Talk:Number/Archive 1

message: blackboard bold does not mesh well with a paragraph. Pizza Puzzle
Eh, I'm used to seeing blackboard bold; plain bold reminds me of a variable, not a set.
I agree that we should use it; however, the current font for it is too big and doesn't mesh with the rest of the page. Perhaps, if somebody submits a slightly smaller .png we can use that. Pizza Puzzle
I plan to upload a whole bunch of <.png>s like this later this month.
Of course, feel free to beat me to it. ^_^
Still we should still prefer markup that renders most directly in HTML, when such a thing works -- for now.
-- Toby Bartels 05:57 12 Jun 2003 (UTC)
----
Could somebody produce a little Venn diagram picture showing the various number sets? I removed this verbal description of the Venn diagram. AxelBoldt 15:23, 29 Sep 2003 (UTC)
replies: text: Rationals are usually defined as equivalence classes of ordered pairs of integers. Saying that they are ratios of integers is reasonable; this is only supposed to be an informal statement. A formal construction is given in the rational number article. --Zundark 08:17, 17 Nov 2003 (UTC)
thread_title: blackboard bold

message: I've added umpteen in the "see also" list, largely to de-orphan it. I'm not absolutely sure this is the right article to link to it, but I can't think of an alternative. Suggestions would be more than welcome. DavidWBrooks 20:01, 16 Feb 2004 (UTC)
thread_title: Umpteen

message: The section "generalizations" should be merged into "extensions" (which could receive subsectioning). I suggest to put it after the nonstandard stuff and before the comment on abstract algebra.
Please feel free to do so. MFH 18:07, 7 Apr 2005 (UTC)
thread_title:  extensions and generalizations 

message: I've removed this:
: Preserving the main ideas of "quantity" except for the total order, one can define numbers as elements of any integral domain.  Unfortunately, the set of polynomials with integer coefficients forms an integral domain, and polynomials are not numbers.
I just don't buy that whether we consider something as a "number" or not has much to do with whether it's in an integral domain. The extended real numbers aren't an integral domain; the ordinal numbers aren't; the cardinal numbers aren't. Matrices in general don't form an integral domain, but nonsingular matrices do, and that doesn't make the latter numbers. --Trovatore 16:45, 27 October 2005 (UTC)
thread_title: Integral domains

message: This is a subject I have thought about carefully for many years.  I am going to attempt a rewrite, a little at a time. Rick Norwood 21:31, 5 December 2005 (UTC)
replies: text: "Decimal", by definition, means base ten. The set is quite well-defined, just not very interesting. I have no objection to its removal. --Trovatore 22:37, 5 December 2005 (UTC)
replies: text: Good point. Rick Norwood 22:45, 5 December 2005 (UTC)
thread_title:  Attempted rewrite. 

message: I think for the list of constants at the bottom of the template, for a symbol for a constant to provide useful information it should have a link.  Would anyone like to undertake a stub, at least, for each of the constants?  I would, but I am unfamiliar with many of them.  Rick Norwood 01:07, 11 December 2005 (UTC)
thread_title:  The Number template 

message: The article states that the symbol for integer comes from the german word "zahlen".
Can anybody check if this is true and give reference?
I'm asking this because
"Zahlen" (=numbers)
seems to be more likely than "(be)zahlen" (=to pay).
There are also other possibilities like "zählen" (=to count)...
--Cyc 17:02, 4 February 2006 (UTC)
thread_title:  Could anybody check this? 

