line as the linear case of an algebraic curve is less fundamental by virtue of being an instance of a more general and therefore more fundamental notion.
Moving beyond Euclidean space, intensionally defined curves, those structured with suitable data appropriate to curves expressed without reference to a higher dimensional embedding space, are arguably a fundamental concept in their own right. A particularly simple example is a curve as a structure endowed with two metrics for respectively arc and chord length, the sort of entity one might run across in a CAD system like Autocad. This is a self-contained yet simple concept of "curve" admitting an equally simple notion of "straight," namely that the two metrics agree! Curves of such a kind are fundamental in the same sense that rings, lattices, etc. are fundamental, making their special cases slightly less fundamental (but only slightly less when the definition is as simple as mere coincidence of the two metrics).
If anything makes lines fundamental it would surely be that they are conceptually simple, being the path referred to in Newton's first law of motion (Newton assumed space was always flat), and encountered early on as one of the simplest instances of a geometric object, only points being simpler (unless a point is defined as the intersection of two nonparallel lines!). But this brings us to the difficult question of what it even means to be "fundamental." I guess this is a big part of what bothers me when I see it in the first sentence of an article, the other equally big part being that even if we all agreed on what it meant it still says very little: few if any concepts list "fundamental" among their defining characteristics. -- Vaughan Pratt ( talk) 23:40, 7 December 2008 (UTC)
I changed it because that is how the formula actually is written, according to the math courses I have taken. —The preceding unsigned comment was added by 75.4.13.98 ( talk) 23:46, 14 May 2007 (UTC).
c is usually used in calculus for the constant. It really could be anything though, there is no rules for choosing the particular letter you use, just guidlines and some loose standards that are always changing accoring to the math course your in or the time your taking it or personal preference. It makes no difference to the equation. Brentt ( talk) 21:41, 9 December 2008 (UTC)
The ray section makes it sound like a ray is half as long as a line, when they are in fact the same length (half of infinity is infinity). M00npirate ( talk) 01:44, 27 January 2009 (UTC)
Tango has argued for retention of the second sentence of the lead, "It is a fundamental object in geometry." I am just as strongly against it. However I don't want to get into an edit war with Tango because these often turn out badly. What do others feel about what this sentence contributes to the lead?
Articles on circles, angles, etc. content themselves with characterizing the concept and its applications without trying to position them in the hierarchy of fundamentality. My feeling is that lines should be described in the same spirit, and that those responsible for the article on them should take a neutral point of view on whether lines deserve to be singled out from other concepts as "fundamental." Otherwise we're going to get into interminable arguments as to whether circles, angles, etc. should also be accorded this special status of "fundamental." I much prefer the terminable kind. -- Vaughan Pratt ( talk) 06:35, 8 December 2008 (UTC)
Hmm. I see Vaughan's point, but I can't say I feel very strongly about it one way or the other. Certainly no one is going to defend the assertion "you can learn geometry just fine without ever bothering about lines", so in that sense they're fundamental. But I don't see the need to say so just here. Are there likely to be readers who are confused on this point? -- Trovatore ( talk) 09:16, 8 December 2008 (UTC)
I'll have to admit that I find this particular controversey rather humorous, since my OR has involved geometries where lines are indeed the fundamental notion and which are rather pointless in the sense that the automorphism group of the geometry does not even preserve the points, so that points are not even definable. -- Ramsey2006 ( talk) 17:13, 8 December 2008 (UTC)
Please see my commentary in Talk:Lineaments about the word lineaments which currently points only to this article. Reply there please. 66.102.204.49 ( talk) 00:49, 8 October 2009 (UTC)
The declaration that the line is the shortest distance between two points is limited to Euclidean Space and further the sitation (number 3) states that it was "sort-of" proved by Euclid and "assumed" by Pytheagoras. Neither of these are proofs and neither should they be stated as such. —Preceding unsigned comment added by 82.68.215.206 ( talk) 11:50, 19 November 2008 (UTC)
Furthermore , such a definition should be explained to have no tangent with the real, physical world,
being a construct that only works on paper or monitor screens.
A straight line might be possible as an entry vector for particles entering a black hole , if those exist, anywhere else in the Universe a line would be a curve , since space is bent and folds with each and any gravity source around.
