Talk:Real number/Archive 3

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reals, defined to non-imaginary ?

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)

I'm afraid the formulation added was not very accurate. All real numbers are also complex numbers (in other words, the set of complex numbers contains the real numbers), that is how the term "complex number" is defined. The number zero is both a real number and an imaginary number, though the latter may be a matter of definition and not accepted by everybody.

Nevertheless, something along the lines you propose should perhaps be added, like "The imaginary unit i, which satisfies the equation i² = −1, is not a real number." I'm not sure what the best place would be. -- Jitse Niesen (talk) 16:14, 27 May 2007 (UTC)

reals definition --> no imaginary components.the BS about what a real is, does not change what thereal is not, the real is not imaginary, please allow non-mathemeticitians to know what a real is, the real has no imaginary parts. CorvetteZ51 16:45, 27 May 2007 (UTC)

There are lots of things a real number is not. The important idea to get across is what a real number is. You seem to be wanting to emphasize the use of "real" as distinctive from complex numbers that aren't real -- but the notion of a complex number is more complicated than, and comes logically after, the notion of a real number. First we have to explain what a real number is, not what it isn't. --Trovatore 18:48, 27 May 2007 (UTC)

my understanding is... any number with non-zero 'i' is non-real. Others are real. Not sure about 0i or 0 + 0i. Please enlighten me, if what I have wrote is not, correct and entirely complete. CorvetteZ51 12:05, 28 May 2007 (UTC)

Well, that's correct, if by "number" you mean "complex number". But then you have to define "complex number" first; how are you going to do that?

here is how Mathworld defines 'complex number', http://mathworld.wolfram.com/ComplexNumber.html , here is how Mathworld defines 'reals', http://mathworld.wolfram.com/RealNumber.html , who dissagrees with that?CorvetteZ51 10:19, 29 May 2007 (UTC):

So first of all you should be aware, in general, that Mathworld, while sometimes a useful resource, is, let's say, "quirky". In the case at hand, though, there's nothing I disagree with in the definitional part of these two articles -- it's just that they never get to an actual definition.

To summarize: The Mathworld articles say that a complex number is x+iy where x and y are real numbers; that's fine. "Real number" is linked and says that a real number is either a rational number or an irrational number; this is true but not a good definition as it appears to make the notion of an irrational number more fundamental than that of a real number, which is backwards. But it could work if "irrational number were suitably defined. Click on the link to "irrational" and you discover that an "irrational number" is a "number that cannot be expressed as p/q for any integers p and q." So what then is a "number"? No link to it, and the chain of definitions has terminated with an undefined term, "number", that as I pointed out above, needs further explanation. --Trovatore 17:39, 29 May 2007 (UTC)

Trovatore, I think there needs to be pointed out that there are multiple definitions for 'real numbers'. For a high school student, the definition would be,,, 'Real' numbers are the union of rational and irrationals, and are essentially the 'number-line' numbers. The term is needed as a contradistinction of imaginary. Somebody else can write the mathmatician's version.CorvetteZ51 11:34, 30 May 2007 (UTC)

The term is in contradistincton to "imaginary"; that's true, and worthy of mention in the lede, I think. However that doesn't define it. The "union of the rationals and the irrationals" would define it if you could define the irrationals first, but how are you going to do that? The "number-line" version is a good intuitive motivation, one that I think deserves more emphasis than it currently has (look back in this page or its archives for the discussion of the "Kantian" perspective, which I actually agree with in this case, as little love as I have for Kant's dour philosophy in general). --Trovatore 17:37, 30 May 2007 (UTC)

You seem to have the idea that the word "number", by itself, is non-problematic, but that isn't so. There are all sorts of things that are or have been occasionally called "numbers" that are not real or complex numbers. A few examples: Transfinite cardinal numbers, transfinite ordinal numbers, hypercomplex numbers, extended real numbers (these include +∞ and −∞), and "numbers" like XBQIRT that identify your reservation at the airport. And even if there weren't all these other meanings of the word "number", there's still the problem that, before it's meaningful to say that a real number is a "number with zero imaginary part", you first have to say what a number is in the first place. --Trovatore 21:09, 28 May 2007 (UTC)

A real number has the property of being a complex number with an imaginary part equal to zero. The formal definition of a real number is a set of numbers that have a bijection with points on a line extending towards infinity in both directions, and is closed under multiplication and addition. From the reals, one can define imaginaries as the square roots of nonpositive reals, and complexes as the sums of reals and imaginaries. 96.229.217.189 (talk) 17:27, 21 February 2012 (UTC) Michael Ejercito

