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Archive 1 | Archive 2 | Archive 3 | Archive 4 |
or am I missing something. Wiki readers who are non-mathmeticians need to know how reals are different. CorvetteZ51 13:50, 22 May 2007 (UTC)
"Any real number can be determined by a possibly infinite decimal representation (such as that of π above), where the consecutive digits indicate the tenth of an interval given by the previous digits to which the real number belongs."
So what does it mean? Number: 3.1415926535. The last consecutive digits 535 indicate the tenth of an interval given by 3.1415926 to which the real number belongs (the whole π)? What the hell? What's an interval given by "previous digits"? And how do I tell which numbers are consecutive and which are previous. inb4 according to definition it's up to you. Well the one who wrote the definition in the lead should've provided with an example, I really can't understand what is written here. 64.134.103.62 ( talk) 22:44, 27 November 2011 (UTC)
Simon Stevin seems to be being used as the basis for categorizing real numbers as a Belgian invention. However what did he really invent? As a separate question, was it also novel?
There's some evidence for a claim that he invented a decimal notation for real non-integers. This may have some novelty to it.
He also seems to have worked on quadratic solutions. It's not clear if these were for rational real solutions, or for complex number solutions.
It's claimed here that he invented real numbers. Is that based on his notation, or his work with quadratics? I'm finding it hard (I'm not a mathematician) to see if this justifies a claim for novel invention with rational real numbers (not merely real numbers) from the notation or else for complex numbers from the quadratics. Either way, I'm finding it hard to see his work as being particularly relevant to real numbers specifically. Andy Dingley ( talk) 16:49, 13 June 2012 (UTC)
I recently added information to the Advanced Properties section regarding the cardinality of the real numbers; specifically that the cardinality of the reals is and that of the natural numbers is . This information was integrated in the first couple of sentences which deals with the cardinality of the reals and how it is strictly larger than that of the natural numbers. However, my edits were reversed. Why? Is this not relevant? — Preceding unsigned comment added by NereusAJ ( talk • contribs) 05:44, 21 December 2011 (UTC)
In any case this point is treated in section "real numbers and logic" and there is no need to consider it twice. It it another question to know if the organization of the article has to be changed for considering the cardinality question only once. D.Lazard ( talk) 08:36, 21 December 2011 (UTC)
My apologies Trovatore. I see now that I am wrong. I was under the illusion that is defined to be . NereusAJ ( talk) 09:42, 21 December 2011 (UTC)
(Fortunately, kids will grow up reading Wikipedia, so hopefully they will learn a correct deviation.) Also, why is "choice" true? and "continuum hypothesis" is neither true or false? -- Taku ( talk)
Hello, I just did a modification to the page, adding the original character that represents the real numbers: ℝ this is a Unicode character called the set of real numbers.
Should all the reference to R be changed to ℝ or this additional note in the article is enough ?
Erik Garres 08:34, 8 January 2007 (UTC)
What is ? This is used for real axes on the argand diagram, so why not in say sets or other references to --150.101.102.188
Would someone please mentions the symbol "ℝ" Unicode number next to it ? -- DynV ( talk) 07:19, 25 October 2009 (UTC)
The article start: "A symbol of the set of real numbers (ℝ)", and the proceeds to use R for ℝ. Weird. Can someone point me to a list of "some browsers won't display" (ℝ), it has been over 5 years since this original sub-heading "symbols used for the set of real number" and (correct me if I am wrong) I suspect that wikipedia has moved on in the mean time and now officially supports Unicode 6.2 fully.
Keep in mind that in the majority?many? of wikipedia's <math> LaTeX sections </math> a ℝ is being displayed. This is producing a weird inconsistency in ℝ notation between wikipedia's text and LaTeX content.
NevilleDNZ ( talk) 08:09, 29 August 2013 (UTC)
To be frank, I have not done a literature search on R for ℝ. But as I said above "This is producing a weird inconsistency in ℝ notation between wikipedia's text and LaTeX content."
Re: "people were using R for the reals long before anyone used blackboard bold" ...
