# Talk:Number

Number was a Mathematics good articles nominee, but did not meet the good article criteria at the time. There are suggestions below for improving the article. Once these issues have been addressed, the article can be renominated. Editors may also seek a reassessment of the decision if they believe there was a mistake.
Article milestones
Date Process Result
July 9, 2010 Peer review Reviewed
July 21, 2010 Good article nominee Not listed
Current status: Former good article nominee

## ?Problems with i ?

Just when I thought it was save in the lee of Euler ...

Evidently, there are problems with the roots. Already square roots, as defined with the reals must be cropped to make sound functions, leaving the trace of ambiguity wrt +/- as roots for equations, they do not exist there for negative arguments, yes, and they invite to antinomies, as elaborated in the article, if not handled with utmost care. I really do not get it, where there might be problems with my edits, besides with wikipedians. :) -Purgy (talk) 13:45, 28 November 2016 (UTC)

Ah, I see, my arguments are not even heard, but simply twinkled away. Talk pages are only for the lower classes, obviously mislead lines are ignored, and sensible remedies (dis-)qualified as personal opinion. I revert one last time, and repeat my petition to discuss here, and not via apodictic edit comments. -Purgy (talk) 14:04, 28 November 2016 (UTC)

I want to explicitely confirm that avoiding the misleading sqrt(-1) really is an improvement, beyond personal opinion. It definitely takes the complex numbers first, before one can talk about complex (square) roots. -Purgy (talk) 14:10, 28 November 2016 (UTC)