Actually I just looked back in the history: Corvette's actual point seems to be about the etymology of "real". I think it is probably accurate to say that "real" is a retronym in contradistinction to "imaginary", and I do think it would be reasonable to say something about that in the lede, if we're sure it's true (anyone have a source?). --Trovatore 18:57, 27 May 2007 (UTC)

I don't think real numbers should be defined as non-imaginary. But I do think that there should be something in this article that gives examples of non-real numbers. It is quite similar to saying an object is a tree if it has leaves and grows. While this is true of a tree, bushes also fit this definition yet are not trees. While it is difficult to make a fully correct definition that does not lead to misnomers, we can still close that gap. I'm mainly pointing this out because the definition says that "real numbers include both rational numbers...and irrational numbers, such as pi and the square root of two" but does not point out that the square root of a negative number (square root of negative two) is not a real number while it can still be irrational. I do see that the next paragraph does note that it is difficult to define a real number, but this seems like something that can help clarify rather than including a statement that needs citation after the definition ("The term "real number" is a retronym coined in response to "imaginary number".").

I will note that I do not have a degree in mathematics, but I think it's still a valid point. I just feel that something so dynamic needs a slightly better definition that may encompass what it isn't instead of only what it is." Xe7al (talk) 04:25, 15 June 2009 (UTC)

unclear definition in the lead

"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)

I agree that the wording is less than clear. It's clear to me what it means, but that's because I'm used to the notion of a nested sequence of closed intervals converging on a point; to a reader not familiar with that, the phrasing may be less than helpful.

Let me explain what it means and then maybe someone can come up with better wording. The idea is that any initial segment of the decimal representation, say 3.14159, represents an interval that you could round off or truncate to that value, for example the interval [3.141590000...,3.141599999...], where 3.141599999... is an alternative notation for 3.141600000..., I'm using truncation rather than rounding, and I'm giving the interval as closed for a reason that may not matter too much to you (see Heine–Borel property if you're curious; the intersection of a nested sequence of compact sets is always nonempty).

Then the next digit, the 2 in 3.141592, means we have now cut the interval down to a tenth of its previous size, to [3.141592000...,3.141592999...].

Continuing forever, we get a nested collection of closed intervals with exactly one point in all of them, namely π

Is the explanation clear, and does it suggest to anyone how better to phrase things in the lead? --Trovatore (talk) 23:52, 27 November 2011 (UTC)

I've been having a look at that lead and been trying to think to myself what would a middle school child make of it since they are taught about real numbers. There are a lots of unnecessary things there which they mightn't have come across yet. Continuum sounds too highfalutin. They probably have never heard of a transcendental or algebraic number. I see no reason to drag in complex lines. Dmcq (talk) 00:04, 26 March 2012 (UTC)

Real numbers as a Belgian invention

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)

There's always people going around claiming things for different countries. This is just silly, the Babylonians if anyone should be credited with the way they were able to add extra places to the end of the approximation of the square root of two. Dmcq (talk) 17:05, 13 June 2012 (UTC)

I read in Simon Stevin: According to van der Waerden (1985, p. 69), Stevin's "general notion of a real number was accepted, tacitly or explicitly, by all later scientists". I have not checked this citation. One may not credit the Babilonian when they did not know the negative real numbers, but they had some idea of this notion. However, although theorems are usually easy to attribute to someone, notions, like the notion of real numbers, evolve progressively. As the first definition of the real numbers, which is correct from the modern point of view, is due to Kronecker, much later than Stevin, one could also say that real numbers was invented by Kronecker. Conclusion: It is wrong to say that the real numbers have been invented by someone. D.Lazard (talk) 17:39, 13 June 2012 (UTC)

The "Belgian invention" thing is just a ludicrous claim; we don't need elaborate arguments about it. Just revert it any time it gets added. --Trovatore (talk) 19:16, 13 June 2012 (UTC)

I'm less interested here in refuting the "invention" claim than simply trying to find out precisely what it was that Stevin discovered. Andy Dingley (talk) 02:07, 14 June 2012 (UTC)

Cardinality

I recently added information to the Advanced Properties section regarding the cardinality of the real numbers; specifically that the cardinality of the reals is $\aleph _{1}$ and that of the natural numbers is $\aleph _{0}$. 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)

Relevant, sure. Unfortunately it's not clear that it's true. --Trovatore (talk) 05:46, 21 December 2011 (UTC)