Even more important ,the lack of such explanation results in confusion for young people as they can't
integrate the preconceptions induced by faulty teaching with the real world .(just think how you first reacted when encountering the reality of measuring the distance between two cities on different continents) —Preceding
unsigned comment added by
Pef333 (
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contribs)
04:04, 7 January 2010 (UTC)
Shouldn't the article give some identities related to lines? I mean stuff like equations for the distance between two lines (as in skew lines) or a point and a line (might merge perpendicular distance here), the angle between two lines, determining if two lines are parallel or perpendicular, etc.? If you look at e.g. the triangle and plane articles, they have a lot more equations in them. -- Coffee2theorems ( talk) 12:19, 29 October 2009 (UTC) a line's degrees is —Preceding unsigned comment added by 67.251.80.240 ( talk) 23:22, 2 March 2010 (UTC)
The page Collinear points redirects to here, yet there was no mention of collinearity in the article. I've added a brief section that needs formatting and possibly expanding. Dbfirs 08:19, 4 April 2010 (UTC)
We seem to have a circular definition here because the definition of curve is "deviation from a straight line". Can we not find a better definition of a straight line, such as the extremely well-known "shortest path between two points" (in Euclidean space)? We could also mention the (possibly later addition by Heron or Diophantus) "definition" of a line in Euclid's Elements as breadthless length that lies equally with respect to the points on itself. How about using Wiktionary's "An infinitely extending one-dimensional figure that has no curvature; one that has length but not breadth or thickness"? Dbfirs 07:04, 29 June 2010 (UTC)
Given a point and a line in a plane, how do you determine what side of the line the point is on?
How about first figuring out where is "front" and where is "back"? And if you live in a 3-dimensional world you should start wondering where is "up" and where is "down". Not not mention the "second up" and "second down" and so forth if your mind is not bound by our common day experience of space and time. Good question, though, since it already lured an answer or two. Lapasotka ( talk) 23:36, 8 June 2011 (UTC)
When you refer to a line do you always mean a straight line? The article curve mentions that a curve can also be called a curved line, so lines aren't necessarily straight? -- 84.119.73.11 ( talk) 09:24, 8 June 2011 (UTC)
The new intro contains four definitions. In my opinion, only the first is ok, but it is informal. The others are useless. Moreover, the second is not consitent with the first.
I can't believe that this is the state of the art. In my opinion an expert editor is urgently needed, to rewrite the introduction.
Paolo.dL ( talk) 14:16, 13 December 2010 (UTC)
I do understand that you can define sets of n-tuples that you can call "straight lines", outside a vector space. These, however, are just sets of n-tuples, not sets of "points", so they are not really "lines", nor "straight lines". For practical applications, they are useless. But as soon as you make it a vector space, by choosing a basis, there's no way to define a basis without a coordinate system. Even without a fixed origin, a basis needs coordinate axes. I did not study affine spaces, but I can't imagine a metric which could build an Euclidean space unless you start from another space in which the coordinate axes have a know curvature or shape, and orientation in space. And how can you know the shape, and how can you define angles, whitout knowing what a straight line is? So, I am not convinced that this definition is non-circular. It may be so complex that you don't realize it is circular, perhaps. Of course, I am not sure, but at least I can say that your explanation does not convince me. Paolo.dL ( talk) 12:20, 14 December 2010 (UTC)
I like to cite: "If the concept of "order" of points of a line is defined, a ray, or half-line, ...." This is totally misleading. You need no concept of order of points to define the ray. :-(
I state: There are 3 types of "lines":
1) of infinite extend in 2 directions : Do you want to call this line?
2) of infinite extend in 1 direction : Do you want to call this ray?
3) of infinite extend in 0 directions : Do you want to call this line segment?
Because in one dimension you have at most 2 directions, this classification is complete.
Why were we not able to state these simple facts clearly in the article? :-(
Achim1999 ( talk) 18:32, 22 June 2012 (UTC)
The ray section contains a nonsense, but not those asserted by Achim1999. It lies only in the conditional statement of the first sentence. In fact a "betweenness" relation on three points is always defined on (Euclidean) lines: Given 3 points on a line, one is between the two others. This relation (or an equivalent one) is either among the axioms of the geometry or, in coordinate geometry, a consequence of the total ordering of the reals. Thus a correct first sentence for this section would be:
"Given two points A and B, the ray with initial point A and passing through B is the set of the points C of the line containing A and B such A is not between B and C."
