Talk:Complex number/Draft: Difference between revisions

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== Definition ==
I reworked the text a bit. So this is why.
* I think <math>\sqrt{-1}</math> is an objectionable notation...
* The definition hardly matches my understanding...  The imaginary unit can be really understood only within the field of complex numbers (defined independently). Otherwise, what is "i"? A square root of (-1)? Then which one? (there are usually two square roots; BTW, have you ever seen an independent definition of a square root of a negative number?). So let's define it by "i^2=1".  Then, does it ''exist''? Does it ''deserve'' to be called a number? (operations are possible?)  The same question arise if we define "i" as a solution of  "x^2+1=0". In practice we can use any of these well known ''properties'', but how can we understand it as a ''definition''?
At best, we can say "i" is "just a formal symbol" with no meaning. We define some operations on formal sums "a+bi". Basically, that's OK. The point is that it explains nothing and it can be done in a more elegant way, where we really define all is needed in terms of elementary well-known objects:


Complex numbers are just ''[[ordered pair]]s of reals'' -as simple as this - with appropriate addition and multiplication. BTW, these operations are enlisted in the article with the "formal" use of "i".
Then i=(0,1). And for computational convenience we discover that i^2=-1, and use it.


:I think your revision is a good one. I had considered using the term "formal expression" for <math>a + bi</math>, but decided not to. But, in truth, I didn't spend a great deal of time on this. It just seemed an obvious omission, giving that there was already an article on real numbers! A possible revision/addition I had considered was adding a section on how the definition can be formalized by saying <math>\mathbb{C}</math> is the splitting field of <math>x^2 + 1</math> over <math>\mathbb{R}</math>. Without context, though, that seems like a bit of overkill. Of course, it's formally the same as the definition of algebraic number fields such as <math>\mathbb{Q}[\sqrt{-1}]</math> or <math>\mathbb{Q}[\sqrt{2}]</math>. But I suppose that's a topic for another article. [[User:Greg Woodhouse|Greg Woodhouse]] 06:21, 2 April 2007 (CDT)
==APPROVED Version 1.0==
<div class="usermessage plainlinks">Discussion for [http://en.citizendium.org/wiki?title=Complex_number&oldid=100097396 Version 1] stopped here. Please continue further discussion under this break. </div>


The bottom line is that I do not object use of "i" in the informal intro, just to give an outline of the idea, there must be, however, a definition that really explains where it logically comes from.
== See comments above ==
--[[User:Aleksander Stos|AlekStos]] 03:01, 2 April 2007 (CDT)


Call me tempramental, but I reworked the opening paragraph a bit. I hope it hasn't changed substantively, but I think the new text flows a bit better with the rest of the article. [[User:Greg Woodhouse|Greg Woodhouse]] 16:27, 3 April 2007 (CDT)
I made some comments in the section [[Talk:Complex number/Draft#What the symbol i means in this article]] above, marked with bullet points, which have not yet been addressed. (I just wanted to mention them after the page break so they won't be forgotten when editing the next version.) --[[User:Catherine Woodgold|Catherine Woodgold]] 20:28, 6 May 2007 (CDT)


== sketch of a plan ==
:I edited in all of the changes I had suggested except for the stuff about quantum physics.  I don't have a textbook on the subject handy.  --[[User:Catherine Woodgold|Catherine Woodgold]] 19:17, 7 May 2007 (CDT)


The status of the <math>\sqrt{-1}</math> notation seems to vary according to different cultures. In French high schools and colleges, it tends to be a taboo, because of the objections pointed out by Alek Stos here. I have heard its usage is far more common in English speaking countries. The problem is that there is a canonical way to choose which square root of a positive real number <math>x</math> we call <math>\sqrt{x}</math> (the positive one), but there is not such a canonical way to choose amongst the two square roots of -1. ''Once'' <math>\mathbb{C}</math> is defined, one can choose some convention, but still a determination of the square root over the complex plane cannot be continuous everywhere. On the other hand, using <math>\sqrt{-1}</math> in an informal way just because it is easy to understand what is meant by it can be defended, as soon as one is warned of not considering it as anything else but a mere notation. As Greg Woodhouse recalls to us, this notation is quite common for algebraic number theory specialists, to denote some quadratic fields. I still think it is a bit dangerous to use it without comment for beginner readers.
I'm not entirely happy with my text in the QM section, either. Trying to interpret superposed states in terms of probabilities is dicey at best, anyway. I'll have to think about this and see if I can come up with something better. In any case, I'm intrigued by what Robert Tito had to say about other uses of complex numbers, particularly in Hamiltonian systems (conjugate coordinates with a factor of i?) Anyway, I was just trying to come up with something that would be recognizable to a wide range of readers (albeit not mathematically naïve ones). If nothing else, the Schrödinger Equation has a certain iconic value. I'm certainly open to other suggestions. [[User:Greg Woodhouse|Greg Woodhouse]] 19:38, 7 May 2007 (CDT)


Now I come to a (somewhat vague) suggestion of structure for the article. I like to introduce complex numbers to my students with the example of the resolution of the cubic equation <math>x^3=15x+4</math> with the so called [[Gerolamo Cardano]]'s method (in fact it is due to [[Scipione del Ferro]] and [[Niccolò Tartaglia]]). Computations are quite easy, and the striking fact is that during them, one has to use some imaginary number which square would be -1, but once the computations are finished, one gets the three real solutions of the equation! At this stage, one can denote the mysterious number by <math>\sqrt{-1}</math>, as we make anyway only purely formal calculations without giving any legitimate sense to them. They just suggest there might be ''something'' which square is -1.
:Wait!  I think the quantum physics stuff is good!  It just needs some editing, as I suggested, e.g. defining the symbols used etc.  --[[User:Catherine Woodgold|Catherine Woodgold]] 19:42, 7 May 2007 (CDT)


Next we need a model to legitimate this mysterious number, and then, Alek Stos's suggestion is best : considering that <math>\mathbb{C}</math> is <math>\mathbb{R}^2</math> with appropriate addition and multiplication laws is the more elementary way to construct complex numbers. Here we can introduce the "i" notation. Moreover, this allows to have a geometric representation of those counterintuitive numbers, with the complex plane. It is still possible to link this with history : the geometrical viewpoint is due to [[Robert Argand]], and the complete construction was achieved by the great [[Carl Friedrich Gauss]]. This section may not only show how complex numbers can be illustrated by geometry, but show too how, reversely, plane geometrical problems can be solved with the power of calculation with complex numbers.
It seems to me that saying things like "<math>\hbar</math> is Planck's constant divided by <math>2\pi</math>" wouldn't really add anything to the article, and I guess that's what bothers me: if you (generic) know what Schrödinger's equation is, this probably doesn't need to be said, and if you don't, the section really doesn't add anything. [[User:Greg Woodhouse|Greg Woodhouse]] 22:40, 7 May 2007 (CDT)


Then, another section may deal with a more abstract point of view, that is <math>\mathbb{C}=\mathbb{R}[X]/\left(X^2+1\right)</math>, and more generally, introduce the notions of [[splitting field]]s, [[algebraic closure]] and so on: thats seems to be Greg Woodhouse's idea. Only an introduction, but it has a legitimate place in our article I think.
== Part of chemistry and physics workgroups? ==


Finally, some applications of complex numbers must be cited : a few words about [[complex analysis]] and [[holomorphic function]]s, etc. Separate articles are needed for the details of course. It also may be emphasized in the applications part than those seemingly purely abstract numbers are very useful in physics.
(I hope no one minds if I move this discussion "below the bar". [[User:Greg Woodhouse|Greg Woodhouse]] 03:34, 7 May 2007 (CDT))


What I like in this sketch of plan for this article is that it begins with a simple, intuitive but not properly formalized idea to end with more precise and more subtle aspects of the theory. Also, I think it is important in this article to stress the historical evolution of the ontological view of complex numbers (how they were little by little accepted from mere calculation artifices to true numbers). Please let me know your opinion. If you think it is a good idea, I can write the cubic equation part quite soon. But if you have better ideas, please share them!
I am just curious why this article's checklist includes it in the chemistry and physics workgroups. It seems that even though this article has applications in those field, including it in every workgroup it applies could get out of hand. - [[User:Jared Grubb|Jared Grubb]] 12:32, 6 May 2007 (CDT)
:that answer is simple: the need for something like a complex number arose from these sciences not from math. Math formalized it, thats all. [[User:Robert Tito|Robert Tito]]&nbsp;|&nbsp;<span style="background:grey">&nbsp;<font color="yellow"><b>[[User talk:Robert Tito|Talk]]</b></font>&nbsp;</span>


--[[User:Sébastien_Moulin|Sébastien Moulin]] <small>[[User_talk:Sébastien_Moulin|(talk me)]]</small> 11:21, 2 April 2007 (CDT)
I disagree. We have an example showing that complex numbers are important in the sciences, too, but complex numbers were introduced in a fundamental way in mathematics (i.e., not just as a notational convenience) long before quantum mechanics had even been thought of. [[User:Greg Woodhouse|Greg Woodhouse]] 16:59, 6 May 2007 (CDT)
: I guess I like the idea! --[[User:Aleksander Stos|AlekStos]] 14:52, 2 April 2007 (CDT)


That seems like an excellent suggestion. Of course, I am hardly qualified to write about the history of the use of complex numbers in mathematics. Writing about applications is a little easier, but it is somewhat difficult to come up with examples that are simultaneously convincing and accessible. Obvious examples of the use of complex numbers include Cauchy's theorem, properties of the Riemann zeta function, Hilbert spaces, quantum mechanics, none of which can be introduced to a non-specialist audience without some preparation. [[User:Greg Woodhouse|Greg Woodhouse]] 19:34, 2 April 2007 (CDT)
Then again, since those workgroups are there, maybe you can sign off on it, too. :-[[User:Greg Woodhouse|Greg Woodhouse]] 17:02, 6 May 2007 (CDT)


I just added an aside on mathematical notation that I hope will address some of the concerns raised here. [[User:Greg Woodhouse|Greg Woodhouse]] 23:07, 2 April 2007 (CDT)
Physics and chemistry used the notion of complex numbers as from the 18th century - when they needed them to describe things. Euler, Gauss, Fourier were not mathematicians but physicists/chemists that needed a solution for their math problems. the complex number by far didn't start with quantum mechanics. I might mention Hamiltonian mechanics as an example, or canonicals. [[User:Robert Tito|Robert Tito]]&nbsp;|&nbsp;<span style="background:grey">&nbsp;<font color="yellow"><b>[[User talk:Robert Tito|Talk]]</b></font>&nbsp;</span> 17:35, 6 May 2007 (CDT)
:You made good work. I wrote the introductory example about the equation <math>x^3=15x+4</math>. I do not know how well it fits with the other sections. Anyway, do not hesitate to modify my text to make it clearer if you like. --[[User:Sébastien_Moulin|Sébastien Moulin]] <small>[[User_talk:Sébastien_Moulin|(talk me)]]</small> 11:15, 4 April 2007 (CDT)


:I think that example is superb! I did rework the English a bit (I hope you don't mind). I also took th liberty (and I hope this wasn't the wrong thing to do) of changing ''X'' to ''x''. I understand the distinction you are making here, but I don't know if it's really necessary to introduce another symbol here. [[User:Greg Woodhouse|Greg Woodhouse]] 11:34, 4 April 2007 (CDT)
: It still seems a little odd to me. But, I suppose [[Ohm's law]] would be more at home in the electrical engineering workgroup than the mathematics, even though it is a mathematic equation... - [[User:Jared Grubb|Jared Grubb]] 23:14, 6 May 2007 (CDT)


Entertainingly written, good job so far --[[User:Larry Sanger|Larry Sanger]] 11:38, 4 April 2007 (CDT)
The historical development of the concept of complex numbers seems like an interesting topic for an article (albeit a challenging one!), but so far as ''this'' article is concerned, I don't think it's really that important. No, that doesn't sound right: I don't mean it's not important, only that I don't think it needs to be addressed in the context of this article. [[User:Greg Woodhouse|Greg Woodhouse]] 03:34, 7 May 2007 (CDT)


:Thank you for those compliments and thanks to Greg Woodhouse for rewriting my awkward English. I agree the X/x distinction was too heavy here and made things harder to understand. --[[User:Sébastien_Moulin|Sébastien Moulin]] <small>[[User_talk:Sébastien_Moulin|(talk me)]]</small> 11:41, 4 April 2007 (CDT)
Does anyone know how to create an archive? Is there an automated, or at least "official" way to do it? I just got the following warning:


== Closing the loop (pun intended) ==
'''WARNING: This page is 87 kilobytes long; some browsers may have problems editing pages approaching or longer than 32kb. Please consider breaking the page into smaller sections.'''


It's just not right to talk about analytic functions without bringing in integration, too. Besides, Cauchy's theorem and Cauchy's integral formula lie at the heart of the reason complex variables are so pervasive in mathematics. Some discussion just had to be included (in my opinion). [[User:Greg Woodhouse|Greg Woodhouse]] 14:38, 6 April 2007 (CDT)
[[User:Greg Woodhouse|Greg Woodhouse]] 03:34, 7 May 2007 (CDT)


== Developing or Developed? ==
:Re archiving:  you might want to discuss with Chris Day or see [[Talk:Biology/Draft]], but since that template is named "Experimental" I suppose procedures haven't been finalized.  (Discussion about it arising from pages like this one may drive the finalization of such procedures.)  Maybe it's being discussed on the forum somewhere, or if not someone could start. --[[User:Catherine Woodgold|Catherine Woodgold]] 07:45, 7 May 2007 (CDT)


This article seems pretty much fleshed out, is it ready to be moved to status 1, or does it need more editing? Also, I'm unfamiliar with the procedure or protocol for advancing an article to this stage. Is there a standard method (such as a template) to request that it be done? [[User:Greg Woodhouse|Greg Woodhouse]] 14:41, 6 April 2007 (CDT)
:Re being part of chemistry and physics workgroups:  Chemistry and physics also need to use 1 + 1 = 2.  They also need to use words with syllables to communicate technical concepts;  that doesn't mean the linguistics Syllable page has to be in the chemistry and physics workgroups. Those sciences use math -- that doesn't mean math is part of the science.  I think perhaps people in the chemistry and physics workgroups should decide whether the article is included or not. It's OK with me either way -- it's not that unreasonable. More justifiable than including a page that presents a proof of 1 + 1 = 2 in those sciences.  --[[User:Catherine Woodgold|Catherine Woodgold]] 18:23, 7 May 2007 (CDT)
:The procedure to advance an article: edit the checklist above according to your liking :-) More seriously, I guess anyone can "asses" the article's level. Generally, if you find that the article more or less covers its scope (as you see it), then why not move it to status 1. In the particular case of 'complex number', I'd not object. Still, I think it needs some further work (I'll try to add my $0.02 too).  --[[User:Aleksander Stos|AlekStos]] 11:53, 11 April 2007 (CDT)


== Comments in footnotes ==
This article was approved by a math editor and currently is listed in the Math Workgroup Approved articles, but not in Chemistry or Physics. Since the article is cross-listed in three workgroups, will there need to be three approval processes? Or will we need editors from all three areas to agree before any one draft gets approved? Or will we declare one "father" workgroup, and the others just raise objections or not... I know this approval process is still in its infancy, but these are questions we really should address at some point. - [[User:Jared Grubb|Jared Grubb]] 02:41, 8 May 2007 (CDT)


I think the use of footnotes is preferable to the "sidebar" comments I used originally. [[User:Greg Woodhouse|Greg Woodhouse]] 15:11, 11 April 2007 (CDT)
== Error in multiplicative property ==


== What now? ==
I've just put the following message on [[User talk:Nancy Sculerati]].
It seems to me like we've pretty much covered Sébastien Moulin's proposed outline. What's the next step? [[User:Greg Woodhouse|Greg Woodhouse]] 11:53, 4 April 2007 (CDT)
<blockquote>
:OK, my $0.02.  The formal definition could be developed in more details. In fact, the meaning of ''i'' is not explained in elementary terms so far; the use of it is not justified. And now for the overall structure. Sebastien Moulin in his excellent plan proposed to do this (i.e. formal definition) after the historical motivation and I do agree it is a good place. A basic geometrical interpretation could fit as the next element, since we still talk about the pairs of reals. Then,  probably a discussion of notation "(a,b) versus a+bi" could be invoked to smoothly pass to "working with complex numbers" section (now I have impression that the notational/formal problems overload the leading section).
Dear Nancy. [[User:Etienne Parizot|Etienne Parizot]] found and fixed an error in [[Complex number/Draft]] which is also present in the approved version, [[Complex number]]. The formula
:As for the scope, I guess the most important things are already presented (and yes, why not move article to status 1). I'd like to see however some more basic notions explicitly defined. I mean e.g. the trigonometric form of complex numbers (i.e. z=r(cos x + i sin x)). And what about introducing the notion of complex roots, i.e. the set of solutions to <math>z^n=a,\quad a\in \mathbb{C}</math>. It fits perfectly in the "algebraic closure" section.  After all, wasn't it (one of) the main motivation(s) for having complex numbers? BTW, I'd prefer to talk about algebraic closure before passing to analysis, which is something of different flavor.
:<math>e^{z_1 z_2} = e^{z_1} e^{z_2} \ </math>
:If you find something of the above logical, I could try to work further on the text. Of course, comments, remarks and collaborators more than welcome. --[[User:Aleksander Stos|AlekStos]] 16:16, 11 April 2007 (CDT)
halfway the section "The complex exponential" should read
:<math>e^{z_1 + z_2} = e^{z_1} e^{z_2} \ </math>
(with a plus sign added on the left-hand side).


::Yes, I think your suggestions are reasonable. The reason the section on algebraic closure ended up where it was is that I was trying to follow the approach of placing material in orde of increasing complexity (and, at the time, I expected the article to be quite a bit shorter). I hadn't originally planned to talk about complex analysis at all (except in passing, when discussing algebraic closure), but included a broad overview of complex analysis (with the obvious exception of Laurent series, a topic that was probably missed out of author fatigue as much as anything) based on reviewer comments. I don't object to writing out the field operations explicitly in terms of ordered pairs if you really think it's important to do so. Oh, and not talking about roots of unity (<math>z^n = 1</math>) is just an oversight on my part, and the reason there are no graphics accompanying the section on the geometric interpretation of complex numbers is just that I'm terrible at that sort of thing. [[User:Greg Woodhouse|Greg Woodhouse]] 19:25, 11 April 2007 (CDT)
This is a big error so I want it to be fixed as soon as possible. I can't imagine any editor would argue with this change. However, I'm not sure what our options are.  
:::OK, let's go then. I'll add (and maybe reshuffle) some text, and I beg you to copy edit. If I try to put some images do not hesitate to express any critical remarks, since I'm not terrible at that either. BTW, I do not think that the formal operations on pairs are important or should be promoted (actually, ''i'' is introduced to avoid this). I just want to have somewhere a complete formal definition, just a math bias :-). --[[User:Aleksander Stos|AlekStos]] 03:08, 12 April 2007 (CDT)
* Some places hint at the possibility to have the constabulary do limited changes to articles without going through the whole approval process (for instance, the section [[#Copyediting matters]] above). I couldn't find anything about rules or procedure though. If such a possibility exists, that would be my preference. For the record, the article [[complex number]] was nominated by [[User:Greg Martin|Greg Martin]] and the nomination was supported by me.
* If this is not possible, I'd like the approval to be revoked. As far as I can see, there is no rule or precedent for this, only an empty section at [[CZ:Approval Process]].
* If neither of the above is possible, or if it would take too long, we can always go for the option of nominating the fixed version for approval. To be honest, it's not that important in the big scheme of things, but it is embarrassing and I feel responsible for it.


Extension and slight reorganization began... Meanwhile, I realized that we need also
Any guidance from you (or anybody who happens to read this) would be much appreciated.  
* perhaps a minor remark on equality of two complex numbers
</blockquote>
* a few words describing the 'meaning' of complex numbers in math and applications. Perhaps something like this: "In math the role of complex numbers is fundamental in as the basic object for complex analysis and a powerful tool elsewhere. In applications, although nothing real corresponds directly to \mathbb{C}, complex numbers are very important tool that allows us to perform a formal manipulation at end of which we arrive at useful conclusions concerning physical quantities". Well, as it stands it is an oversimplification to be refined; now just a note for future reference. --[[User:Aleksander Stos|AlekStos]] 07:26, 12 April 2007 (CDT)
-- [[User:Jitse Niesen|Jitse Niesen]] 08:37, 10 May 2007 (CDT)


== Complex numbers in physics ==
Hi All, I have commented out the Approval tag per nominating editor [[User:Jitse Niesen|Jitse Niesen]] who has revoked his approval.  During this time, our Approval editor, [[User:Nancy Sculerati|Nancy Sculerati]] can make the appropriate changes and she can replace the Approval tag.  If more chances are made, then I would suggest giving yourself an additional 24 hours before re-approval to give others a chance to review the changes. --[[User:D. Matt Innis|Matt Innis]] [[User talk:D. Matt Innis|(Talk)]] 08:46, 10 May 2007 (CDT)


I suppose the most obvious example of an area in physics where complex numbers seem ragther fundamental is in quantum mechanics, where it is actually quite crucial that the wave functions are complex valued and not merely real valued. I've actually been thinking about heuristic arguments for motivating the Schrödinger equation and, in particular, why <math>\psi</math> must be a complex function, but I don't want to go to far afield, either. In my opinion, it's possible to go overboard when arguing that complex quantities are not really fundamental. In fact, I'm not so sure I even agree that they are not. [[User:Greg Woodhouse|Greg Woodhouse]] 22:58, 16 April 2007 (CDT)
: I think this was handled very well. What I saw was: Etienne correctly decided that this problem should be corrected very quickly; Nancy sent me (presumably other editors) an email to alert me to it; by the time I came to CZ, it had already been decided that the change was appropriate and needed, and was made to the approved version as well as to the draft.  
:It would be great to insert a hint why wave functions are complex! I already mentioned that the article should not only state the basic definitions and some "how to", but also, "why" and "what for". Perhaps the latter is even more important than the former, according to the spirit of [[CZ:Article Mechanics]]. On the other hand, the article should be kept reasonably long and of limited scope, so perhaps an extensive chapter on quantum mechanics does not belong in.  
:Your suggestion "''I'm not so sure I even agree that they are not''" can be interpreted as disagreement with my claim that in applications \mathbb{C} is just a tool that is 'unreal' :-) If so, I do not object making complex numbers 'fundamental' here and there (and I've never said it is _only_ a tool). Of course, 'fundamentality' should not go unexplained and your quantum mechanics example fits perfectly here. --[[User:Aleksander Stos|AlekStos]] 02:59, 17 April 2007 (CDT)


Well, in a naïve way, there is the obvious fact that <math>e^{it}</math> is an eigenfunction of a complex operator but not a real one, and eigenstates are the only thing we can "observe". There is, of course, the formal similarity of the Schrödinger equatiion to the ordinary wave equation and other hints, but I want to keep the article focused, too (though that's hardly apparent from what I've written so far!) When you get right down to it, I think i do have something of a distaste for worrying overmuch about the ontological status of complex numbers. They are mathematical abstractions, but so are real numbers, and integerers, too. [[User:Greg Woodhouse|Greg Woodhouse]] 07:09, 17 April 2007 (CDT)
: Although this might be "outside the rules", I think that here judgment wins the day, concerning what we might call "clear factual mistakes or obvious typos". I certainly take responsibility for nominating the article for approval without seeing this mistake. And sharp eyes [[User:Etienne Parizot|Etienne]]! - [[User:Greg Martin|Greg Martin]] 14:29, 10 May 2007 (CDT)


===New section===
There will ''always'' be such mistakes, and a good approval process can take care of them.It is understandable that Jitse, who was shocked by the sudden recognition of such a mistake, wanted it fixed IMMEDIATELY. It was understandable that Matt acted to accomodate him, my only point is -in the future we now know that the approvals editor could have done the copyedit at Jitse's say so.  Approval cannot be "revoked" in this manner. Think about it. If it could be, that sets a terrible precedent, you can imagine how in a different circumstance such a precedent could be misused. I cannot add details to the approval process policy without the Editorial Council (of which I am a member) being up and running, with a voting process in place. Right now we are figuring out the process. I said several times that I would take responsibilty for copyedits at this stage with any of the nominating editors.Maybe we should add that, in an emergency the constable can put up a note saying that there is a copyediting problem that is being corrected- in progress. [[User:Nancy Sculerati|Nancy Sculerati]] 12:10, 11 May 2007 (CDT)
Well, I've added a little section on complex numbers in quantum mechanics (a topic I think really ''has'' to be included in an article on complex numbers). This was all pretty much off the top of my head while I sit here listening to the Science Channel. [[User:Greg Woodhouse|Greg Woodhouse]] 23:19, 17 April 2007 (CDT)


== Request for approval ==
== Style issues ==


Editors: Could you take a look at this article and, if you think it's ready, initiate the approval process? [[User:Greg Woodhouse|Greg Woodhouse]] 13:24, 22 April 2007 (CDT)
Quoting some examples of style that I consider a bit too informal. In particular, there are many phrases/clauses that make the article verbose. In the first paragraph, there are: ''Of course'', ''As it happens'', ''At first glance'', ''perhas more importantly''.