What do you mean? Of course it is true. $\aleph _{0}$ represents the cardinality of the natural numbers. $\aleph _{1}$ represents the cardinality of the power set of the natural numbers. The cardinality of the continuum, ${\mathfrak {c}}$, is the same as $\aleph _{1}$ by the Continuum Hypothesis. Perhaps you should consult the Wikipedia page on Cardinality. NereusAJ (talk) 06:36, 21 December 2011 (UTC)

This is exactly the point — what is not clear is precisely whether the continuum hypothesis is true. --Trovatore (talk) 06:51, 21 December 2011 (UTC)

(To summarize: The continuum hypothesis is known to be independent of ZFC. ZFC can neither prove that it's true, nor that it's false. That does not by itself close the issue; realists believe that it's either really true or really false, whether or not we can find out which is the case, and some propose various ideas by which we might hope to find out.) --Trovatore (talk) 06:54, 21 December 2011 (UTC)

That ${\mathfrak {C}}=\aleph _{1}$ is not just a consequence of the Continuum Hypothesis. It was proven by Cantor in his 1874 uncountability proof. See Cardinality of the continuum.NereusAJ (talk) 07:17, 21 December 2011 (UTC)

No, sorry, you're mistaken on this. Take a look at the continuum hypothesis article and search for "independence" or "independent". --Trovatore (talk) 07:20, 21 December 2011 (UTC)

Sorry. Your right. I should wait until we resolve this matter before editing the page again. I don't understand your objection. I believe we can agree than the cardinality of the reals is ${\mathfrak {c}}$. The article already states that the cardinality of the reals is equal to the cardinality of the power set of the natural numbers. I assume you don't have a problem with this. Now the cardinality of the natural numbers is $\aleph _{0}$ (right?). Furthermore, the cardinality of the power set of the natural numbers is $\aleph _{1}=2^{\aleph _{0}}$. Surely, this means ${\mathfrak {c}}=\aleph _{1}$? Or am I missing something obvious? NereusAJ (talk) 07:34, 21 December 2011 (UTC)

Oh, I think you may have misunderstood a previous post of mine. When I spoke of Cantor's 1874 proof, I did not mean to imply that Cantor proved the Continuum Hypothesis. I meant that he proved the equivalence ${\mathfrak {c}}=\aleph _{1}$ without using the Continuum Hypothesis. NereusAJ (talk) 08:04, 21 December 2011 (UTC)

• The cardinality of the reals is ${\mathfrak {c}}$, yes. In fact I think that's the definition of ${\mathfrak {c}}$.
• The cardinality of the reals is also $2^{\aleph _{0}}$, and $2^{\aleph _{0}}={\mathfrak {c}}$
• The cardinality of the powerset of the naturals is $2^{\aleph _{0}}={\mathfrak {c}}$
• None of those points is controversial, and none involves CH
• But now when you claim that the cardinality of the powerset of the naturals is $\aleph _{1}$, or that $\aleph _{1}=2^{\aleph _{0}}$, well, those are both equivalent to CH.
• Finally, no, Cantor did not prove in 1874 that ${\mathfrak {c}}=\aleph _{1}$. The statement ${\mathfrak {c}}=\aleph _{1}$ is equivalent to CH, so if he had proved that, he would ipso facto have proved CH. --Trovatore (talk) 08:22, 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 $\aleph _{1}$ is defined to be $2^{\aleph _{0}}$. NereusAJ (talk) 09:42, 21 December 2011 (UTC)

Well, you're not alone. Unfortunately the popularizers frequently make this mistake, and even many people who go into mathematics have read the popularizers, and if they never take a set theory course (which most don't) they may never be disabused of the misimpression. --Trovatore (talk) 09:47, 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)

Choice is normally assumed to be true because most mathematicians take it as an axiom as it makes life much easier, whereas the continuum hypothesis doesn't affect most normal maths and under some defensible axioms it would be wrong so for instance ${\mathfrak {c}}=\aleph _{2}$ could also be quite a reasonable conclusion. Dmcq (talk) 16:20, 14 June 2012 (UTC)

A little clarification: Choice is usually assumed to be true by pure mathematicians. But, in computational mathematics and in particular for automated theorem proving choice is usually not assumed. D.Lazard (talk) 16:45, 14 June 2012 (UTC)

No, I really think these both miss the point. Choice is assumed to be true because it's just obviously true. Once you've understood the motivating picture, you can accept the picture or not, but if you do accept it, you have to really go out of your way to avoid giving assent to choice.

AC is probably the most complicated axiom that is accepted on the basis of self-evidence. For other axioms (say large-cardinal axioms) there are other forms of evidence and justification, but not self-evidence.