However this sentence is too long and should be split, and the remainder of the section should be modified accordingly D.Lazard ( talk) 07:33, 25 June 2012 (UTC)
Having looked at the reader feedback comments on this article, it is clear to me that some segment of readers are not understanding the lead section. I am referring to the constant call for a "definition" of a line. What is said in the lead, and repeated in the first paragraphs of the Euclidean geometry section state clearly, at least to me, that there will not be a "definition". Perhaps this message is too subtlely delivered. I will attempt to address this problem head on with a section that I'll call Definitions versus descriptions in which some (but not all) of the discussion in the earlier section of this talk page will appear. This new section may at first appear to be too redundant of other material in the article, but I am hoping that other editors will help smooth out that problem. Bill Cherowitzo ( talk) 16:39, 19 November 2012 (UTC)
I am somewhat sorry that I missed last year's discussions in the previous two sections (NOT!). It seems to me that the reason folks were going round and round without coming to any conclusion is a lack of clarity about what a definition means. In any axiomatic system (and geometry is certainly such, Euclidean or otherwise) there must exist primitive notions, objects or relationships that have no definition. Every definition in the system can ultimately be traced back to rest on the primitive objects and the relations between them which are given by the axioms of the system. When you have different axiom systems that describe the same subject, they do not have to have the same set of primitive notions. So it is possible that a primitive notion in one system can be a definition in a second system, because the second system has a different set of primitives. This is relevant to the previous discussions in the following way: The concept of a line is a primitive notion in most axiomatic treatments of Euclidean geometry - specifically, and very emphatically, in Hilbert's treatment and Euclid's treatment (although he didn't realize it). In coordinate geometry you can define a line by means of a linear equation because you have changed the axiom system and line is no longer a primitive object. An axiom system for coordinate geometry will generally have an axiom that says that the points of a line are in one-to-one correspondence with the real numbers (for example, G.D. Birkhoff's treatment, circa 1936) a statement that you would not find in Hilbert or Euclid. In Artin's Geometric Algebra which has already been mentioned, a new set of axioms for Euclidean geometry is given, and he can, with respect to this new set, define a line ... because it is no longer a primitive notion. These are not examples where there are two (or more) definitions for the same object because the definitions are with respect to different axiom systems and in essence you have changed the groundrules and are now comparing apples and oranges.
The only reason that I have not gone in and edited the lead here is that I do not know how to do it in such a way that would remain faithful to what I have just written (without being as preachy as I have been) and yet be at the level that this article is trying to achieve. Any suggestions would be welcome. Wcherowi ( talk) 19:56, 16 September 2011 (UTC)
This may be an appropriate applied maths point of view, but a Formalist (and on Sundays I am one) sees no connection between a mathematical theory and whatever it is that you call "reality". It doesn't keep me up at night, but I would wonder what the physical referent of a Klein quadric in 5-dimensional projective space would be. I think it does matter whether something is a primitive or not. In the '50's Ma Bell (AT&T) had designed some telephone switching boxes which were models of the projective plane of order 5. In this model switches were lines. I would like to know how thinking about "breadthless widths" or any variant of that will help anyone understand the sentence I wrote before this one. I am not advocating dropping descriptions, but I am concerned about the limitations on our thinking processes that inappropriate descriptions can foster.
I fully agree with your outline of what the flow of an article should be. What I am grappling with, as an editor, is how to simplify something that I might understand from an advanced viewpoint without distorting it or providing a false impression. I believe this to be a very difficult task, but one that we need to master for good WP articles. Certainly one aspect of this task is to be very careful with the language that is used. So, when I see utter nonsense like defining something to be a primitive, which appears in the lead of this article, I tend to get upset and want to do something about it. Wcherowi ( talk) 19:07, 17 September 2011 (UTC)
I see that I am not making myself clear, so let me go ahead with the edit I had in mind and I'll respond to any comments about it. Wcherowi ( talk) 18:55, 19 September 2011 (UTC)
The article lede has a long quotation, translated from French, the first part of which applies equally well to curved line as to straight line. Indeed, the original starts speaking of fr:ligne, not of fr:droite! Similarly, Euclid's "breadthless length" would appear to apply to curved lines as well. It is only later in the quotation that it mentions fr:ligne droite, when referring to "equally extended between its points" (i.e., having zero curvature -- arc length differences equal Euclidean distances); I think this is the essential second half of Euclide's (striaght) line definition. Fgnievinski ( talk) 03:46, 30 June 2015 (UTC)
In this newly added section, to be consistent with the rest of the article, mention should be made of whether or not this is a formal definition of Lobachevsky and if it is, what are the primitives that he is using. If it is informal, and I suspect that it may be, it does not quite fit with the intro in this section. Bill Cherowitzo ( talk) 21:43, 22 August 2015 (UTC)
This subsection has been introduced in August 2015. Although the heading is plural, it contains only one example, which is a line is the locus of points in a plane that are equidistant to two distinct given points
. This is either misleading or misplaced. In fact, the enclosing section is about the definition of the concept. This example does not define the concept, as remarked by a recent edit pointing that this would lead to a circular definition. In fact, this is an example of a theorem asserting that some set of points (locus) is a line. The article could contain a section "Example of properties that define lines", but this is misleading to place such a section as a subsection of the section "Definitions versus descriptions", as such properties are not definitions nor descriptions. Therefore, I'll remove this subsection.