== Comments ==
Is this as per the policy of Citizendium? Should the number of such phrases/clauses be reduced? [[User:Vipul Naik|Vipul Naik]] 02:19, 8 June 2007 (CDT)


I'm not an editor, but:  excellent articleI particularly like the cubic equation used as a motivation for defining the complex numbers. In general, I like the motivation and enthusiasm throughout the article.
Hi Vipul, and welcome to CZ!, the answer to your first question is "yes, there are style differences here" - see [http://locke.citizendium.org:8080/wiki/CZ:Article_Mechanics#Narrative_coherence_and_flow this section of the article mechanics article concerning style]. Your input is welcome. [[User:D. Matt Innis|Matt Innis]] [[User talk:D. Matt Innis|(Talk)]] 07:56, 8 June 2007 (CDT)


:Thank you. The historical motivation (cubic equations) was contributed by Sebastién Moulin, and is most appreciated. [[User:Greg Woodhouse|Greg Woodhouse]] 21:02, 22 April 2007 (CDT)
Personally, I prefer  more casual or informal style. Of course, this doesn't mean the articles need be any less precise or rigorous, only more readable, and maybe a little less intimidating.
[[User:Greg Woodhouse|Greg Woodhouse]] 11:39, 8 June 2007 (CDT)


Would it be OK if I go through the article putting html math tags around all the little math formulas, e.g. changing ''a + bi'' to <math>a + bi</math>?
==Remaining errors in this approved article?==
 
Hi, I came across this [http://en.wikipedia.org/wiki/Wikipedia_talk:Wikipedia_Signpost/2007-07-30/Citizendium_analysis#Citizendium.27s_approval_process_already_overlooked_a_.22big_error.22 Error claims on WP Signpost].
:Please do. [[User:Greg Woodhouse|Greg Woodhouse]] 21:02, 22 April 2007 (CDT)
I think there are valid points there, especially regarding the interpretation of 1/z and the comment on the potential function (clearly it can't represent some force since it is a scalar). These should be looked at closer. Are there plans to have this article revised in the near future? Thanks. [[User:Hendra I. Nurdin|Hendra I. Nurdin]] 13:55, 20 October 2007 (CDT)
 
:Hi, did you look at the draft? ;-) Except for the potential thing, deleted from the draft some time ago, I don't really think this is as problematic as suggested. But if you feel like, we could find a better wording for some text. Then re-approving looks like a good idea. [[User:Aleksander Stos|Aleksander Stos]] 14:41, 20 October 2007 (CDT)
::I added some math html tags near the beginning of the article.  I also added "scriptstyle" &mdash; I'm not sure if that's a good idea or not. On my browser, with "scriptstyle" or "textstyle" the math symbols come out too big, but without them they come out too small. Anyway, I think it's good to put the math tags because they're meaningful -- better software might display the same text better if it's marked as being math. --[[User:Catherine Woodgold|Catherine Woodgold]] 19:05, 24 April 2007 (CDT)
:PS. You may also have a look at [[User:Aleksander_Stos/ComplexNumberAdvanced|my "advanced" draft]]. At present I gave up the idea behind that work -- but some portions of the article might be useful here. I don't know. [[User:Aleksander Stos|Aleksander Stos]] 16:02, 20 October 2007 (CDT)
 
::I'm merely passing on some criticisms I happened to stumble onto (I don't know how many people have read it before). Of course, if there is anything valid in them then they should be considered. As for "complex division amounting to conjugation with scaling", well it does sound a bit misleading to me (I don't know about other people, which is why I brought it up here :-)). Consider <math>z_1/z_2 = z_1/|z_2|^2 \bar z_2</math> then it does not have anything to do with the conjugate of <math>z_1 z_2</math> which is <math>\bar z_1 \bar z_2 </math> -- so what does "conjugation" in this part of the article refer to? Compare this for example, with the discussion of <math>z_1 z_2</math> in the article. As for your "advanced" draft (such as the section on roots of complex numbers), perhaps parts of it can go as subpages of the article? [[User:Hendra I. Nurdin|Hendra I. Nurdin]] 19:35, 20 October 2007 (CDT)
:::This is a difficult matter because it depends a lot on the browser. However, scriptstyle should go before the formula. The main possibilities that I see are
::P.S. Does the removal of the assertion that the potential function represents some force from the article not warrant a re-approval process? This gives rise again to the issue that some relatively "minor" changes like this to an approved article should be possible to do with ease. [[User:Hendra I. Nurdin|Hendra I. Nurdin]] 19:44, 20 October 2007 (CDT)
:::# no math tags, e.g., <tt>of the form a+bi obtained by</tt> which renders as
:::#:&hellip; of the form a+bi obtained by &hellip;
:::# emulate maths formatting in HTML, e.g., <tt><nowiki>of the form ''a'' + ''bi'' obtained by</nowiki></tt> which renders as
:::#:&hellip; of the form ''a'' + ''bi'' obtained by &hellip;
:::# use math tags, e.g., <tt>of the form &lt;math>a+bi&lt;/math> obtained by</tt> which renders as
:::#:&hellip; of the form <math>a+bi</math> obtained by &hellip;
:::# use math tags and textstyle, e.g., <tt>of the form &lt;math>\textstyle a+bi&lt;/math> obtained by</tt> which renders as
:::#:&hellip; of the form <math>\textstyle a+bi</math> obtained by &hellip;
:::# use math tags and scriptstyle, e.g., <tt>of the form &lt;math>\scriptstyle a+bi&lt;/math> obtained by</tt> which renders as
:::#:&hellip; of the form <math>\scriptstyle a+bi</math> obtained by &hellip;
:::Options 1 and 2 are not always possible. For instance, <math>x_1^2</math> cannot be displayed in HTML; the best you can do is ''x''<sub>1</sub><sup>2</sup>. Option 3 roughly amounts to choosing option 2 if the formula can be displayed in HTML and option 4 otherwise, but the software behind it needs some fine-tuning. Option 5 is similar to option 4 except that the symbols come out smaller. -- [[User:Jitse Niesen|Jitse Niesen]] 21:08, 24 April 2007 (CDT)
::::\scriptstyle formula looks _very_ good in my firefox(windows). It makes the symbols quite naturally fit the line. Without it (bare 'math' tags) the symbols are awkwardly big if rendered as images (I do not know why this is not always the case). I guess it is a problem of a more general nature. Perhaps we could discuss the advantages of different options on Math forum and try to establish some guidelines in this regard? I hope Catherine and Jitse wouldn't mind if I  [http://forum.citizendium.org/index.php/topic,860.0.html post] what is written above. Hope to see you [http://forum.citizendium.org/index.php/topic,860.0.html there]. --[[User:Aleksander Stos|AlekStos]] 07:01, 26 April 2007 (CDT)
 
Here are some suggestions for minor changes.  I'm not putting them in directly at the moment because I see there's a possible approval process going on so I thought I'd better get opinions first.
 
In the second sentence, 1st paragraph of "Historical example":  ''"This is so even for equations with three real solutions, as the method they used sometimes requires calculations with numbers which squares are negative. "''  To me, the phrase "this is so" lacks an antecedent;  I'm wondering "what is so?".  So I suggest changing it to ''"This need is present even for..."'' or ''"Even for equations with three real solutions, the method they used..."''  Also, near the end of the sentence, ''"which"'' doesn't seem to quite fit in grammatically.  I would change it to ''"...numbers whose squares are negative"'' or ''"numbers of which the squares are negative"''.
 
A few lines later in the "Historical example" section, a minor point:  I think it would sound better to put a comma after "that is" in ''"Now we choose the second condition on u and v, that is 3uv − 15 = 0, or uv = 5."'', or to change "that is" to "as" or to change it to ''"Now we choose 3uv − 15 = 0, or uv = 5 as the second condition on u and v."''
 
A little further down:  It says ''"...the usual formulae giving the solutions require to take the square root of the discriminant,"'' Well, in my dialect, the words "require to take" wouldn't appear.  I would tend to change it to "require taking".  But maybe it's grammatically correct in another dialect.
 
A little further: ''"denotes an hypothetical number which square would be  − 1"'' Again, I would change this to "whose square" or "of which the square".
 
''"square of real numbers are always nonnegative"'' Here, "square" should be plural.
 
In the two lines of equations after values for u and v are selected, the left-hand-sides are the same in bothI would prefer to delete the left-hand-side of the second line and begin it with an equals sign to show it as a continuation of the calculation from the previous line.  Otherwise, if the reader isn't careful with details it looks like two different calculations, one for u and one for v.
 
At the beginning of the section "Formal definition" it says ''"Formally, complex numbers are ordered pairs of real numbers."''  I think this is not the only way to define complex numbers.  I think they could be defined as polynomials, or as points on a plane, or probably in a number of other ways.  (For example, a possible, though awkward and probably not useful, definition would be to define them as sets each of which contain three elements: two real numbers and another set which is either the null set or the set containing zero;  I believe this definition could be used just as formally, though requiring more effort.)  So I would prefer to change this to ''"Formally, complex numbers can be defined as ordered pairs of real numbers."''  Even better:  if this is the way Gauss defined them, I would like to see something like ''"Complex numbers were defined formally by Gauss as ordered pairs of real numbers."''  I would like to see Gauss mentioned in the formal definition section, or  else at the end of the previous section, to the end of the last sentence, ''"A rigorous construction of this set was given much later by Carl Friedrich Gauss in 1831."'', tack on ''"...which is described in the next section."''  Or something, to let the reader know the historical context of the material which "we" understand in the formal definition section.  Otherwise, the reader doesn't know whether Gauss defined them a different way.
 
:I like your suggestion here. The matter of how to ''define'' the complex numbers was actually a point of contention here, and so what you see is kind of an "nth iteration", and I certainly appreciate a fresh perspective here. [[User:Greg Woodhouse|Greg Woodhouse]] 21:02, 22 April 2007 (CDT)
 
::ThanksSo, is that the way Gauss defined them? --[[User:Catherine Woodgold|Catherine Woodgold]] 18:53, 23 April 2007 (CDT)
 
:I don't know just how Gauss defined them. Maybe someone knowing more about the history of mathematics can answer that one. [[User:Greg Woodhouse|Greg Woodhouse]] 23:42, 23 April 2007 (CDT)


::According to Burton's ''History of Mathematics'', the definition of complex numbers as pairs of real numbers is due to Hamilton. I added this to the article. Aside: the approval process hasn't started (this needs the involvement of an editor), and my impression is that even during the process copyediting can be done freely, so I feel no qualms about changing the article, and I also urge Catherine (and others) to work on it.
Hendra, all it would take is a math editor who has not worked on it to nominate it for re-approval, or three editors who have worked on it to re-approve. If we can get that together, I will be glad to make the draft the approved version.  --[[User:D. Matt Innis|Matt Innis]] [[User talk:D. Matt Innis|(Talk)]] 20:40, 20 October 2007 (CDT)
::I've no idea what Gauss did in 1831. Perhaps whoever wrote this can clarify? -- [[User:Jitse Niesen|Jitse Niesen]] 02:08, 24 April 2007 (CDT)
::Well, let's see what others think about this first, as it could be that it's just me being pedantic and perhaps in view of others these changes may not be necessary :-) Anyways, any further changes need to be considered carefully, so that if it does have to go through re-approval, no further minor changes would need to be made afterwards. [[User:Hendra I. Nurdin|Hendra I. Nurdin]] 21:08, 20 October 2007 (CDT)
::Okay, sounds like a plan.  I was  going to leave a message on Jitse's page but you beat me to it!  I'll wait and see what develops.  If you have any questions, just stop by my talk page. --[[User:D. Matt Innis|Matt Innis]] [[User talk:D. Matt Innis|(Talk)]] 21:53, 20 October 2007 (CDT)
:::I hadn't seen the criticisms before. I think they are valid points and that we should revise the article accordingly. Hendra, please change the draft as you see fit (I'm rather busy now so I can't be of much help at the moment, sorry). The approval process is not that much effort, so you shouldn't worry about that. -- [[User:Jitse Niesen|Jitse Niesen]] 22:07, 20 October 2007 (CDT)
::::Done. I have also added a remark that division by c+di is only defined if c and d are not simultaneously zero in the part of the article that discusses operation on complex numbers. Therefore I invite all authors and editors who had been previously involved in the approved article to check my edits and make any modifications and corrections as deemed necessary. However, I think that the work is not all done yet. There is a bit more to be done on the section about complex numbers in physics. The sentence


::Reading the forums, I see that Catherine has a much better idea of what the approval process entails. -- [[User:Jitse Niesen|Jitse Niesen]] 21:41, 24 April 2007 (CDT)
:::::"Now, there is some subtlety in the interpretation of ψ because a system can be affected by observation, and the functions ψ we "see" must be eigenstates of the operator defined by the Schrödinger equation, but when we do measure, say, the position of a particle, the probability of finding it in a small region R is just ..."


:::Thank you, but it was certainly not my intention to set a new fashion of not making any copyediting changes (or content changes?) during an approval process.  I was likely overly cautious here in not making some of these changes directly without discussion.  People need to develop a balance between "be bold" and "don't be too bold". The details of the approval process have not been worked out yet. --[[User:Catherine Woodgold|Catherine Woodgold]] 08:00, 25 April 2007 (CDT)
::::is quite vague and is likely to cause misunderstanding. I guess I know a fair bit about the mathematical formalisms of quantum mechanics, but I'd rather not delete things nor make substantial changes without first soliciting the opinions of those who have worked on this part, and other authors who know the subject quite well, and get their input on what is meant exactly by this sentence and whether it needs to be further elaborated upon for clarity, or changed to avoid misinterpretations. [[User:Hendra I. Nurdin|Hendra I. Nurdin]] 00:25, 21 October 2007 (CDT)


At the end of the first paragraph of "Beyond the notation" it says ''"...and we discuss it in more details."''  I would change "details" to the singular form "detail" because that's the way this idiom is usually used.
== QM again ==


2nd paragraph of "Beyond the notation" section: ''"There is a well established tradition in mathematics..."'' I would hyphenate "well-established", following the rule that multi-word phrases are usually hyphenated when used as adjectives.
The article states:


Later in that section, in the part about modular arithmetic, around where it says ''"And by the same token,"'' I didn't follow the reasoning at first.  I didn't see the connection between the polynomial example and the imaginary-number example until I'd studied it for a while.  Suggestions to help other readers there:  Use numbers that aren't quite so simple, so the analogy is more obvious;  e.g. use (5x + 2)(3x + 1) instead of (x + 1)(x + 2);  and/or state that the order of the numbers is reversed, and/or say "5x + 2 is analogous to 2 + 5i", or reverse the order in the polynomials, i.e. (2 + 5x)(1 + 3x); and/or say "analogously" instead of "by the same token";  and/or say "where the modulus is <math>1 + i^2</math>" or "modulo <math>1 + i^2</math>".
: ''Now, there is some subtlety in the interpretation of ψ because a system can be affected by observation, and the functions ψ we "see" must be eigenstates of the operator defined by the Schrödinger equation, but when we do measure, say, the position of a particle, the probability of finding it in a small region R is just ... ''


End of first paragraph of "geometric interpretation":  "both ... but also" doesn't sound right to meI would delete "both" or change "but" to "and".
In my view this sentence should be deleted because it has '''absolutely nothing to do with complex numbers.''' It gives me the unpleasant WP experience of something that is added by somebody somewhere with some time on his hands, which is why many WP articles are headache-causing kinds of patchwork. At most one could do in this article is a link to quantum mechanics, where the Born postulate for probability of observation can be put in proper setting--[[User:Paul Wormer|Paul Wormer]] 07:42, 21 October 2007 (CDT)


''" Translation corresponds, to complex addition"'' I really like this section, about how the operations are interpreted geometrically;  there's a lot of energy and excitement here.  I would just delete the one comma after "corresponds".
PS Most interactions in QM are invariant under time reversal. It can be shown that &psi; can be chosen to be real in that case. And indeed, 95% of quantum chemistry deals with real functions. --[[User:Paul Wormer|Paul Wormer]] 07:45, 21 October 2007 (CDT)


"Algebraic closure" section:  My dictionary defines "holomorphic" as "having a derivative at each point in its domain". I suggest putting this in parentheses after the word.
:Yes, deletion of the whole section would be one solution. As a replacement application we could instead insert the Laplace and Fourier transforms which use complex numbers in an essential way, or perhaps something on phasors. Let's see what the editors think would be best. [[User:Hendra I. Nurdin|Hendra I. Nurdin]] 08:03, 21 October 2007 (CDT)


''"But, by the triangle inequality, we know that outside a neighborhood of the origin..."'' OK, maybe I should have understood this. But I didn'tI was imagining a small neighbourhood of the origin. It would be clearer if it said "there exists a neighbourhood of the origin such that outside that neighbourhood..."  Possibly it would be an improvement if it said "some neighbourhood" rather than "a neighbourhood".
::There are also certain irreducible representations of some (physically important) groups that inherently  are complex (Wigner, Am. J. Math. vol 62, p. 57 1941). These could be mentioned as examples of complex numbers in physics. --[[User:Paul Wormer|Paul Wormer]] 09:45, 21 October 2007 (CDT)
:::Paul, would you be interested in putting this in the article to replace what is currently there? Btw, which group does this paper talk about? Perhaps we can work on this section together, I could insert some additional engineering applications. Let me know what you think. Thanks. [[User:Hendra I. Nurdin|Hendra I. Nurdin]] 07:30, 23 October 2007 (CDT)


I can't follow the second paragraph of "Algebraic closure".  Maybe if it were explained to me I could help modify it to be a little more easily understandable by others?  I suggest after ''"is the splitting field of <math>x^2 + 1</math>, "'' inserting ''"(i.e. the set of polynomials with real coefficients modulo <math>x^2 + 1</math>)"'', if that's correct.  ''" so if we can show that \mathbb{C} has no finite extensions, then we are done."'' I wonder what a finite extension is, and why we would be done if we knew that?  ''"Suppose <math>K/\mathbb{C}</math> is a finite normal extension"''  I don't understand the notation "<math>K/\mathbb{C}</math>".  ''"...with Galois group ''G''."'' OK, maybe here I should just give up until another Citizendium page is done explaining what a Galois group is.   ''"A Sylow 2-subgroup ''H'' must correspond to an intermediate field ''L'',"'' Hmm.  Does this mean there must exist a Sylow 2-subgroup with those properties, or does it mean all Sylow 2-subgroups will have those properties?  (Again, I guess I'll wait until there's a page explaining what a Sylow 2-subgroup is, but I should be able to at least follow the there-exists part of the language.)   ''"such that ''L'' is an extension of <math>\mathbb{R}</math> of ''odd'' degree,"'' Is it clear here that "extension" means a field, not just a multidimensional vector space?  Maybe this is the standard definition of "extension".  ''"but we know no such extensions exist."'' I think this means there are quaternions (dimension 4) and octonions (dimension 8) but no similar fields of odd-numbered dimension.  But how do we know this?  Are we sure we didn't use this result (fundamental theorem of algebra) when we were establishing that there are no such odd-numbered fields?  It would be good to at least name the theorem being used here, or state when it was proven or something about the proof such as what fields of mathematics are used in it.
::::Dear Hendra, don't you think it would be a good idea to leave it to the approving editors to correct the article? We can signal what we don't like. For instance, the following sentence in the article
:::::''the functions ψ we "see" must be eigenstates of the operator defined by the Schrödinger equation''
:::: is bordering on being wrong;  a wave function can be a superposition of eigenstates, see [[particle in a box]] for a graphic example. Maybe "see" refers to a collapse of the wave function, but that would be a collapse to an eigenstate of the position operator. Further, the Schrödinger equation mentioned (time-dependent) is not an eigenvalue equation, so the term "operator defined by" is pretty inconclusive.  