Re: "some browsers won't display it"
Maybe this is a favorite vs centre kind of issue? The "inconsistency in ℝ notation" is at the discretion of the individual editor...?
NevilleDNZ ( talk) 22:58, 29 August 2013 (UTC)
A recent discussion I saw on a user talk page on my watchlist reminded me of this: It's very odd that Zeno's paradoxes are not mentioned on this page. In some sense they are a central part of the reason that the notion of real numbers is important in the first place. Once you have internalized the reals, it can be hard to understand why anyone would have ever thought they were paradoxical — but that's because you have the notion of the reals, and the notion of infinitely many points in an interval of finite length is clearly just true, not a paradox.
Of course, by itself, that doesn't differentiate the reals from (say) the rationals, but the paradoxes lead naturally to the notion of a limit point, and from there, the reals are the next natural stop.
I don't know offhand where to find a good source, but surely there must be one. I would think this should be treated fairly centrally in the exposition of the motivation for the concept. -- Trovatore ( talk) 19:57, 5 July 2016 (UTC)
In section § Axiomatic approach, the Archimedean property of the reals is not mentioned. I wonder if this is true that a Dedekind-complete ordered field is necessarily Archimedean. If not, Archimedean property must be added to the axioms. If yes, the proof is certainly not immediate, and a hint of the proof or a citation must be provided in this section, because Archimedean property is not a consequence of other notions of completion. D.Lazard ( talk) 13:05, 23 July 2017 (UTC)
CorvetteZ51 ( talk) 09:36, 10 June 2015 (UTC)
I wanted, and have included a note on notation as I was looking for R++. I am not sure what the best way to include this is. We can I think have:
Is what is on at the moment OK or does anyone have any suggestions? ( Msrasnw ( talk) 14:46, 4 February 2014 (UTC))
The introductory paragraph reads "The real numbers include all the rational numbers..."
The first section "Basic Properties" reads "More formally, real numbers have ... the least upper bound property." and then "hence the rational numbers do not satisfy the least upper bound property."
Is this a contradiction or am I even dumber than I realize? Drienstra ( talk) 00:09, 8 October 2014 (UTC)
I read and commented on this article while trying to understand a book on mathematical infinities. I believe you when you argue there is no contradiction. Thank you for the explanation. Drienstra ( talk) 20:02, 26 November 2017 (UTC)
in modern usage, a real number is a contradistinction to an imaginary number. CorvetteZ51 ( talk) 09:12, 9 June 2015 (UTC)
About the "slow motion edit warring" ( Deacon Vorbis, Rebelyis): two wikiprojects notified, math and phys. Boris Tsirelson ( talk) 19:22, 12 February 2018 (UTC)
I do not oppose to adding a short description, but I have slight reservations to mentioning " numberline" (even when this is defined as being simply all the reals). I would not have these reservations along an article about " constructible numbers", but I have, e.g., no immediate (without rectification?) access to locate on the numberline, i.e., to pick up a corresponding length. Maybe, I am off track, but I wanted to articulate my provisos, based on fundamental opposition to the sloppiness in elementary math education. Purgy ( talk) 07:27, 22 February 2018 (UTC)
Number representing a continuous quantity. D.Lazard ( talk) 23:09, 22 February 2018 (UTC)
Thanks for ridding the number line. Maybe it is just my native language, which lets me prefer "numbers forming a continuum", because of allowing "continuity" also on discrete topologies. Purgy ( talk) 09:47, 23 February 2018 (UTC)
Hi, I would like to suggest an important correction to the number line you displayed. Please remove PI, sqrt(2), gamma and e as they are not numbers but algorithms. These algorithms have no definite values only approximations. The latter may be depicted on the number line, but algorithms which produce endless sequences of approximations have no home there. — Preceding unsigned comment added by Counting floats ( talk • contribs) 18:37, 2 March 2018 (UTC)
Thank you very much for responding and explaining your point of view.