As to the last question, well, it's just not so that CH is "neither true nor false". From the POV of a mathematical realist, CH is either true or false, but we probably don't yet know which one. We may or may not ever know in the future. But in any case, certainly neither CH nor ~CH has the kind of self-evidence that AC does. --Trovatore (talk) 19:47, 14 June 2012 (UTC)

This is not the place for this discussion. But Choice axiom is exactly as "obviously true" and as "self-evident" as Euclides parallel postulate is. The proofs that they are independent axioms are very similar: constructing a model of the theory with the negation of the axiom inside the theory with the axiom. By the way, I did not know "self-evidence" and "obviously true" as mathematical notions. D.Lazard (talk) 21:08, 14 June 2012 (UTC)

The comparison with Euclidean/non-Euclidean geometry is complicated; I agree in some ways and disagree in others, but obviously disagree with your ultimate conclusion. As you say it's not the place. If you like I'll explain on your talk page. --Trovatore (talk) 21:27, 14 June 2012 (UTC)

Symbols used for the set of real number

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)

It's not a good idea to use ℝ, as some browsers won't display it. Also, it's not the "original character" for the reals - people were using R for the reals long before anyone used blackboard bold, and many people still prefer R (except, perhaps, on blackboards). --Zundark 10:56, 8 January 2007 (UTC)

What is $\Re$? This is used for real axes on the argand diagram, so why not in say sets or other references to $\mathbb {R}$ --150.101.102.188

$\Re$ is sometimes used for denoting the real part of a complex number (although $\operatorname {Re}$ is more common for this purpose), which is why you've seen it used to label the real axis of the Argand diagram. It's sometimes used for other (unrelated) things as well. I'm not sure why you think it should be used in references to $\mathbb {R}$. --Zundark 14:30, 12 March 2007 (UTC)

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)

Zundark's more important point is the second one — blackboard bold is an expedient to put bold on blackboards. Some workers do use it as "the symbol for the reals" even in books and papers, but I do not think this is the majority usage. My preference is just plain R. There's no reason that can't be used in LaTeX as well. --Trovatore (talk) 16:07, 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" ...

Interesting to encounter this line of argument: To paraphrase "people were using stones for the counting long before anyone used computers"...

Re: "some browsers won't display it"

ℂ, ℍ, ℕ, ℙ, ℚ, ℝ & ℤ are widely supported. c.f. Blackboard bold

FYI: Here is the complete list of bold characters: 𝔸 𝔹 ℂ 𝔻 𝔼 𝔽 𝔾 ℍ 𝕀 𝕁 𝕂 𝕃 ℕ 𝕆 ℙ ℚ ℝ 𝕊 𝕋 𝕌 𝕍 𝕎 𝕏 𝕐 ℤ

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)

Refs and notes

• The first Note could be split into a separate footnotes
• There's inconsistent mix of inline refs and full refs (tagged)
• Reference columns User:D.Lazard, User:CBM "Pls. don't add columns for no reason; columns aren't mandatory in any way, esp. not for full refs, and not really an improvement. There is no lack of vertical space in a browser". Sorry adding twice was an edit conflict, so hadn't seen it had been undone. There is a reason we use them, and after about 10 refs is common. Here we're at 15 refs. As you say it's optional. There is a whitespace issue, I selected 37em so it was quite wide columns, hardly no reason, but as in a vocal minority I'll leave it for you guys. Widefox; talk 09:49, 11 April 2016 (UTC)

Note that this is the current style (both fixed numbers or use of colwidth= are both deprecated). Thanks Izno for fixing the ref style. If they're all consistent now, pls remove the maintenance template. Regards Widefox; talk 09:26, 15 April 2016 (UTC)

@Widefox: colwidth == no parameter name per Template:Reflist. Colwidth happens to be more explicit in intent. To call it deprecated is incorrect also; Reflist's documentation says nothing to that effect. Since it's the same output, I have no issue with it currently though. Feel free also to remove the maintenance template yourself. --Izno (talk) 11:14, 15 April 2016 (UTC)

Can't remember where it says it's deprecated, but it's there somewhere (maybe the talk, or in a MOS, or in communication with me). Widefox; talk 11:56, 15 April 2016 (UTC)

Zeno's paradoxes

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)

Axiomatic approach

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)

I got the answer, which is "yes". Nevertheless Archimedean property is sufficiently important for appearing in the field definition, I'll add it. D.Lazard (talk) 15:23, 23 July 2017 (UTC)

D. what grade level are you aiming this article at?

CorvetteZ51 (talk) 09:36, 10 June 2015 (UTC)

• (This section relocated from head of page to chronological order by me.)
  --Jerzy•t 04:30, 26 November 2017 (UTC)
  

Notation

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:

The set of positive numbers is often denoted by R₊, or R⁺ or R_≥0.