message: Unless someone objects, I am going to remove from the list of constants all constants with red links.  So, act quickly to save your favorite constant. Rick Norwood 16:09, 5 February 2006 (UTC)
thread_title: List of constants

message: Pfafrich -- I notice that some people use rv for reversion and others use rw.  Is there any difference that I should be aware of? Rick Norwood 13:17, 6 February 2006 (UTC)
thread_title: Trivia 

message: * JA: I'll be rummaging through my pack for some suitable stuffing as the day/week wears on.  Jon Awbrey 14:32, 6 February 2006 (UTC)
thread_title:  References & Bibliography 

message: * JA: In places where the longstanding equivocality of the term "natural number" has becomes a pain-in-the-[insert your favorite anatomical location here] on a second-by-second basis, like Ninja Sloane's OEIS, it has become standard to end all the fussn'-&-fightin' by using the dab terms non(-)negative integer and positive integer.  Those who miss the excitement can still go tete2tete over the hyphen in non(-)negative.  Jon Awbrey 16:18, 12 February 2006 (UTC)
replies: text: Oh, were it so easy!  The problem, both historically and pedagogically, is that the natural numbers come before the integers, so your definiton defines the simpler concept in terms of the more advanced concept, which is, of course, then in turn defined in terms of the simpler concept. Rick Norwood 19:40, 12 February 2006 (UTC)

text: The problem is that there are two current usages.  Each, individually, avoids ambiguity.  As long as both are in use, ambiguity is inevitable.  All, then, that is necessary to avoid ambiguity is for one side or the other to give in.  That hasn't happened yet.  Rick Norwood 22:16, 13 February 2006 (UTC)
thread_title: Non(-)negative integer & Positve integer are becoming standard

message: I noticed that much of the information in the history section has little to do with the content of this article.  For example, is this really the page to mention the history of solutions to quadratic equations?  I do think that there needs to be some discussion regarding the scope of this page before major changes are made. Grokmoo 18:15, 13 February 2006 (UTC)
replies: text: Yes as pointed out above, the history section could do with a good copy edit. As to why quintics are important to the development of number: they are the lowest degree polynomials whose solutions cannot allways be expressed by radicals (algebraic expressions involving roots), hence opening the way to transendental numbers. They were also an important step in development of Galois theory, the main technique for proving transendentalism of pi and e. --Salix alba (talk) 22:17, 13 February 2006 (UTC)
replies: text:  Yes, but perhaps these sorts of things belong in number theory.  If not, then there should at least be some mention of how these things relate to the main topic. Grokmoo 02:10, 14 February 2006 (UTC)
replies: text: The definitely do not belong in number theory, that field is about properties of integers, especially primes. This article is about the different number systems and how they are related. --Salix alba (talk) 09:40, 14 February 2006 (UTC)
replies: text: Number theory is not only about integers, but also about algebraic and transcendental numbers (see algebraic number theory). Of course, it doesn't necessarily follow that the connection quintics → algebraic numbers should not be mentioned here. -- Jitse Niesen (talk) 11:30, 14 February 2006 (UTC)
thread_title: Relatedness of items in history section

message: But actually, i is not the square root of -1, since negative numbers can't have a square root. In reality, it should be said thus: i^2 = -1, which is not the same thing. This is actually said in the article about the imaginary unit.
:The square root function is extended from the real numbers to the complex numbers, after which the principle square root of minus one is indeed i.  Still, it might be better in the article to say i squared is minus one. Rick Norwood 14:16, 11 April 2006 (UTC)
thread_title: Imaginary unit

message: Perhaps it would be worth adding some reference to theories about the origins of counting, such as that put forward in A.  Seidenberg, 'The ritual origin of counting', Archive for History of Exact Sciences, Volume 2, Issue 1, Jan 1975, Pages 1 - 40.  I don't have a copy of this paper lying around, so am not best placed to add the material. — Stumps 13:38, 15 February 2006 (UTC)
thread_title:  Origins of Number 

message: Good work removing bullet points and shortening.  I hope you, or somebody, tackles the rest of the bullet points.  
In particular, what is already covered in Numeral should not be repeated here unless very briefly. Rick Norwood 13:59, 15 February 2006 (UTC)
thread_title:  Ben Standeven's edit 

message: Here is how the introduction to this important article read before my recent edit.  Below, I mention some of the problems with this version.
"Originally the word number meant a count, or tally, of discrete objects. These days this application is referred to as cardinal number. Later use of the word came to include quantifying measurements of basic physical quantities like mass, length, area, volume and time.  
    