:I included this proof to illustrate how a very different(?) sort of argument could be used to show that <math>\mathbb{C}</math> is algebraically closed. I don't expect that the argument would be understood by a reader not having had a university level course in algebra, but here's the idea: By the intermediate value theorem, any polynomial of odd degree must have a root, and so extensions built up from odd degree polynomials must contain (real) roots for those polynomials. Now, the Galois group of a normal field extension (roughly, one that arises through adjunction of all roots of a set of polynomials) has some order. If n is the largest integer such that <math>2^n</math> divides th order of the group, there must be a normal subgroup of that order (i.e, a subgroup left invariant by congugation by elements of G) by the Sylow theorem for p = 2. Now, the subset of K fixed by this subgroup is, in fact a subfield which we may call L. Since n is maximal, L must be an odd degree extension, meaning any element of L must be root of an odd degree polynomial, but they ''must'' have roots in the base field, a contadiction. [[User:Greg Woodhouse|Greg Woodhouse]] 21:02, 22 April 2007 (CDT)
:::: I am of the opinion that it is better to spend our energy on '''new''' articles, given the present vast emptiness of CZ. In [[Legendre polynomial]] I linked to [[orthogonal polynomials]]. I saw that you wrote Gram-Schmidt, so for you it would be a piece of cake to write a nice article about general orthogonal polynomials, with links to Laguerre, Hermite, Legendre, Jacobi, etc. Best wishes, --[[User:Paul Wormer|Paul Wormer]] 06:59, 25 October 2007 (CDT)


I think it would be helpful to put some reassuring words into this paragraph such as "less advanced readers may not be able to follow this, but ..." or "some readers may wish to skip to the beginning of the next paragraph" or "readers sufficiently familiar with field theory will be able to follow this".  (2nd paragraph of "Algebraic closure".)
::::PS. Upon rereading the Wigner article that I mentioned earlier, I noticed that Wigner does not mention any specific groups, only characteristics of groups. But, complex numbers are essential for irreducible representations of cyclic groups and for the even-dimensional irreps of SU(2). Schur's second lemma requires the solution of a polynomial equation and hence an algebraically closed field.  --[[User:Paul Wormer|Paul Wormer]] 06:59, 25 October 2007 (CDT)


Under "What about complex analysis?" it says at the end of the first paragraph ''"(The more interesting question is why we would want to avoid using it!)"''  I have mixed feelings about this sentence. It is an interesting question, and is the sort of thing that makes this article interesting -- gives it zing. On the other hand, it seems to contradict the flow of what had just been said earlier in the paragraph.  It's sort-of like saying "let's prove this theorem" and then proving it and then saying "why would anybody want to prove a theorem like that?"  Seems jarring or derogatory of the article.  I'm not sure how to fix this and keep the zing.
:::::I removed the whole QM section; there are simply too many problems with it. It would be nice if somebody could write a section on applications of complex numbers outside maths. You don't need permission of the approving editors to do so (that's why it's called a draft), but I hereby do give you permission in case you feel happier with it.
:::::I'm not so sure what the best application would be to put in that section. Phasors is relatively easy to explain, but I think it's mainly an organizational tool and it's not essential to use complex numbers - one can just use sine and cosine. However, Laplace transforms may be too difficult, given that we tried hard to make the page understandable with a minimum of prior knowledge. Or perhaps QM is a good example after all when written up properly; we can just show the Schrodinger equation and say that it has an i in there.
:::::By the way, I moved from Australia to England and that's why I haven't been around much lately. Still settling in, and all my books are still en route, but I should be able to spend some more time here soon. -- [[User:Jitse Niesen|Jitse Niesen]] 07:51, 26 October 2007 (CDT)


In the section called "Differentiation", it talks about whether it's meaningful to differentiate complex functions, which is fine except that earlier in the article we already used the concept of holomorphic, which I thought used the concept of differentiation in its definition.  So it seems that perhaps things are not being done in a rigourous order.
== Representation via matrices? ==


''"This seemingly innocuous difference actually has far reaching implications."'' I would hyphenate "far-reaching" when used as an adjective phrase.
Just wondering if anyone had already considered an alternative version of the formal definition by defining complex numbers as being a subset of GL_2(R)?  Many "university level" people will have seen the basic definition of matrix multiplication, and as such, it may seem less foreign than the definition of multiplication for ordered pairs of real numbers. {{unsigned|Barry R. Smith}}


Just after Cauchy-Riemann equations are introduced, ''"They may be obtained by noting that if the approach path is on x-axis"'', insert "the" before "x-axis".
== Move some topics to advanced page? ==


In the 2nd paragraph of the "Complex numbers in physics" section, I think the html math tags have been forgotten around one of the psi symbols.  Also in that section:  ''"It's not hard to see that these functions must be complex waves, but it can be demonstrated experimentally that this must be so. "'' I had a course in quantum mechanics and I don't see why they must be complex wavesCould the argument be fleshed out a bit?  The double-slit experiment demonstrates a wave nature of the particles, but how does it demonstrate a ''complex'' wave nature in particular?
Near the top of the discussion, where a plan for the "complex number" page was sketched, the following comment was made:  "I like to introduce complex numbers to my students with the example of the resolution of the cubic equation <math>x^3=15x+4</math> with the so called [[Gerolamo Cardano]]'s method (in fact it is due to [[Scipione del Ferro]] and [[Niccolò Tartaglia]]). Computations are quite easy, and the striking fact is that during them, one has to use some imaginary number which square would be -1, but once the computations are finished, one gets the three real solutions of the equation!"  While it is true that this is probably the earliest example where it became clear that complex numbers were necessary even for the study of real quantities, I definitely disagree with the statement "computations are quite easy".  I am also surprised that you introduce complex numbers to students with this example.  I have given a project to students of working through this example, and I would not say that they found it easyIn another section of this discussion is the comment, "I certainly agree that articles, especially articles about basic topics like complex numbers, shouldn't scare the reader away right off the bat, but perhaps we need to temper our desire to make the article start out slowly and in a non-intimidating fashion with a bit of logical coherence."  It seems to me that our first example might scare many readers away right off the bat.  What does a mathematician think about this example?  Is it a struggle to get through?  Does it make you not want to continue?


Thanks for an enjoyable read.  --[[User:Catherine Woodgold|Catherine Woodgold]] 19:01, 22 April 2007 (CDT)
The initial idea of writing x=u+v where u and v will be specified later is offputting to many people.  This is followed by an application of the binomial theorem, and then an unmotivated factoring step.  Then, it is stated that "we only required that x = u + v. Hence, we can choose another condition on u and v. We pick this condition to be 3uv − 15 = 0"Again, the average non-specialist, I would imagine, would wonder, why this condition? And why ''are'' we allowed to choose another condition?  I could go on listing more potential difficulties that I see in this example.


:Well, I suppose what I had in mind was that if <math>\psi</math> is an eigenstate corresponding to a real eigenvalue (for <math>\Delta</math>, at least) it will not be periodic. If you write out a Fourier expansion, you've got to have complex terms. Another argument is that for probabilities to make sense, you've got to have interference. [[User:Greg Woodhouse|Greg Woodhouse]] 21:09, 22 April 2007 (CDT)
As we now have the advanced subpage option, why not move this example to an advanced subpage, and then refer to it in the actual article.  Perhaps say something like, "A common question is why bother with complex numbers when real numbers almost always seem sufficient for applications. Indeed, the ancients would ignore complex solutions to quadratic equations. It wasn't until the 16th century that it began to be clear that sometimes, complex numbers were indispensable even in problems that seemingly only involve real numbers.  An example of this can be found on the "advanced subpage".


::Like Catherine, I don't find the argument in the article that convincing as it stands. The double-slit experiment with light is usually explained without using complex numbers. The article should explain why you need complex numbers if you do the same experiment with electron beams. In other words, why is the quantum-mechanical wave function e^(x-ct) while light beams are modelled with sin(x-ct)?
Even if this approach isn't taken, might I suggest an alternative way to formulate the current example that is probably more palatable to the average reader:  do not derive the solution of the "reduced cubic" by introducing u,v, etc. Instead, just give them the formula for the roots of a reduced cubic -- it isn't very complicated, and an analogy can be made with the quadratic formula. Then show that 4 is a solution, but that the formula gives an expression involving a complex number. Finally, show that this complex can be written as 4.
::An other argument, if you're willing to consider Schrodinger's equation as given, is simply to point out the factor i in the equation. Perhaps that already proves that complex numbers are essential in quantum mechanics. -- [[User:Jitse Niesen|Jitse Niesen]] 21:08, 24 April 2007 (CDT)


== Quantum Mechanics ==
Also, I think that no matter what, some reference should be made to the fact that although complex numbers are typically introduced these days in high school, when the quadratic formula comes up, that the ancients had versions of the quadratic formula but still didn't accept complex numbers.  [[User:Barry R. Smith|Barry R. Smith]] 13:34, 30 April 2008 (CDT)


I started a new section because the previous section is getting long and difficult to edit. I agree that the example of electron diffraction, at least as presented, isn't a terribly convincing argument for the necesseity of dealing with complex numbers, and I may just delete it. I guess that what I have in mind was that the operator corressponding to momentum is <math>-i\hbar\Delta</math> and, in terms of matrices, this going to mean complex eigenvalues. Frankly, I don't have a very good feel for the mathematics behind the correspondence between observables and operators - it just seems like a kind of "black box" to me. [[User:Greg Woodhouse|Greg Woodhouse]] 22:42, 24 April 2007 (CDT)
: All sounds good. Go for it! [[User:J. Noel Chiappa|J. Noel Chiappa]] 22:26, 4 May 2008 (CDT)


:I found [http://www.scottaaronson.com/democritus/lec9.html this exposition] by Scott Aaronson enlightening when it made rounds on the interweb a few weeks ago (scroll down to "Real vs. Complex Numbers"). [[User:Fredrik Johansson|Fredrik Johansson]] 02:15, 25 April 2007 (CDT)
== New philosophical addition ==


What a thought provoking article! Unfortunately, I barely have time to glance at it right now, but I shall return to it soon. [[User:Greg Woodhouse|Greg Woodhouse]] 06:49, 25 April 2007 (CDT)
With regards to Christopher Reiss's new philosophical addition, I believe that this material definitely should be moved out of the introductory paragraph -- as far as I know, an introductory paragraph should usually serve as an abstract for the article, giving a concise non-technical summary of what most would consider the most salient features of the topic? 


== Can we make it even better? ==
Now the issues raised are partially addressed already in the history section, notably, the fact that ancients did not believe that complex numbers were "real".  Furthermore, the long example in the "advanced" subpage describes the first instance in which complex numbers seemed to be necessary for something.  If this material could be improved using some of Christopher's ideas, then I propose that the material be integrated into a whole within to history section.


Great work, everyone! Certainly a great article on this important topic is a significant plus for the Mathematics section of CZ.
The only content I would take issue with is the statement that "It is now understood that arithmetic is a pure abstraction which we are free to modify", and related declarations.  This is probably the most common philosophical view taken by current mathematicians, and probably many people well versed in mathematics.  However, this idea is much older -- Platonic realism is an early example, due to Plato millennia ago.  So the phrase "it is now" is misleading.  On the other hand, the statement "understood that arithmetic is a pure abstraction" is also misleading.  Better would be to say that the majority of current mathematicians believe this.  But there is still a raging debate about this in the philosophy of mathematics.  For alternatives, see "empiricism" and "fictionalism", for example, at the [http://en.wikipedia.org/wiki/Philosophy_of_mathematics philosophy of math] wikipedia site.  Even if we add this material to what Christopher wrote, I believe that content would more properly be put in a philosophy of math site, as the result would probably be quite lengthy and too much of a tangent from the topic of "complex numbers".


I'm proposing a list of modifications below. All of them are (of course) debatable.
Any thoughts?[[User:Barry R. Smith|Barry R. Smith]] 12:20, 22 May 2008 (CDT)


; Move the formal definitions (the sections "Formal definition" and "Beyond the notation") much later in the article.
:I completely agree. I did not see any reply from Christopher, so I simply removed the text. Here it is for future reference:
: When we teach complex numbers to math majors, first we simply teach them as "a+bi"s together with how arithmetic works on them; only later do we teach formal definitions. And CZ articles aren't written for math majors even - more like general college-educated people. I think it's preferable to just explain how they work first - this is more valuable general information by far than formal constructions (just as is the case for integers, real numbers, ...!). As an aside, I think the "Beyond the notation" section right now is undecided between two goals: justifying the "a+bi" notation, and describing the construction of the complex numbers as R[x]/(x^2+1).
::''Complex numbers were once considered 'fictitous' on the grounds "there is no square root of negative one." This misconception is rooted in a philosophical conception of number which is now seen as misguided. The notion behind this 'predjudice' is : arithmetic exists in the physical world, or is an attribute of physical reality. This notion has repeatedly proven a stumbling block in the history of mathematics. It is now understood that arithmetic is a pure abstraction which we are free to modify. It is legitimate to experiment with the abtract system first and then seek real world mechanisms which the abstraction can model. Rather suprisingly, by freeing the abstract system so that it is no longer "real", the abstraction became a much broader and more powerful model of the physical world.''
; Move the basic operations and polar coordinate operations very close to the top of the article.
::''This has happened repeatedly in the development of mathematics. Originally, only the counting numbers, 1,2,3 .. were considered 'real'. This left the result of certain division operations - fractions - 'unreal'. But one can define an arithmetic of fractions which is immensely useful in the physical world, and which also describes the counting numbers as a special case. Similarly, the result of certain subtraction operations yielded 'unreal', negative results. Arithmetic was expanded again to include negative numbers. Yet again, it was found that the square root of two has no solution among the fractions. These 'unreal' entities were eventually admitted into arithmetic as it continued to grow in power.''
: The person on the street will have more use for that information than any of the rest, including the Cardano's method example, tantalizing though it is. ''Appended later - [[User:Greg Woodhouse|Greg W.]]'s suggestion to keep the Cardano's method example first seems reasonable; the basic operations could come right after. - [[User:Greg Martin|Greg Martin]] 03:43, 26 April 2007 (CDT)''
:As Barry says, parts of this may, in less absolute form, be integrated in the History section (or perhaps elsewhere); in fact, I think the question of whether complex numbers exist probably should be treated in the article. I would include a link to an article about philosophical aspects of mathematics where this is discussed in more detail. -- [[User:Jitse Niesen|Jitse Niesen]] 07:19, 2 June 2008 (CDT)
; Rewrite the first two sentences very carefully.
: The first sentence, standing alone, should constitute an excellent definition of "complex number". At the moment the second sentence is a crucial part of the definition. The way it's written right now is good, don't get me wrong ... but we want that first sentence to be great, fantastic, beyond reproach!
; Take out the ''proofs'' that the complex numbers are algebraically closed.
: In an introduction to the subject of complex numbers, I think it's much more important to describe ''what'' we know about them then ''how'' we know it. To use an analogy, an introductory article about DNA would certainly describe the double-helix structure and nucleotide pairs, but I suspect it wouldn't go into the details of how that structure was discovered and verified. We mathematicians love proofs, and the field of mathematics would be ludicrous without them; but proofs are our version of ''how'' we know, not ''what'' we know. (Of course these levels can recurse upon one another....)
; Take out all of the stuff about complex analysis and start a new article with it.
: self-explanatory
; Create a new section titled something like "Why bother using complex numbers?" and gather relevant material thereunder. (Did I just use the word "thereunder"?!)
: Examples of such relevant material would be the Cardano's method example, the fact that the complex numbers are algebraically closed, physical applications ... perhaps the ease of writing down the solutions to constant-coefficient differential equations (not sure about the relative importance of that example) ... things like this. People who think that complex numbers aren't worth the bother (a reasonable position, on the face of it) should be converted by this section.
; Do put math tags around all the in-line pieces of math, even single variables.
: This refers to the Comments made in Section 9 of this discussion page. Right now the software isn't perfect at melding math with prose, but that could change in the future; we want the structure of the article source to be accurate so that future improvements to the software automatically improve this article.


Keep up the great work, authors! - [[User:Greg Martin|Greg Martin]] 00:08, 25 April 2007 (CDT)
== Division and conjugation ==


::I am no mathematician, and you may use the above to improve this article, but - as the famous surgeon [[William Stewart Halsted|Halsted]] once said- Better is the enemy of the good. You can imagine how that might apply to doing a surgical operation. For an article to be approved it should be reasonably complete, true, and well-written. After approval, more work will continue on the draft. In other words- it can always be better- but is it adequate? Complete as it stands? That's the question for approval. [[User:Nancy Sculerati|Nancy Sculerati]] 00:17, 25 April 2007 (CDT)
''Discussion moved from [[User talk:Jitse Niesen#Complex number page]]'' BEGIN [[User:Peter Schmitt|Peter Schmitt]] 23:44, 19 July 2009 (UTC)


:::Many of the points that Greg mentions boil down to the appropriate level for the article. As it stands, the article is tough going for the average reader. For instance, I have never learnt Galois theory, so I don't understand the details of the algebraic proof of the fundamental theory of algebra. Of course, I can skip the section and it won't cause any problems, but we should avoid forcing many readers to skip sections. Similarly, I greatly enjoy the historical section, but it is rather demanding, so I can see the value in the suggestion to move the "basic operations" and "geometric interpretation" up.  
The [[Complex number]] page still contains what I consider to be a blatant error: "In other words, up to a scaling factor, division by z is just complex conjugation."  I think this would be correct if it said, "In other words, up to a scaling factor, division by z is just multiplication by the complex conjugate of z" or if it said "In other words, up to a scaling factor, taking the reciprocal of z is just complex conjugation"; but as it stands (according to the only reasonable interpretation I can see) it's equating two operations which in general involve completely different changes to the angle on the complex plane. As you know, this problem was pointed out on a Wikipedia discussion page in 2007. As a math editor, would you please either ask a constable to correct just this one sentence in the current article, or arrange to have the draft approved?  (I haven't looked at the latest draft; I'm just concerned about this particular error.) Thanks. [[User:Catherine Woodgold|Catherine Woodgold]] 15:28, 11 July 2009 (UTC)
:::However, Nancy also has a point. Looking at the [[CZ:Approval Standards|Approval Standards]] (points a-e), I don't see any grave problems. The article needs some vigorous copyediting, along the lines that Catherine mentioned. Furthermore, there are a couple of sentences that I'd like to see clarified:
:::# "A rigorous construction of this set was given much later by Carl Friedrich Gauss in 1831" (at the end of the historical section): as mentioned above.
:::# "we could have avoided the use of the exponential function here, but only at the cost of more complicated algebra" (at the start of the section ''What about complex analysis?''): what does this refer to?
:::# "this is just the condition for the existence of a scalar potential" (in the complex integration section): needs some elaboration.
:::# the physical section needs a rethink (or be removed for the time being).
:::As I see it, there are two possibilities. Either we aim to have the article approved as soon as possible; if somebody can vouch for the Galois theory section, this shouldn't take too long in my opinion. We can then discuss the issue on the level we should aim at. The other possibility is to start with discussing the appropriate level, possibly rewrite the article, and then approve it. Both are acceptable to me. Greg, and others, what do you think? -- [[User:Jitse Niesen|Jitse Niesen]] 09:28, 25 April 2007 (CDT)


If it can be rewritten so that it is clearly understandable to an intelligent layman at the university level- that is optimal. [[User:Nancy Sculerati|Nancy Sculerati]] 09:31, 25 April 2007 (CDT)
: The formulation may be unfortunate, but it is correct. The "scaling factor" is 1/|z|^2, a real number, and the angles (the argument) of the conjugate and the inverse are the same. Probably it would be better to write "In other words, up to the scaling factor 1/|z|^2, division by z is just complex conjugation." I don't know if in such a case approved version can be corrected. [[User:Peter Schmitt|Peter Schmitt]] 22:07, 11 July 2009 (UTC)


:I've been trying to write for university level students who are mathematically knowledgeable, but who haven't necessarily taken any upper division or graduate level courses in mathematics. I've also tried to write so that anyone having had just basic calculus would be able to understand most, if not all, of the article. [[User:Greg Woodhouse|Greg Woodhouse]] 10:14, 25 April 2007 (CDT)
::Why don't the two of you, and possibly anyone else you can rope in, work out an *exact* replacement phrase and then put it into this discussion area. If all of you agree that it should replace the Approved version, either I'll change it myself or I'll ask Matt what he things about it. [[User:Hayford Peirce|Hayford Peirce]] 22:30, 11 July 2009 (UTC)


I've found these comments very thoughtful and helpful - thank you! In particular, [[User:Nancy Sculerati|Nancy]] and [[User:Jitse Niesen|Jitse]] put in focus for me that the following question is really what's under discussion here: '''what does approving an article signify?'''
::: My suggestion:
:::: In other words, up to the scaling factor <math>\frac 1 {\left|z\right|^2}</math> (a real number), division by ''z'' is just complex conjugation.
::: (Unfortunately, the fraction looks awful in text.) [[User:Peter Schmitt|Peter Schmitt]] 22:57, 11 July 2009 (UTC)


I agree that a read through [[CZ:Approval Standards|Approval Standards]] doesn't unearth much to complain about (some things perhaps, but all debatably). ''(For example I think we're doing a great job presenting complex numbers in a neutral fashion!)'' On the other hand, look at the three paragraphs in the section of [[CZ:Article_Mechanics]] called [[CZ:Article_Mechanics#The_nature_or_purpose_of_an_encyclopedia_article|The nature or purpose of an encyclopedia article]]. I think this paradigm is what I'm hoping to adhere more closely to, with my suggestions above.
If we use our current approval (and re-approval) rules: since Peter is a mathematics editor, I would suggest that Peter refrain from making any changes to the article and let Jitse or Catherine make the change on the Draft.  Then (assuming Peter agrees with the change), he can can re-nominate the draft for approval using the single editor process (since he has not made any content edits to the article)... HELLO CATHERINE! :) 


Moreover, I see also that there are respectfully competing motivations for approving articles in this early stage of CZ. On the one hand, we want articles approved early to prove that CZ is flourishing (and also to show how effective our own workgroup is being). On the other hand, early approved articles probably bear the weight of much scrutiny and ought, ideally, to be exemplary role models for CZ articles.
Check with the [[User:Approvals Manager]] (Joe) if you want to be sure.  
[[User:D. Matt Innis|D. Matt Innis]] 23:21, 11 July 2009 (UTC)


I welcome feedback on both these aspects of the approval decision. If someone points me to a place on the forums where this might be under discussion, I'd be happy to catch up there. - [[User:Greg Martin|Greg Martin]] 13:31, 25 April 2007 (CDT)
:Matt's suggestions sound v. feasible to me. [[User:Hayford Peirce|Hayford Peirce]] 23:36, 11 July 2009 (UTC)


::Greg, the truth is that it's up to you (and Jitse) as Editor(s). I do think that your concerns are exactly correct and that the crux here has to do with the "nature or purpose of an encyclopedia article". At what point does this article qualify as meeting those guidelines? That's your call. The Approved version is just a stable version that is true and accurate, and a "good-enough" encyclopedia article. That phrase "good-enough" is borrowed from the parenting literature, Pediatricians use it reassure good mothers and fathers that they don't have to be perfect or better to raise their children, they do have to be... Once approval is made, more work continues on the draft - so it's not like a print edition decision, there s more leeway.Obviously, though, you want to be proud of the article. I'd say the best way to get an idea of where we are in the process is to go to the Main page and click through the Approved Articles. They are all on different levels, some are more complete than others, the writing in some is better than in others, but all are decent. As the Approvals Management Editor, and as somebody who worked on most of the approved articles as an author, I'd say that we continue to argue among ourselves over both those aspects of the approvals decision and nobody can really settle it for you. Here are some pragmatic questions: '''Is the subject covered so that the reader knows - by the end of the article -what the title of the article means? Can the reader learn from the article without having to already know what the article covers? Is the article a narrative that can be followed from start to finish?''' Can you as Editor feel confident that it is not plagerized and that references are appropriate? Is it nicely illustrated? Are spelling and grammar correct? Are there typos in the math? There will be an opportunity after Approval to fix minor errors- copyedit by contacting me - but the article should be in good shape by approval. When the article is nominated for approval, the editor can choose between 48 hours and 1 week between the time the approval nomination template goes up and approval will occur. During that time others will be invited to look. Other Mathematics Editors could remove the template if they think the article is really problematic, you yourself might if problems of an unsuspected magnitude or a large number of small problems are pointed out such that the article isn't smooth by the approval date. Hope this long winded comment helps. By the way- the reason that there is the new "copyedit through the Approvals Editor" rule is that, often, as Approval nears the article gets swarmed by editors/authors trying to get it better. This often results in big improvement, but leaves a rough edge. You probably have to experience it to understand what I mean, once you get a couple of articles approved, you'll see- that is, if Mathematicians are anything like Biologists. We'' know'' you are better, of course - we just don't know how much. The end of the approval process for us has always been asymptotic. :) [[User:Nancy Sculerati|Nancy Sculerati]] 20:50, 25 April 2007 (CDT)
:: As far as I see, the draft differs much from the approved version (a direct comparison seems to be difficult). I thought there is a possibility to edit such things without (re)approval? This is not a correction but only a clarification, and certainly not a change of content. (By the way, I think something is wrong - much too difficult -- if an editor is disqualified to make an approval even after such cosmetic edits. Even some minor edits should be allowed. I think it is simply cheating if suggesting a change is allowed, but doing the same edit is not.) [[User:Peter Schmitt|Peter Schmitt]] 23:41, 11 July 2009 (UTC)