But I haven't seen R^≥0.

The set of negative numbers is often denoted by R_-, or R_≤0

But I haven't seen R^≤0.

The set of strictly positive numbers is often denoted by R₊₊, R⁺⁺ or R_>0.

But I haven't seen R^>0.

The set of strictly negative numbers is often denoted by R_-- or R_<0.

But I haven't seen R^-- or R^<0.

Is what is on at the moment OK or does anyone have any suggestions? (Msrasnw (talk) 14:46, 4 February 2014 (UTC))

For the strictly positive numbers, one can also see $\mathbf {R} _{+}^{*}$, deriving from the fact that the positive reals are the nonzero and nonnegative ones. This notation is rather common in Number theory where $\mathbf {R} ^{*}$ is standard for the nonzero reals. D.Lazard (talk) 15:20, 4 February 2014 (UTC)

Zero is not a positive number (see sign (mathematics)) so R_≥0 is incorrect for the positive numbers. What distinction are you making between "positive" and "strictly positive"? R^-- can be found in the reference I gave. While the "++" notation makes sense, I have never seen it used and so would like to see a reference for it. In my experience, the superscripted notation (of single symbols) predominates and subscripts are resorted to when other conventions clash (as D.Lazard has pointed out above or when a power notation is to be used significantly). However, there is no single standard notation and there may be areas in mathematics which use conventions that I am not familiar with - hence the need for citations. Bill Cherowitzo (talk) 18:56, 4 February 2014 (UTC)

We can't possibly include every notation so my take is we should just include the most common ones. R , R⁺, and Rⁿ seem sufficient to me. Mathematical objects such as R^-, R^ῳ, R[x], etc. are probably beyond the scope of this introductory article. Mr. Swordfish (talk) 19:34, 4 February 2014 (UTC)

Sorry about the fuss and my confusion. I came accross this in the conext of convex optimisation in economics but looking here [1] there seem quite a few refs for R₊ and R₊₊ (the later which I was looking for here.) One of the sources Anandalingam, G., S. Raghavan, Subramanian Raghavan eds. (2003)Telecommunications Network Design and Management Springer has the line

"We will use standard notations R and R₊ for the sets of real and real non- negative numbers, respectively; and a not quite standard notation R₊₊ for the set of strictly positive real numbers."

Best wishes (Msrasnw (talk) 23:37, 4 February 2014 (UTC))

In my experience, none of these notations is used much and I don't think it's clear that they are standardized. If you come across R⁺ in a paper, and it matters whether it includes 0 or not, then you'd better check what the author said about it. If the author didn't say, well, that's his fault; he should have, because there is not actually a standard meaning.

Most authors avoid the problem by writing explicitly [0,∞) or (0,∞) or something like that. Personally I would prefer to remove all reference to these R⁺-type notations. They aren't used enough to be worth the trouble of trying to track down whether they're standard or not. --Trovatore (talk) 23:46, 4 February 2014 (UTC)

I agree with this. Do we keep Rⁿ ? I think this is pretty standard notation, but perhaps out-of-place here? Mr. Swordfish (talk) 17:06, 5 February 2014 (UTC)

   User:John Baez removed, at 01:50, 3 September 2014, his five-minute-old contrib at this pt on the talk page, summarizing that it had resulted from his confusion.--Jerzy•t 05:36, 26 November 2017 (UTC)

Do real numbers include the rational numbers or not?

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)

If a set has the least upper bound property, a proper subset of that set doesn't have to have that property (we say that a proper subset doesn't inherit the property in this case). You are pointing to the classical example of this phenomenon. The set of rational numbers less than √2 has lots of rational upper bounds, but no rational least upper bound. The same set, thought of as a set of real numbers has the real number √2 as the least upper bound. To put it another way, given a set with the least upper bound property, if you toss out some elements to get a proper subset, some of the things that you toss could be least upper bounds of some subsets of the proper subset you are left with. This proper subset would then not have the least upper bound property. I hope this helps. Bill Cherowitzo (talk) 02:47, 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)