D.Lazard (
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00:00, 3 October 2015 (UTC)
The notion of ray occurs in different areas of mathematics, e.g. in topology and in the theory of Hilbert spaces. In this revision there are both notion but David Eppstein has thrown out them saying that they are out of place here. Then where is their place? — Preceding unsigned comment added by 89.135.79.17 ( talk) 06:27, 29 October 2020 (UTC)
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D.Lazard (
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16:37, 29 October 2020 (UTC)The presentation uses polar equations that is not frequently used or taught in high school or college courses or used by mathematicians. In particular, dependency on the point-slope form is not a good starting point to introduce the equation because vertical lines have undefined slope! The most common polar equation seen in textbooks is derived from the normal form in Cartesian coordinates. In addition other Wikipedia articles depend on the common polar equation form. The article needs to present the equation of the form and discuss the pedal distance, pedal angle and Harmonic Addition Theorem used to derive the equation using a single sinusoid. 2601:140:8980:4B20:ADA7:C5B3:B1CD:B655 ( talk) 18:47, 23 February 2021 (UTC)
Also, given the very close relationship between the normal form in Cartesian coordinates, the notation used in the "normal form section" and the "lines in polar coordinates section" need to be common. Use of for the polar angle of the line's pedal is not acceptable because is the polar angle dimension for the polar coordinate system. A contributor to the normal form section is opposed to the using a different Greek letter to represent the angle. (It seems very odd that after 25 years of work on this article we're still dealing with this.) Please chime in with your view! 2601:140:8980:4B20:86:3A7F:35DE:74DD ( talk) 00:44, 25 February 2021 (UTC)
If you actually read the edit, I replaced with . As I stated above, theta is not a good choice due to its use in polar coordinates. Any text that does this is NOT a good one because it invites confusion. (Given all the letters in the Greek alphabet, why choose theta?) My edits are mathetmatically correct, there is no need for professional mathematicians (I'm an applied mathematician myself), and I WILL insert citations for them -- that is the Wikipedia way! (The discussions are all high school level anyway). The current article omits geometric interpretation of the equation, a crucial element for the article. PLEASE read a good math text on the normal form and lines in polar coordinates: Here's a thourgh one written by an Iowa State University professor: https://orion.math.iastate.edu/alex/166H/polar_lines_tangents.pdf. The professor uses theta sub zero for the angle of the pedal rather than theta. (An ok choice. Open up a Schaum's on "Analytic Geometry" and you'll see what college students are reading.)
By reading the Wikipedia article, a high school student should be able to understand and write the equation for the normal line to the line is, identity the normal's angle with the x-axis or polar axis, the distance the line is from the origin (or pole) sometimes called the pedal distance, and convert back-and-forth between the normal and polar forms using their understanding of the geometry of the line. In polar coordinates, the student should be able to to identify the coordinates of the point of intersection of the pedal with the line by inspection; and a 30 second scribble to give the coordinates in Cartesian coordinates using the normal form. I think this should take a few hours of writing rather than a couple of decades. We shouldn't let the readers down. If you'd like to initiate with Professor Alexander's text or Schaum's please -- the kids are waiting. 2601:140:8980:4B20:3124:B672:E0D2:96B1 ( talk) 16:15, 25 February 2021 (UTC)
Not to duplicate the article on the real line, the whole notion about the geometry of needs to be addressed in some way. Some essential topics for encyclopedic converage would include postulates related to lines, distance, the absolute value function, ordering, the triangle inequalities, dense sets, and continuity. 69.138.197.204 ( talk) 09:35, 3 March 2021 (UTC)