Well, in the case of PI for example they have generated 5 billion digits or so of it. Each which can be appended to 3.14 to form the next better approximation. Which one of these 5 billion are you going to depict on the number line? The first couple, or all of them ? You can't stick the Greek letter Pi onto the real number line anywhere because it stands for an algorithm not for a real number. But let's say that when you annotate a position on the number line with the string "Pi" you meant only the best known approximation of it which is 3.14 Fine. But 3.14 can also be produced by an uncountable number of other algorithms or tricks e.g. 31.4/10 or sqrt(9.8596) or 3 + 0.1 + 0.04, and so on. Thus 3.14 cannot possibly be paired with or reserved for a single well-known algorithm. The only thing 3.14 can stand for is 314 beads each which has the size 1/100 of the unity bead.
There is a reason we named the number line the "NUMBER" line and not the "any algorithm" line. It should have only real numbers with an exceptionally tight set of syntax rules. These should not be open for interpretation or haphazard change and should cover both the integers and the floating point numbers. But that is a subject for another day.
As to where I got the idea, thank you for asking. I am a smart guy, can think for myself and figured it out.
I hope this helps to clarify my position. — Preceding unsigned comment added by Counting floats ( talk • contribs) 22:02, 2 March 2018 (UTC)
Well, confining myself to positive base-10 decimal case (to keep it simple ) : My understanding of integers is any string which has some mix of ten digits from 0 to 9, provided that there is no leading zero. For decimal floats the string may have the 10 digits and the decimal point in any mix provided that a few syntax rules are followed. Won't elaborate them here, except to say that you will recognize an incorrect float when you see one (e.g. : 00..0.5 and so on ). Note that floats existed long before computers, for the sole purpose to increase the resolution of computing and measurement without limits. That is what the meaning of the implied digits on the right side of the decimal point representing negative powers of the base.
Yes, I am aware of the real number definition with integers, floats and an open-ended mix of algorithms added in. I fully understand that, however this will have to change, not just on Wikipedia but in the entire mathematical community. Algorithms are a vast open collection of mathematics in action whose outputs are recorded as integers, floats, musical notes, colors, spatial measurements, behaviors you name it they do it. Floats are just one of many ways of expressing the logic of human thought. To lump algorithms side by side with floats or colors simply makes no sense as they must respect the cause and effect hierarchy.
Thank you for your time, it was an interesting exchange, cleared up some things for me.
Tamas Varhegyi — Preceding unsigned comment added by Counting floats ( talk • contribs) 23:29, 2 March 2018 (UTC)
No problem I was just testing an idea, you are right, it is not going to be decided here on Wiki. Making the change is not going to matter one-way or another.
I appreciate that you only insinuated that I am a novice about reals, floats and algorithms but did not call me names. I have seen worse on your Talk pages. You did help me out considerably and that I appreciate.
However, if you don't have anything better to do I have a question for you : Pretend that you are given a job of placing the algorithmic output(s) of PI on the number line. Where would you put it ? I am not testing just curious. That was my original question I meant to ask but somehow I got sidetracked a bit.
I — Preceding
unsigned comment added by
Counting floats (
talk •
contribs)
00:42, 3 March 2018 (UTC)
Well, didn't I say so that "number line" might not be as undisputed a metaphor for reals as is generally assumed, especially in elementary math revised for Tsirel's below education? Apologies for the following aside to Counting floats: naively, you cannot localize ANY SINGLE number on a line because of non-zero breadth of any marking (compare to probability of hitting even any rational), so π and e have the same rights to be embossed there as any real has. Connectedness, constructability, and computability are just higher finesses.
Purgy (
talk)
08:03, 3 March 2018 (UTC)
Every mathematical discourse consists of two parts. One is the intuitive part, aimed to support intuition and to explain what is done. Every figure belongs to this part. The second part is the formal part, consisting of axioms, definitions, statement of theorems and proofs. Mathematical texts that are reduced to their formal part are boring and can be understood only by people who know the subject. On the other hand, a mathematical text reduced to its intuitive part is non-scientific, as there is no way for validate or invalidate its content.
