Talk:Number

Rational/irrational

Surely the picture next to the real numbers is wrong, as any real number must be either rational or irrational, and the little Venn diagram shows the rationals and irrationals as disjoint subsets of the reals when one should be the complement of the other?

i.e. the label for "irrational" should go in the whole space that's in the reals but not in the rationals, NOT have its own little circle inside the reals (leaving a large section of the reals that is apparently neither rational nor irrational). —Preceding unsigned comment added by 81.159.20.37 (talk) 23:54, 7 April 2009 (UTC)

Agreed, I came here to make the same comment 202.36.179.66 (talk) 07:14, 3 July 2009 (UTC)

I removed the incorrect image. It had been up there for long enough. It is not only wrong because it implies there are real numbers that are neither rational or irrational, it also could mislead one about the relative "sizes" of the sets (i.e. irrational numbers are far more numerous than rational ones, etc.) —Preceding unsigned comment added by 202.36.179.66 (talk) 02:13, 20 July 2009 (UTC)

Sorry, but the diagram is still wrong; it still implies that there are real numbers of a different type, neither rational nor irrational, with all that free space around both sets. Maybe using a "squarish" diagram you could put the irrational side-by-side with the rational, this one containing all the pertaining number sets. Then draw a thick line around both, forming a rectangle containing both and just it, meaning the real numbers set. Just my ten cents as a Math teacher.189.62.116.193 (talk) — Preceding undated comment added 21:03, 22 February 2014 (UTC)

What diagram are you talking about? If you mean this one, the criticism is valid, but the image doesn't appear in the article, so I'm not sure what the problem is. If you think it's so misleading that it shouldn't be hosted by WP at all, on the grounds that someone might stumble on it and be misled, well, you can make that argument at Wikipedia:Files for deletion. --Trovatore (talk) 21:53, 22 February 2014 (UTC)

Definition of i

I believe that defining i as "the square root of negative one" is a wrong approach. (e.g. because $i^{2}=-1=(-i)^{2}$ - so it is not unambiguous) Shouldn't be the complex numbers defined as ℝ×ℝ with following operations:

• (a; b) + (x; y) := (a + x; b + y);
• (a;b)⋅(x;y):=(ax-by; bx+ay);

and than denote i:=(0,1) ?

Chilkes (talk) 22:21, 26 February 2012 (UTC)

There are some advantages in defining complex numbers in that way, but for the purposes of this particular article I think the informal definition "the square root of negative one" is fine. The interested reader can get more details by reading the linked articles such as imaginary unit. Gandalf61 (talk) 11:31, 27 February 2012 (UTC)

−i is the negative square root of minus one, just like 1 is the square root of 1 and −1 is the negative square root. Dmcq (talk) 00:15, 27 June 2012 (UTC)

No. Sorry, but i is neither positive nor negative. The symbol i represents one of the square roots of −1. Once that symbol is introduced, it is easy to show that −i is the other. Rick Norwood (talk) 12:07, 27 June 2012 (UTC)

i is not a number and cannot be defined as a number. It is exactly a 2 dimensional vector with the operations you stated. Incompetent mathematicians think of it as a number because the real part can be used with "real numbers" (also non-existent). Complex theory is inherently unsound, but then again, those who advocated its use were not very smart mathematicians - much like the mathematicians you encounter today. 98.194.121.63 (talk) 20:41, 11 July 2012 (UTC)

You are still pushing the "new calculus", which has no place in Wikipedia unless published. However, this statement, although clearly seen to be false by any competent mathematician, is not obviously wrong to a layman, so further discussion might be productive.

But probably not. — Arthur Rubin (talk) 21:32, 11 July 2012 (UTC)

i can be very well defined as a number. You first should ask your self: what could be a number? And this is a philosophic question -- probably far too tricky to be solved in agreement on wikipedia audience. :)

But now to the original problem. I cite: "the square root of negative one". The basic problem lies in ill-definition / usage of english words, IMHO! There is a unique square root function (per definition a function must be unique) and a certain value in its range is called the square root of ... (e.g. -1). On the other hand the solution(s) of the equation "x^2= -1" are called square root(s) of -1 and there are two different. As often, the real problem lies in bad or not existing precise definition of wordings. — Preceding unsigned comment added by 2001:638:504:C00E:214:22FF:FE49:D786 (talk) 16:06, 20 September 2012 (UTC)

I agree with what was said so far and would like to add that the formulation "i is one of the two solutions of the equation x^2= -1", although it may at first not seem to solve the problem, accurate. It is not important, "which" of the two solutions you define to be i. The other, as Rick states earlier, can be shown to be -i then, meaning that if i is a solution (whatever i is), then -i must also be a solution. One of the effects that this symmetry has is that complex solutions to polynomial equations always appear in conjuagte pairs. By the way, 98.194.121.63's statement sounds very similar to what was said in the past about negative numbers (e.g. negative solutions to polynomial equations) when they were still uncommon. --Doubaer (talk) 11:42, 26 June 2014 (UTC)

What is a number?