I think part of the problem here, might have been that the wording "More formally, real numbers have ... the least upper bound property" could be read to mean that each real number has this property, when instead the least upper bound property, is a property of the entire set. I've edited the sentence to try and make this more clear. Paul August ☎ 20:16, 26 November 2017 (UTC)

misleading article, unhelpful to those who want to learn

in modern usage, a real number is a contradistinction to an imaginary number. CorvetteZ51 (talk) 09:12, 9 June 2015 (UTC)

Maybe true, if "contradistinction" would be defined in mathematics (this is an article about mathematics). In any case, this cannot define the mathematical concept of "real number", as imaginary numbers need real numbers to be defined. D.Lazard (talk) 09:47, 9 June 2015 (UTC)

In the history section we say that Descartes distinguished between real roots of a polynomial and imaginary roots. Is it worth mentioning this in the lede as a way of explaining the adjective? Tkuvho (talk) 17:23, 9 June 2015 (UTC)

I agree, and I have introduced such a sentence in the lead. This has the advantage to make the lead less WP:TECHNICAL by providing an informal description of what is a real number. D.Lazard (talk) 09:49, 10 June 2015 (UTC)

Thanks but I think we should stick to roots of polynomials, as Descartes did, rather than putting words in his mouth. When you claim that real numbers are encountered in the real world, do you include the almost all of them that are not definable? Tkuvho (talk) 12:01, 10 June 2015 (UTC)

About "not definable", do not be sure; see Talk:Integer sequence#Definable sequences. Boris Tsirelson (talk) 19:28, 12 February 2018 (UTC)

In physics...or not?

About the "slow motion edit warring" (Deacon Vorbis, Rebelyis): two wikiprojects notified, math and phys. Boris Tsirelson (talk) 19:22, 12 February 2018 (UTC)

Probably a good idea. My main concern was that the removal appeared to be based on the editor's own views on the nature of reality, and not how it's treated in the literature, regardless. It may well be that this mention gives undue weight, or that it's just not worth keeping for another reason. –Deacon Vorbis (carbon • videos) 19:30, 12 February 2018 (UTC)

Came here from the projects. The paragraph misrepresents the holographic principle--in particular, I have never seen in relaible sources the leap from finite information on the boundary to "ones and zeros". The finiteness comes from non-commutative effects akin to the Heisenberg uncertainty principle, but the underlying quantum model is based on a continuous space. Daives' article [2] has a good less-technical explanation of implications of the principle for physical law. Even an axiomatization effort of QM using qubits postulates an underlying continuous space, e.g., [3]. There do exist discrete spacetime theories, e.g., theories based on the Regge calculus, Discrete Spacetime Quantum Field Theory, trying the model the universe as a Von Neumann universe, etc., but I don't know of any that are mainstream. --Mark viking (talk) 23:32, 12 February 2018 (UTC)

I generally agree with Mark viking here. XOR'easter (talk) 00:30, 13 February 2018 (UTC)

Why should it matter whether the space-time model is continuous or discrete? Aren't real numbers already required for the formulation of quantum mechanics? Sławomir Biały (talk) 01:22, 13 February 2018 (UTC)

That's a good point. There exist theories of, for example, Discrete quantum mechanics and p-adic quantum mechanics; in these the space is discretized or sort of discretized, but the value of the wave function remains complex. --Mark viking (talk) 03:58, 13 February 2018 (UTC)

Per the discussion here and at the WikiProjects, I've replaced that paragraph with a shorter one. XOR'easter (talk) 21:04, 14 February 2018 (UTC)

Looks good to me. Sławomir Biały (talk) 21:13, 14 February 2018 (UTC)

Short description

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 $\pi$ 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)

Well, "position along a line" represents the reals much better than it does the constructible reals. A line, intuitively, is connected. The constructible reals are not connected. Arguably (and I would very much take this position) the geometric concept of the line is the motivation for the real numbers in the first place, rather than the other way around. --Trovatore (talk) 10:28, 22 February 2018 (UTC)

Yes; on the line all points are created equal, while numbers are rational or not, algebraic or nor, etc. For Euclid a number was the ratio of two lengths, not a single length (I think so). Boris Tsirelson (talk) 10:48, 22 February 2018 (UTC)

Hmm, yeah, you have a point there, as it were. I like to mention the "line" because it evokes a couple of things that are distinctive about the reals — their connectedness, which distinguishes them from the integers or the rationals (or the constructible numbers), and their dimensionality, which distinguishes them from the complex numbers.

But you're correct that there are objections that could be made, and we can't handle them in a "short" description. Short descriptions have a suggested "soft limit" of 40 characters, whereas "kind of number representing a point on a line" is already at 49. We could lose "kind of" to get down to 41.