The Real line cannot be properly defined, and one can prove nothing about it. It belongs to the intuitive part of the mathematical discourse, and has been introduced only for helping intuition and motivation. As the real line may not be properly defined, it is not a mathematical object, and nothing can be proven about it. On the other hand, the set of real numbers can be formally defined (the definition is not really elementary, but this is life :-). Thus, it belongs to the formal part of the discourse.
IMO, the main flaw in Counting floats's post, is that it does not make any distinction between the intuitive and the formal discourse, and tries to prove (or disprove) assertions on something for which this is a nonsense. D.Lazard ( talk) 16:15, 3 March 2018 (UTC)
This article is currently inconsistent in notation. The first half uses blackboard bold ℝ throughout, but the second half uses bold R almost everywhere. The article ought to be self-consistent. My preference is for blackboard bold on the grounds that the symbol ℝ is never, to my knowledge, used with any meaning other than the set of reals, whereas R might be used to represent any vector or matrix. -- Dr Greg talk 14:45, 21 April 2018 (UTC)
I attempted to improve the opening sentence of the article by replacing the hand-waving phrase, "quantity along the line," with the more precise phrase, "position on the real line." Editor Purgy Purgatorio then reverted my edit without giving a specific reason, saying only, "rather no improvement," (what is "rather" supposed to mean here?). So let me explain my edit here in more detail, in preparation for restoring it.
First, the word line has many meanings in mathematics. The phrase the line is generally understood by math-literate readers to refer to the real line, but the present article is addressed to general readers, so it seems reasonable to use the more specific term here. Second, the idea of "quantity along a line" could be taken to mean any function of position, but here we simply mean the position, so why not just say it that way? Eleuther ( talk) 09:05, 9 May 2018 (UTC)
OK, taking a deep breath here. I have to object to User:D.Lazard removing the longstanding language about the reals being quantities along a line. That is the essence of the reals. It's what the ancient Greeks had in mind (though they didn't know it at the time). It can be found in all sorts of sources as the first description of the reals. By comparison, "measurable quantity" doesn't mean anything. ("Continously varying quantity" would be slightly better, I guess, but I really think we should start with the line, which is the ur-example of the reals.) -- Trovatore ( talk) 09:59, 9 May 2018 (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 | Archive 4 |
or am I missing something. Wiki readers who are non-mathmeticians need to know how reals are different. CorvetteZ51 13:50, 22 May 2007 (UTC)
"Any real number can be determined by a possibly infinite decimal representation (such as that of π above), where the consecutive digits indicate the tenth of an interval given by the previous digits to which the real number belongs."
So what does it mean? Number: 3.1415926535. The last consecutive digits 535 indicate the tenth of an interval given by 3.1415926 to which the real number belongs (the whole π)? What the hell? What's an interval given by "previous digits"? And how do I tell which numbers are consecutive and which are previous. inb4 according to definition it's up to you. Well the one who wrote the definition in the lead should've provided with an example, I really can't understand what is written here. 64.134.103.62 ( talk) 22:44, 27 November 2011 (UTC)
Simon Stevin seems to be being used as the basis for categorizing real numbers as a Belgian invention. However what did he really invent? As a separate question, was it also novel?
There's some evidence for a claim that he invented a decimal notation for real non-integers. This may have some novelty to it.
He also seems to have worked on quadratic solutions. It's not clear if these were for rational real solutions, or for complex number solutions.
It's claimed here that he invented real numbers. Is that based on his notation, or his work with quadratics? I'm finding it hard (I'm not a mathematician) to see if this justifies a claim for novel invention with rational real numbers (not merely real numbers) from the notation or else for complex numbers from the quadratics. Either way, I'm finding it hard to see his work as being particularly relevant to real numbers specifically. Andy Dingley ( talk) 16:49, 13 June 2012 (UTC)
I recently added information to the Advanced Properties section regarding the cardinality of the real numbers; specifically that the cardinality of the reals is and that of the natural numbers is . This information was integrated in the first couple of sentences which deals with the cardinality of the reals and how it is strictly larger than that of the natural numbers. However, my edits were reversed. Why? Is this not relevant? — Preceding unsigned comment added by NereusAJ ( talk • contribs) 05:44, 21 December 2011 (UTC)
In any case this point is treated in section "real numbers and logic" and there is no need to consider it twice. It it another question to know if the organization of the article has to be changed for considering the cardinality question only once. D.Lazard ( talk) 08:36, 21 December 2011 (UTC)
My apologies Trovatore. I see now that I am wrong. I was under the illusion that is defined to be . NereusAJ ( talk) 09:42, 21 December 2011 (UTC)
(Fortunately, kids will grow up reading Wikipedia, so hopefully they will learn a correct deviation.) Also, why is "choice" true? and "continuum hypothesis" is neither true or false? -- Taku ( talk)
Hello, I just did a modification to the page, adding the original character that represents the real numbers: ℝ this is a Unicode character called the set of real numbers.