Close "the new calculus". Nothing to see here. Move along.
The following discussion has been closed. Please do not modify it.
Popular misconceptions: 1. Number came before measure. FALSE 2. Ratios came from natural numbers. FALSE 3. Irrational numbers are numbers. FALSE 4. Real numbers exist. FALSE 5. Complex numbers are numbers. FALSE All one has to do is study Euclid's Elements from which all these misconceptions are easily debunked. http://thenewcalculus.weebly.com/uploads/5/6/7/4/5674177/magnitude_and_number.pdf 98.194.121.63 (talk) 16:52, 26 June 2012 (UTC) What does that have to do with this article, even if they were at all plausible. Which they are not. — Arthur Rubin (talk) 00:42, 27 June 2012 (UTC) Rubin, the fact that you have to ask proves you are not a mathematician. What does Euclid have to do with number? Answer: Everything. 98.194.121.63 (talk) 17:54, 8 July 2012 (UTC) I'd forgotten you were here. I wish I could still forget. As for your assertions, they are not published anywhere, so cannot be used as a reference in Wikipedia. If they referred to a published work which said those things, that work might be usable, but, as there is no such published work .... Don't feed the trolls. Rick Norwood (talk) 12:01, 27 June 2012 (UTC) Norwood, you are a troll and it shows. 98.194.121.63 (talk) 17:54, 8 July 2012 (UTC)

Blacklisting URLS?

http://johngabrie1.wix.com/newcalculus

How number is derived: https://www.filesanywhere.com/fs/v.aspx?v=8b6e62875e636d7bb399 197.79.38.87 (talk) 11:01, 8 September 2014 (UTC)

If the symbol for a number is called a numeral, what is the word for a number called?

Quite simply, the heading says it all. -- — T13   ( C • M • Click to learn how to view this signature as intended ) 16:09, 13 February 2013 (UTC)

It is also called a numeral - see numeral system - "A numeral system (or system of numeration) is a writing system for expressing numbers" - and numeral (linguistics) - "In linguistics, a numeral is a word class (part of speech) designating numbers". This ambiguity sometimes causes confusion. Gandalf61 (talk) 09:48, 14 February 2013 (UTC)

Why should there be a special word? One can either say list ten animals or list ten words for animals. There isn't a special word for word for animal. Dmcq (talk) 14:11, 14 February 2013 (UTC)

Probably, historically, mathematicians were fussier than biologists. Today the distinction between number and numeral is seldom made. However, if you want to get technical, there is a difference between the word "cat" and the animal "cat". Thus a cat has whiskers, but "cat" has three letters (and no whiskers). Rick Norwood (talk) 15:26, 14 February 2013 (UTC)

After some research, I have concluded that numbers such as "one" "two" "three" are cardinal and ones such as "first" "second" "third" are ordinal. Further than that, in word form is verbose and otherwise it is in numeric form. -- — T13   ( C • M • Click to learn how to view this signature as intended ) 16:59, 14 February 2013 (UTC)

I'll mention (because it came up in an article edit) that a "numeral" is generally understood to be a single symbol (or "digit"), while a "number" is written using one or more numerals/symbols/digits. So, the only way "symbol for a number" makes sense is when the number in question can be represented by a single symbol/numeral/digit (i.e. 0-9 in base-10, Arabic). —[AlanM1(talk)]— 07:26, 8 March 2013 (UTC)

Alan, I do not understand what your point is. In the heading clearly states "a number" which is singular in of itself. I was simply trying to determine what the noun for the verbose/long name of a number is. — T13   ( C • M • Click to learn how to view this signature as intended ) 12:44, 8 March 2013 (UTC)

No, a numeral may have more than one symbol. 123 is a numeral. XVII is a Roman numeral. Rick Norwood (talk) 15:40, 8 March 2013 (UTC)

I disagree. M-W defined numeral (noun) as "1: a conventional symbol that represents a number" (singular symbol). OED, though, says "a figure, symbol, or group of figures or symbols denoting a number. a word expressing a number." I still think my usage is more common. "123" is a number, consisting of three numerals. "XVII" is a number composed of 4 Roman numerals (i.e. one might write "Convert 17 to Roman numerals, not a Roman numeral). Time to find some published sources, I guess. —[AlanM1(talk)]— 00:52, 9 March 2013 (UTC)

M-W is using the word "symbol" in the common sense that would be understood if I said "cat" is a symbol for a feline. There is no intent there to limit the symbol to a single character. OED is more careful, noting that a numeral can be a group of symbols. 123 is a numeral, which expresses in three digits the same number as CXXIII. The word for a single symbol numeral is "digit".Rick Norwood (talk) 20:09, 30 October 2014 (UTC)

Arabic versus Veyselic Numbers

Arabic numbers are written LTR, most significant digits are on the left, however Veyselic Numbers are arabic numbers written from right to left, RTL, like number ten is 01 (or in arabic .1). More info can be found in http://en.wikipedia.org/wiki/User_talk:Veyselperu Thanks 213.74.174.138 (talk) 08:57, 22 August 2013 (UTC)