(edit conflict) Please read WP:BRD. I agree that some of the sentences that you have edited need improvement. But, in most cases, you replace a badly written sentence by something that is worse or misleading or containing a personal POV. As several independent sentences are concerned by the same edit, it is too time consuming for other editors to improve your edits, and the only solution, recommended by WP:BRD, is too keep the old version until a consensus is obtained by discussing here. Thus please, explain the motivation of your edit. D.Lazard (talk) 14:25, 28 November 2016 (UTC)
I think the article would be better off with the sqrt(-1) in the lead removed okay. It would be much better to make a clear case for what you think is actually wrong rather than attacking people for reverting your badly phrased English. I do not think this is an article where 'extreme care' in dealing with the mathematics would be helpful, it should be more of a popular article and made easy to read, though if one can avoid problems so much the better. I think something like '..., and complex numbers, which extend the real numbers by including the imaginary unit ${\displaystyle i}$ such that its square ${\displaystyle i^{2}=-1}$." might do the trick and I'll try putting that in. Dmcq (talk) 17:18, 28 November 2016 (UTC)
It never came to mind that would have attacked someone by my phrasings, but rather -not righteously!- felt myself attacked by D.Lazard's bureaucratic triptychon of "worse, misleading, PPOV", while I had already presented my arguments before. I want to apologize for any offense, and for not being able to make my case clear, which I -unintentionally!- caused by my suboptimal use of the English language. BTW, I was firstly reverted by Paul August for "problematic" content, not inapt language use.
On the case itself, it appears evident to me that you share at least a slight preference for i² = -1 with me, and therefore I want to hint to the rather disputable introduction of a "formal square root", which replaced my "squares to -1" on one occasion. Is "to square to" really that bad a verb? I heard it on many occasions in lectures.
I share the opinion that this is not an article where unconditional math rigidity were a sine qua non, but -as said- the sqrt(-1) is -to me evidently, and as demonstrated in the article itself- more capable of leading to misconceptions, than the offered alternative, even when not that much wide spread in all communities. Under the given circumstances I do not consider it very fruitful to discuss the respective advantages and flaws. Just let me know if otherwise. I consider this a quite readable article with interspersed nice remarks for an interested non-professional.
For the time being I cannot imagine that i² = -1 makes the article harder to read than in the sqrt-variant, but I will refrain from editing in this improvement. -Purgy (talk) 09:28, 29 November 2016 (UTC)
I agree that it is better to avoid, as far as possible, to write ${\displaystyle {\sqrt {-1}}.}$ However we must remember that this article is intended for readers, that may know nothing about algebra, equations, factorization of polynomials (introduced in one of your edits). For a beginner, and historically, complex number have been introduced for making always possible the arithmetic operation of computing square roots, exactly as integers and rational numbers were introduced for making always possible subtraction and division. From this point of view, the old version with ${\displaystyle {\sqrt {-1}}}$ is better than the version of Dmcq: "which extend the real numbers by including the imaginary unit ${\displaystyle i}$ such that its square ${\displaystyle i^{2}=-1}$". Therefore, I'll replace it by "which extend the real numbers by adding a square root of −1". This avoids both ${\displaystyle {\sqrt {-1}}}$ and unnecessary technicalities. D.Lazard (talk) 10:51, 29 November 2016 (UTC)
I most humbly apologize in advance, but it is simply not true that I introduced the notion "factorization of polynomials" into this article. To the contrary, my effort was to connect the original occurrence of this term more obviously to the notion of "algebraic closure", which caused the original mentioning of factorization. Imho, connecting "not algebraically closed" with missing solutions of algebraic equations (I introduced this!), is pedagogically more straight forward, than immediately associating it with factorizability (the original occurrence).
It is in some pervert way amusing that all three people involved in contributing here, agree on some notation to be best avoided, but nevertheless, the one, who claims this most misleading view especially for this here article, supersedes the two others and forces some traditional High School level(?) view into the article. There are no reasons made explicit, why newbies would understand some rubbish better than correct, equally simple definitions. Paul August, who started this dispute by considering my edits not "badly phrased", but "problematic"(sic!) did not comment on this at all, yet. An other remarkable haut gout of this debate is the mother tongue of the ("sqrt of -1" instead of "sqrt(-1)")-advocate: FRENCH! The french mathematicians, world famous for their exquisite formal power and access to math, rotate in their respective graves, when reading about "formal square roots" to introduce complex numbers, and the claim i := sqrt(-1) were "meaningful" in beginners' education and "easier to grasp" than i² := -1.
This article was once proposed for featuring it calling it good, now it is under some neighbourhood watch. -Purgy (talk) 09:18, 30 November 2016 (UTC)
Please stop the personal attacks mixed with apologies and stick to the subject matter.
If a student sees i^2=-1 I don't see why they wouldn't immediately turn that into sqrt(-1) in their minds so whilst I can see there are problems with sqrt(-1) I don't see that avoiding any mention of it will fix anything. As to High School level, the obvious question is why is closure so important and useful that one adds unreal quantities, isn't it like adding the inverse of zero? So I don';t see a reason why the idea of closure should be added before sqrt(-1) in an article about numbers. Dmcq (talk) 10:02, 30 November 2016 (UTC)
OK, no more efforts to signal a cooperative mindset:
• Please, stop that libel and slander by repeatedly, unsourcedly accusing me of attacking persons.
• I strongly claim that the reverted formulations in the article are certainly not below the linguistic average in Wikipedia (I do not refer here to my angered texts in the talks), and do not deserve the standard rebuffing verdict "badly phrased".
• As one can easily verify, my starting statement rotates about arguments against sqrt(-1) and pro i²=-1. That the consensus, the latter being the better introduction, nevertheless lead to rubbish (formal square root!) prevailing, without any reflexions on why not change, naturally and necessarily drew my replies to the embedding environment.
• Even when senselessly repeated, it is not true that I introduced the term "algebraic completeness" in this article.
I want to avoid looking like trolling against the petrified Wikipedia establishment (too much rock), so I will stop commenting here (until further notice, at my discretion). Nevertheless, I really(!) cordially invite anyone to my talkpage for an exchange on a fair footing. -Purgy (talk) 10:48, 1 December 2016 (UTC)
I am sorry you take that attitude. I will not interact with you except where necessary unless you confine your comments to the subject matter. Dmcq (talk) 13:05, 1 December 2016 (UTC)

────────────────────────────────────────────────────────────────────────────────────────────────────Since you're sorry, take this! Nothing is new in the biggest part, all is compiled from above:

Just when I thought it was save in the lee of Euler ... (to throw in a reputable source)
Arguments against roots:
- as defined with the reals, (they) must be cropped to make sound functions (doing awy with the lower branch of the relation)
- leave a trace of ambiguity wrt +/- (whereever they appear)
- they do not exist within the reals for negative arguments (why should they for i ?)
- invite to antinomies, if not handled with utmost care. (as elaborated in the article)
- It definitely takes the complex numbers first, before one can talk about complex (square) roots