:::Thank you so much Nancy for your helpful thoughts. I've bolded a few high-level criteria so I can easily find them again later. So in terms of copyediting, I should contact you once the Approval warning-time elapses? - [[User:Greg Martin|Greg Martin]] 03:43, 26 April 2007 (CDT)
:: I have put a link on the Approval Manager talk page. [[User:Peter Schmitt|Peter Schmitt]] 23:49, 11 July 2009 (UTC)


Greg Martin, I agree that the CZ style of creating a narrative style could help the beginning of this article so that we don't lose the audience in the first two sentencesIt would be nice to have a prose type intro or lead that first explained what the imaginary number is all about; what is it? why do we need it?  what does it help us to do?  Something like that. Am I thinking right?  --[[User:D. Matt Innis|Matt Innis]] [[User talk:D. Matt Innis|(Talk)]] 21:59, 25 April 2007 (CDT)
:::I looked at the draft and the approved version, they are completely different on [[Complex_number/Draft#Geometric_interpretation|this point]]. Somebody made some drastic changes. Further, I would say division of ''c'' by ''z'' is ''multiplication'' of  ''c'' by the complex conjugate of ''z'' (and division by the <strike>modulus</strike> square of modulus of ''z''). In the polar representation of complex numbers the issue is completely trivial, as we will all agree. It is <math> 1/(re^{a}) = e^{-a}/r</math>, which holds for all real and complex <math>a</math>. --[[User:Paul Wormer|Paul Wormer]] 08:09, 12 July 2009 (UTC)


::Yes in my mind, Matt. Of course the question is how best to do it ... and there's always the chicken and the egg problem: how can we explain why we need it if we don't start by saying what it is? but why will people care if we start with a "dry" definition" - [[User:Greg Martin|Greg Martin]] 03:43, 26 April 2007 (CDT)
:::: "and division by the square of the modulus". Yes, this would avoid the displayed fraction. [[User:Peter Schmitt|Peter Schmitt]] 09:53, 12 July 2009 (UTC)
::::: If we want to spell it out completely, then with <math> z = r\exp(i\theta)\;</math>
::::::<math>c/z = c/(r\exp(i\theta)) = (c/r)\exp(-i\theta)= (c/r^2)\;r\exp(-i\theta)= (c/r^2)\;\overline{z}\;</math>,


:It's been awhile, but I believe I started out by saying outright that complex numbers are formal sums of the form <math>a + bi</math> where <math>a</math> and <math>b</math> are real. A reviewer/editor immediately responded that this approach was "objectionable" and, after some negotiation, we ended up where we are now, with a short digression into just what this might mean prior to any real examples. (That's where the early reference to <math>\mathbb{C}</math> as the splitting field of <math>x^2 + 1</math> came from.) Frankly, I think that starting out by introducing complex numbers as formal sums of real numbers and then presenting the solution of cubics as a motivating example is entirely reasonable. Talking about complex numbers as ordered pairs at this stage seems to me to just burden the reader with a digression in the name of mathematical rigor - and one that isn't really needed, anyway. [[User:Greg Woodhouse|Greg Woodhouse]] 22:42, 25 April 2007 (CDT)
::::: --[[User:Paul Wormer|Paul Wormer]] 11:13, 12 July 2009 (UTC)


::I agree completely that leading with a definition of complex numbers as ordered pairs is not the way to go; moving them much later in the article (if they're kept at all) is surely preferable. ... Hmm. I can see your point that (with a slightly sharpened version of the current first paragraph) having the Cardano's method example start off the narrative is as good a choice as any. I've appended the above list of comments accordingly. - [[User:Greg Martin|Greg Martin]] 03:43, 26 April 2007 (CDT)
::::::It seems that you are saying that replacing the approved version with the draft would add more errors than it would fix.  Your choices then would be to 1) fix the errors in the draft and use either the individual editor approval or three editor approval method to change the approved version or 2) revert the draft version to the version you like, then make the change that Catherine and Peter are looking to make and then use the individual or three editor approval methods as above.  Does anyone see any other choices.  


:::A historical remark, just to point out that to my understanding there was no objection concerning the ''formal sums''. To the contrary, the editor/reviewer, who stepped in, supposedly me(?), a CZ-author in this case, didn't like the ''notation'' \sqrt{-1} in the lead and called it "objectionable" at least in definition (see an [http://en.citizendium.org/wiki?title=Complex_number&direction=prev&oldid=100066614 older version] and the top of this talk page). At the same time the "reviewer" used the formal sum instead (see [http://en.citizendium.org/wiki?title=Complex_number&direction=next&oldid=100066614 here]). Then, indeed, it evolved to what we see. For example, the ordered pair definition was put for a while in the lead, then it was moved down and elaborated on. A paragraph "aside on notation" written by one author[http://en.citizendium.org/wiki?title=Complex_number&diff=100067756&oldid=100067717], became a regular and important section in another place by an action of another author... While we had some ideas (a 'plan' outlined by Sebastien), the realization was not straightforward. In general, this is how the wiki ''bazaar'' works, unlike the ''cathedral'' of a single-authored piece. Isn't it fun? Isn't it surprising that given the method the results are not too bad? :-) --[[User:Aleksander Stos|AlekStos]] 10:07, 26 April 2007 (CDT)
::::::The idea of the [[CZ:Approval Process|approval rules]] is to make us work together to come up with the most accurate article possible while, at the same time, allowing the article to remain stable while we do. Hopefully, this reduces the workload on our experts. The errors in the draft are an example of why we want to have an approved version that is difficult to change - without editorial input. Changing our rules for something like this that can be managed within the same rules seems only a means to weaken them. However, it is possible, but would require community input from all the workgroups to consider all the ramifications of such a change.


== Complex analysis section ==
::::::'' (By the way, I think something is wrong - much too difficult -- if an editor is disqualified to make an approval even after such cosmetic edits. Even some minor edits should be allowed. I think it is simply cheating if suggesting a change is allowed, but doing the same edit is not.)''  It's not so much about cheating, it shows that more than one editor agrees to the change, thus increasing the likelihood that the change is more accurate - while at the same time allowing only two editors to make a difference (which is easier than finding three - something that you are also asking for).  It is a way to both make it easier to make a change and keep one fallible editor from approving his/her own work. I hope that makes sense.


So, should this section be deleted? I can certainly do that. In fact, I only added this section (which was meant to be a 30,000 foot view, anyway) because someone thought it needed to be included. [[User:Greg Woodhouse|Greg Woodhouse]] 23:05, 25 April 2007 (CDT)
::::::[[User:D. Matt Innis|D. Matt Innis]] 12:33, 12 July 2009 (UTC)


:I think it's a good idea to delete the section (although leaving behind, of course, some brief references to complex analysis) from this article, but we don't need to throw the material away entirely - why don't we start a new [[complex analysis]] page? The same thing would work for the material on the [[Fundamental Theorem of Algebra]] - that would an article where it would make good sense to present the proofs in the current "Algebraic closure" section. - [[User:Greg Martin|Greg Martin]] 03:47, 26 April 2007 (CDT)
I think I have been misunderstood here, and I'll try to clarify:
I noticed the message of Catherine (not addressed at me) and answered it.
The challenged sentence - in the approved version - is '''correct''',  
but might indeed be confusing for some readers. Therefore I suggested a minor edit to the approved version
(thinking of [[CZ:Approval_Process#Overview]], last paragraph) because approving the draft version
would - in view of the major changes - require more checking and possibly a lot of discussion.
If this is not thought as adequate or allowed, then the approved version can stay as it is.


==Approval==
The remark on the approval process was a reaction on the suggestion:
: "since Peter is a mathematics editor, I would suggest that Peter refrain from making any changes to the article and let Jitse or Catherine make the change on the Draft".
What is the difference between an explicit suggestion by an editor
which is dutifully incorporated by some author (possibly a non-editor),
and the same change made by the editor himself? The difference is only a formal one -- that was what I meant by cheating.
(There need not be another editor involved!) Moreover, I thought that copyedit changes are allowed --
and this I would classify as copyediting.
<br> [[User:Peter Schmitt|Peter Schmitt]] 13:11, 12 July 2009 (UTC)


since this article is governed by maths, physics and chemistry, I second Greg's appeal to move this article to approval status. If nobody objects I will add the approval tag on his behalf on the page with an approval date one week after placement of the tag. [[User:Robert Tito|Robert Tito]]&nbsp;|&nbsp;<span style="background:grey">&nbsp;<font color="yellow"><b>[[User talk:Robert Tito|Talk]]</b></font>&nbsp;</span> 23:42, 25 April 2007 (CDT)
:That does clarify some. I did not understand that the discussion concerned something that was basically correct on the approved page. Unfortunately, I'm not sure that I could have known that if you had not told me :) - which I think makes it different than a copyedit - which anyone could recognize does not change meaning.  Because it is not really self-evident that it does not make a content change, I don't think this is something that a constable can or should do without the approvals manager seeing things through (which he very well might do).  It's more to protect the editor that has endorsed the article than anything else.


::A mathematics editor needs to make that appeal, Rob. This is a Math article. ''I'' object. Let the Mathematics Editors ask us for a template or let them put it on themselves. It's Greg Martin's and Jitse's call. When they are ready, one or both will indicate it. Perhaps, meanwhile, you can help me with the [[Rottweiler]] article? [[User:Nancy Sculerati|Nancy Sculerati]] 00:42, 26 April 2007 (CDT)
:Concerning the addendum, Jitse knows very well where that came from.  I actually revoked the approval of one of his articles when he realized it had a math error in it.  He'll tell you that I took a pretty good beating over that one!  And I think they were right to do that, approved articles need to be hard to change. The addendum makes it clear that the ''nominating'' editor can change it ''with the help of the approvals manager''. It is the nominating editor who has his name on the article and therefore has endorsed it. That is why we have given him/her more leeway to make a change.  Jitse could still make that change, as you note, I think - with the help of the approvals manager. Of course, the other choices still remain - to re-approve using your credentials if Jitse does not respond.
::sorry Nancy, I am no biologist and personally do not like dogs. I see not one single point of interest or knowledge regarding dogs that might be of any relevance to Rottweilers. [[User:Robert Tito|Robert Tito]]&nbsp;|&nbsp;<span style="background:grey">&nbsp;<font color="yellow"><b>[[User talk:Robert Tito|Talk]]</b></font>&nbsp;</span>


== graphic for geometrical/polar interpretations? ==
:You make another good point about an author being able to make a change that an editor cannot, but it still requires two heads.  Remember that we deal with controversial articles that have competing views even among editors.  The concept is to keep one view from eliminating the other view without some oversight.  Whether this is successful at keeping that from happening, or if it keeps us from making more important corrections, or if we might be able to come up with a better way, is something that might need discussion elsewhere if it is causing problems. [[User:D. Matt Innis|D. Matt Innis]] 14:33, 12 July 2009 (UTC)


Not much else to say: inspired by one of Nancy's comments, I think it would be nice to include a graphic or two that illustrated the geometrical interpretation and the polar-coordinate form of complex numbers. - [[User:Greg Martin|Greg Martin]] 03:53, 26 April 2007 (CDT)
::Peter says that the approved version is essentially correct, but I side with Catherine and say that the word '''multiplication''' is missing. --[[User:Paul Wormer|Paul Wormer]] 15:23, 12 July 2009 (UTC)


:Yes, please! I'm no artist, but I've been looking for free or inexpensive software I might use to create appropriate graphics. But if anyone else has graphics they'd be willing to contribute, I'd be most grateful. [[User:Greg Woodhouse|Greg Woodhouse]] 09:12, 26 April 2007 (CDT)
::: Because of the formula above the sentence I read this only as "division (of 1) by z" ... if you read it as "division of some u by z, then this would certainly be wrong. Perhaps I did not want to see a serious mistake? [[User:Peter Schmitt|Peter Schmitt]] 15:35, 12 July 2009 (UTC)


Here is a first attempt. Please do not hesitate to comment on that (how to improve the image). Ideally, we'd have a couple of images in the same graphical style. First, however, this style is to be determined.--[[User:Aleksander Stos|AlekStos]] 14:10, 26 April 2007 (CDT)
::::Looks like you're getting closer to a more accurate description. Once you've decided what you want to do, let the approvals manager know and we'll go from there.  I think our options remain the same. [[User:D. Matt Innis|D. Matt Innis]] 17:07, 12 July 2009 (UTC)


:I think it looks great. [[User:Greg Woodhouse|Greg Woodhouse]] 17:04, 26 April 2007 (CDT)
Since no further comments have been added, I suggest to replace
: In other words, up to a scaling factor, division by z is just complex conjugation.
by
: In other words, up to a real scaling factor (the square of the reciprocal of the modulus), division of 1 by ''z'' is just complex conjugation.
because this seems to be the least change which repairs the sentence in the approved version
(and still fits into the style of the paragraph).
<br> For a reapproval of a new draft certainly more changes (and discussions) will be made.
<br> [[User:Peter Schmitt|Peter Schmitt]] 00:21, 19 July 2009 (UTC)


:Looks good, but the text should be readable in the thumbnail. [[User:Fredrik Johansson|Fredrik Johansson]] 17:14, 26 April 2007 (CDT)
::I suggest that we proceed through a re-approval process anyway. I don't see three editors on that approval tag. [[User:D. Matt Innis|D. Matt Innis]] 18:30, 19 July 2009 (UTC)
::Right. I'll fix proportions. --[[User:Aleksander Stos|AlekStos]] 17:41, 26 April 2007 (CDT)


== Roots of polynomials? ==
''Discussion moved from [[User talk:Jitse Niesen#Complex number page]]'' END [[User:Peter Schmitt|Peter Schmitt]] 23:44, 19 July 2009 (UTC)


Currently it says ''"We need to show that any polynomial <math>p(x) \in \mathbb{R}[x]</math> has a root in<math>\mathbb{R}</math>."''  Um, I think this is not true? --[[User:Catherine Woodgold|Catherine Woodgold]] 08:00, 26 April 2007 (CDT)
=== How to proceed? ===


:It's fixed now. Thanks. [[User:Greg Woodhouse|Greg Woodhouse]] 09:09, 26 April 2007 (CDT)
Ok, but I do not know what I should do.
I do '''not''' want to approve the draft as it is now. <br>
But I want to change one sentence of the approved version in order to either
: correct an error in a sentence, or
: to clarify a sentence that can be misunderstood
(depending how one interprets this sentence in context).<br>
I would approve such a corrected version (and I would make the changes myself, it this is allowed)
&mdash; but how is this done correctly?
<br>[[User:Peter Schmitt|Peter Schmitt]] 00:04, 20 July 2009 (UTC)


The algebraic closure section probably should talk about polynomials with complex coefficients (as far as I remember, a field is algebraically closed if any polynomial with coefficients in _this_ very field has a root). So corrected. The problem is that admitting complex coefficients influences "intution" subsection. The values of the polynomial function are complex (even for the argument in \mathbb{R}) and  we do not need to pass through 0 on the plane, there is no +infinity nor -infinity. Moreover, we would need a proof for p\in\mathbb C[x]. We should probably rethink it. --[[User:Aleksander Stos|AlekStos]] 12:28, 26 April 2007 (CDT)
::Okay, you can't correct it yourself and approve it yourself. You can still use Jitse, or the other editors as noted above. However, since you are an editor and since there are lots of other changes on the draft, I think it would be possible to revert the entire draft to the approved version then have someone make a change that you can agree to and nominate that version as 2.0. <s>Once the version is approved</s>, you can then revert the draft to the older draft (if you want) or start over again from there - as long as you discuss why on the talk page.  If someone disagrees, that gives them plenty of opportunity and options to discuss any issues before it gets re-approved. Again, though, the [[User:Approvals Manager]] needs to be kept abreast of this and had veto power if something doesn't look right - because he is my redundant failsafe system!  [[User:D. Matt Innis|D. Matt Innis]] 04:16, 20 July 2009 (UTC)


I'll see if I can reword it, but the point is that if K/R is an extension of degree not a power of 2, there is an intermediate field L such that L/R is an extension of odd degree. [[User:Greg Woodhouse|Greg Woodhouse]] 17:58, 26 April 2007 (CDT)
:::If it's really just fixing an error, then I don't see any reason we can't just insert the correction into the approved version of the article; we've done that before, though mainly just with grammatical or spelling errors.  Since this would be a content correction, we would need to make sure that the original approving editors still approve or we'll have to remove their names from the approval.  Jitse is definitely still around to agree or disagree with a suggested change but Greg Martin hasn't made a contribution for over 2 years, so he'll be harder to track down. 


== examples ==
:::So these are our options, as I see them: '''(1)''' If we have a specific proposal for a change on the table, we can ask Jitse to review it.  Then, if he agrees to the change, we'll make the change on the approved version and replace Greg Martin's name with Peter's, assuming that Peter approves of the rest of the article.  But then I think we still need one more editor. '''(2)''' We can do the same thing and try to track down Greg Martin. '''(3)''' We can re-initiate the approval process for some later version of the article.  That could be the current version or it could be any other version since the time of approval or it could be a wholly new version.  We would need either one un-involved editor or three who have contributed. '''(4)''' We can revoke approval based on the error that the current approved version contains.  This is the least preferable of our options, and we should avoid it if we can.


QM is a nice example for the "practical" application of i. However topics in chemistry and physics such as NMR, Statistical chemistry and physics, Fourier analysis used in many topics, and even problems in classical mechanics all use complex numbers. Even though QM is a nice topic it lacks for instance the need for (the introduction of) complex numbers. As in not maths was the driving force to ''create'' the concept of complex numbers but the wish to describe ''nature'' in a more complete way. [[User:Robert Tito|Robert Tito]]&nbsp;|&nbsp;<span style="background:grey">&nbsp;<font color="yellow"><b>[[User talk:Robert Tito|Talk]]</b></font>&nbsp;</span>
:::By the way, does this topic fall within the field of number theory?  If so, [[User:Barry_R._Smith|Barry Smith]] might be willing to sign off on approval, either for the old draft with the proposed change or on a new draft.


== Ugh! ==
:::Apologies for taking so long to jump into the conversation. --Joe ([[User:Approvals Manager|Approvals Manager]]) 17:48, 22 July 2009 (UTC)


I don't know whether I'm more surpised that I put the superscripts in the wrong place when writing out the Laplacian (I know where they go, really!) or that I missed it so many times. I  can't read a map, either. [[User:Greg Woodhouse|Greg Woodhouse]] 18:12, 26 April 2007 (CDT)
::::Number theory is notoriously difficult to define.  Although "complex number" sounds like it might fall under the field of number theory, I think most mathematicians would agree that the entire scope of the study of complex numbers is not classified as number theory.[[User:Barry R. Smith|Barry R. Smith]] 18:44, 22 July 2009 (UTC)


== bad \scriptstyles, bad ==
::::::The ''elementary'' theory of complex numbers (which means, not the advanced theory of analytic functions) is within the scope of ''every'' mathematician. And moreover, even the advanced (if not too much advanced, maybe) theory of analytic functions is usually within the scope of number theorists. I think so. (Sorry for the late intervention.) [[User:Boris Tsirelson|Boris Tsirelson]] 19:02, 23 July 2009 (UTC)


I've taken out all the \scriptstyles from this article. On my browser it causes problems - sometimes the embedded math doesn't appear in its correct spot but rather floats somewhere nearby on the page (overlaid on other text). More importantly, though, what should ("morally") happen is that the math-display software should figure out whether the math is inline or not and display it accordingly (or, perhaps, the math tag will split into two separate tags, one for inline math and one for displayed math). I believe we should format articles to reflect structure correctly, even if the software hasn't quite caught up to that structure. - [[User:Greg Martin|Greg Martin]] 19:31, 29 April 2007 (CDT)
:::::The topic doesn't necessarily have to fall ''wholly'' within your field in order for you to approve it. In fact, most of our approved articles span several fields of inquiry and each individual editor who participates in the approval process usually comes from only one or two of the fields that are relevant. Working together, the editors from different fields approve a single topicIf number theory has something important to say about complex numbers, then that is enough for you to be an approving editor. --Joe ([[User:Approvals Manager|Approvals Manager]]) 16:16, 23 July 2009 (UTC)
:I agreeI was going to say something like that.  We should just say "math", and the browsers should figure out how to display it. --[[User:Catherine Woodgold|Catherine Woodgold]] 19:37, 29 April 2007 (CDT)


== The first sentence ==
:::::I agree with the choices that Joe gives above with the exception of 4, which seems that we should have editors decide to revoke approval, and if we have that, we might as well re-approve it.  After looking at the amount of additional edits that have been made to the draft, and considering that most were made by well informed authors and editors, I think it might be a better choice to keep working on this draft until we have something that Peter can agree to endorse. Othewise, it is likely that we will be re-approving again and again until all of the changes have been incorporated anyway. How close are we Peter? [[User:D. Matt Innis|D. Matt Innis]] 23:35, 22 July 2009 (UTC)


Here's a try at defining them completely in one sentence as Greg Martin suggests:
(unindent)
:''The '''complex numbers''' <math>\mathbb{C}</math> are numbers of the form <math>a+bi\scriptstyle</math>, where <math>a</math> and <math>b</math> are [[real number]]s and <math>i</math> is the imaginary unit, which is a solution of the equation <math>x^2+1=0</math>.
Allow me to clarify my position in some detail:
--[[User:Catherine Woodgold|Catherine Woodgold]] 19:35, 29 April 2007 (CDT)
This discussion was not started by me, but by a concern of [[User:Catherine Woodgold|Catherine Woodgold]] placed on Jitse's talk page. I noticed it, and that Jitse did not react. (He is online with some edits on WP, so I am not sure if we may count on him.)  


== less notation? ==
I probably would not have looked at the (approved) Complex number article for a long time, and if so, I would probably not have noticed the sentence under discussion because I would have read it as correct by (unconciously) ''assuming''
what the author (probably) had in mind when writing it.
(That certainly is also the reason why it was not caught at approval time.)
Checking the article, I still was not looking thoroughly enough, and thought that clarifying "scaling factor" would suffice.
However, Paul pointed out that I was wrong in reading "division" as "division of 1" -- taken literally, this was not said.


Here's a thesis to argue (about this article and about math articles in general): we should use as little mathematics notation as possible. For example, in place of
Since this sentence lacks clarity, it should be corrected. I do not see this change as "a content correction",
but I won't fight your ruling.
And I am with Catherine that it should be corrected as quickly as possible.
Since I have not yet thoroughly read the complete article I have no opinion how close to (re)approval the current draft is.
Anyway, I think this (and similar cases) should be corrected independently, and be recognized as corrections to the first approved version (I suppose that earlier approved versions will still be accessible as such? They should!) and not be hidden among a lot of changes between two approved versions.