As many elementary math books note, what we call Arabic numbers are really Arabic numerals. And Verselic Numbers are really numerals as well. The quantity indicated does not change whether the digits are written right to left or left to right. Both use base ten.Rick Norwood (talk) 20:12, 30 October 2014 (UTC)

Hatnote

An editor is starting edit warring for including in the hatnote a direct link to Book of Numbers. He has already been reverted by two different users, and he reverts immediately these reverts. This direct link in inappropriate for at least three reasons. Firstly, it is aimed for "disambiguating an article name that is not ambiguous" (see WP:NAMB). Secondly, this article is linked to in Number (disambiguation), to which refers the hatnote {{other uses}}. Thirdly, it breaks the policy of neutral point of view, by emphasizing without good reasons a particular use of the word "number". For these three reasons, I'll revert this editor again. D.Lazard (talk) 18:22, 30 October 2014 (UTC)

Note that this page is named "Number" (singular) rather than "Numbers". So strictly speaking it is not an issue of disambiguation at all. Tkuvho (talk) 18:30, 30 October 2014 (UTC)

It is not quite that simple, as "Numbers" redirects here. Sławomir Biały (talk) 18:50, 30 October 2014 (UTC)

Further, Number (disambiguation) disambiguates both "number" and "numbers". --50.53.38.33 (talk) 20:05, 30 October 2014 (UTC)

There are two redirects in play: Numbers redirects to "Number", and Numbers (disambiguation) redirects to Number (disambiguation). --50.53.38.33 (talk) 20:34, 30 October 2014 (UTC)

This hatnote does not belong here. I fully agree with D.Lazard's original edit. There's no such hatnote in our Book article either. - DVdm (talk) 20:16, 30 October 2014 (UTC)

I agree with all of that, except it's not so much a POV issue as undue weight. The Book of Numbers is hardly an alternative primary topic; it may seem so to some Christians but only some, and Christians are only a small minority of English speakers worldwide. Rather than put undue weight on that use it should just link to the disambiguation page, where it's rightly listed as just one item among many.--JohnBlackburne^words_deeds 20:18, 30 October 2014 (UTC)

• I don't necessarily disagree, except that it had to have been added for a reason. I'm currently going through the history one revision at a time (I'm up to Revision as of 17:49, May 6, 2010) trying to find when it was added and by whom. My guess is that it was a heavily searched for topic and because "Numbers" (short for "The Book of Numbers") was redirected to this page, people were having difficulty in finding it. Please be patient, and as soon as I have found the edit (and hopefully some discussion about this someplace), I'll report back and we can move forward. In the mean time, please leave the page in the condition it was in BEFORE the BOLD edit was made to remove this hatnote until AFTER this DISCUSSion has concluded and there is consensus for the change. Thank you for your patience. :) Happy editing in the mean time! — {{U|Technical 13}} ^{(e • t • c)} 20:24, 30 October 2014 (UTC)

I made my edit (and several other edits) before I read your comment, else I would have waited. Still, I can't imagine anybody searching for the Book of Numbers having trouble with the disambiguation page.Rick Norwood (talk) 20:36, 30 October 2014 (UTC)

I moved Book of Numbers to the top of the "Literature" list on Number (disambiguation). --50.53.38.33 (talk) 20:46, 30 October 2014 (UTC)

One solution might be to have "numbers" redirect to the disambiguation page instead of here.Rick Norwood (talk) 20:53, 30 October 2014 (UTC)

Having Numbers redirect to Number (disambiguation) is a good idea. --50.53.38.33 (talk) 21:06, 30 October 2014 (UTC)

• (edit conflict) So, I finally found it... I suppose I should have worked down from the most recent edits instead of up from the oldest as it was about six weeks ago it was introduced in this edit by Red Slash which was immediately REVERTed in this revision by D.Lazard and then rereverted by Red Slash in this change citing Wikipedia:Hatnote in his edit summary. Reading that editing guideline, I see there are a couple subsections that seem to support it should be there and I see other sections that maybe suggest otherwise. It's not entirely clear to me at this time which section of that page Red Slash was suggesting supports inclusion. One potential compromise I can possibly think of is a discussion of whether or not Numbers should redirect to Number, Book of Numbers, or Number (disambiguation). I'm tempted to think that perhaps it should redirect to the disambig. I'd like to hear if RS can clarify his position on why it should be included and I'd like to hear others opinions about fixing what I consider to be the biggest part of the problem, which is the redirect. — {{U|Technical 13}} ^{(e • t • c)} 21:19, 30 October 2014 (UTC)

I don't think changing the target of Numbers makes sense. Almost always the singular and plural of a title go to the same place. Otherwise editors will find themselves repeatedly getting it wrong, writing e.g. Numbers or Numbers interchangeably and expecting then to do the same thing so not checking where they actually go, and then being surprised when they go to different places. The guideline is WP:POFRED, and one of the main reasons for creating redirects to an article is from its plural.--JohnBlackburne^words_deeds 21:32, 30 October 2014 (UTC)