Arguments for introducing i² = -1
- avoiding the misleading sqrt(-1) really is already an improvement (squaring is a well behaved function)
- this sensible remedy (is)(dis-)qualified as "personal opinion" (as defence, appealing to Euler above)
Questioning the reasons for reverting (is this already considered inappropriate???)
- Where are the (postulated) problems with my edits?
---
All the above was written even before(!) you tried to insert the i² = -1
---
Immediately afterwards I
- critisized the rather disputable introduction of a "formal square root" (not defined for reals, still not applicable to complex numbers)
- stated agreement to certain views (I share the opinion ...) on the article,
- reinforced the above mentioned drawbacks of the roots-formulation and
- documented my inability to perceive any increase of conceptual hardship with the i² = -1 variant, which is just postulated without any supporting argument.
---
As something new I add in reply to your opinion on students, that I certainly would not require students to forgo a (to be introduced) complex square root, but I would not squeeze in inappropriate tools, like the real square root, to fit in an undefined environment. I prefer to admit to the reader that at the stage of introducing i, the concept of a complex root is not yet formalized. I prefer the caveat to the "trust me"-promises, and the complex root is fundamentally different to the real root in some way.

BTW, how to deal with the untenable claim that I had introduced the notions of "factorizing of polynomials" and of "algebraic closure" in the article?

Besides some timing reference, and my personal need for sincerity, this is strictly on the subject matter, at least imho. -Purgy (talk) 10:22, 2 December 2016 (UTC)

I just did a Google of "how should complex numbers be introduced" and got some interesting results. There seemed to be no great preference for i or sqrt first in the individual introductions, but do tend to show both i^2+1=0 or i^2=-1 and sqrt(-1) early on, so I think the most relevant one I can point to for ideas about this is at stackexchange How can I introduce complex numbers to precalculus students? plus some of the links it references. I tried also "How best to define a complex number" and got a different set. Wikipedia is not supposed to be a text book and is more on the define side though it is not a dictionary either. I do think there is a preference for throwing a couple of definitions and examples at people rather than trying to give just a single definition and exclude others. Dmcq (talk) 11:10, 2 December 2016 (UTC)
Since it does not seem to be over yet (please, see below), I ask for your appreciation of me withholding my comments to your search for the time being. -Purgy (talk) 17:05, 2 December 2016 (UTC)
OMG!, can't you be satisfied that your consensually deprecated version still dominates the article, and for whom might it be necessary to give a line of evidence that I'm correct, and you insist on battling?
Looking a few paragraphs below my partially reverted edit you will find the sentence:
In abstract algebra, the complex numbers are an example of an algebraically closed field, meaning hat every polynomial with complex coefficients can be factored into linear factors.

I do not know when this sentence was introduced, but it definitely takes precedence to my edit, in which I tried to put the also already existing sentence
The real numbers are not, however, an algebraically closed field, because they do not include the square root of minus one.

in a better formal interrelation, closing a chain of notions from "alg. closed" -> "missing solution" -> "factoring of polynomials" -> "alg. closed" by editing it to:
The real numbers are not, however, an algebraically closed field, because they do not include a solution to the algebraic equation x²+1=0, often addressed as the square root of minus one, which would allow for a factorization of the associated polynomial in linear factors.

Please note, how delicately I tried to edit, and still, I never said a single word against the reversion of these -imho very well explainable and explained(!)- edits, even when I strongly disagree with the given arguments for the reversion, but I protested and I still protest against the claim that I had introduced the notions of factorization and closure into this article.
Therefore,
- my claims hold wrt "factorization of polynomials (introduced in one of your edits)" by D. Lazard, and wrt "I don't see a reason why the idea of closure should be added" by Dmcq.
- I never disputed having introduced the term "solution to the algebraic equation"
- I never "complained about inexistent accusations", so I cannot stop this
I do, however, strongly doubt that there are many readers of this article, "who know nothing about algebra, equations, ...." There may well be many readers who have not heard about Gauß Fundamental Theorem, and about "closure" in general context, but as said already, I consider these remarks as little highlights, not impeding it's readability, but generating curiosity.
I also wonder why a concept, consensually deprecated, still subjugates the article, and why the obvious flaw of "formal square root" is not considered as detrimental, both to Wikipedia's reputation, as to readers' possible gain from this article, and why this is therefore not reverted. -Purgy (talk) 16:43, 2 December 2016 (UTC)

## Skimming through stackexchange

When looking at the efforts there, I could not destill that i²=-1 and i=sqrt(-1) were treated on equal footing, not even close. There is one particularly nice article, hailed to more than 700 upvotes, which (also!) does not mention the sqrt in introducing the complex numbers. The whole collection of remarks is bantered by the laudable search for a most visually accessible introduction of the complex numbers. Obviously this lead to representing them in the plane, with multiplcation as rotating and stretching, identifying the i-fold as a 90°-rotation, not in the minimum relying on sqrts. What a nice coincidence when heuristics and pure math intentions meet in one view : i²=-1 !