- Mathematicians have extended the meaning of number to include abstractions such as negative numbers, transcendental numbers and the \sqrt{-1}, also known as i. In common usage, one may find number symbols used as labels (telephone and highway numbers) They are also used together with, or instead of, alphabet letters to indicate ordering (serial numbers and letters ). The latter usage is encapsulated in the notion of ordinal number — first, second, third, etc.  + In common usage, one may find number symbols used as labels (telephone and highway numbers). They are also used together with, or instead of, alphabet letters to indicate ordering (serial numbers and letters ). The latter usage is encapsulated in the notion of ordinal number — first, second, third, etc."
  
"Originally the word number..."  The word "number" is of relatively recent origin -- it is the meaning of the concept, not the word, that is the subject of this article.
"...referred to as cardinal number".  No, ordinals are also used for counts or tallies: "first, second, third".
"...later use of the word..." we do not know enough about the origins of numbers to say definitively that counting came earlier than measuring.  This is an assumption.
"...used together with, or instead of, alphabet letters..."  Rather, occasionally alphabet letters are used in serial numbers, but this is a minor point, not one necessary to cover in the article.
The article, especially the introduction, needs to concentrate on the concept number.  There is already an article on numeral. Rick Norwood 14:57, 18 February 2006 (UTC)
thread_title:  problems with the introduction 

message: Good work removing bullet points.  I'm going to change the present tense to the past in the material you added, though.
The difference between References and Bibliography is this, and it is essential.  References are works actually used in writing the text, or works that verify statements in the text.  Bibliography is much more general, and can include other works on the same subject and also related works, even if nothing in them appears in the article. Rick Norwood 23:44, 28 February 2006 (UTC)
thread_title:  Ben Standeven's edit 

message: Not sure where to put this, so I put it here. Many of the x (number) pages for 3-digit numbers simply redirect to the previous 100, i.e. abb (number) redirects to a00 (number). Is this really desirable? I think it's confusing. ~iNVERTED | Rob (Talk) 22:52, 26 August 2006 (UTC)
thread_title:  Redirects 

message: I'm undecided about this, but I might like to replace "decimal numeral", which is a nonstandard term, by "decimal representation", which is a standard term that some think applies only to positive (or non-negative) real numbers.  An alternative would be to use "signed decimal representation", or a similar neologism.
Opinions?
RandomP 19:14, 25 September 2006 (UTC)
replies: text: Where do our Arabic numerals 0 1 2 3 4 5 6 7 8 9 come from?  Which people invented our excellent decimal positional system?
If not someone documents other, more modern usage, I'd be inclined to retain the limited van der Waerden usage of 'numerals'. JoergenB 09:46, 26 September 2006 (UTC)

text: The former. I'd like to retain the older (v. d. W.) usage, which implies changing this article; unless someone presents support for what you call 'the current usage'.  In other words, is this a current usage also outside wikipedia? JoergenB 15:31, 4 October 2006 (UTC)
thread_title:  Replace "decimal numeral" by "decimal representation"? 

message: :Numbers should be distinguished from numerals, the symbols used to represent numbers
Really?  Why?  It's perfectly possible (if really really awkward) to define a number to be (the mathematical representation of) a symbol used to represent that number.   I think that one of the things that makes mathematics hard to understand for some people is that, ultimately, the mathematical definition of most number systems is a string of symbol to be formally manipulated according to certain rules.  That's not the intuition I (and, I would think, most other mathematicians) have, but it might be a bit of a letdown for people first to be told that a number isn't really a string of symbols, only to be then taught otherwise by a moderately eccentric maths professor.
I'm not sure how to fix this. "can be distinguished"?  "are distinguished"?
RandomP 19:24, 25 September 2006 (UTC)
thread_title:  "should" is not a good word for wikipedia 