: ... if <math>\alpha \in \mathbb{R}</math> is real ...
I am not sure if I understand what "(speciality) editor" for number theory means and implies.
Basic arithmetic with complex numbers is common knowledge for all mathematicians (and most natural scientists)
and should be known by many non-mathematicians, as well.
On the other hand, number theory also uses complex numbers, e.g., when discussing algebraic number fields.
What I want to say by this: This is not an issue for which any special knowledge is needed.


let's prefer to write
[[User:Peter Schmitt|Peter Schmitt]] 11:58, 23 July 2009 (UTC)


: ... if <math>\alpha</math> is a real number ...
:In my view the needed correction is almost equal to correcting a typographical error, it is really minute. Let's not waste any more time on it, give Peter (or me) the right to fix it in the approved article as follows
::Old: "In other words, up to a scaling factor, division by z is just complex conjugation"
::New: "In other words, up to a scaling factor, division of one by z is just complex conjugation"
:Addition  of 5 letters, 2 short words.
:--[[User:Paul Wormer|Paul Wormer]] 12:12, 23 July 2009 (UTC)


In fact, I bet we could compose this entire article without even using the symbols <math>\mathbb{C}</math> and <math>\mathbb{R}</math>! That isn't an end in itself, but it would reflect our commitment to keeping the article as accessible to non-professionals (to whom mathematical notation can be as off-putting as random Greek words in a history article might be to me) as possible. - [[User:Greg Martin|Greg Martin]] 19:55, 29 April 2007 (CDT)
::Okay, the two of you agree that it is not really a change in content but only a correction/clarification of the presentation.  Let's make the change in the approved version and then if the draft develops to the point that we need to reapprove, we will.  Matt (or Hayford, if you're watching) could you make the change Paul wrote above?  Thanks much.


::It might be educational to do both. Perhaps in the article do it as suggested above, but write an addendum-either to each section or to the article that has the "real" math. Just define the symbols, franky-I've forgotten how to read them and would like to. Is that alpha intersects the set of real numbers? Remind me, I'm not an idiot but its been 30 years. [[User:Nancy Sculerati|Nancy Sculerati]] 20:00, 29 April 2007 (CDT)
::By the way, there is a precedent for what I'm calling "ex post facto approvals". If there are not already three approving editors listed in the metadata of an approved article and another editor from the relevant workgroup(s) wishes to add his or her name to the approval, he or she may do so.  Simply add your name below the other approving editors in the "required for Approved template" section of the metadata and then notify me on the Approval Manager's talk page. So if either Peter or Barry (Barry, see my response above) would like to add his name to the approval of the current version after we make the correction, I would encourage it. --Joe ([[User:Approvals Manager|Approvals Manager]]) 16:16, 23 July 2009 (UTC)


:::It means alpha is an element of the real numbers;  in other words, alpha is a real number.    Maybe we need an article "mathematical notation" that acts like a glossary, and in each math article whenever math notation is first used there can be a footnote or something with a link to that articleOr whereever some particularly unusual notation is used there could be such a footnoteI'm thinking two such articles:  One for math notation (symbols), and another for math words, called "mathematical terminology" or "glossary of mathematical terms" or something, with definitions of words such as "countable" or "integer" for which there isn't a whole article yet or for which we might never have a whole separate article; for example we might end up with one article that talks about both the integers and the natural numbers, but it might be confusing if a link called "natural numbers" redirected to an article called "integers":  people might draw the conclusion that the two sets are the same. So a link to a glossary might be more appropriate.  Sorry it took a while to get around to answering your questionPerhaps there exists a math editor who will comment on these ideas?  :-)  --[[User:Catherine Woodgold|Catherine Woodgold]] 20:11, 3 May 2007 (CDT)
:::You are asking me to change an article that was approved by Greg MArtin and Jitse Niesen to something that Peter Schmitt and Paul Wormer think is wrong, but you want me to leave Greg and Jitse's names on itWhether I agree that the change is a good one is not an argument from an administrative perspectiveWe have two choices, get Jitse to change it or re-approve a draft version. So, if I can, I will revert all the changes on the draft and then make the change in the new draft (which anyone could have done) and it will be up to Peter to decide if he wants to use the single editor approval for it. That's as far as a constable can go, it is up to you guys to pull the rest together.  [[User:D. Matt Innis|D. Matt Innis]] 23:40, 23 July 2009 (UTC)


== Comments II ==
::::Okay, [http://en.citizendium.org/wiki?title=Complex_number/Draft&oldid=100533989 this is the new draft version] that includes reverting to the original and then changing the sentence in question. While it is possible that other changes may occur, I won't replace the changes that have been made since until after it is approved.  Then I will return the changes that were made since the first approval to the new draft version. I do not expect this to be a precedent for how things should be doen in the future without further discussion from the rest of the community and consideration is made for how it would affect other more controversial articles, but at least it doesn't give attribution to Jitse and Greg for something they did not approve, which I think is important from the standpoint of reliability and liability. [[User:D. Matt Innis|D. Matt Innis]] 23:53, 23 July 2009 (UTC)


In "algebraic closure" it says ''"We offer two separate proofs,"'' but I think the proofs are no longer included in the article.
:::::What you describe is what I was worried about but not what I thought I was asking you to do.  The approval process is in place precisely because it stops the content from slipping and sliding away from the approved version.  But if this is really just fixing an error and not changing the content, this situation isn't different from other corrections that have been made.  I get the impression from the mathematicians here that this is akin to fixing a sentence that mistakenly says "3 divided by 6 is 2" when the truth is that "6 divided by 3 is 2".
:::::I don't want the situation to become acrimonious though.  So let's see if we can't simply reapprove.  That means we need an editor to start the process. --Joe ([[User:Approvals Manager|Approvals Manager]]) 00:31, 24 July 2009 (UTC)


:Fixed. [[User:Greg Woodhouse|Greg Woodhouse]] 08:14, 30 April 2007 (CDT)
:::::: Sorry, I shall not be able to join in again before tomorrow (in about 30 hours) -- I say this to avoid the impression that I do no longer follow this. For the moment, just a short remark: The change is similar to changing a "this is" in a "the last mentioned is" to avoid that the "this" is misunderstood. Is this (only) so complicated because it is mathematics? [[User:Peter Schmitt|Peter Schmitt]] 10:08, 24 July 2009 (UTC)


In "historical example" where it says ''"Now we choose the second condition on <math>u</math> and <math>v</math>, namely <math>3uv-15=0</math>, or <math>uv=5</math>. "'' I don't like "namely" here because it suggests something that had previously been defined (or decided). I would replace "namely" with "as".
:::::::Not exactly because it is mathematics but because it is a topic that neither Matt nor I are expert in.  We both want to make sure that we aren't making a change to the approved article that the original approving editors would '''dis'''approve of.  It seems that that is the case here, but the only way for us to be ''absolutely'' sure is to get word from the original approving editors.  Spelling or grammar mistakes are easier because anyone can recognize them. I hope we can re-approve the article and improve it even more in other ways while we're at it; that's what re-approval is all about, after all.  --Joe ([[User:Approvals Manager|Approvals Manager]]) 14:19, 24 July 2009 (UTC)
:::::::: Now you have chosen to let the article persist (for who knows how long) with&mdash; what [[user: Catherine Woodgold|Catherine Woodgold]] considers to be&mdash;a blatant error.--[[User:Paul Wormer|Paul Wormer]] 15:56, 24 July 2009 (UTC)


:I'm not yet sure how to rephrase this. [[User:Greg Woodhouse|Greg Woodhouse]] 08:14, 30 April 2007 (CDT)
::::::::: Just to be clear, all that needs to be done is for a math editor who has not made an edit to [http://en.citizendium.org/wiki?title=Complex_number/Draft&oldid=100533989 approve this version of the draft]. The only thing that is different than the current approved version is the edit that was requested - "In other words, up to a scaling factor, division '''of one''' by z is just complex conjugation". [[User:D. Matt Innis|D. Matt Innis]] 01:28, 25 July 2009 (UTC)
::Replacing "namely" by "as" looks good to me. The point is that the reader may ask ''why'' we ''can'' impose that simplifying condition. This is not explained and I'm not sure whether it should be (do you see a short convincing answer? If so, maybe just a short footnote could be added). --[[User:Aleksander Stos|AlekStos]] 10:20, 30 April 2007 (CDT)
:::Well, I'd give the following motivation for the condition. "''At first glance, the condition is quite strong and may rule out some solutions. But at the moment we need only one solution and the condition simplifies the way to find it''". If you find this explanation useful, some form of it could be put into a footnote.--[[User:Aleksander Stos|AlekStos]] 10:33, 30 April 2007 (CDT)
::::I'm not sure whether more explanation is required than what's already in the article, but I might explain it like this: 
:::::''[[Cardano's method]] for solving it suggests looking for a solution by writing it as a sum <math>x=u+v</math>, which leaves us free to choose another condition on <math>u</math> and <math>v</math> later;  for example, we could assign any value to <math>u</math> and then <math>v</math> would be determined. ... Now we choose the second condition on <math>u</math> and <math>v</math> as <math>3uv-15=0</math>, or <math>uv=5</math>.  We can do this because one of the two variables can be any number of our choice.''
::::Here's another try:
:::::''[[Cardano's method]] for solving it suggests looking for a solution by introducing a number <math>u</math> which we will choose later, and letting <math>v</math> be <math>x - u</math>. ...  Now we choose <math>u</math> to be the number such that <math>3uv-15=0</math>, or <math>uv=5</math>.''
::::Or yet another try:
:::::''[[Cardano's method]] for solving it suggests looking for a solution by writing it as a sum <math>x=u+v</math>, which introduces an additional degree of freedom, allowing us to add another equation of our choice later as a constraint on the two variables. ... Now we choose the additional equation constraining the two variables as <math>3uv-15=0</math>, or <math>uv=5</math>.''
::::--[[User:Catherine Woodgold|Catherine Woodgold]] 19:50, 30 April 2007 (CDT)
:::::Jitse Niesen has found a clear, concise way of saying this.  Well done:
::::::''Now we recall that we did not completely specify <math>u</math> and <math>v</math>; we only required that <math>x=u+v</math>. Hence, we can choose another condition on <math>u</math> and <math>v</math>. We pick this condition to be...''
:::::--[[User:Catherine Woodgold|Catherine Woodgold]] 08:06, 5 May 2007 (CDT)


In "formal definition", ''"It follows that <math>\sqrt{-1}</math>, the hypothetical number whose square root gives -1, is well-defined as (0,1)"'' This sounds wrong to me.  How about "''"It follows that a candidate for <math>\sqrt{-1}</math>, the hypothetical number whose square root gives -1, is well-defined as (0,1)"''  (or maybe ''has been constructed'' instead of ''is well-defined''.)
Well, I have just filled out the template (hopefully, correct -- is it true that the date should be given in this form???).
I have read the article and discovered nothing serious. (Some historical remarks should perhaps be checked.)
I also see some things I would change or extend, but this would take time -- and may be a cause for long discussions.
Furthermore, if I would start to edit, I would not be able to approve it. So it seems best to leave the article as it is.
[[User:Peter Schmitt|Peter Schmitt]] 00:42, 26 July 2009 (UTC)


:That should just be "defined". (I think thi is a remnant of previous edits.) [[User:Greg Woodhouse|Greg Woodhouse]] 08:14, 30 April 2007 (CDT)
:Peter, thanks for taking the lead. After re-approval, we can keep working on any minor adjustments on the draft version.  --Joe ([[User:Approvals Manager|Approvals Manager]]) 00:50, 26 July 2009 (UTC)
::No. It's just moderate command of English of the author ;-) Be aware, please, that some text just needs copy editing and be bold... Otherwise I'd end up by finding that I bring more problems than help. --[[User:Aleksander Stos|AlekStos]] 10:05, 30 April 2007 (CDT)
:::I inserted "a candidate for"Either "defined" or "well-defined" looks fine to me (it now says "defined").  Actually, "constructed" might be better than "defined". --[[User:Catherine Woodgold|Catherine Woodgold]] 17:37, 1 May 2007 (CDT)
Re multiline equations:  The following syntax works on Wikipedia but does not work here:
<nowiki>
<math>
\begin{align}
f(x) & = (a+b)^2 \\
      & = a^2+2ab+b^2 \\
\end{align}
</math>
</nowiki>
--[[User:Catherine Woodgold|Catherine Woodgold]] 07:58, 30 April 2007 (CDT)


:I [http://forum.citizendium.org/index.php/topic,892.0.html reported] this issue on the forums. -- [[User:Jitse Niesen|Jitse Niesen]] 08:33, 3 May 2007 (CDT)
::Excellent. I'll wait to get the go-ahead from Joe ([[User:Approvals Manager|Approvals Manager]]) tomorrow. Sorry for any undue delay, but all in all, this way we have a legitimate single editor approval that has been duly scrutinized and agreed upon. Peter will also then be able to make any 'corrections' to the approved article once his name is on it. Meanwhile, if anyone would like to discuss changing the process to make these things easier, feel free to bring your ideas to the forums or the [[CZ Talk:Approval Process]]. [[User:D. Matt Innis|D. Matt Innis]] 02:00, 26 July 2009 (UTC)


== General structure revisited ==
:::Looks good to me.  Go ahead with the mechanics!
:::If other editors still wish to lend their support to the newly approved version, they may do so by following the steps I described above for "ex post facto approvals". --Joe ([[User:Approvals Manager|Approvals Manager]]) 14:19, 27 July 2009 (UTC)
==APPROVED Version 2.0==
<div class="usermessage plainlinks">Discussion for [http://en.citizendium.org/wiki?title=Complex_number/Draft&oldid=100533989 Version 2.0] stopped here. Please continue further discussion under this break. </div>


The present version is significantly improved, refined and 'cleaned up' as compared to what it was a few days ago. Still, I miss something from the past, something I'd venture to label as a logical flow. For example, there were quite natural passages from one section to another and some logical order of appearance. Now, each section looks like a bit isolated issue. For example,  the geometric interpretation begins with "''Since complex number z = x + iy corresponds to an ordered pair of real numbers(...)''". We have never introduced those pairs  before! (the formal definition is at the end). Also, the sequence of headings "algebraic closure - complex numbers in physics - formal definition" does not look very natural. In particular, our formal definition was designed to be not very abstract (nor long) and to make a smooth link from a "historical motivation" to "working with complex numbers". At the end of article the section is quite pointless as it makes direct references to both "historical" and "working" sections. Paradoxically, I think it'd be better to suppress the "formal" section as it stands than to keep it at the end. But IMHO it would be a _big_ pity to delete it, just think about possibly most interested readers here, i.e. math students.
This version includes one of many changes that were made to the draft since the first approval by other editors. I'll replace the old draft changes now so that they may be considered in any future versions. [[User:D. Matt Innis|D. Matt Innis]] 02:51, 28 July 2009 (UTC)


I do not claim that the former version was 'better'; just one aspect of it better matched my understanding of [[CZ:Article Mechanics]]. If I am the only one to think like this, then forget it. If not, surely something positive can be done with the present structure too.
:I've restored the draft.  One important thing to remember is that the sentence that was changed in the new version: "In other words, up to a scaling factor, division of one by z is just complex conjugation" does not appear in the draft because the sentence was essentially reworked too much for me to replace.  It woul dbe nice for someone to take a look and make sure that what is there is accurate.  Thanks all! [[User:D. Matt Innis|D. Matt Innis]] 03:00, 28 July 2009 (UTC)


The point is that if we present just isolated issues, it would be not that different from Wikipedia, where everybody inserts his two cents and the result resembles a (good) "reference manual" instead of  some introductory text. For me, a main advantage of CZ would be an integration of presented ideas at the top level. A general thought present from the beginning to the end. A more continuous flow instead of hedgehog of facts. Can it be achieved in a system of wiki? Sometimes I see CZ as an experiment to prove that the answer is 'yes'.  --[[User:Aleksander Stos|AlekStos]] 09:55, 30 April 2007 (CDT)
== correcting approved version ==


:I took another look at the opening section and reworded it somewhat. I hope this goes at least some of the way to addressing your concerns. Frankly, though, I don't see a problem with the "ordered pair" issue. The reason is that the section on geometric intepretation says that the complex number <math>x + iy</math> ''corresponds'' to the the ordered pair <math>(x, y)</math>, it doesn't say that it ''is'' the ordered pair. Later on, in the "Formal definition" section, the text says that ''Hamilton'' defined complex numbers as ordered pairs. Okay, so now we're introducing a new point of view. Making note of this explicitly seems to me to be, well, pedantic. I won't argue that th general sense of coherence and logical flow in the article hasn't suffered through the process of editing and re-editing, but I really don't think things are so bad as you seem to suggest. [[User:Greg Woodhouse|Greg Woodhouse]] 18:04, 30 April 2007 (CDT)
Concerning the problems with [[Complex number#Complex numbers in physics]] in the approved version of this page
::Well, I do not claim it's bad (and to make it clear: I do not think there is something that prevents the article from being approved). But I'd argue that it just misses a little something that is worth 'fighting' for. A general synthetic perspective. As e.g. that provided by Sébastien's plan and further discussions. Or virtually _any_ other. I thought CZ encourages this in its [[CZ:Article Mechanics]] and e.g. after approval we could work it out (as well as many other 'local' improvements). Maybe I used some strong formulations in my previous post but this is because a consensus about using a synthetic perspective is quite important to me. I'd like to help establish an "style manual" in math that would accent e.g. explanatory side of our texts and this 'general synthetic' approach. But this is a more general issue not that much concerning the present article; this could be discussed in more detail on the forum. --[[User:Aleksander Stos|Aleksander Stos]] 02:43, 1 May 2007 (CDT)
(see [[#QM again]] Paul Wormer has suggested [http://forum.citizendium.org/index.php/topic,2968.msg26446.html#msg26446]
to delete it completely.
After some thinking about the options to resolve the problem, I consider it best to remove this section from the approved version and '''not''' to approve the current draft which contains several major revisions including the removal of another section, too.
The article is nicely written but lacks some characteristics that an encyclopedia's (main) page should have.
Therefore it is best to keep it as it is as an approved page (but correct errors, of course), instead of rewriting it partially.


It seems to me is that their is a fundamental tension between what might be called the natural structure of an article and our tendency to try and move more abstract or difficult material to the end of the article (or delete it altogether). I don't know how many times I was asked to move something to "later on" or "much later on". In many cases, that's appropriate, but I don't think it is ''always'' so. How can we not disrupt the flow of a mathematics article when formal arguments are removed, definitions are pushed to the end of the article, and key ideas are not introduced early on? I certainly agree that articles, especially articles about basic topics like complex numbers, shouldn't scare the reader away right off the bat, but perhaps we need to temper our desire to make the article start out slowly and in a non-intimidating fashion with a bit of logical coherence.
May I do the necessary steps without being disqualified as approving editor?
[[User:Peter Schmitt|Peter Schmitt]] 23:58, 18 December 2009 (UTC)


Truth be told, one of the things I've hoped to accomplish with Citizendium is making difficult ideas more accessible, and I think I can write reasonably well (if I can force myself to do it!) but we can't avoid asking the reader to ''think''. Now, we're all accustomed to the traditional theorem and proof approach to presenting mathematics, and that may not always be the best style to use. But proofs are the "why" of mathematics. If our articles contain nothing but definitions and statements of results, then I believe we have failed. Exposition can take the form of detailed examples, exploratory questions, computations with commentary, and probably more. But, at some level, I think we need to provide the reader with some of what the basic ideas are and how they fit together. Who knows? Maybe people will come a little less mathematics averse if it isn't presented to them as a series of definitions and techniques to be learned by rote. [[User:Greg Woodhouse|Greg Woodhouse]] 13:50, 1 May 2007 (CDT)
:Peter, BECAUSE you are the editor whose name is on this article, I think it is appropriate that you should be able to change information that you *now* feel is improper. I have reverted the draft to the current approved version. To continue to be within our [[CZ:Approval Process]] guidelines on single editor approval, how about asking Paul if he would clean up the physics section in any way that he feels necessary (as we would want a physics special expert to do) and then decide if you want to re-approve that version. [[User:D. Matt Innis|D. Matt Innis]] 01:11, 19 December 2009 (UTC)


:Many thanks for your thoughtful post. Concerning "moving something later on" etc., I do not think that the formal definition is absolutely needed near the beginning. I do not think that without it the article can not be naturally constructed. Formal definition at the end is just _fine_ with me. My only concern was the text flow: the 'formal' section was written to logically fit a particular position in the presentation and bound to the surrounding text. Moving it requires reworking both the section itself and some other parts of the article (well, something has been done).
::I haven't seen any movement here. Is this solution not working for you, or is it just a low priority? [[User:D. Matt Innis|D. Matt Innis]] 16:48, 22 December 2009 (UTC)
:::Yes, Peter what's happening?--[[User:Paul Wormer|Paul Wormer]] 17:56, 22 December 2009 (UTC)


:Generally, I'd like too see a text that presents the issue passing smoothly from one section to another (and this does not seem to imply the 'traditional' ''definition-theorem'' presentation at all). For example, I'd venture to say that despite some problems it was at least partially achieved in the previous version in the sequence "historical example -> formal definition -> working with complex numbers". There was 'general idea': we need \sqrt{-1} - but what the hell can it be, if it can not be 'real'? - so we make a new definition - ok, what do we get, in fact? how do we work with this? and then, are there other benefits?. But I _do_not_ think it is the only natural configuration. I agree with your "explanatory approach". Clearly, this can be done --even more naturally than in "formal" approach-- with some 'general idea' too: a part of the text gives rise to a question that is answered by what follows and so on.
:::: Sorry, if I kept you waiting. During the last days I did not do much here. (And today I suddenly lost contact to CZ for several hours.) But the main point is: Since I am expected not to touch the draft I have waited for someone to do the edits (remove the physics section, and - perhaps - make the already corrected "division sentence" still more explicit.
:::: Paul, I have seen your new picture, and I have some comments, but I think that this need not concern us here. (A theoretical question: If I edit the caption, and the picture is used - would this disqualify me as approving single editor?) Since we both think that a new start will be best it is better to have the article (and the draft) as untouched as possible. After correcting its factual errors we can without time pressure think about a good place for it (subpage?) and a replacement. (I shall probably prepare something to start with offline first.)
:::: Off topic: I have corrected the typo in [[Literature/Draft]]. There should be now problem to correct it on the Approved page. And the other open cases should also be possible, even if Editors are needed. There are History and Biology Editors active.
:::: [[User:Peter Schmitt|Peter Schmitt]] 00:20, 23 December 2009 (UTC)


: BTW, concerning the formalism I think it should not be used unless it is inevitable (well, often it is..) or when it is needed for accuracy reasons or, last but not least, when we want help students --a large and important part of our 'audience', after all--  understand the very formalism as well. --[[User:Aleksander Stos|Aleksander Stos]] 16:46, 1 May 2007 (CDT)
:::::I took care of the spelling error on Literature/Draft, thanks!  My understanding is that any corrections to content on this page would have to be made by Paul (including image captions) in order for the single editor process to remain a viable option. [[User:D. Matt Innis|D. Matt Innis]] 15:11, 23 December 2009 (UTC)


::I think that the order in the current section is quite natural and that a good narrative can be constructed out of it. After all, this is more or less the historical order: cubic equations (Cardano et al.), working with complex numbers (e.g., Euler), fundamental theorem of algebra (d'Alembert/Gauss), formal construction (Hamilton). Only the section on quantum mechanics is out of order historically. In fact, I thought that the previous version suffered from a sudden transition where it introduced the formal definition; the multiplication came quite out of the blue.
== Nomination for reapproval ==
::Nevertheless, Aleksander makes a good point. We moved some sections of the article around and this often introduces jarring transitions. Some of them have been smoothened out, but more work is needed. -- [[User:Jitse Niesen|Jitse Niesen]] 09:34, 3 May 2007 (CDT)


== Approval discussion ==
Thanks, Paul, for making the corrections! I have now nominated the draft for reapproval.
I have set the date to today, so that this step can be performed whenever a Constable feels ready.
(For the reasons for this reapproval see above.)