Numbers (TV series) is a closer title match and gets about twice as many page views as Book of Numbers. It's best to leave all uses for the disambiguation page. I noticed that Red Slash promotes Jesus on their user page. The hatnote shouldn't link to a religious topic which may not even be the main use of "Numbers" as a title. I think Numbers should continue to redirect to the singular as we usually do. PrimeHunter (talk) 21:52, 30 October 2014 (UTC)

• Now that I've researched and found the actual chain of events, I've removed the hatnote from the article as that is the proper condition per the BRD process. I apologize if it seemed I was for having that content there, I honestly am as unbiased as can be about it as long as process is followed. I also apologize for putting a couple of you in a knee-jerk "quick-revert" situation, but I didn't realize at the time that removing the content was actually the revert and the BRD process wasn't properly being followed to begin with. Anyways, now to discuss what to do with Numbers. I believe that changing the redirect to the disambiguation page is the best option here to prevent undue preference of any one term over another and I don't see there as being a clear primary topic. While "numbers" may be a plural form of "number" it's also a proper noun for at least a TV series (which is no longer in production) and a book of the bible. It's also apparently a magazine, a band, multiple music album titles, and a few other things. So, all of that said, lacking a definite PRIMARYTOPIC and to prevent UNDUE weight being attributed to any of the topics, I'd say the best option is to change the redirect to point to the disambiguation page. — {{U|Technical 13}} ^{(e • t • c)} 22:54, 30 October 2014 (UTC)

JohnBlackburne: "… writing e.g. Numbers or Numbers interchangeably …"

To be clear, the wiki markup is: [[Number]]s and [[Numbers]].

--50.53.38.33 (talk) 07:22, 31 October 2014 (UTC)

• What does that ahve to do with anything here? You can write numbered, numbering, numberular, numbertasticular, or numbers. They all point directly to Number. It's no different than piped links... — {{U|Technical 13}} ^{(e • t • c)} 12:12, 31 October 2014 (UTC)

The problem is that many editors expect the singular and plural to lead to the same article, and if they write a sentence with a plural then they often wikilink the whole word. For example, Distributive property currently includes: "allows a type theory to add concepts like [[numbers]]". PrimeHunter (talk) 13:23, 31 October 2014 (UTC)

JohnBlackburne: "… editors will find themselves repeatedly getting it wrong …"

Redirects are for the convenience of readers, not editors — editors should be testing their links before saving. Anyway, there is a bot that notifies editors who link to a disambiguation page. --50.53.46.109 (talk) 17:52, 31 October 2014 (UTC)

Redirect

I think it is clear from the comments above that while not everybody agrees, if we put it to a vote then linking numbers to number (redirect) is the solution with the most support. Nobody is likely to type in "numbers" looking for this page, and if they do the redirect will quickly lead them here. Not that it is a big deal one way or the other, but I don't think always linking a plural to a singular is Wikipedia official policy. If I'm wrong, I'm sure somebody will supply the appropriate quotation. Rick Norwood (talk) 23:07, 30 October 2014 (UTC)

I linked to the guideline above, and it's overwhelmingly the practice. I can't think of any instance where the singular goes to an article and the plural goes to another. The only instances I've come across are separate disambiguation pages, one for singular and one for plural, because together they'd be too long. If you still think it should be changed probably the best place is at WP:RFD, which is not just for deletion but discussion.--JohnBlackburne^words_deeds 01:33, 31 October 2014 (UTC)

How about Blues? Tkuvho (talk) 09:17, 31 October 2014 (UTC)

Good example. There are pages for Blue, Blues, Blue (disambiguation), and Blues (disambiguation). --50.53.46.109 (talk) 14:13, 31 October 2014 (UTC)

Yes, good example. In that case there are two primary topics, the colour and the music type ("The Blues"). There are probably others. But that is an exceptions where the plural has a distinct meaning. Number and numbers are the singular and plural of the same thing, the plural doesn't have a distinct meaning.--JohnBlackburne^words_deeds 14:37, 31 October 2014 (UTC)

Well, but it does — it means the Book of Numbers, for example; that's the whole reason this is being discussed. "Blues" can also be a straightforward plural of "blue", so the cases are actually reasonably parallel, I think.

I support retargeting numbers to somewhere other than this page, probably number (disambiguation). --Trovatore (talk) 14:51, 31 October 2014 (UTC)

Root is an article. Roots and Roots (disambiguation) are redirects to Root (disambiguation). See also, Statistic and Statistics. --50.53.46.109 (talk) 14:56, 31 October 2014 (UTC)

Dot and Dots are different disambiguation pages. --50.53.46.109 (talk) 15:21, 31 October 2014 (UTC)

Compare Job (article), Jobs (dab), and Job (disambiguation). --50.53.46.109 (talk) 15:38, 31 October 2014 (UTC)

Pen is an article, and Pens is a disambiguation page. --50.53.46.109 (talk) 20:12, 31 October 2014 (UTC)

Sun is an article, and Suns is a disambiguation page. --50.53.46.109 (talk) 20:18, 31 October 2014 (UTC)

This seems pretty definitive. I'll make the change if nobody else does. But I'll wait until tomorrow.Rick Norwood (talk) 22:45, 31 October 2014 (UTC)

Retargeting numbers to numbers (disambiguation) seems to make more sense than retargeting it to number (disambiguation). Tkuvho (talk) 20:38, 1 November 2014 (UTC)

(ec)The problem with that is that numbers (disambiguation) is already a redirect. I do think that it might make sense to split them, though. I think the current setup makes it too hard for readers looking for the Book of Numbers.