It would be ridiculous to claim that i²=-1 would not call for the question about a square-root-function in the complex domain, applicable even to -1, strongly disallowed with the real square root, but I consider the answer to this to be a welcome consequence of introducing i via its square, and not the rhizom of its introduction. Up to now I missed the chance to get to know why a sqrt-introduction of i could be seen as methodologically, didactically, or mathematically advantageous.

Looking a bit beyond the currrent horizon, at the construction of the quaternions, there the famous Hamilton carvings i²=j²=k²=-1 show up, documenting a principle of construction. Is this a reason for hiding i²=-1 behind i=sqrt(-1)?

I cannot assume that sufficient qualified reliable sources do not prefer the i²=-1, I have not seen any evidence of disadvantages, nor of any advantage of the sqrt, I even meant to perceive consensus on the sqrt being problematic, so, please, why is there still this strange claim alive that the number-article must preserve the sqrt-introduction, and keep up unsourced constructions of "formal square roots", and what are the "problems" announced with the first revert of my edits here. -Purgy (talk) 11:21, 6 December 2016 (UTC)

What was upvoted more than 700 times? You're not confusing with bronze badges are you? Please be specific about what you are referring to rather than expecting me to search around trying to figure it out from the long list of possibilities. Referring to specific instances to illustrate what you are saying would be very helpful. Dmcq (talk) 12:14, 6 December 2016 (UTC)
Following the first link in your reference to here, and scrolling to the second(?) entry, leads to the mentioned article, which I assume to have this number of upvotes. Sorry, I do not know that much about stackexchange to be aware of bronze badges, their looks and places and meanings.
I'm certainly prepared to show you all the ways to places I know, and, honestly, I expected you had been there, already, and I certainly did not expect, that you would search for it. May I, in exchange, also ask for places, offering answers to my questions for preferring the sqrt-introduction, and to be more specific about the items I should illustrate? -Purgy (talk) 14:22, 6 December 2016 (UTC)

NO! I am not going to expose myself any further to a never ending story of offense and humiliation, just for trying to be allowed to discuss improvements of this article. I declare that I am prepared and willing to try to contribute to an improvement of this article, and discuss this with everyone (really everyone!), who is willing to do so on equal footing (not just: it's me, who's right), but not as a petitioner, being talked down and rebuffed for intransparent reasons. Recall, that I still wait for being given a rational reason, why my primary edits were considered problematic, and therefore reverted first hand. -Purgy (talk) 13:10, 7 December 2016 (UTC)

## Number article names

The articles about the numbers from 0 to 9 are called 0 to 9. However, the article about the number 10 is called 10 (number). I believe that it should be called 10. Is it possible to change the name of the article? Bobby Jacobs (talk) 23:25, 5 March 2017 (UTC)

I noticed that 10 (number) was just moved to 10. However, there are more number articles that should be moved. Bobby Jacobs (talk) 15:05, 8 March 2017 (UTC)
Just for reference: there was an RFC about the titles of these articles, see Talk:AD 1/Archive 1#RFC: Should articles "1" to "100" be about numbers instead of years?. -- Tea2min (talk) 15:12, 8 March 2017 (UTC)

## Irrational Numbers

Why are irrational numbers not included in the 2nd paragraph. The phrases "In mathematics, the notion of number has been extended over the centuries to include 0, negative numbers, rational numbers such as 1/2 and −2/3, real numbers such as √2 and π," implies that the rational numbers are the real numbers whereas the real numbers include the rational numbers, the irrational numbers, the natural numbers, the negative integers and 0. Ruskin (talk) 12:17, 3 April 2017 (UTC)

To include the irrational numbers in that sentence you would need to answer the question–what are the irrational numbers an extension of? Since there is no answer to this question, the irrationals don't belong in a discussion about extending number sets. To put this into perspective consider this. There are rational numbers which are not integers, yet you are not claiming that they should be included as a separate set of numbers in this progression. --Bill Cherowitzo (talk) 17:48, 3 April 2017 (UTC)