message: I remember that I read a long time ago that primitive cultures didn't have a name for 3 or more, equating anything above 2 with infinite. Also, IIRC in Greece the first number was 2, and 1 was not considered a number (it's not necessary in language). Does anyone have a source on this? This should be History of Three and History of One :-) Albmont 16:44, 4 January 2007 (UTC)
replies: text: It is true that some languages did/do run out of names for numbers quite early. I'm not exactly sure of how this relates to the article, as the concept of numbers exists, the extension of the concept to larger and larger values is not more significant at one stage than another. The article already refers to the Greeks questioning of 1, but doesn't that have more to do with philosophy than language? JPD (talk) 17:34, 4 January 2007 (UTC)
thread_title:  Before History of Zero 

message: in the article section about the History of irrational numbers, 
the article mentions that Hippasus was killed by Pythagoras himself (...However Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers. He could not disprove their existence through logic, but his beliefs would not accept the existence of irrational numbers and so he sentenced Hippasus to death by drowning.)
however, in the wiki about Hippasus of Metapontum, it is mentioned otherwise. he is supposed to have been killed by fellow pythgoreans.
please verify and correct —The preceding unsigned comment was added by 59.94.112.63 (talk) 19:20, 26 March 2007 (UTC)
replies: text: First, this is not inconsistant, since the person who passes sentence is not the person who carries that sentence out.  Second, it is a story, and there is no particular reason to think that it is a true story.  I'll do some reserch and make the facts clear in both places. Rick Norwood 19:44, 26 March 2007 (UTC)
thread_title:  Inconsitent Information about 'Hippasus of Metapontum' 

message: What is this? Looks like a bad German translation. Literally it means "determinant of continued fraction", but I have never heard about such a thing. :pl:Wikipedysta:Olaf —The preceding unsigned comment was added by 83.5.253.53 (talk) 07:15, 9 April 2007 (UTC)
thread_title: Kettenbruchdeterminanten

message: I remember reading years ago in a maths textbook about the general formulae that solve (I think) quartic polynomials in the real number domain. Unfortunately I cannot find the book. The point was that these equations contain square roots and for certain types of polynomials as you perform the calculations a subtraction occurs between the square root of a negative number and the square root of the same negative number. The results are real numbers, but two complex numbers briefly appear while applying the formula.
I know this sounds very vague but maybe it could be pertinent. Anybody who knows better please consider adding it to the article.
151.20.157.195 11:09, 12 April 2007 (UTC)
replies: text: There are a number of example where computations involving complex numbers lead to real answers, but more important, the complex numbers are a splitting field for every polynomial with real (or complex) coefficients.  That is, every polynomial factors into linear factors over the complex numbers. Rick Norwood 12:45, 12 April 2007 (UTC)
thread_title: Complex numbers are not an arbitrary extension of the reals

message: If there is a serious point behind this flag, please explain it.  Rick Norwood 13:01, 16 April 2007 (UTC)
thread_title:  neutrality flag 

message: Jan Z seems to have a substantial knowledge of the subject of zero, but I have trouble understanding what he has written.  In case anybody else is having similar problems:
The ancients loved to discuss what is or isn't a number.  For some, one wasn't a number -- the numbers began with two.  For others, the square root of two wasn't a number, because it was geometric rather than algebraic.
We now understand that definitions are arbitrary, to some extent, and a number is anything we say is a number.  Today, zero is a number.  It stands for the quantity "none at all".  (Resist temptation to quote Hitchhiker's Guide to the Galaxy.)
Zero is also a digit and is used, in decimal notation, as a placeholder, as in 208, which is read "two hundred eight".
So, there are really three things under consideration here.  Of course, every people have a way of saying "none at all", so that is moot.  The three questions are: Did ancient people have a symbol (as opposed to a word) for "none at all"?  Did ancient people consider this symbol a "number", like 2, or something else?  Did ancient people ever use a symbol, either the same symbol as their 0 or a different symbol, as a placeholder.
Jan Z seems to be saying that they did, but his examples don't have placeholders in them, so it is not clear to me what he is saying.
My understanding is that the Babylonians, at least, wrote (using I for an upright wedge and > for a sideways wedge) II>>>I for two sixties, three tens, and one = 151, but that they wrote II I for two sixties and one = 121, with a space rather than a placeholder, and also that II could be 2 or 120 depending on the context.  Of course, they may have done one thing at one time and another thing at another time, they had a long history, and most of their tablets have never been transcribed, much less translated.
Does anyone have more information about this?
Rick Norwood 20:02, 19 April 2007 (UTC)
thread_title:  I got plenty of nothing. 