Here's the thing: Usually we consider two criteria, the first being what readers are likely to mean when they enter a term in the search box, and the second being what editors are likely to mean when they wikilink a term.

I think the first criterion is likely to point fairly strongly to the Book of Numbers (and perhaps also to Numb3rs the TV show). There isn't too much reason to type the s if you're looking for number.

The second criterion, on the other hand, is more aligned with what John Blackburne has been saying; my intuition is that most editors typing [[numbers]] are likely to intend to link to this article.

But here's the thing: they probably shouldn't be. Extremely general topics like this one are a bit paradoxical; they're very important articles, but they're usually poor targets for wikilinks. There aren't very many contexts where it's natural for a reader to want to say "oh, at this point I want to go read about numbers in general". We should usually discourage wikilinks to extremely general topics, except occasionally from other articles about extremely general topics, of which there aren't very many. --Trovatore (talk) 20:47, 1 November 2014 (UTC)

That would not accomplish anything, since numbers (disambiguation) automatically redirects to number (disambiguation). However, I can't figure out how to change where "numbers" automatically redirects. Does anyone know how to do this?Rick Norwood (talk) 20:41, 1 November 2014 (UTC)

Trovatore, I am not sure what the final conclusion of your train of thought above was but if it is that number (disambiguation) and numbers (disambiguation) should be split, I agree. There are several items there that appear in the plural in addition to the book of Numbers. Incidentally, the book of Numbers is occasionally referred to as "Numbers" just as one might say "Genesis" rather than "book of Genesis". Tkuvho (talk) 21:10, 1 November 2014 (UTC)

I've restored the Numbers redirect to point to this page and created an RfD for it. I should have noticed it before but the list of exceptions to the plural rule our SPA IP found is hardly definitive. There are 8,870 redirects in Category:Redirects from plurals, so the few found are about 0.1% of them. There are very few exceptions, and they usually only exist where there's good reason, such as the singular not having a plural (such as Sun) or the singular and plural meaning quite different things (such as statistic and statisctics). But in this case Numbers is just the plural of Number, the primary topic.--JohnBlackburne^words_deeds 10:23, 4 November 2014 (UTC)

There are not that many examples of the plural form not redirecting to the singular because adding the final "s" usually results in the meaning of the plural rather than a different meaning. In those cases when the meaning is different, it is appropriate to redirect accordingly. Redirecting Numbers to Numbers (disambiguation) (a new page) is a good illustration of WP:IAR. Other editors are invited to comment. Tkuvho (talk) 10:41, 4 November 2014 (UTC)

I've already given my opinion above, as have a number of others, so there is no need to repeat it. If JohnBlackburne is the only person here who knows how to change a redirect, then I guess what he wants wins. By the way, "suns" is a perfectly good plural. See, for example, "A Thousand Splendid Suns", by the author of "The Kite Runner". And "Numbers" is a common name for a book of the Bible, so it has a meaning unrelated to "number".Rick Norwood (talk) 13:07, 4 November 2014 (UTC)

I am not sure what you mean by your question about changing a redirect. You edit a redirect page the way you would edit any other page, though I don't recommend any further changes until there is a consensus. Note that there is a separate discussion page now for these redirects at https://en.wikipedia.org/wiki/Wikipedia:Redirects_for_discussion/Log/2014_November_4 Tkuvho (talk) 13:34, 4 November 2014 (UTC)

For the record, I'm against a redirect, and therefore think this article ought to have a hatnote. --Sammy1339 (talk) 14:27, 4 November 2014 (UTC)

The article has a hatnote already. The issue is what kind of hatnote it should have. Tkuvho (talk) 14:58, 4 November 2014 (UTC)

Try to vote at the bottom of https://en.wikipedia.org/wiki/Wikipedia:Redirects_for_discussion/Log/2014_November_4#Numbers Tkuvho (talk) 16:04, 11 November 2014 (UTC)

Is number theory really called "arithmetic".