Should all the reference to R be changed to ℝ or this additional note in the article is enough ?
Erik Garres 08:34, 8 January 2007 (UTC)
What is ? This is used for real axes on the argand diagram, so why not in say sets or other references to --150.101.102.188
Would someone please mentions the symbol "ℝ" Unicode number next to it ? -- DynV ( talk) 07:19, 25 October 2009 (UTC)
The article start: "A symbol of the set of real numbers (ℝ)", and the proceeds to use R for ℝ. Weird. Can someone point me to a list of "some browsers won't display" (ℝ), it has been over 5 years since this original sub-heading "symbols used for the set of real number" and (correct me if I am wrong) I suspect that wikipedia has moved on in the mean time and now officially supports Unicode 6.2 fully.
Keep in mind that in the majority?many? of wikipedia's <math> LaTeX sections </math> a ℝ is being displayed. This is producing a weird inconsistency in ℝ notation between wikipedia's text and LaTeX content.
NevilleDNZ ( talk) 08:09, 29 August 2013 (UTC)
To be frank, I have not done a literature search on R for ℝ. But as I said above "This is producing a weird inconsistency in ℝ notation between wikipedia's text and LaTeX content."
Re: "people were using R for the reals long before anyone used blackboard bold" ...
Re: "some browsers won't display it"
Maybe this is a favorite vs centre kind of issue? The "inconsistency in ℝ notation" is at the discretion of the individual editor...?
NevilleDNZ ( talk) 22:58, 29 August 2013 (UTC)
A recent discussion I saw on a user talk page on my watchlist reminded me of this: It's very odd that Zeno's paradoxes are not mentioned on this page. In some sense they are a central part of the reason that the notion of real numbers is important in the first place. Once you have internalized the reals, it can be hard to understand why anyone would have ever thought they were paradoxical — but that's because you have the notion of the reals, and the notion of infinitely many points in an interval of finite length is clearly just true, not a paradox.
Of course, by itself, that doesn't differentiate the reals from (say) the rationals, but the paradoxes lead naturally to the notion of a limit point, and from there, the reals are the next natural stop.
I don't know offhand where to find a good source, but surely there must be one. I would think this should be treated fairly centrally in the exposition of the motivation for the concept. -- Trovatore ( talk) 19:57, 5 July 2016 (UTC)
In section § Axiomatic approach, the Archimedean property of the reals is not mentioned. I wonder if this is true that a Dedekind-complete ordered field is necessarily Archimedean. If not, Archimedean property must be added to the axioms. If yes, the proof is certainly not immediate, and a hint of the proof or a citation must be provided in this section, because Archimedean property is not a consequence of other notions of completion. D.Lazard ( talk) 13:05, 23 July 2017 (UTC)
CorvetteZ51 ( talk) 09:36, 10 June 2015 (UTC)
I wanted, and have included a note on notation as I was looking for R++. I am not sure what the best way to include this is. We can I think have:
Is what is on at the moment OK or does anyone have any suggestions? ( Msrasnw ( talk) 14:46, 4 February 2014 (UTC))
The introductory paragraph reads "The real numbers include all the rational numbers..."
The first section "Basic Properties" reads "More formally, real numbers have ... the least upper bound property." and then "hence the rational numbers do not satisfy the least upper bound property."