message: Typographically, this section is a little bit of a mess. Is there value in unifying the presentation of the numbers and sets of numbers to all be in the usual typeface (for example, −1234.56 instead of -1234.56\, and N instead of \mathbb{N}? Xantharius 18:18, 4 June 2007 (UTC)
replies: text: Unless there are objections I shall proceed with the above plan. Xantharius 17:55, 5 June 2007 (UTC)
replies: text:  After a link to Blackboard bold appeared on the page, I discovered that it was possible to have the Unicode symbols for the most popular number systems instead of the PNG-style characters: thus ℚ instead of \mathbb{Q}. I think the Unicode ones are far nicer. Thoughts? Xantharius 16:59, 11 August 2007 (UTC)

text: It was just that the inline use of the PNG characters disrupts the line spacing: the Unicode ones don't. They also don't stand out as much (actually preferable, in my opinion!). Xantharius 20:08, 11 August 2007 (UTC)
thread_title: Types of numbers

message: A number is a word(concept) that represents and contains a sequence of patterns or data.  One word, equals one object.
For instance the numeral one, represents the mental object of one, as well as the geometric-symbol of one.
The numeral one could be considered the first letter of the mathematical alphabet.  I think it's best to think of mathematics as a language that describes shapes and patterns. A number would be considered both data and a shape at the same time, a data-shape. In fact all numerals are geometric shapes. -- Yours truly BeExcellent2every1 (talk) 12:24, 21 November 2007 (UTC)
replies: text: This is original research and not suitable for Wikipedia. Rick Norwood (talk) 15:30, 19 November 2007 (UTC)
thread_title:  A number is 

message: :The figure "File:REAL NUMBERS.svg|thumb|300 px|upright|The real numbers contain the irrational, rational, integers and natural numbers, and transcendental numbers." has been removed by 202.36.179.66 , on the grounds that it suggests that: The image implies there are real numbers that are neither rational or irrational. Could also mislead one about the relative "sizes" of these sets.
:I agree that the figure risks misleading, although it comes down to how one interprets the figure; the caption gives no clue, such as saying that it is a Venn diagram. — Charles Stewart (talk) 07:57, 20 July 2009 (UTC)
thread_title: Removal

message: "speculated that the first known"  Did you really write that?  —Preceding unsigned comment added by 64.85.211.7 (talk) 18:55, 11 August 2009 (UTC)
thread_title:  History 

message: Strictly speaking the following statement under complex numbers is wrong:
:Each of the number systems mentioned above is a proper subset of the next number system. Symbolically, \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} .
for instance natural numbers are defined by sets containing sets, integers by pairs of naturanl numbers with an equivalence relation under subtraction, rational numbers by a pair of integers again, real numbers by a pair of sets defining a Dedekind cut, or perhaps other definitions but they are not subsets in the set sense.
However we all know what it means, has anyone got a wording that would be more accurate without being too pedantic and silly? Dmcq (talk) 12:43, 19 July 2010 (UTC)
replies: text: The statement as it stands reflects major reference books.  Your objection is technically correct, but we say "A is a subset of B" even when what we really mean is "There is a subset of B isomorphic to A."  The isomorphism is understood.  This situation is similar to saying "1 + 1 = 2" instead of saying "the number represented by the numeral 1 added to the number represented by the numeral 1 equals the number represented by the numeral 2." Rick Norwood (talk) 13:01, 19 July 2010 (UTC)
replies: text: Actually, by "A is a subset of B" we mean that "The canonical map from A to B is 1-1,", rather than "There is a subset of B isomorphic to A."  Still, the mathematical convention should stand.  — Arthur Rubin  (talk) 22:38, 19 July 2010 (UTC)
replies: text: It is a point that bothers me slightly, because I do think the reals are different enough in kind from the simpler structures to justify thinking of, say, the real number zero, as a different object from the natural number zero, and this at a more fundamental level than, say, the choice of a coding via Cauchy sequences or Dedekind cuts.  But I doubt there's any useful way to talk about that at the level of an article like number. --Trovatore (talk) 22:48, 19 July 2010 (UTC)
replies: text: I think it would be counterproductive at number – perhaps I am mistaken, but it should be a general introduction. That is to say, almost everyone knows (or rather thinks they know) what a number is and so this article shows it is a bit more complicated than that, but it doesn't have to dot every I or cross every T, referring to other articles is more appropriate. It should, of course, be correct. Si Trew (talk) 20:59, 30 July 2010 (UTC)