## Help on edits of History

Could someone knowledgeable, please, check the authenticity and the removal of a picture by edits starting here? Purgy (talk) 12:18, 12 April 2017 (UTC)

The main changes I see were first to combine two paragraphs on Bhramagupta, and second the (I would guess accidental) deletion of a valuable image. Earlier, someone deleted the whole section on zero, but that was quickly reverted. I think the section looks ok now, but I'll reread it and see if I catch anything. Rick Norwood (talk) 12:31, 12 April 2017 (UTC)

## Strong reservations

I want to express distinct objection to the last edits by Gireen, at least as far as content beyond references (which I cannot judge) is concerned. In my opinion the statements of what is defined as what require revision, and especially the claim of "irrationals" being "decimals" is flawed. Perhaps some guardian angels should have a look at this. Purgy (talk) 05:46, 23 May 2017 (UTC)

Specifically on the edit to the irrationals section, I think you read my revision incorrectly. The (cited) definition I gave for irrational numbers was "Real numbers that cannot be represented as the fraction ${\displaystyle a \over b}$ for any integers ${\displaystyle a}$ and ${\displaystyle b}$". I then, later in the paragraph, said that "A decimal represents an irrational number if and only if it has an infinite number of digits and does not eventually repeat", which is not only true, it can be logically derived from the sentence you reverted the document back to ("A decimal represents a rational number if and only if it has a finite number of digits or eventually repeats forever, after any initial finite string of digits"). All I did was turn the sentences from "P is sufficient for Q" to "¬P is necessary for ¬Q". If you have a big problem with that statement you should consider editing the current document aswell, because it says the same thing.
Regardless, what else did you see that you disagreed with, so I can amend the original edit?— Preceding unsigned comment added by Gireen (talk (talkcontribs) 01:14, 24 May 2017 (UTC)
I agree with Purgy Purgatorio, and I have reverted your edit. Feel free to reinsert the references that you have added, if they are relevant for the present state of the article. Here are the detailed reasons of my revert:
• Natural numbers ... are formally defined ...: Normally, the normal user of this article ignore the meaning of "formally defined". Moreover, you introduce a circular definition, as whole numbers are usually defined from natural numbers.
• Proof in section "Rational numbers"": useless here, as this section is a summary of another article, and confusing, as possibly hiding the main points of this summary
• Section "Real numbers": circular definition again, the reals are defined in terms of rationals and irrationals, and later the irrationals are defined in terms of reals and rationals.
D.Lazard (talk) 06:52, 24 May 2017 (UTC)
Edit conflict, perhaps some duplications contained:
Line 52: Not that I am a fan of " most familiar numbers", but "formally defined as as the set of ... whole numbers" without "formal" definition of whole numbers is definitely worse to me. (In my world the whole numbers are constructed from the naturals.) Removing the parens around "sometimes called ... " gives even more, instead of less importance to a deprecated naming, assumedly perpetuated in elementary school. Supressing a "reason" for including the 0, (strangely) imported from a group structure and not inherent to the naturals per se, is imho no improvement to this article.
Line 81: Your footnote mistakes the "definition" of "equality of rationals" for a theorem in need of a proof.
Line 91: Assuming the "irrationals" to get a hold on the reals by combining them with the rationals is circular reasoning, therefore I object to this edit. Again, "measuring numbers" are not my particular favourites, but they do not pretend to be "formally" defined. You are right that the propositions about non-rationals and irrationals within the reals are equivalent, but even though I did not oppose to this very edit, I do not consider it an improvement. "Irrational" is more of a naming convention but a useful definition, they are a continuum and there are many variants to be irrational ;) . I deny the a priori ability of decimals for "having infinitely many digits". This requires considering limits and gets deeply involved in the definition of reals. I strongly plead for restricting all these didactic troubles associated with the equality 1=0.999... within this rather elementary article to an absolute minimum. I consider "infinitely repeating" as less embarrassing than "infinitely many digits". Decimals are best left outside of abstract considerations, imho.
Line 102: The deleted sentence is correct and more appropriate than referring to non-repeating or infinitely many decimals.
I hope you take from this that it is tedious to selectively revert and to discuss ex post some perhaps inappropriately bold edits, and that it is advisable to discuss such things in advance. Finally, I want to remark that the reversion by D.Lazard is certainly not based on personal amity.
Please, do not amend your original edits, they are refuted. Purgy (talk) 08:28, 24 May 2017 (UTC)

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