Is this a contradiction or am I even dumber than I realize? Drienstra ( talk) 00:09, 8 October 2014 (UTC)
I read and commented on this article while trying to understand a book on mathematical infinities. I believe you when you argue there is no contradiction. Thank you for the explanation. Drienstra ( talk) 20:02, 26 November 2017 (UTC)
in modern usage, a real number is a contradistinction to an imaginary number. CorvetteZ51 ( talk) 09:12, 9 June 2015 (UTC)
About the "slow motion edit warring" ( Deacon Vorbis, Rebelyis): two wikiprojects notified, math and phys. Boris Tsirelson ( talk) 19:22, 12 February 2018 (UTC)
I do not oppose to adding a short description, but I have slight reservations to mentioning " numberline" (even when this is defined as being simply all the reals). I would not have these reservations along an article about " constructible numbers", but I have, e.g., no immediate (without rectification?) access to locate on the numberline, i.e., to pick up a corresponding length. Maybe, I am off track, but I wanted to articulate my provisos, based on fundamental opposition to the sloppiness in elementary math education. Purgy ( talk) 07:27, 22 February 2018 (UTC)
Number representing a continuous quantity. D.Lazard ( talk) 23:09, 22 February 2018 (UTC)
Thanks for ridding the number line. Maybe it is just my native language, which lets me prefer "numbers forming a continuum", because of allowing "continuity" also on discrete topologies. Purgy ( talk) 09:47, 23 February 2018 (UTC)
Hi, I would like to suggest an important correction to the number line you displayed. Please remove PI, sqrt(2), gamma and e as they are not numbers but algorithms. These algorithms have no definite values only approximations. The latter may be depicted on the number line, but algorithms which produce endless sequences of approximations have no home there. — Preceding unsigned comment added by Counting floats ( talk • contribs) 18:37, 2 March 2018 (UTC)
Thank you very much for responding and explaining your point of view.
Well, in the case of PI for example they have generated 5 billion digits or so of it. Each which can be appended to 3.14 to form the next better approximation. Which one of these 5 billion are you going to depict on the number line? The first couple, or all of them ? You can't stick the Greek letter Pi onto the real number line anywhere because it stands for an algorithm not for a real number. But let's say that when you annotate a position on the number line with the string "Pi" you meant only the best known approximation of it which is 3.14 Fine. But 3.14 can also be produced by an uncountable number of other algorithms or tricks e.g. 31.4/10 or sqrt(9.8596) or 3 + 0.1 + 0.04, and so on. Thus 3.14 cannot possibly be paired with or reserved for a single well-known algorithm. The only thing 3.14 can stand for is 314 beads each which has the size 1/100 of the unity bead.
There is a reason we named the number line the "NUMBER" line and not the "any algorithm" line. It should have only real numbers with an exceptionally tight set of syntax rules. These should not be open for interpretation or haphazard change and should cover both the integers and the floating point numbers. But that is a subject for another day.
As to where I got the idea, thank you for asking. I am a smart guy, can think for myself and figured it out.
I hope this helps to clarify my position. — Preceding unsigned comment added by Counting floats ( talk • contribs) 22:02, 2 March 2018 (UTC)
Well, confining myself to positive base-10 decimal case (to keep it simple ) : My understanding of integers is any string which has some mix of ten digits from 0 to 9, provided that there is no leading zero. For decimal floats the string may have the 10 digits and the decimal point in any mix provided that a few syntax rules are followed. Won't elaborate them here, except to say that you will recognize an incorrect float when you see one (e.g. : 00..0.5 and so on ). Note that floats existed long before computers, for the sole purpose to increase the resolution of computing and measurement without limits. That is what the meaning of the implied digits on the right side of the decimal point representing negative powers of the base.
Yes, I am aware of the real number definition with integers, floats and an open-ended mix of algorithms added in. I fully understand that, however this will have to change, not just on Wikipedia but in the entire mathematical community. Algorithms are a vast open collection of mathematics in action whose outputs are recorded as integers, floats, musical notes, colors, spatial measurements, behaviors you name it they do it. Floats are just one of many ways of expressing the logic of human thought. To lump algorithms side by side with floats or colors simply makes no sense as they must respect the cause and effect hierarchy.