How about "number representing a continuous quantity"? --Trovatore (talk) 19:34, 22 February 2018 (UTC)

About its location on the numberline see File:Pi-unrolled-720 new.gif.   :-)   Boris Tsirelson (talk) 11:00, 22 February 2018 (UTC)

The present description is "kind of number representing a position on a line". IMO, it is not convenient, as presently people are less accustomed to geometry than to calculus. Thus reducing reals to a geometric concept is a bad idea. On the other hand, the introduction of reals in 19th century were motivated essentially for allowing a rigorous treatment of limits and continuity. Therefore, I fully support Trovatore's suggestion of Number representing a continuous quantity. D.Lazard (talk) 23:09, 22 February 2018 (UTC)

Support "Number representing a continuous quantity". Paul August ☎ 23:14, 22 February 2018 (UTC)

Done. --Trovatore (talk) 23:20, 22 February 2018 (UTC)

"Continuous" seems wrong, since it includes a position on a line segment or circle or plane or sphere or R^n. Limits and continuity apply there, too, after all. Even if "quantity" somehow excludes multi-dimensional cases (which I don't think it does), it still covers line segments and circles. --Macrakis (talk) 23:46, 22 February 2018 (UTC)

Remember, we're just trying to establish context for people looking at the mobile app. We don't have to give an exact demarcation of the subject matter. --Trovatore (talk) 00:02, 23 February 2018 (UTC)

"Continuous" is not wrong. The objection of Macrakis would be valid if "representing" would mean "characterizing", which is not the case. "Measuring" could be better than "representing" (an area is a real number). However, it could be confusing, as suggesting implicitly a physical process and a limited accuracy. Therefore I suggest to keep "representing", but I would not oppose to change it to "measuring". D.Lazard (talk) 07:42, 23 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)

can't depict algorithms on the number line, only their approximate instantiations

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)

They are most definitely (real) numbers. That there are algorithms to compute them to arbitrary accuracy means they are computable numbers (more or less). Where did you get the idea that they aren't real numbers, or that they can't be displayed on a number line? –Deacon Vorbis (carbon • videos) 19:06, 2 March 2018 (UTC)

About the philosophy of User:Counting floats, see also b:Numerical calculations and rigorous mathematics#Introduction. Boris Tsirelson (talk) 19:40, 2 March 2018 (UTC)

Very nice! Sławomir Biały (talk) 22:32, 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)

It must be nice to know that you are what you call a "smart guy", but the entities you refer to are very good examples of what in mathematics is defined as the concept of "real number". Certainly there are algorithms for calculating successively better approximate values for π, but that does not mean that π is one of those algorithms. It is not entirely easy to understand what you are trying to say, but it looks very much as though you are conceiving the real number line as consisting as a discrete set of points, each representable by a finite number of decimal figures, but the real number line as understood in mathematics is not discrete. Your comment about floating point numbers is completely off the point, because "floating point" is merely a way of representing numbers in a computer, and not part of the mathematical concept of real number. The editor who uses the pseudonym "JamesBWatson" (talk) 22:36, 2 March 2018 (UTC)

(ec) Hi User:Counting floats. The viewpoint presented in this article is the standard one in mathematics, and the article, appropriately, presents the standard view. That is not going to change.

There are alternative views that have something in common with the notions that I suspect you have in mind. You might be interested in ultrafinitism. It is appropriate for the article to discuss heterodox views such as ultrafinitism, but not as the main thrust of the article. I haven't checked recently whether the article discusses ultrafinitist approaches to the real numbers. If it does not, or if the coverage is inadequate, you are welcome to present reliable sources and suggest changes (or even make them yourself, but I don't really recommend making them yourself, because I think it needs a deeper understanding of the topic than I judge you to have, apologies if I'm incorrect about that).

If you are interested in learning about these issues, I would invite you to ask a question at the mathematics reference desk. This talk page, unfortunately, is not the right place to discuss them. --Trovatore (talk) 22:41, 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)

OK, well, we aren't going to argue about it here. As it stands there is zero chance that your wishes will be accommodated. As I say, if you can find reliable sources, it might be possible to cover views you would find amenable, in the same way that we would cover other fringe views. --Trovatore (talk) 00:23, 3 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)

Per WP:TPG, article talk pages are not meant for answering questions about the subject matter. Feel free to ask this question at WP:RD/Math. --Trovatore (talk) 00:48, 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)