I'm not sure I understand the approval process correctly, but I believe that if nothing happens the [http://en.citizendium.org/wiki?title=Complex_number&oldid=100091926 version] mentioned by Greg Martin in the approval notice becomes the approved version. I'm quite keen to remove the reference to Gauss (1831), which I did in the latest edit. As far as I can find out, Gauss was the first to use the term "complex number" in 1831, but I didn't find anything about a rigorous definition.
For more information see also my talk page: [[User talk:Peter Schmitt#Complex numbers again|Complex numbers again]].
I hope that no further problems will surface in this article.
However, as discussed there, Paul and I think that eventually this article should be replaced by a more encyclopedic one,
while this colloquial introduction should find another home.


:I believe Gauss published his proof of the fundamental theorem of algebra in his doctoral dissertation, which appears in "modern form" in appendix A of Benjamin Fine and Gerhard Rosenbetrger, ''The Fundamental Theorem of Algebra''. The history of mathematics isn't even close to being my strong point, but I think Gauss spoke in terms of "multiply extended quantities". On the other hand, the ring <math>\mathbb{Z}[i]</math> is known as the Gaussian integers. Again, I don't know how Gauss described them (in his own notation). Something to look into, I suppose. [[User:Greg Woodhouse|Greg Woodhouse]] 10:44, 3 May 2007 (CDT)
[[User:Peter Schmitt|Peter Schmitt]] 12:46, 25 December 2009 (UTC)


::I only know a bit about the history because I spent 15 minutes in the library last week. As I remember it, Gauss did indeed prove the fundamental theorem in his thesis but his proof was not rigorous. Gauss wrote his thesis before 1831, I think (around 1800?). He published several other proofs of increasing rigour.  
==APPROVED Version 2.1==
::It seems that Sébastien Moulin might know more on this, so I tried to e-mail him to ask for clarification. -- [[User:Jitse Niesen|Jitse Niesen]] 10:48, 5 May 2007 (CDT)
<div class="usermessage plainlinks">Discussion for [http://en.citizendium.org/wiki?title=Complex_number/Draft&oldid=100613198 Version 2.1] stopped here. Please continue further discussion under this break. </div>


Greg Martin, if you change the approval template to mention [http://en.citizendium.org/wiki?title=Complex_number&oldid=100094888 the current revision] (as I'm writing), I support the approval. If you change it to a later version, I'll also support, unless there are some substantial edits that make the article worse. -- [[User:Jitse Niesen|Jitse Niesen]] 09:22, 3 May 2007 (CDT)
== Phasors ==


::[[User:Greg Woodhouse|Greg W.]] makes an excellent point, one that would be easily overlooked by us editors new to the approval process: when changes are made to "ToApprove" articles, we need to update the template accordingly (not necessarily every update, but at least before the approval kicks in). I'm about to do so for this article (and [[Prime number]] too). Actually any mathematics editor would be free to do so, by my understanding - being bold and making reasonable assumptions on what I as the original editor would find a reasonable updated version. I can always protest later in strange cases. - [[User:Greg Martin|Greg Martin]] 00:26, 5 May 2007 (CDT)
I think there should be a section in this article about complex numbers expressed as [[phasor]]s, which are commonly used in electrical engineering calculations involving [[alternating current]] [[voltage]]s, [[Electric current|currents]], and [[impedance]]. Phasors are complex numbers expressed in terms of magnitude and phase angle. [[User:Henry A. Padleckas|Henry A. Padleckas]] 11:29, 28 February 2011 (UTC)


== discovered algebraically closed? ==
: You are talking about the polar form, aren't you? Calling this "phasor" in the context of the complex number would be confusing.
 
: It is rather the other way: Phasors are described with the help of complex numbers (and not: complex numbers are expressed as phasors). --[[User:Peter Schmitt|Peter Schmitt]] 22:24, 28 February 2011 (UTC)
In the current article: ''(As a final comment in this analysis, we discover that <math>\mathbb{C}</math> has no finite [[field extension|extension]] and must therefore be [[algebraic closure|algebraically closed]].)'' The fact itself is true of course, but I don't see how the preceding discussion allows us to conclude this "discovery". - [[User:Greg Martin|Greg Martin]] 00:30, 5 May 2007 (CDT)
::Yes, the polar form, as in &nbsp;<math>  r \ang \theta . \,</math>.  [[User:Henry A. Padleckas|Henry A. Padleckas]] 22:41, 28 February 2011 (UTC)
 
: I used "discover" because I was trying to avoid using "can" again (as in "can show"). It seemed good at the time, but I see your point. I changed it to "could next show"... How does that sound? - [[User:Jared Grubb|Jared Grubb]] 11:14, 5 May 2007 (CDT)
 
== What the symbol i means in this article ==
 
I'd like to raise the level of mathematical rigour in the first two paragraphs of "working with complex numbers".  The most important change, I think, would be to replace
:''In the remainder of the article, we will use the letter ''i'' to denote the "number" <math>\sqrt{-1}</math>.''
with this:
:''In the remainder of the article, we will use the letter <math>i</math> to denote a solution to the equation <math>i^2 = -1</math>.
This would bring it up to the level of rigour shown in the first sentence of the article, which is good enough here, I think.  It's OK to use <math>\sqrt{-1}</math> in the cubic-equation section, since it's made clear there that the symbol may not have meaning.  In "working with complex numbers" we have to be a little more careful with the problem that <math>-i</math> is also a square root of -1.  Just stating in a footnote that there is a problem is not as good as giving a better definition for a symbol that the rest of the article is going to depend heavily on.
 
I'm not thrilled with the sentence ''"The first step in giving some legitimacy to the "number" <math>\sqrt{-1}</math> is to explain how to compute with it."''  That is not the only possible first step.  Some people might prefer to see a rigourous definition as a first step.  How about changing "The" to "A" or "Our"? --[[User:Catherine Woodgold|Catherine Woodgold]] 08:27, 5 May 2007 (CDT)
 
:I see what you mean, and I implemented your suggestions with some changes. It seems important to me to state explicitly that the symbol <math>i</math> in that section is the same as the symbol <math>\sqrt{-1}</math> in the previous section. -- [[User:Jitse Niesen|Jitse Niesen]] 10:27, 5 May 2007 (CDT)
 
:*I like your changes.  Here's another possible edit, mainly to remove redundancy: at the end of the 2nd paragraph of the "working with complex numbers" section, to change
:::''Complex numbers whose imaginary part is <math>0</math> are of the form <math>a+0i</math>. In this way, the real number <math>a</math> is considered as the complex number <math>a + 0i</math> whose imaginary part is zero.''
::to this:
:::''The real number <math>a</math> is considered to be the complex number <math>a + 0i</math> whose imaginary part is zero.''
::Besides deleting more than a sentence, I'm suggesting changing "as" to "to be";  or maybe to "to be equivalent to" or "to be identical to" or "to be the same thing is".
::Actually, even better perhaps would be to turn it around the other way:
:::''The complex number <math>a + 0i</math> whose imaginary part is zero is considered to be the same thing as the real number <math>a</math>.''
::Or maybe not.  Mentioning the real number first implies that this can be done for any real number.
:*Another thing:  Near the bottom of "the complex exponential" section it says
:::''Of course, there is no reason to assume this identity.  We only need note that...''
::However, I think the identity had just been proven.  Better wording might be:
:::''There is another simple way to establish this identity.  We need only note that ...''
::or
:::''Another way to establish this identity is to note that...''
:*Oh-oh -- I think the quantum physics section could use a bit of editing. <math>m</math> and <math>\hbar</math> should be defined, for example. Is <math>\hbar</math> Planck's constant?  (Or maybe a reduced Planck's constant, divided by 2 pi?)  Presumably <math>m</math> is mass, e.g. the mass of the electron in the example stated.  I would like to see <math>\psi</math> defined earlier in the section:  it could be introduced by saying that the equation is an equation for the "probability amplitude" <math>\psi</math> (and possibly add that it's defined as a scalar at each point (x,y,z) in space).  Otherwise one sees this equation with a bunch of symbols and doesn't know which one is considered the variable.  An alternative way to define x, y and z would be to add "at position (x,y,x)" after "and an electron".  I would delete "per unit mass" because if an electron is specified then mass is a constant, and because the equation seems to be dividing by mass anyway, so saying "per unit mass" doesn't seem correct to me here. I think it might be better to give the equation relating the probability amplitude to the probability much earlier in the section, as a definition of <math>\psi</math> before giving the equation for <math>\psi</math>. Maybe more could be said about why it would be hard to do this without complex numbers, e.g. ''"In this way, using <math>\psi</math> which takes on complex values, a relatively simple and useful equation for the real-valued probability can be established.  There is no known neat, simple way to do this without using complex numbers."'' --[[User:Catherine Woodgold|Catherine Woodgold]] 12:35, 5 May 2007 (CDT)
 
:*I think that the following in "formal definition" is misleading:
:::''Positive real numbers satisfy the identity
:::::<math> \sqrt{a}\times\sqrt{b} = \sqrt{a \times b}, </math>
:::but this identity does not hold for negative real numbers, whose square roots are not real.''
::I would say that the identity holds if the symbol <math>\sqrt{a}</math> for example is taken to mean a solution of <math>x^2 = a</math> rather than the positive solution.  In this case I see the problem as lying with the third, not the second, equals sign.  Oh, well, maybe it's fine as it is.  One way to fix it up could be to append ''"because the square root symbol denotes the positive solution to <math>x^2 = a</math>, not just any solution"''  after ''"whose square roots are not real"''.
:*One of those dreaded "this's" as pronouns appears at the beginning of the "formal definition" section but I'm not sure it can easily be reworded.  Maybe ''"The above all shows"'' or ''We've seen that'' or something.  Actually, it seems somewhat inaccurate as well as vague:  ''"This all shows that complex numbers behave very much like real numbers"'' seems to mean that we've shown that they can be used without running into inconsistencies, but we haven't proven that yet.  I added a "this" for "This approach was taken by Hamilton", but that's "this" as an adjective, which is OK, I believe.
:*In the "formal definition" section it says ''"Such pairs can be added and multiplied as follows"''.  I think it would be better to say something like ''"Addition and multiplication for such pairs can be defined as follows."'' (or "he defined them" or "we define" or something.)
:*I would prefer to see the terms "commutative", "associative" and "distributive" appear in the text.  Explanations of them can appear too, but I would at least put these technical terms in parentheses after each explanation.  That way, people familiar with the terms can read faster, skimming over the explanations.  Putting them in footnotes doesn't help anyone read faster. I was taught these terms in elementary school.  I think it would help to say something about the proofs that these properties hold, e.g. "It can easily be shown that...".  How about like this:
:::''These definitions satisfy most of the basic properties of addition and multiplication of real numbers, and we can employ many formulas from the elementary algebra we are accustomed to. More specifically, it can easily be shown that addition and multiplication as defined above are commutative, associative, and that multiplication is distributive over addition;  in other words,...''
:*It says ''" In other words, we can define <math>i</math>, the number satisfying <math>i^2=-1</math>, as the pair (0,1)."'' which seems to me to be saying that there is only one number satisfying that equation ("the"). How about this instead: ''"In other words, we can define <math>i</math>, the symbol we've been using, as the pair (0,1).  In this way we have a way of indicating which one we mean of the two solutions of the equation <math>i^2=-1</math>;  the other is now denoted (0,-1)."''  --[[User:Catherine Woodgold|Catherine Woodgold]] 13:18, 5 May 2007 (CDT)
 
 
== Are we there yet? ==
 
Shall a constable place the approval tag? It is May 6, unless I hear otherwise, I will contact a constable to do so. Should the editors wish to do so, on their own, I encourage them to proceed. [[User:Nancy Sculerati|Nancy Sculerati]] 16:22, 6 May 2007 (CDT)
:I think we are there Nancy. [[User:Robert Tito|Robert Tito]]&nbsp;|&nbsp;<span style="background:grey">&nbsp;<font color="yellow"><b>[[User talk:Robert Tito|Talk]]</b></font>&nbsp;</span>
 
 
==APPROVED Version 1.0==
<div class="usermessage plainlinks">Discussion for [http://en.citizendium.org/wiki?title=Complex_number&oldid=100097396 Version 1] stopped here. Please continue further discussion under this break. </div>
 
== See comments above ==
 
I made some comments in the section [[Talk:Complex number/Draft#What the symbol i means in this article]] above, marked with bullet points, which have not yet been addressed.  (I just wanted to mention them after the page break so they won't be forgotten when editing the next version.) --[[User:Catherine Woodgold|Catherine Woodgold]] 20:28, 6 May 2007 (CDT)
 
:I edited in all of the changes I had suggested except for the stuff about quantum physics.  I don't have a textbook on the subject handy.  --[[User:Catherine Woodgold|Catherine Woodgold]] 19:17, 7 May 2007 (CDT)
 
I'm not entirely happy with my text in the QM section, either. Trying to interpret superposed states in terms of probabilities is dicey at best, anyway. I'll have to think about this and see if I can come up with something better. In any case, I'm intrigued by what Robert Tito had to say about other uses of complex numbers, particularly in Hamiltonian systems (conjugate coordinates with a factor of i?) Anyway, I was just trying to come up with something that would be recognizable to a wide range of readers (albeit not mathematically naïve ones). If nothing else, the Schrödinger Equation has a certain iconic value. I'm certainly open to other suggestions. [[User:Greg Woodhouse|Greg Woodhouse]] 19:38, 7 May 2007 (CDT)
 
:Wait!  I think the quantum physics stuff is good!  It just needs some editing, as I suggested, e.g. defining the symbols used etc.  --[[User:Catherine Woodgold|Catherine Woodgold]] 19:42, 7 May 2007 (CDT)
 
It seems to me that saying things like "<math>\hbar</math> is Planck's constant divided by <math>2\pi</math>" wouldn't really add anything to the article, and I guess that's what bothers me: if you (generic) know what Schrödinger's equation is, this probably doesn't need to be said, and if you don't, the section really doesn't add anything. [[User:Greg Woodhouse|Greg Woodhouse]] 22:40, 7 May 2007 (CDT)
 
== Part of chemistry and physics workgroups? ==
 
(I hope no one minds if I move this discussion "below the bar". [[User:Greg Woodhouse|Greg Woodhouse]] 03:34, 7 May 2007 (CDT))
 
I am just curious why this article's checklist includes it in the chemistry and physics workgroups. It seems that even though this article has applications in those field, including it in every workgroup it applies could get out of hand. - [[User:Jared Grubb|Jared Grubb]] 12:32, 6 May 2007 (CDT)
:that answer is simple: the need for something like a complex number arose from these sciences not from math. Math formalized it, thats all. [[User:Robert Tito|Robert Tito]]&nbsp;|&nbsp;<span style="background:grey">&nbsp;<font color="yellow"><b>[[User talk:Robert Tito|Talk]]</b></font>&nbsp;</span>
 
I disagree. We have an example showing that complerx numbers are important in the sciences, too, but complex numbers were intoduced in a fundamental way in mathematics (i.e., not just as a notational convenience) long before quantum mechanics had even been thought of. [[User:Greg Woodhouse|Greg Woodhouse]] 16:59, 6 May 2007 (CDT)
 
Then again, since those workgroups are there, maybe you can sign off on it, too. :-)  [[User:Greg Woodhouse|Greg Woodhouse]] 17:02, 6 May 2007 (CDT)
 
Physics and chemistry used the notion of compplex numbers as from the 18th century - when they needed them to describe things. Euler, Gauss, Fourier are not mathematicians but physicists/chemists that needed a solution for their math problems. the complext number by far didn't start with quantum mechanics. I might mention Hamiltonian mechanics as an example, or canonicals. [[User:Robert Tito|Robert Tito]]&nbsp;|&nbsp;<span style="background:grey">&nbsp;<font color="yellow"><b>[[User talk:Robert Tito|Talk]]</b></font>&nbsp;</span> 17:35, 6 May 2007 (CDT)
 
: It still seems a little odd to me. But, I suppose [[Ohm's law]] would be more at home in the electrical engineering workgroup than the mathematics, even though it is a mathematic equation... - [[User:Jared Grubb|Jared Grubb]] 23:14, 6 May 2007 (CDT)
 
The historical development of the concept of complex numbers seems like an interesting topic for an article (albeit a challenging one!), but so far as ''this'' article is concerned, I don't think it's really that important. No, that doesn't sound right: I don't mean it's not important, only that I don't think it needs to be addressed in the context of this article. [[User:Greg Woodhouse|Greg Woodhouse]] 03:34, 7 May 2007 (CDT)
 
Does anyone know how to create an archive? Is there an automated, or at least "official" way to do it? I just got the following warning:
 
'''WARNING: This page is 87 kilobytes long; some browsers may have problems editing pages approaching or longer than 32kb. Please consider breaking the page into smaller sections.'''
 
[[User:Greg Woodhouse|Greg Woodhouse]] 03:34, 7 May 2007 (CDT)
 
:Re archiving:  you might want to discuss with Chris Day or see [[Talk:Biology/Draft]], but since that template is named "Experimental" I suppose procedures haven't been finalized.  (Discussion about it arising from pages like this one may drive the finalization of such procedures.)  Maybe it's being discussed on the forum somewhere, or if not someone could start. --[[User:Catherine Woodgold|Catherine Woodgold]] 07:45, 7 May 2007 (CDT)
 
:Re being part of chemistry and physics workgroups:  Chemistry and physics also need to use 1 + 1 = 2.  They also need to use words with syllables to communicate technical concepts;  that doesn't mean the linguistics Syllable page has to be in the chemistry and physics workgroups.  Those sciences use math -- that doesn't mean math is part of the science. I think perhaps people in the chemistry and physics workgroups should decide whether the article is included or not.  It's OK with me either way -- it's not that unreasonable.  More justifiable than including a page that presents a proof of 1 + 1 = 2 in those sciences.  --[[User:Catherine Woodgold|Catherine Woodgold]] 18:23, 7 May 2007 (CDT)
 
This article was approved by a math editor and currently is listed in the Math Workgroup Approved articles, but not in Chemistry or Physics. Since the article is cross-listed in three workgroups, will there need to be three approval processes? Or will we need editors from all three areas to agree before any one draft gets approved? Or will we declare one "father" workgroup, and the others just raise objections or not... I know this approval process is still in its infancy, but these are questions we really should address at some point. - [[User:Jared Grubb|Jared Grubb]] 02:41, 8 May 2007 (CDT)
 
== Error in multiplicative property ==
 
I've just put the following message on [[User talk:Nancy Sculerati]].
<blockquote>
Dear Nancy. [[User:Etienne Parizot|Etienne Parizot]] found and fixed an error in [[Complex number/Draft]] which is also present in the approved version, [[Complex number]]. The formula
:<math>e^{z_1 z_2} = e^{z_1} e^{z_2} \ </math>
halfway the section "The complex exponential" should read
:<math>e^{z_1 + z_2} = e^{z_1} e^{z_2} \ </math>
(with a plus sign added on the left-hand side).
 
This is a big error so I want it to be fixed as soon as possible. I can't imagine any editor would argue with this change. However, I'm not sure what our options are.
* Some places hint at the possibility to have the constabulary do limited changes to articles without going through the whole approval process (for instance, the section [[#Copyediting matters]] above). I couldn't find anything about rules or procedure though. If such a possibility exists, that would be my preference. For the record, the article [[complex number]] was nominated by [[User:Greg Martin|Greg Martin]] and the nomination was supported by me.
* If this is not possible, I'd like the approval to be revoked. As far as I can see, there is no rule or precedent for this, only an empty section at [[CZ:Approval Process]].
* If neither of the above is possible, or if it would take too long, we can always go for the option of nominating the fixed version for approval. To be honest, it's not that important in the big scheme of things, but it is embarrassing and I feel responsible for it.
 
Any guidance from you (or anybody who happens to read this) would be much appreciated.
</blockquote>
-- [[User:Jitse Niesen|Jitse Niesen]] 08:37, 10 May 2007 (CDT)
 
Hi All, I have commented out the Approval tag per nominating editor [[User:Jitse Niesen|Jitse Niesen]] who has revoked his approvalDuring this time, our Approval editor, [[User:Nancy Sculerati|Nancy Sculerati]] can make the appropriate changes and she can replace the Approval tag.  If more chances are made, then I would suggest giving yourself an additional 24 hours before re-approval to give others a chance to review the changes. --[[User:D. Matt Innis|Matt Innis]] [[User talk:D. Matt Innis|(Talk)]] 08:46, 10 May 2007 (CDT)
 
: I think this was handled very well. What I saw was: Etienne correctly decided that this problem should be corrected very quickly; Nancy sent me (presumably other editors) an email to alert me to it; by the time I came to CZ, it had already been decided that the change was appropriate and needed, and was made to the approved version as well as to the draft.
 
: Although this might be "outside the rules", I think that here judgment wins the day, concerning what we might call "clear factual mistakes or obvious typos". I certainly take responsibility for nominating the article for approval without seeing this mistake. And sharp eyes [[User:Etienne Parizot|Etienne]]! - [[User:Greg Martin|Greg Martin]] 14:29, 10 May 2007 (CDT)
 
There will ''always'' be such mistakes, and a good approval process can take care of them.It is understandable that Jitse, who was shocked by the sudden recognition of such a mistake, wanted it fixed IMMEDIATELY. It was understandable that Matt acted to accomodate him, my only point is -in the future we now know that the approvals editor could have done the copyedit at Jitse's say so.  Approval cannot be "revoked" in this manner. Think about it. If it could be, that sets a terrible precedent, you can imagine how in a different circumstance such a precedent could be misused. I cannot add details to the approval process policy without the Editorial Council (of which I am a member) being up and running, with a voting process in place. Right now we are figuring out the process. I said several times that I would take responsibilty for copyedits at this stage with any of the nominating editors.Maybe we should add that, in an emergency the constable can put up a note saying that there is a copyediting problem that is being corrected- in progress. [[User:Nancy Sculerati|Nancy Sculerati]] 12:10, 11 May 2007 (CDT)
 
== Style issues ==
 
Quoting some examples of style that I consider a bit too informal. In particular, there are many phrases/clauses that make the article verbose. In the first paragraph, there are: ''Of course'', ''As it happens'', ''At first glance'', ''perhas more importantly''.
 