Thank you for your time, it was an interesting exchange, cleared up some things for me.
Tamas Varhegyi — Preceding unsigned comment added by Counting floats ( talk • contribs) 23:29, 2 March 2018 (UTC)
No problem I was just testing an idea, you are right, it is not going to be decided here on Wiki. Making the change is not going to matter one-way or another.
I appreciate that you only insinuated that I am a novice about reals, floats and algorithms but did not call me names. I have seen worse on your Talk pages. You did help me out considerably and that I appreciate.
However, if you don't have anything better to do I have a question for you : Pretend that you are given a job of placing the algorithmic output(s) of PI on the number line. Where would you put it ? I am not testing just curious. That was my original question I meant to ask but somehow I got sidetracked a bit.
I — Preceding
unsigned comment added by
Counting floats (
talk •
contribs)
00:42, 3 March 2018 (UTC)
Well, didn't I say so that "number line" might not be as undisputed a metaphor for reals as is generally assumed, especially in elementary math revised for Tsirel's below education? Apologies for the following aside to Counting floats: naively, you cannot localize ANY SINGLE number on a line because of non-zero breadth of any marking (compare to probability of hitting even any rational), so π and e have the same rights to be embossed there as any real has. Connectedness, constructability, and computability are just higher finesses.
Purgy (
talk)
08:03, 3 March 2018 (UTC)
Every mathematical discourse consists of two parts. One is the intuitive part, aimed to support intuition and to explain what is done. Every figure belongs to this part. The second part is the formal part, consisting of axioms, definitions, statement of theorems and proofs. Mathematical texts that are reduced to their formal part are boring and can be understood only by people who know the subject. On the other hand, a mathematical text reduced to its intuitive part is non-scientific, as there is no way for validate or invalidate its content.
The Real line cannot be properly defined, and one can prove nothing about it. It belongs to the intuitive part of the mathematical discourse, and has been introduced only for helping intuition and motivation. As the real line may not be properly defined, it is not a mathematical object, and nothing can be proven about it. On the other hand, the set of real numbers can be formally defined (the definition is not really elementary, but this is life :-). Thus, it belongs to the formal part of the discourse.
IMO, the main flaw in Counting floats's post, is that it does not make any distinction between the intuitive and the formal discourse, and tries to prove (or disprove) assertions on something for which this is a nonsense. D.Lazard ( talk) 16:15, 3 March 2018 (UTC)
This article is currently inconsistent in notation. The first half uses blackboard bold ℝ throughout, but the second half uses bold R almost everywhere. The article ought to be self-consistent. My preference is for blackboard bold on the grounds that the symbol ℝ is never, to my knowledge, used with any meaning other than the set of reals, whereas R might be used to represent any vector or matrix. -- Dr Greg talk 14:45, 21 April 2018 (UTC)
I attempted to improve the opening sentence of the article by replacing the hand-waving phrase, "quantity along the line," with the more precise phrase, "position on the real line." Editor Purgy Purgatorio then reverted my edit without giving a specific reason, saying only, "rather no improvement," (what is "rather" supposed to mean here?). So let me explain my edit here in more detail, in preparation for restoring it.
First, the word line has many meanings in mathematics. The phrase the line is generally understood by math-literate readers to refer to the real line, but the present article is addressed to general readers, so it seems reasonable to use the more specific term here. Second, the idea of "quantity along a line" could be taken to mean any function of position, but here we simply mean the position, so why not just say it that way? Eleuther ( talk) 09:05, 9 May 2018 (UTC)
OK, taking a deep breath here. I have to object to User:D.Lazard removing the longstanding language about the reals being quantities along a line. That is the essence of the reals. It's what the ancient Greeks had in mind (though they didn't know it at the time). It can be found in all sorts of sources as the first description of the reals. By comparison, "measurable quantity" doesn't mean anything. ("Continously varying quantity" would be slightly better, I guess, but I really think we should start with the line, which is the ur-example of the reals.) -- Trovatore ( talk) 09:59, 9 May 2018 (UTC)