Wow... Is a line in general, and the number line in particular, in elementary math education, an ideal geometric line (of zero width)? Or rather, in elementary math education, "three points are always on a line, provided that points and lines are thick enough" (this is a well-know joke about "the main theorem of applied geometry")? Boris Tsirelson (talk) 08:23, 3 March 2018 (UTC)

And (again), does not this picture define the point $\pi$ on the number line, in elementary math education? Boris Tsirelson (talk) 09:24, 3 March 2018 (UTC)

Thanks, this really useful theorem did not belong to my knowledge! :) Well, I tried to argue that "elementary" intuition about an object, vaguely introduced as "number line", does not suffice as a base for introducing concepts like "dense" rationals, for "interjacent" algebraic numbers, and also not for "completion" by the rest, and -all together- does not suffice for making up the "connected continuum" of reals. This is the basis for me objecting to the "number line" as a good means to "introduce" or to shortly "describe" reals. Re the π-marking I stated already in my first remark on short descriptions that rectification (of the wheels circumference) does not provide an "immediate access" to a point on a line. Purgy (talk) 15:11, 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)

Blackboard bold or just bold?

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)

Until quite recently, the article used ordinary bold, and someone changed a few of these the blackboard bold. The blackboard bold unicorn character is not well-supported, even in current browsers, so it is discouraged by WP:MSM. Sławomir Biały (talk) 15:15, 21 April 2018 (UTC)

Quantity along the line

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)

Re-reverting a new edit is generally not the recommended way, taking it to the TP is good.

"rather" is to mean that I do not like the previous wording very much either, your suggestion is no noticeable improvement, imho.

Introducing "real numbers" via referring to a "real line" is a logical no-no.

I deny that "real number" is about a position, but is rather (again) about a quantity in first sight.

I stop reverting with this. Purgy (talk) 09:16, 9 May 2018 (UTC)

No, I think you're quite wrong. The line is historically and conceptually prior to the reals. The line is the motivation for the reals in the first place. --Trovatore (talk) 09:34, 9 May 2018 (UTC)

Oh, sorry, I missed the context here. I don't have any immediate preference between "quantity along a line" and "position along a line". Maybe "quantity along a line" is better. But we should definitely mention the line. --Trovatore (talk) 09:37, 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)

Yes, I vaguely recall seeing in some text on the history of math, that the idea of "continuously varying quantity" (in the context of the emergent notion of function) was the reason to drop the former idea that each number should have individual name. Boris Tsirelson (talk) 10:46, 9 May 2018 (UTC)

I agree that measurable quantity is a poor phrase here. Sławomir Biały (talk) 13:33, 9 May 2018 (UTC)

"Longstanding" is not a valid argument for Wikipedia discussions. The historical and philosophical relationship between real numbers and positions on a line is an interesting question, but this would require sources, and, in any case does belong to the first sentence of the lead. By the way Euclides did not use numbers for marking a position on a line—this was invented by Descartes—, but he used them for measuring distances, which is clearly not the same. Similarly, Trovatore's opinion about the essence of the reals has nothing to do here.

The possible readers of this article include all physicists, engineers, and computer scientist. For them, the reference to a line is not useful and possibly confusing as their use of the reals is not generally related to geometry (for example, they use reals for the time and the energy). As WP is an encyclopedia, the first sentence must be significant for almost every reader. Such a common background is clearly provided by the use of reals for measurement. I agree that "measurable quantity" is not a best choice for that. "Continously varying quantity" is worse, as many measured quantities do not vary. On the other hand the phrase "continuous quantity" appears many times in Quantity. Therefore, I'll replace "measurable quantity" by "continuous quantity". D.Lazard (talk) 14:09, 9 May 2018 (UTC)

It seems clear to me that distance (along a line) was the first "continuous quantity" of concern. Certainly this was the case for the Greeks. That fact seems important here. I prefer the original language. Paul August ☎ 16:53, 9 May 2018 (UTC)

For all physicists and engineers every measurement has a finite resolution. Given that the second paragraph of the lead deals with rational, algebraic and transcendental numbers, I am skeptical about the use of reals for measurement. On the other hand, the first irrational "quantity" was of course geometric (${\sqrt {2}}$). Boris Tsirelson (talk) 20:02, 9 May 2018 (UTC)

I agree with Paul August - I think that the original language was better than "continuous quantity", which seems to be a much less common way to talk about things. We should emphasize that the real numbers are exactly the distances along a line. I don't think that the term "continuous quantity" is significant for many readers (and aren't complex numbers also 'continuous' quantities? If I don't know the answer to that, how many readers will know it? ) — Carl (CBM · talk) 22:07, 9 May 2018 (UTC)