The article says the word "arithmetic" is also used for number theory. That's not a usage I've ever heard? Is anybody aware of such a usage?Rick Norwood (talk) 12:29, 3 November 2014 (UTC)

Try this: Serre, Jean-Pierre Cours d'arithmétique. (French) Collection SUP: "Le Mathématicien", 2 Presses Universitaires de France, Paris 1970 188 pp. Tkuvho (talk) 12:48, 3 November 2014 (UTC)

It would probably be better to write "the word arithmetic has also been used for number theory". There are many old references that use "arithmetic" or "higher arithmetic". See the last paragraph of the lead of Number theory for details. D.Lazard (talk) 12:52, 3 November 2014 (UTC)

I think "has been used" is an understatement. The term is still in widespread use and also appears in the names of other subjects like arithmetic geometry. --Sammy1339 (talk) 15:30, 3 November 2014 (UTC)

Thanks.Rick Norwood (talk) 14:27, 3 November 2014 (UTC)

Factual accuracy of the History section

I added a disputed tag to that section of the article. Sources are woefully inadequate and there are many dubious claims. For example, the proof that ${\sqrt {2}}$ is not rational is called an "existence proof," the apocryphal story about Hippasus seems to be described as fact, Lambert's proof that $\pi$ is irrational is weirdly listed as the first result about transcendental numbers, about Cantor's work it is falsely claimed "this was the first mathematical model that represented infinity by numbers and gave rules for operating with these infinite numbers," it is claimed that Tartaglia had a hand in discovering imaginary numbers (which as far as I know he never used - they were briefly mentioned in Cardano's Ars Magna whose only connection to Tartaglia was that it plagiarized him, and not on that subject), and generally I can't verify hardly anything in this section for extreme lack of references. I'm planning on making a lot of changes to this article when I have time, but this will be the last section I will work on, because I regard it as lowest priority. (Math before math history - top priority in my view is a formal definition of the main number systems, which is entirely lacking right now.) To anyone who worked on this section: please provide references if you can. To allow an opportunity for this I'm going to leave the section untouched for a week or two, but without references I'm inclined to outright remove most of it (replacing what I can of course), since I'd rather not include all these questionable claims and I'm not qualified to rewrite the whole section by myself. Also please note that mathematics textbooks are not reliable sources for mathematics history before the 20th century - for example, in just one paragraph, Jacobson's famous graduate textbook Basic Algebra manages not only to bungle the history of the cubic equation but to get Tartaglia's name wrong. And of course things we all heard in lectures - which seems to be the source of some of the information here - are even less reliable. --Sammy1339 (talk) 17:40, 3 November 2014 (UTC)

Good luck with your project.Rick Norwood (talk) 21:57, 3 November 2014 (UTC)

I don't think wholesale deletion of the section is appropriate. The work on any given wiki page is incremental. There is material there that's legitimate; other passages need to be worked on. One could try identifying the contributor who contributed the questionable passages by examining the page's history, and inviting him to provide references, etc. Tkuvho (talk) 15:00, 4 November 2014 (UTC)

@Tkuvho:The section has been there in a similar state for years and is almost totally unsourced, and about every third sentence is dubious. The editors who contributed this material may not be around anymore, but I absolutely encourage anybody who can to provide references. --Sammy1339 (talk) 15:51, 4 November 2014 (UTC)

Sammy1339's edit

Compare "A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, and so forth. A notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, the term number may refer to a symbol, a word, or a mathematical abstraction."

with

"A number is one of several related types of mathematical objects. The original examples are the natural numbers 1, 2, 3, and so forth, which are commonly used for counting and ordering. Among many other uses, two of the most important are measurement and the description of probability. Fractions are well-suited to these purposes. Numerals, which represent numbers symbolically, are often used for labels (as with telephone numbers) and codes (as with ISBNs). In common usage, the term number may refer to a symbol, a word, or a mathematical abstraction."

Rick Norwood (talk) 15:15, 5 November 2014 (UTC)

Lede

(edit conflict)

The rearrangement of the lead by Sammy1339 has been reverted by Rick Norwood, with edit summary rv Good faith edit. Number should be an entry level article. To rewrite it in a more abstract style is not an improvement.

I do not understand this edit summary, and I do not see any "more abstract style": the main difference between the two leads, lies in the order of the sentences. The advanced mathematics (real and complex numbers) have been pushed by Sammy toward the end of the lead, while the common usage of the word number and the basic arithmetic operations are moved toward the beginning. Both changes improve readability for the layman.

Beside this reordering, the main change that I have noted id the change of

A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, and so forth.

into

A number is one of several related types of mathematical objects. The original examples are the natural numbers 1, 2, 3, and so forth, which are commonly used for counting and ordering. Among many other uses, two of the most important are measurement and the description of probability. Fractions are well-suited to these purposes.

I do not see anything "more abstract" in the latter formulation. The difference lies only in the introduction of the fundamental idea that there are several kinds of numbers, and the mentions of fractions and probabilities. This is certainly not too abstract for the layman, as its suffices to open a newspaper encounter these two concepts. IMO, the mention in the lead of the existence of several types of numbers and the fact that fractions are numbers is fundamental. This could help to avoid confusions such the one that led to the strange discussion at Talk:Integer#Percentages.