text: I think the statement above, "we all know what it means", indicates the problem in this article about trying to decide what audience it is aiming at. Si Trew (talk) 21:01, 30 July 2010 (UTC)

text: I would like to mention two points that did not come out clearly enough in the discussion above.  1. numbers are arguably more "primitive" objects than set theory.  Thus, the construction of the integers in terms of pairs of natural numbers is just that: a particular construction, which does not change the fact that natural numbers are found among the integers. 2. While there is not much argument about the status of natural, integer, and rational numbers, there is a bit of a controversy about what constitutes the reals.  The intuitionist perspective is quite different from the classical one.  The current state of the page does not reflect this at all.  Perhaps this is appropriate for the level the page is aiming at, or perhaps at least a brief mention should be included.  Tkuvho (talk) 10:08, 1 August 2010 (UTC)
thread_title:  Not subsets 

message: I saw an edit that suggested it's unnecessary in this article to mention zero in distinction from natural numbers, because zero is a natural number. But that does not seem to be an invariant property of the term "natural number" in the several sources I have at hand. Many sources start the set of natural numbers at 1, and I think that this article should reflect that uncertainty of definition, as the article Natural_number does. -- WeijiBaikeBianji (talk) 04:14, 24 September 2010 (UTC)
replies: text: Yes, I agree with you.  I looked at that issue briefly a while back but the correct fix wasn't immediately obvious.  Shouldn't be too hard to find reasonable language; go for it. --Trovatore (talk) 07:41, 24 September 2010 (UTC)
replies: text: The current article already reflects this in the section Number#Natural numbers: "Traditionally, the sequence of natural numbers started with 1. However, in the 19th century, set theorists and other mathematicians started the convention of including 0 in the set of natural numbers". DVdm (talk) 14:36, 24 September 2010 (UTC)
replies: text: The article did not reflect what User DVdm indicates it did. The use of the word "convention" means that it is generally agreed upon to include zero. If I take my books off the shelf, about half will include zero, and about half will not. I fixed this yesterday. The article now indicates how "Natural" may be used to describe two different sets. If you can word it better, feel free of course. Cliff (talk) 16:49, 28 March 2011 (UTC)
replies: text: Automated theorem proving tends to use the definition with them starting at 1 so we might be heading back to the old ways again! 20:28, 28 March 2011 (UTC)
thread_title:  Article [[Natural_number]] says zero may or may not be included in set of natural numbers. 

message: Is there a reason that the histories of these two different sets of numbers are discussed in the same section?Cliff (talk) 00:06, 15 February 2011 (UTC)
thread_title:  Transcendental numbers and reals  

message: The first three in the table describe sets of numbers, but the last three describe individual elements from the respective sets. If anyone can think about how to fix this, please do. I'll think about it.  Cliff (talk) 20:34, 3 April 2011 (UTC)
replies: text: Describing a general element from a set is one of the standard ways of describing a set see 'special sets' in set (mathematics).Dmcq (talk) 20:45, 3 April 2011 (UTC)
replies: text: I'm not expressing confusion about what is going on, I'm expressing concern over the clarity of this article as it pertains to the "average" reader, not editor.  Cliff (talk) 04:17, 4 April 2011 (UTC)
thread_title: Table in classification section