Is this as per the policy of Citizendium? Should the number of such phrases/clauses be reduced? [[User:Vipul Naik|Vipul Naik]] 02:19, 8 June 2007 (CDT)
 
Hi Vipul, and welcome to CZ!, the answer to your first question is "yes, there are style  differences here" - see [http://locke.citizendium.org:8080/wiki/CZ:Article_Mechanics#Narrative_coherence_and_flow this section of the article mechanics article concerning style]. Your input is welcome. [[User:D. Matt Innis|Matt Innis]] [[User talk:D. Matt Innis|(Talk)]] 07:56, 8 June 2007 (CDT)
 
Personally, I prefer  more casual or informal style. Of course, this doesn't mean the articles need be any less precise or rigorous, only more readable, and maybe a little less intimidating.
[[User:Greg Woodhouse|Greg Woodhouse]] 11:39, 8 June 2007 (CDT)
 
==Remaining errors in this approved article?==
Hi, I came across this [http://en.wikipedia.org/wiki/Wikipedia_talk:Wikipedia_Signpost/2007-07-30/Citizendium_analysis#Citizendium.27s_approval_process_already_overlooked_a_.22big_error.22 Error claims on WP Signpost].
I think there are valid points there, especially regarding the interpretation of 1/z and the comment on the potential function (clearly it can't represent some force since it is a scalar). These should be looked at closer. Are there plans to have this article revised in the near future? Thanks. [[User:Hendra I. Nurdin|Hendra I. Nurdin]] 13:55, 20 October 2007 (CDT)
:Hi, did you look at the draft? ;-) Except for the potential thing, deleted from the draft some time ago, I don't really think this is as problematic as suggested. But if you feel like, we could find a better wording for some text. Then re-approving looks like a good idea. [[User:Aleksander Stos|Aleksander Stos]] 14:41, 20 October 2007 (CDT)
:PS. You may also have a look at [[User:Aleksander_Stos/ComplexNumberAdvanced|my "advanced" draft]]. At present I gave up the idea behind that work -- but some portions of the article might be useful here. I don't know. [[User:Aleksander Stos|Aleksander Stos]] 16:02, 20 October 2007 (CDT)
::I'm merely passing on some criticisms I happened to stumble onto (I don't know how many people have read it before). Of course, if there is anything valid in them then they should be considered. As for "complex division amounting to conjugation with scaling", well it does sound a bit misleading to me (I don't know about other people, which is why I brought it up here :-)). Consider <math>z_1/z_2 = z_1/|z_2|^2  \bar z_2</math> then it does not have anything to do with the conjugate of <math>z_1 z_2</math> which is <math>\bar z_1 \bar z_2 </math> -- so what does "conjugation" in this part of the article refer to? Compare this for example, with the discussion of <math>z_1 z_2</math> in the article. As for your "advanced" draft (such as the section on roots of complex numbers), perhaps parts of it can go as subpages of the article? [[User:Hendra I. Nurdin|Hendra I. Nurdin]] 19:35, 20 October 2007 (CDT)
::P.S. Does the removal of the assertion  that the potential function represents some force from the article not warrant a re-approval process? This gives rise again to the issue that some relatively "minor" changes like this to an approved article should be possible to do with ease. [[User:Hendra I. Nurdin|Hendra I. Nurdin]] 19:44, 20 October 2007 (CDT)

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APPROVED Version 1.0

See comments above

I made some comments in the section Talk:Complex number/Draft#What the symbol i means in this article above, marked with bullet points, which have not yet been addressed. (I just wanted to mention them after the page break so they won't be forgotten when editing the next version.) --Catherine Woodgold 20:28, 6 May 2007 (CDT)

I edited in all of the changes I had suggested except for the stuff about quantum physics. I don't have a textbook on the subject handy. --Catherine Woodgold 19:17, 7 May 2007 (CDT)

I'm not entirely happy with my text in the QM section, either. Trying to interpret superposed states in terms of probabilities is dicey at best, anyway. I'll have to think about this and see if I can come up with something better. In any case, I'm intrigued by what Robert Tito had to say about other uses of complex numbers, particularly in Hamiltonian systems (conjugate coordinates with a factor of i?) Anyway, I was just trying to come up with something that would be recognizable to a wide range of readers (albeit not mathematically naïve ones). If nothing else, the Schrödinger Equation has a certain iconic value. I'm certainly open to other suggestions. Greg Woodhouse 19:38, 7 May 2007 (CDT)

Wait! I think the quantum physics stuff is good! It just needs some editing, as I suggested, e.g. defining the symbols used etc. --Catherine Woodgold 19:42, 7 May 2007 (CDT)

It seems to me that saying things like " is Planck's constant divided by " wouldn't really add anything to the article, and I guess that's what bothers me: if you (generic) know what Schrödinger's equation is, this probably doesn't need to be said, and if you don't, the section really doesn't add anything. Greg Woodhouse 22:40, 7 May 2007 (CDT)

Part of chemistry and physics workgroups?

(I hope no one minds if I move this discussion "below the bar". Greg Woodhouse 03:34, 7 May 2007 (CDT))

I am just curious why this article's checklist includes it in the chemistry and physics workgroups. It seems that even though this article has applications in those field, including it in every workgroup it applies could get out of hand. - Jared Grubb 12:32, 6 May 2007 (CDT)

that answer is simple: the need for something like a complex number arose from these sciences not from math. Math formalized it, thats all. Robert Tito |  Talk 

I disagree. We have an example showing that complex numbers are important in the sciences, too, but complex numbers were introduced in a fundamental way in mathematics (i.e., not just as a notational convenience) long before quantum mechanics had even been thought of. Greg Woodhouse 16:59, 6 May 2007 (CDT)

Then again, since those workgroups are there, maybe you can sign off on it, too. :-) Greg Woodhouse 17:02, 6 May 2007 (CDT)

Physics and chemistry used the notion of complex numbers as from the 18th century - when they needed them to describe things. Euler, Gauss, Fourier were not mathematicians but physicists/chemists that needed a solution for their math problems. the complex number by far didn't start with quantum mechanics. I might mention Hamiltonian mechanics as an example, or canonicals. Robert Tito |  Talk  17:35, 6 May 2007 (CDT)

It still seems a little odd to me. But, I suppose Ohm's law would be more at home in the electrical engineering workgroup than the mathematics, even though it is a mathematic equation... - Jared Grubb 23:14, 6 May 2007 (CDT)

The historical development of the concept of complex numbers seems like an interesting topic for an article (albeit a challenging one!), but so far as this article is concerned, I don't think it's really that important. No, that doesn't sound right: I don't mean it's not important, only that I don't think it needs to be addressed in the context of this article. Greg Woodhouse 03:34, 7 May 2007 (CDT)

Does anyone know how to create an archive? Is there an automated, or at least "official" way to do it? I just got the following warning:

WARNING: This page is 87 kilobytes long; some browsers may have problems editing pages approaching or longer than 32kb. Please consider breaking the page into smaller sections.

Greg Woodhouse 03:34, 7 May 2007 (CDT)

Re archiving: you might want to discuss with Chris Day or see Talk:Biology/Draft, but since that template is named "Experimental" I suppose procedures haven't been finalized. (Discussion about it arising from pages like this one may drive the finalization of such procedures.) Maybe it's being discussed on the forum somewhere, or if not someone could start. --Catherine Woodgold 07:45, 7 May 2007 (CDT)
Re being part of chemistry and physics workgroups: Chemistry and physics also need to use 1 + 1 = 2. They also need to use words with syllables to communicate technical concepts; that doesn't mean the linguistics Syllable page has to be in the chemistry and physics workgroups. Those sciences use math -- that doesn't mean math is part of the science. I think perhaps people in the chemistry and physics workgroups should decide whether the article is included or not. It's OK with me either way -- it's not that unreasonable. More justifiable than including a page that presents a proof of 1 + 1 = 2 in those sciences. --Catherine Woodgold 18:23, 7 May 2007 (CDT)

This article was approved by a math editor and currently is listed in the Math Workgroup Approved articles, but not in Chemistry or Physics. Since the article is cross-listed in three workgroups, will there need to be three approval processes? Or will we need editors from all three areas to agree before any one draft gets approved? Or will we declare one "father" workgroup, and the others just raise objections or not... I know this approval process is still in its infancy, but these are questions we really should address at some point. - Jared Grubb 02:41, 8 May 2007 (CDT)

Error in multiplicative property

I've just put the following message on User talk:Nancy Sculerati.

Dear Nancy. Etienne Parizot found and fixed an error in Complex number/Draft which is also present in the approved version, Complex number. The formula

halfway the section "The complex exponential" should read

(with a plus sign added on the left-hand side).

This is a big error so I want it to be fixed as soon as possible. I can't imagine any editor would argue with this change. However, I'm not sure what our options are.

  • Some places hint at the possibility to have the constabulary do limited changes to articles without going through the whole approval process (for instance, the section #Copyediting matters above). I couldn't find anything about rules or procedure though. If such a possibility exists, that would be my preference. For the record, the article complex number was nominated by Greg Martin and the nomination was supported by me.
  • If this is not possible, I'd like the approval to be revoked. As far as I can see, there is no rule or precedent for this, only an empty section at CZ:Approval Process.
  • If neither of the above is possible, or if it would take too long, we can always go for the option of nominating the fixed version for approval. To be honest, it's not that important in the big scheme of things, but it is embarrassing and I feel responsible for it.

Any guidance from you (or anybody who happens to read this) would be much appreciated.

-- Jitse Niesen 08:37, 10 May 2007 (CDT)

Hi All, I have commented out the Approval tag per nominating editor Jitse Niesen who has revoked his approval. During this time, our Approval editor, Nancy Sculerati can make the appropriate changes and she can replace the Approval tag. If more chances are made, then I would suggest giving yourself an additional 24 hours before re-approval to give others a chance to review the changes. --Matt Innis (Talk) 08:46, 10 May 2007 (CDT)

I think this was handled very well. What I saw was: Etienne correctly decided that this problem should be corrected very quickly; Nancy sent me (presumably other editors) an email to alert me to it; by the time I came to CZ, it had already been decided that the change was appropriate and needed, and was made to the approved version as well as to the draft.
Although this might be "outside the rules", I think that here judgment wins the day, concerning what we might call "clear factual mistakes or obvious typos". I certainly take responsibility for nominating the article for approval without seeing this mistake. And sharp eyes Etienne! - Greg Martin 14:29, 10 May 2007 (CDT)

There will always be such mistakes, and a good approval process can take care of them.It is understandable that Jitse, who was shocked by the sudden recognition of such a mistake, wanted it fixed IMMEDIATELY. It was understandable that Matt acted to accomodate him, my only point is -in the future we now know that the approvals editor could have done the copyedit at Jitse's say so. Approval cannot be "revoked" in this manner. Think about it. If it could be, that sets a terrible precedent, you can imagine how in a different circumstance such a precedent could be misused. I cannot add details to the approval process policy without the Editorial Council (of which I am a member) being up and running, with a voting process in place. Right now we are figuring out the process. I said several times that I would take responsibilty for copyedits at this stage with any of the nominating editors.Maybe we should add that, in an emergency the constable can put up a note saying that there is a copyediting problem that is being corrected- in progress. Nancy Sculerati 12:10, 11 May 2007 (CDT)

Style issues

Quoting some examples of style that I consider a bit too informal. In particular, there are many phrases/clauses that make the article verbose. In the first paragraph, there are: Of course, As it happens, At first glance, perhas more importantly.

Is this as per the policy of Citizendium? Should the number of such phrases/clauses be reduced? Vipul Naik 02:19, 8 June 2007 (CDT)

Hi Vipul, and welcome to CZ!, the answer to your first question is "yes, there are style differences here" - see this section of the article mechanics article concerning style. Your input is welcome. Matt Innis (Talk) 07:56, 8 June 2007 (CDT)

Personally, I prefer more casual or informal style. Of course, this doesn't mean the articles need be any less precise or rigorous, only more readable, and maybe a little less intimidating. Greg Woodhouse 11:39, 8 June 2007 (CDT)

Remaining errors in this approved article?

Hi, I came across this Error claims on WP Signpost. I think there are valid points there, especially regarding the interpretation of 1/z and the comment on the potential function (clearly it can't represent some force since it is a scalar). These should be looked at closer. Are there plans to have this article revised in the near future? Thanks. Hendra I. Nurdin 13:55, 20 October 2007 (CDT)

Hi, did you look at the draft? ;-) Except for the potential thing, deleted from the draft some time ago, I don't really think this is as problematic as suggested. But if you feel like, we could find a better wording for some text. Then re-approving looks like a good idea. Aleksander Stos 14:41, 20 October 2007 (CDT)
PS. You may also have a look at my "advanced" draft. At present I gave up the idea behind that work -- but some portions of the article might be useful here. I don't know. Aleksander Stos 16:02, 20 October 2007 (CDT)
I'm merely passing on some criticisms I happened to stumble onto (I don't know how many people have read it before). Of course, if there is anything valid in them then they should be considered. As for "complex division amounting to conjugation with scaling", well it does sound a bit misleading to me (I don't know about other people, which is why I brought it up here :-)). Consider then it does not have anything to do with the conjugate of which is -- so what does "conjugation" in this part of the article refer to? Compare this for example, with the discussion of in the article. As for your "advanced" draft (such as the section on roots of complex numbers), perhaps parts of it can go as subpages of the article? Hendra I. Nurdin 19:35, 20 October 2007 (CDT)
P.S. Does the removal of the assertion that the potential function represents some force from the article not warrant a re-approval process? This gives rise again to the issue that some relatively "minor" changes like this to an approved article should be possible to do with ease. Hendra I. Nurdin 19:44, 20 October 2007 (CDT)

Hendra, all it would take is a math editor who has not worked on it to nominate it for re-approval, or three editors who have worked on it to re-approve. If we can get that together, I will be glad to make the draft the approved version. --Matt Innis (Talk) 20:40, 20 October 2007 (CDT)

Well, let's see what others think about this first, as it could be that it's just me being pedantic and perhaps in view of others these changes may not be necessary :-) Anyways, any further changes need to be considered carefully, so that if it does have to go through re-approval, no further minor changes would need to be made afterwards. Hendra I. Nurdin 21:08, 20 October 2007 (CDT)
Okay, sounds like a plan. I was going to leave a message on Jitse's page but you beat me to it! I'll wait and see what develops. If you have any questions, just stop by my talk page. --Matt Innis (Talk) 21:53, 20 October 2007 (CDT)
I hadn't seen the criticisms before. I think they are valid points and that we should revise the article accordingly. Hendra, please change the draft as you see fit (I'm rather busy now so I can't be of much help at the moment, sorry). The approval process is not that much effort, so you shouldn't worry about that. -- Jitse Niesen 22:07, 20 October 2007 (CDT)
Done. I have also added a remark that division by c+di is only defined if c and d are not simultaneously zero in the part of the article that discusses operation on complex numbers. Therefore I invite all authors and editors who had been previously involved in the approved article to check my edits and make any modifications and corrections as deemed necessary. However, I think that the work is not all done yet. There is a bit more to be done on the section about complex numbers in physics. The sentence
"Now, there is some subtlety in the interpretation of ψ because a system can be affected by observation, and the functions ψ we "see" must be eigenstates of the operator defined by the Schrödinger equation, but when we do measure, say, the position of a particle, the probability of finding it in a small region R is just ..."
is quite vague and is likely to cause misunderstanding. I guess I know a fair bit about the mathematical formalisms of quantum mechanics, but I'd rather not delete things nor make substantial changes without first soliciting the opinions of those who have worked on this part, and other authors who know the subject quite well, and get their input on what is meant exactly by this sentence and whether it needs to be further elaborated upon for clarity, or changed to avoid misinterpretations. Hendra I. Nurdin 00:25, 21 October 2007 (CDT)

QM again

The article states:

Now, there is some subtlety in the interpretation of ψ because a system can be affected by observation, and the functions ψ we "see" must be eigenstates of the operator defined by the Schrödinger equation, but when we do measure, say, the position of a particle, the probability of finding it in a small region R is just ...

In my view this sentence should be deleted because it has absolutely nothing to do with complex numbers. It gives me the unpleasant WP experience of something that is added by somebody somewhere with some time on his hands, which is why many WP articles are headache-causing kinds of patchwork. At most one could do in this article is a link to quantum mechanics, where the Born postulate for probability of observation can be put in proper setting. --Paul Wormer 07:42, 21 October 2007 (CDT)

PS Most interactions in QM are invariant under time reversal. It can be shown that ψ can be chosen to be real in that case. And indeed, 95% of quantum chemistry deals with real functions. --Paul Wormer 07:45, 21 October 2007 (CDT)

Yes, deletion of the whole section would be one solution. As a replacement application we could instead insert the Laplace and Fourier transforms which use complex numbers in an essential way, or perhaps something on phasors. Let's see what the editors think would be best. Hendra I. Nurdin 08:03, 21 October 2007 (CDT)
There are also certain irreducible representations of some (physically important) groups that inherently are complex (Wigner, Am. J. Math. vol 62, p. 57 1941). These could be mentioned as examples of complex numbers in physics. --Paul Wormer 09:45, 21 October 2007 (CDT)
Paul, would you be interested in putting this in the article to replace what is currently there? Btw, which group does this paper talk about? Perhaps we can work on this section together, I could insert some additional engineering applications. Let me know what you think. Thanks. Hendra I. Nurdin 07:30, 23 October 2007 (CDT)
Dear Hendra, don't you think it would be a good idea to leave it to the approving editors to correct the article? We can signal what we don't like. For instance, the following sentence in the article
the functions ψ we "see" must be eigenstates of the operator defined by the Schrödinger equation
is bordering on being wrong; a wave function can be a superposition of eigenstates, see particle in a box for a graphic example. Maybe "see" refers to a collapse of the wave function, but that would be a collapse to an eigenstate of the position operator. Further, the Schrödinger equation mentioned (time-dependent) is not an eigenvalue equation, so the term "operator defined by" is pretty inconclusive.
I am of the opinion that it is better to spend our energy on new articles, given the present vast emptiness of CZ. In Legendre polynomial I linked to orthogonal polynomials. I saw that you wrote Gram-Schmidt, so for you it would be a piece of cake to write a nice article about general orthogonal polynomials, with links to Laguerre, Hermite, Legendre, Jacobi, etc. Best wishes, --Paul Wormer 06:59, 25 October 2007 (CDT)
PS. Upon rereading the Wigner article that I mentioned earlier, I noticed that Wigner does not mention any specific groups, only characteristics of groups. But, complex numbers are essential for irreducible representations of cyclic groups and for the even-dimensional irreps of SU(2). Schur's second lemma requires the solution of a polynomial equation and hence an algebraically closed field. --Paul Wormer 06:59, 25 October 2007 (CDT)
I removed the whole QM section; there are simply too many problems with it. It would be nice if somebody could write a section on applications of complex numbers outside maths. You don't need permission of the approving editors to do so (that's why it's called a draft), but I hereby do give you permission in case you feel happier with it.
I'm not so sure what the best application would be to put in that section. Phasors is relatively easy to explain, but I think it's mainly an organizational tool and it's not essential to use complex numbers - one can just use sine and cosine. However, Laplace transforms may be too difficult, given that we tried hard to make the page understandable with a minimum of prior knowledge. Or perhaps QM is a good example after all when written up properly; we can just show the Schrodinger equation and say that it has an i in there.
By the way, I moved from Australia to England and that's why I haven't been around much lately. Still settling in, and all my books are still en route, but I should be able to spend some more time here soon. -- Jitse Niesen 07:51, 26 October 2007 (CDT)

Representation via matrices?

Just wondering if anyone had already considered an alternative version of the formal definition by defining complex numbers as being a subset of GL_2(R)? Many "university level" people will have seen the basic definition of matrix multiplication, and as such, it may seem less foreign than the definition of multiplication for ordered pairs of real numbers. ...said Barry R. Smith (talk) (Please sign your talk page posts by simply adding four tildes, ~~~~.)

Move some topics to advanced page?

Near the top of the discussion, where a plan for the "complex number" page was sketched, the following comment was made: "I like to introduce complex numbers to my students with the example of the resolution of the cubic equation with the so called Gerolamo Cardano's method (in fact it is due to Scipione del Ferro and Niccolò Tartaglia). Computations are quite easy, and the striking fact is that during them, one has to use some imaginary number which square would be -1, but once the computations are finished, one gets the three real solutions of the equation!" While it is true that this is probably the earliest example where it became clear that complex numbers were necessary even for the study of real quantities, I definitely disagree with the statement "computations are quite easy". I am also surprised that you introduce complex numbers to students with this example. I have given a project to students of working through this example, and I would not say that they found it easy. In another section of this discussion is the comment, "I certainly agree that articles, especially articles about basic topics like complex numbers, shouldn't scare the reader away right off the bat, but perhaps we need to temper our desire to make the article start out slowly and in a non-intimidating fashion with a bit of logical coherence." It seems to me that our first example might scare many readers away right off the bat. What does a mathematician think about this example? Is it a struggle to get through? Does it make you not want to continue?

The initial idea of writing x=u+v where u and v will be specified later is offputting to many people. This is followed by an application of the binomial theorem, and then an unmotivated factoring step. Then, it is stated that "we only required that x = u + v. Hence, we can choose another condition on u and v. We pick this condition to be 3uv − 15 = 0". Again, the average non-specialist, I would imagine, would wonder, why this condition? And why are we allowed to choose another condition? I could go on listing more potential difficulties that I see in this example.

As we now have the advanced subpage option, why not move this example to an advanced subpage, and then refer to it in the actual article. Perhaps say something like, "A common question is why bother with complex numbers when real numbers almost always seem sufficient for applications. Indeed, the ancients would ignore complex solutions to quadratic equations. It wasn't until the 16th century that it began to be clear that sometimes, complex numbers were indispensable even in problems that seemingly only involve real numbers. An example of this can be found on the "advanced subpage".

Even if this approach isn't taken, might I suggest an alternative way to formulate the current example that is probably more palatable to the average reader: do not derive the solution of the "reduced cubic" by introducing u,v, etc. Instead, just give them the formula for the roots of a reduced cubic -- it isn't very complicated, and an analogy can be made with the quadratic formula. Then show that 4 is a solution, but that the formula gives an expression involving a complex number. Finally, show that this complex can be written as 4.

Also, I think that no matter what, some reference should be made to the fact that although complex numbers are typically introduced these days in high school, when the quadratic formula comes up, that the ancients had versions of the quadratic formula but still didn't accept complex numbers. Barry R. Smith 13:34, 30 April 2008 (CDT)

All sounds good. Go for it! J. Noel Chiappa 22:26, 4 May 2008 (CDT)

New philosophical addition

With regards to Christopher Reiss's new philosophical addition, I believe that this material definitely should be moved out of the introductory paragraph -- as far as I know, an introductory paragraph should usually serve as an abstract for the article, giving a concise non-technical summary of what most would consider the most salient features of the topic?

Now the issues raised are partially addressed already in the history section, notably, the fact that ancients did not believe that complex numbers were "real". Furthermore, the long example in the "advanced" subpage describes the first instance in which complex numbers seemed to be necessary for something. If this material could be improved using some of Christopher's ideas, then I propose that the material be integrated into a whole within to history section.

The only content I would take issue with is the statement that "It is now understood that arithmetic is a pure abstraction which we are free to modify", and related declarations. This is probably the most common philosophical view taken by current mathematicians, and probably many people well versed in mathematics. However, this idea is much older -- Platonic realism is an early example, due to Plato millennia ago. So the phrase "it is now" is misleading. On the other hand, the statement "understood that arithmetic is a pure abstraction" is also misleading. Better would be to say that the majority of current mathematicians believe this. But there is still a raging debate about this in the philosophy of mathematics. For alternatives, see "empiricism" and "fictionalism", for example, at the philosophy of math wikipedia site. Even if we add this material to what Christopher wrote, I believe that content would more properly be put in a philosophy of math site, as the result would probably be quite lengthy and too much of a tangent from the topic of "complex numbers".