In summary, I disagree with Rick, when he says that Sammy's edit is not an improvement: remaining at the same starting level, it better follows the guidelines of MOS:LEAD, by sorting the sentences by increasing level of technicality. Moreover, this is done by increasing the accuracy of the wordings. Therefore, I'll reinstall Sammy's lead, and suggest to discuss further improvements from this version. D.Lazard (talk) 14:55, 5 November 2014 (UTC)

While I was typing the section above, D. Lazard restored Sammy1339's version and added this section. I'm going to move the two versions here for ease of comparison.Rick Norwood (talk) 15:18, 5 November 2014 (UTC)

Version One

Compare "A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, and so forth. A notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, the term number may refer to a symbol, a word, or a mathematical abstraction."

Version Two

A number is one of several related types of mathematical objects. The original examples are the natural numbers 1, 2, 3, and so forth, which are commonly used for counting and ordering. Among many other uses, two of the most important are measurement and the description of probability. Fractions are well-suited to these purposes. Numerals, which represent numbers symbolically, are often used for labels (as with telephone numbers) and codes (as with ISBNs). In common usage, the term number may refer to a symbol, a word, or a mathematical abstraction.

Here is why I prefer Version One. It begins with a simple declarative sentence that tells the reader what they want to know. The first sentence of version two does not tell the reader anything. If the first sentence doesn't tell the reader anything, they are unlikely to read the second sentence. The only change in the second sentence is the use of 1 instead of 1. The problem with the third sentence in version two is that, so far, only natural numbers have been introduced, which are not used for measurement and probability. I know the sentence doesn't actually say 1, 2, 3 are used for measurement and probability, but one sentence following the other suggests a connection which is not intended. Version Two then says "fractions are well-suited to these purposes". If fractions really were well-suited to measurement and probability, then we should introduce them before the two examples of their use. Actually, real numbers are better suited than fractions, especially for measurement. No problem with the other two sentences.

I'll pause here for comments. Rick Norwood (talk) 16:07, 5 November 2014 (UTC)

I too prefer the first version. It's the only version that agrees with the guidelines for the first sentence, that it should wherever possible say what the subject is. So "A number is a..." should be how it starts whatever the definition that follows. It is less technical, mentioning natural numbers and numeral but no other mathematical concepts, while the second version adds fractions and probability, not only more technical but unnecessarily specific – numbers appear in almost all areas of maths and singling out those two is odd. Also stating that probability (with measurement) is important is a value judgement which seems even more dubious – one could argue that using numbers in engineering and physics is as important, or using them in arithmetic is even more so.--JohnBlackburne^words_deeds 16:57, 5 November 2014 (UTC)

I would like the first sentence to say what the subject is, too, but there is no brief definition. The first line in the first version is simply wrong. Real numbers are not used for counting, and complex numbers are not generally used for measuring or labeling, neither are any of these things really defining properties of numbers. So I changed it for the sake of accuracy. --Sammy1339 (talk) 20:38, 5 November 2014 (UTC)

There's nothing wrong with it as a brief definition. You use numbers to count, measure and label many things. They are used for other things too, but those are perhaps the most obvious and familiar applications. These different applications often require different numbers; natural numbers for counting, real numbers for most measurements, etc. But they're all numbers.--JohnBlackburne^words_deeds 21:01, 5 November 2014 (UTC)

But complex numbers are not for any of those things, so it's not a definition so much as a selection of things some numbers are used for, which is why it doesn't belong in the first sentence where it is likely to be misinterpreted as defining the concept of a number. --Sammy1339 (talk) 21:22, 5 November 2014 (UTC)

Complex numbers are used for measuring. Electrical impedance is one example. In any case, the first sentence in Version Two is empty of content, since almost any word can replace "numbers". An integral is one of several related types of mathematical objects. A ring is one of several related types of mathematical objects. A function is one of several related types of mathematical objects. The first sentence should tell the reader something they want to know.Rick Norwood (talk) 22:03, 5 November 2014 (UTC)

Well, various physical quantities are represented by complex numbers, but these are aggregated from real data (like resistance and reactance in your example, or phase and probability amplitude for a particle in quantum mechanics). It seems to me that the results of measurements are always real, but even that might be a little naive. We can probably have a philosophical discussion about what measurement is. More to the point though, the issue of whether numbers can be used in this way says nothing about what they are - the original first sentence was hardly descriptive. If I wanted to start with a definition, I would say "A number is an element of the complex number field," and obviously that is too technical, besides still being incomplete. I can't think of a good way to express what a number is to a layperson in less than two or three paragraphs. --Sammy1339 (talk) 22:48, 5 November 2014 (UTC)

After having read above comments, my opinion is that both versions have some advantages. Also, both versions omit the most important fact about the context of the topic: numbers are at the basis of almost all mathematics. Thus, I have tried to take everything into account, and this leads to the following proposition for the first paragraph.

Numbers are mathematical objects that are at the basis of almost all mathematics, and, among many uses, are used to count and measure. The original examples are the natural numbers 1, 2, 3, and so forth, which are well suited for counting. For measuring, one generally use rational numbers, which are commonly represented by fractions. A notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, the term number may refer to a symbol, a word, or a mathematical abstraction.

D.Lazard (talk) 23:41, 5 November 2014 (UTC)

Sorry if I seem contrarian, but I definitely can't agree with the statement that "numbers are the basis of almost all mathematics." Even in the earliest days they were not needed for synthetic geometry, which made up the better part of higher mathematics until the Renaissance, and today they have only a minor role if any in most of abstract algebra, most of topology, most of mathematical logic ... in fact I've rarely heard the word "number" in a math talk. They're everywhere in analysis of course, but even then, they can easily be replaced by more general constructions; it's just that most analysts don't care about that sort of thing. Anyway the super-specialness of numbers is a popular misconception I definitely don't want this page to perpetuate. It's enough to say that numbers have huge importance in mathematics - but that's not a defining feature either. --Sammy1339 (talk) 00:38, 6 November 2014 (UTC)

Trying to resolve this issue I looked at the 1911 Britannica definition, which unfortunately is not very descriptive either: "a word generally expressive of quantity, the fundamental meaning of which leads on analysis to some of the most difficult problems of higher mathematics." And still not too accurate - are complex numbers expressive of quantity? Are negative numbers, even? I agree with the concern that the first sentence in the current version is rather empty and I'm really trying to think of a way to improve it, but the term simply might not admit a succinct definition, as apparently was noticed 100 years ago. --Sammy1339 (talk) 01:16, 6 November 2014 (UTC)

Now we have three proposals. I'll call D.Lazard's proposal Version Three. I like it better than Two, less than One. It has the virtue of saying something in the first sentence to the people who visit this page and are not mathematicians. I agree with Sammy1339 that "the basis of almost all mathematics" is an overstatement. Euclid proved theorems about numbers, but they were not as basic to Euclid's Elements as points and lines. And choosing the rational numbers instead of the real numbers for measurement is non-standard. Is the square root of two not the measure of the hypotenuse of a right triangle whose base and height are one? Is pi not the measurement of the ratio of circumference to diameter?

It seems to me that all the things we call "numbers" are either subsets or extensions of the real numbers, but that doesn't really solve our problem. We seem to be at an impasse. I hope more people weigh in on this important question.Rick Norwood (talk) 01:19, 6 November 2014 (UTC)

My rationale for naming fractions instead of real numbers as "suitable for measurement" is that actual measurements are approximate, leaving the reals no practical advantage. (i.e. I meant measurement in the sense of rulers, not in the sense of Lebesgue.) The purpose of the reals is doing calculus, but that's a rather abstract place to start. The definition of numbers, of course, is the second-to-last paragraph. It could be moved to the top, but then this conflicts with the convention of proceeding from more concrete to more abstract. I'm also a little concerned about giving top billing to "number" as understood in pure mathematics, a rare and perverse occupation which shouldn't be allowed to dominate a concept that rightly belongs to everybody. --Sammy1339 (talk) 01:34, 6 November 2014 (UTC)

Oh and by the way, p-adic numbers are neither subsets nor extensions of the reals, but per my comment above perhaps nobody should care. Anyway "real number" is much too esoteric the first sentence anyhow - most undergrads don't know what it means. --Sammy1339 (talk) 01:40, 6 November 2014 (UTC)

Rick Norwood I agree with your ordering: three is an improvement on two but I still prefer one. There's nothing incorrect in three but I much prefer one's straightforward and direct definition. Sure it leaves out all sorts of mathematical detail and subtlety but that's not needed in the first sentence, which has to be readily understood by all levels of readership, i.e. all ages and abilities.--JohnBlackburne^words_deeds 03:59, 6 November 2014 (UTC)

I had hoped others would weigh in, but that hasn't happened, so essentially we have three versions. Two only have the support of the person who wrote them. Version One is older and has two supporters. I propose we go back to Version One, at least until we get some definite consensus. Rick Norwood (talk) 22:54, 7 November 2014 (UTC)

I posted what I felt was a compromise (I still prefer the current version to that one) but JohnBlackburne reverted it. Personally I'd rather move on and revisit this some other time. --Sammy1339 (talk) 23:27, 7 November 2014 (UTC)

While we're trying to decide among three versions it isn't really helpful to modify it to yet another version without discussion. Per Rick Norwood the only version there's any consensus over is the first one, which was the original. so we should return to that, until we arrive at consensus on a new version.--JohnBlackburne^words_deeds 01:39, 8 November 2014 (UTC)

Good.Rick Norwood (talk) 13:17, 8 November 2014 (UTC)

archiving the talk page

Resolved

The posts on this page go back to 2007. I know there is a way to archive the older ones, but I don't know what it is. If anyone knows how to do this, or how to set up auto-archiving of threads that have not changed in more than a year, it would probably be an improvement. Rick Norwood (talk) 13:13, 11 December 2014 (UTC)

• Hey Rick, you are right. I've set up archiving and archived some of the oldest messages into the first archive to it's maxarchivesize. A bot will come clean up the rest of them that are more than a year old in 24-48 hours. It can be adjusted after that if the page is still too long. Happy editing! — {{U|Technical 13}} ^{(e • t • c)} 15:00, 11 December 2014 (UTC)

Suggested correction

English language 66.87.82.174 (talk) 01:36, 27 January 2015 (UTC)