Any thoughts?Barry R. Smith 12:20, 22 May 2008 (CDT)

I completely agree. I did not see any reply from Christopher, so I simply removed the text. Here it is for future reference:
Complex numbers were once considered 'fictitous' on the grounds "there is no square root of negative one." This misconception is rooted in a philosophical conception of number which is now seen as misguided. The notion behind this 'predjudice' is : arithmetic exists in the physical world, or is an attribute of physical reality. This notion has repeatedly proven a stumbling block in the history of mathematics. It is now understood that arithmetic is a pure abstraction which we are free to modify. It is legitimate to experiment with the abtract system first and then seek real world mechanisms which the abstraction can model. Rather suprisingly, by freeing the abstract system so that it is no longer "real", the abstraction became a much broader and more powerful model of the physical world.
This has happened repeatedly in the development of mathematics. Originally, only the counting numbers, 1,2,3 .. were considered 'real'. This left the result of certain division operations - fractions - 'unreal'. But one can define an arithmetic of fractions which is immensely useful in the physical world, and which also describes the counting numbers as a special case. Similarly, the result of certain subtraction operations yielded 'unreal', negative results. Arithmetic was expanded again to include negative numbers. Yet again, it was found that the square root of two has no solution among the fractions. These 'unreal' entities were eventually admitted into arithmetic as it continued to grow in power.
As Barry says, parts of this may, in less absolute form, be integrated in the History section (or perhaps elsewhere); in fact, I think the question of whether complex numbers exist probably should be treated in the article. I would include a link to an article about philosophical aspects of mathematics where this is discussed in more detail. -- Jitse Niesen 07:19, 2 June 2008 (CDT)

Division and conjugation

Discussion moved from User talk:Jitse Niesen#Complex number page BEGIN Peter Schmitt 23:44, 19 July 2009 (UTC)

The Complex number page still contains what I consider to be a blatant error: "In other words, up to a scaling factor, division by z is just complex conjugation." I think this would be correct if it said, "In other words, up to a scaling factor, division by z is just multiplication by the complex conjugate of z" or if it said "In other words, up to a scaling factor, taking the reciprocal of z is just complex conjugation"; but as it stands (according to the only reasonable interpretation I can see) it's equating two operations which in general involve completely different changes to the angle on the complex plane. As you know, this problem was pointed out on a Wikipedia discussion page in 2007. As a math editor, would you please either ask a constable to correct just this one sentence in the current article, or arrange to have the draft approved? (I haven't looked at the latest draft; I'm just concerned about this particular error.) Thanks. Catherine Woodgold 15:28, 11 July 2009 (UTC)

The formulation may be unfortunate, but it is correct. The "scaling factor" is 1/|z|^2, a real number, and the angles (the argument) of the conjugate and the inverse are the same. Probably it would be better to write "In other words, up to the scaling factor 1/|z|^2, division by z is just complex conjugation." I don't know if in such a case approved version can be corrected. Peter Schmitt 22:07, 11 July 2009 (UTC)
Why don't the two of you, and possibly anyone else you can rope in, work out an *exact* replacement phrase and then put it into this discussion area. If all of you agree that it should replace the Approved version, either I'll change it myself or I'll ask Matt what he things about it. Hayford Peirce 22:30, 11 July 2009 (UTC)
My suggestion:
In other words, up to the scaling factor (a real number), division by z is just complex conjugation.
(Unfortunately, the fraction looks awful in text.) Peter Schmitt 22:57, 11 July 2009 (UTC)

If we use our current approval (and re-approval) rules: since Peter is a mathematics editor, I would suggest that Peter refrain from making any changes to the article and let Jitse or Catherine make the change on the Draft. Then (assuming Peter agrees with the change), he can can re-nominate the draft for approval using the single editor process (since he has not made any content edits to the article)... HELLO CATHERINE! :)

Check with the User:Approvals Manager (Joe) if you want to be sure. D. Matt Innis 23:21, 11 July 2009 (UTC)

Matt's suggestions sound v. feasible to me. Hayford Peirce 23:36, 11 July 2009 (UTC)
As far as I see, the draft differs much from the approved version (a direct comparison seems to be difficult). I thought there is a possibility to edit such things without (re)approval? This is not a correction but only a clarification, and certainly not a change of content. (By the way, I think something is wrong - much too difficult -- if an editor is disqualified to make an approval even after such cosmetic edits. Even some minor edits should be allowed. I think it is simply cheating if suggesting a change is allowed, but doing the same edit is not.) Peter Schmitt 23:41, 11 July 2009 (UTC)
I have put a link on the Approval Manager talk page. Peter Schmitt 23:49, 11 July 2009 (UTC)
I looked at the draft and the approved version, they are completely different on this point. Somebody made some drastic changes. Further, I would say division of c by z is multiplication of c by the complex conjugate of z (and division by the modulus square of modulus of z). In the polar representation of complex numbers the issue is completely trivial, as we will all agree. It is , which holds for all real and complex . --Paul Wormer 08:09, 12 July 2009 (UTC)
"and division by the square of the modulus". Yes, this would avoid the displayed fraction. Peter Schmitt 09:53, 12 July 2009 (UTC)
If we want to spell it out completely, then with
,
--Paul Wormer 11:13, 12 July 2009 (UTC)
It seems that you are saying that replacing the approved version with the draft would add more errors than it would fix. Your choices then would be to 1) fix the errors in the draft and use either the individual editor approval or three editor approval method to change the approved version or 2) revert the draft version to the version you like, then make the change that Catherine and Peter are looking to make and then use the individual or three editor approval methods as above. Does anyone see any other choices.
The idea of the approval rules is to make us work together to come up with the most accurate article possible while, at the same time, allowing the article to remain stable while we do. Hopefully, this reduces the workload on our experts. The errors in the draft are an example of why we want to have an approved version that is difficult to change - without editorial input. Changing our rules for something like this that can be managed within the same rules seems only a means to weaken them. However, it is possible, but would require community input from all the workgroups to consider all the ramifications of such a change.
(By the way, I think something is wrong - much too difficult -- if an editor is disqualified to make an approval even after such cosmetic edits. Even some minor edits should be allowed. I think it is simply cheating if suggesting a change is allowed, but doing the same edit is not.) It's not so much about cheating, it shows that more than one editor agrees to the change, thus increasing the likelihood that the change is more accurate - while at the same time allowing only two editors to make a difference (which is easier than finding three - something that you are also asking for). It is a way to both make it easier to make a change and keep one fallible editor from approving his/her own work. I hope that makes sense.
D. Matt Innis 12:33, 12 July 2009 (UTC)

I think I have been misunderstood here, and I'll try to clarify: I noticed the message of Catherine (not addressed at me) and answered it. The challenged sentence - in the approved version - is correct, but might indeed be confusing for some readers. Therefore I suggested a minor edit to the approved version (thinking of CZ:Approval_Process#Overview, last paragraph) because approving the draft version would - in view of the major changes - require more checking and possibly a lot of discussion. If this is not thought as adequate or allowed, then the approved version can stay as it is.

The remark on the approval process was a reaction on the suggestion:

"since Peter is a mathematics editor, I would suggest that Peter refrain from making any changes to the article and let Jitse or Catherine make the change on the Draft".

What is the difference between an explicit suggestion by an editor which is dutifully incorporated by some author (possibly a non-editor), and the same change made by the editor himself? The difference is only a formal one -- that was what I meant by cheating. (There need not be another editor involved!) Moreover, I thought that copyedit changes are allowed -- and this I would classify as copyediting.
Peter Schmitt 13:11, 12 July 2009 (UTC)

That does clarify some. I did not understand that the discussion concerned something that was basically correct on the approved page. Unfortunately, I'm not sure that I could have known that if you had not told me :) - which I think makes it different than a copyedit - which anyone could recognize does not change meaning. Because it is not really self-evident that it does not make a content change, I don't think this is something that a constable can or should do without the approvals manager seeing things through (which he very well might do). It's more to protect the editor that has endorsed the article than anything else.
Concerning the addendum, Jitse knows very well where that came from. I actually revoked the approval of one of his articles when he realized it had a math error in it. He'll tell you that I took a pretty good beating over that one! And I think they were right to do that, approved articles need to be hard to change. The addendum makes it clear that the nominating editor can change it with the help of the approvals manager. It is the nominating editor who has his name on the article and therefore has endorsed it. That is why we have given him/her more leeway to make a change. Jitse could still make that change, as you note, I think - with the help of the approvals manager. Of course, the other choices still remain - to re-approve using your credentials if Jitse does not respond.
You make another good point about an author being able to make a change that an editor cannot, but it still requires two heads. Remember that we deal with controversial articles that have competing views even among editors. The concept is to keep one view from eliminating the other view without some oversight. Whether this is successful at keeping that from happening, or if it keeps us from making more important corrections, or if we might be able to come up with a better way, is something that might need discussion elsewhere if it is causing problems. D. Matt Innis 14:33, 12 July 2009 (UTC)
Peter says that the approved version is essentially correct, but I side with Catherine and say that the word multiplication is missing. --Paul Wormer 15:23, 12 July 2009 (UTC)
Because of the formula above the sentence I read this only as "division (of 1) by z" ... if you read it as "division of some u by z, then this would certainly be wrong. Perhaps I did not want to see a serious mistake? Peter Schmitt 15:35, 12 July 2009 (UTC)
Looks like you're getting closer to a more accurate description. Once you've decided what you want to do, let the approvals manager know and we'll go from there. I think our options remain the same. D. Matt Innis 17:07, 12 July 2009 (UTC)

Since no further comments have been added, I suggest to replace

In other words, up to a scaling factor, division by z is just complex conjugation.

by

In other words, up to a real scaling factor (the square of the reciprocal of the modulus), division of 1 by z is just complex conjugation.

because this seems to be the least change which repairs the sentence in the approved version (and still fits into the style of the paragraph).
For a reapproval of a new draft certainly more changes (and discussions) will be made.
Peter Schmitt 00:21, 19 July 2009 (UTC)

I suggest that we proceed through a re-approval process anyway. I don't see three editors on that approval tag. D. Matt Innis 18:30, 19 July 2009 (UTC)

Discussion moved from User talk:Jitse Niesen#Complex number page END Peter Schmitt 23:44, 19 July 2009 (UTC)

How to proceed?

Ok, but I do not know what I should do. I do not want to approve the draft as it is now.
But I want to change one sentence of the approved version in order to either

correct an error in a sentence, or
to clarify a sentence that can be misunderstood

(depending how one interprets this sentence in context).
I would approve such a corrected version (and I would make the changes myself, it this is allowed) — but how is this done correctly?
Peter Schmitt 00:04, 20 July 2009 (UTC)

Okay, you can't correct it yourself and approve it yourself. You can still use Jitse, or the other editors as noted above. However, since you are an editor and since there are lots of other changes on the draft, I think it would be possible to revert the entire draft to the approved version then have someone make a change that you can agree to and nominate that version as 2.0. Once the version is approved, you can then revert the draft to the older draft (if you want) or start over again from there - as long as you discuss why on the talk page. If someone disagrees, that gives them plenty of opportunity and options to discuss any issues before it gets re-approved. Again, though, the User:Approvals Manager needs to be kept abreast of this and had veto power if something doesn't look right - because he is my redundant failsafe system! D. Matt Innis 04:16, 20 July 2009 (UTC)
If it's really just fixing an error, then I don't see any reason we can't just insert the correction into the approved version of the article; we've done that before, though mainly just with grammatical or spelling errors. Since this would be a content correction, we would need to make sure that the original approving editors still approve or we'll have to remove their names from the approval. Jitse is definitely still around to agree or disagree with a suggested change but Greg Martin hasn't made a contribution for over 2 years, so he'll be harder to track down.
So these are our options, as I see them: (1) If we have a specific proposal for a change on the table, we can ask Jitse to review it. Then, if he agrees to the change, we'll make the change on the approved version and replace Greg Martin's name with Peter's, assuming that Peter approves of the rest of the article. But then I think we still need one more editor. (2) We can do the same thing and try to track down Greg Martin. (3) We can re-initiate the approval process for some later version of the article. That could be the current version or it could be any other version since the time of approval or it could be a wholly new version. We would need either one un-involved editor or three who have contributed. (4) We can revoke approval based on the error that the current approved version contains. This is the least preferable of our options, and we should avoid it if we can.
By the way, does this topic fall within the field of number theory? If so, Barry Smith might be willing to sign off on approval, either for the old draft with the proposed change or on a new draft.
Apologies for taking so long to jump into the conversation. --Joe (Approvals Manager) 17:48, 22 July 2009 (UTC)
Number theory is notoriously difficult to define. Although "complex number" sounds like it might fall under the field of number theory, I think most mathematicians would agree that the entire scope of the study of complex numbers is not classified as number theory.Barry R. Smith 18:44, 22 July 2009 (UTC)
The elementary theory of complex numbers (which means, not the advanced theory of analytic functions) is within the scope of every mathematician. And moreover, even the advanced (if not too much advanced, maybe) theory of analytic functions is usually within the scope of number theorists. I think so. (Sorry for the late intervention.) Boris Tsirelson 19:02, 23 July 2009 (UTC)
The topic doesn't necessarily have to fall wholly within your field in order for you to approve it. In fact, most of our approved articles span several fields of inquiry and each individual editor who participates in the approval process usually comes from only one or two of the fields that are relevant. Working together, the editors from different fields approve a single topic. If number theory has something important to say about complex numbers, then that is enough for you to be an approving editor. --Joe (Approvals Manager) 16:16, 23 July 2009 (UTC)
I agree with the choices that Joe gives above with the exception of 4, which seems that we should have editors decide to revoke approval, and if we have that, we might as well re-approve it. After looking at the amount of additional edits that have been made to the draft, and considering that most were made by well informed authors and editors, I think it might be a better choice to keep working on this draft until we have something that Peter can agree to endorse. Othewise, it is likely that we will be re-approving again and again until all of the changes have been incorporated anyway. How close are we Peter? D. Matt Innis 23:35, 22 July 2009 (UTC)

(unindent) Allow me to clarify my position in some detail: This discussion was not started by me, but by a concern of Catherine Woodgold placed on Jitse's talk page. I noticed it, and that Jitse did not react. (He is online with some edits on WP, so I am not sure if we may count on him.)

I probably would not have looked at the (approved) Complex number article for a long time, and if so, I would probably not have noticed the sentence under discussion because I would have read it as correct by (unconciously) assuming what the author (probably) had in mind when writing it. (That certainly is also the reason why it was not caught at approval time.) Checking the article, I still was not looking thoroughly enough, and thought that clarifying "scaling factor" would suffice. However, Paul pointed out that I was wrong in reading "division" as "division of 1" -- taken literally, this was not said.

Since this sentence lacks clarity, it should be corrected. I do not see this change as "a content correction", but I won't fight your ruling. And I am with Catherine that it should be corrected as quickly as possible. Since I have not yet thoroughly read the complete article I have no opinion how close to (re)approval the current draft is. Anyway, I think this (and similar cases) should be corrected independently, and be recognized as corrections to the first approved version (I suppose that earlier approved versions will still be accessible as such? They should!) and not be hidden among a lot of changes between two approved versions.

I am not sure if I understand what "(speciality) editor" for number theory means and implies. Basic arithmetic with complex numbers is common knowledge for all mathematicians (and most natural scientists) and should be known by many non-mathematicians, as well. On the other hand, number theory also uses complex numbers, e.g., when discussing algebraic number fields. What I want to say by this: This is not an issue for which any special knowledge is needed.

Peter Schmitt 11:58, 23 July 2009 (UTC)

In my view the needed correction is almost equal to correcting a typographical error, it is really minute. Let's not waste any more time on it, give Peter (or me) the right to fix it in the approved article as follows
Old: "In other words, up to a scaling factor, division by z is just complex conjugation"
New: "In other words, up to a scaling factor, division of one by z is just complex conjugation"
Addition of 5 letters, 2 short words.
--Paul Wormer 12:12, 23 July 2009 (UTC)
Okay, the two of you agree that it is not really a change in content but only a correction/clarification of the presentation. Let's make the change in the approved version and then if the draft develops to the point that we need to reapprove, we will. Matt (or Hayford, if you're watching) could you make the change Paul wrote above? Thanks much.
By the way, there is a precedent for what I'm calling "ex post facto approvals". If there are not already three approving editors listed in the metadata of an approved article and another editor from the relevant workgroup(s) wishes to add his or her name to the approval, he or she may do so. Simply add your name below the other approving editors in the "required for Approved template" section of the metadata and then notify me on the Approval Manager's talk page. So if either Peter or Barry (Barry, see my response above) would like to add his name to the approval of the current version after we make the correction, I would encourage it. --Joe (Approvals Manager) 16:16, 23 July 2009 (UTC)
You are asking me to change an article that was approved by Greg MArtin and Jitse Niesen to something that Peter Schmitt and Paul Wormer think is wrong, but you want me to leave Greg and Jitse's names on it. Whether I agree that the change is a good one is not an argument from an administrative perspective. We have two choices, get Jitse to change it or re-approve a draft version. So, if I can, I will revert all the changes on the draft and then make the change in the new draft (which anyone could have done) and it will be up to Peter to decide if he wants to use the single editor approval for it. That's as far as a constable can go, it is up to you guys to pull the rest together. D. Matt Innis 23:40, 23 July 2009 (UTC)
Okay, this is the new draft version that includes reverting to the original and then changing the sentence in question. While it is possible that other changes may occur, I won't replace the changes that have been made since until after it is approved. Then I will return the changes that were made since the first approval to the new draft version. I do not expect this to be a precedent for how things should be doen in the future without further discussion from the rest of the community and consideration is made for how it would affect other more controversial articles, but at least it doesn't give attribution to Jitse and Greg for something they did not approve, which I think is important from the standpoint of reliability and liability. D. Matt Innis 23:53, 23 July 2009 (UTC)
What you describe is what I was worried about but not what I thought I was asking you to do. The approval process is in place precisely because it stops the content from slipping and sliding away from the approved version. But if this is really just fixing an error and not changing the content, this situation isn't different from other corrections that have been made. I get the impression from the mathematicians here that this is akin to fixing a sentence that mistakenly says "3 divided by 6 is 2" when the truth is that "6 divided by 3 is 2".
I don't want the situation to become acrimonious though. So let's see if we can't simply reapprove. That means we need an editor to start the process. --Joe (Approvals Manager) 00:31, 24 July 2009 (UTC)
Sorry, I shall not be able to join in again before tomorrow (in about 30 hours) -- I say this to avoid the impression that I do no longer follow this. For the moment, just a short remark: The change is similar to changing a "this is" in a "the last mentioned is" to avoid that the "this" is misunderstood. Is this (only) so complicated because it is mathematics? Peter Schmitt 10:08, 24 July 2009 (UTC)
Not exactly because it is mathematics but because it is a topic that neither Matt nor I are expert in. We both want to make sure that we aren't making a change to the approved article that the original approving editors would disapprove of. It seems that that is the case here, but the only way for us to be absolutely sure is to get word from the original approving editors. Spelling or grammar mistakes are easier because anyone can recognize them. I hope we can re-approve the article and improve it even more in other ways while we're at it; that's what re-approval is all about, after all. --Joe (Approvals Manager) 14:19, 24 July 2009 (UTC)
Now you have chosen to let the article persist (for who knows how long) with— what Catherine Woodgold considers to be—a blatant error.--Paul Wormer 15:56, 24 July 2009 (UTC)
Just to be clear, all that needs to be done is for a math editor who has not made an edit to approve this version of the draft. The only thing that is different than the current approved version is the edit that was requested - "In other words, up to a scaling factor, division of one by z is just complex conjugation". D. Matt Innis 01:28, 25 July 2009 (UTC)

Well, I have just filled out the template (hopefully, correct -- is it true that the date should be given in this form???). I have read the article and discovered nothing serious. (Some historical remarks should perhaps be checked.) I also see some things I would change or extend, but this would take time -- and may be a cause for long discussions. Furthermore, if I would start to edit, I would not be able to approve it. So it seems best to leave the article as it is. Peter Schmitt 00:42, 26 July 2009 (UTC)

Peter, thanks for taking the lead. After re-approval, we can keep working on any minor adjustments on the draft version. --Joe (Approvals Manager) 00:50, 26 July 2009 (UTC)
Excellent. I'll wait to get the go-ahead from Joe (Approvals Manager) tomorrow. Sorry for any undue delay, but all in all, this way we have a legitimate single editor approval that has been duly scrutinized and agreed upon. Peter will also then be able to make any 'corrections' to the approved article once his name is on it. Meanwhile, if anyone would like to discuss changing the process to make these things easier, feel free to bring your ideas to the forums or the CZ Talk:Approval Process. D. Matt Innis 02:00, 26 July 2009 (UTC)
Looks good to me. Go ahead with the mechanics!
If other editors still wish to lend their support to the newly approved version, they may do so by following the steps I described above for "ex post facto approvals". --Joe (Approvals Manager) 14:19, 27 July 2009 (UTC)

APPROVED Version 2.0

This version includes one of many changes that were made to the draft since the first approval by other editors. I'll replace the old draft changes now so that they may be considered in any future versions. D. Matt Innis 02:51, 28 July 2009 (UTC)

I've restored the draft. One important thing to remember is that the sentence that was changed in the new version: "In other words, up to a scaling factor, division of one by z is just complex conjugation" does not appear in the draft because the sentence was essentially reworked too much for me to replace. It woul dbe nice for someone to take a look and make sure that what is there is accurate. Thanks all! D. Matt Innis 03:00, 28 July 2009 (UTC)

correcting approved version

Concerning the problems with Complex number#Complex numbers in physics in the approved version of this page (see #QM again Paul Wormer has suggested [1] to delete it completely. After some thinking about the options to resolve the problem, I consider it best to remove this section from the approved version and not to approve the current draft which contains several major revisions including the removal of another section, too. The article is nicely written but lacks some characteristics that an encyclopedia's (main) page should have. Therefore it is best to keep it as it is as an approved page (but correct errors, of course), instead of rewriting it partially.

May I do the necessary steps without being disqualified as approving editor? Peter Schmitt 23:58, 18 December 2009 (UTC)

Peter, BECAUSE you are the editor whose name is on this article, I think it is appropriate that you should be able to change information that you *now* feel is improper. I have reverted the draft to the current approved version. To continue to be within our CZ:Approval Process guidelines on single editor approval, how about asking Paul if he would clean up the physics section in any way that he feels necessary (as we would want a physics special expert to do) and then decide if you want to re-approve that version. D. Matt Innis 01:11, 19 December 2009 (UTC)
I haven't seen any movement here. Is this solution not working for you, or is it just a low priority? D. Matt Innis 16:48, 22 December 2009 (UTC)
Yes, Peter what's happening?--Paul Wormer 17:56, 22 December 2009 (UTC)
Sorry, if I kept you waiting. During the last days I did not do much here. (And today I suddenly lost contact to CZ for several hours.) But the main point is: Since I am expected not to touch the draft I have waited for someone to do the edits (remove the physics section, and - perhaps - make the already corrected "division sentence" still more explicit.
Paul, I have seen your new picture, and I have some comments, but I think that this need not concern us here. (A theoretical question: If I edit the caption, and the picture is used - would this disqualify me as approving single editor?) Since we both think that a new start will be best it is better to have the article (and the draft) as untouched as possible. After correcting its factual errors we can without time pressure think about a good place for it (subpage?) and a replacement. (I shall probably prepare something to start with offline first.)
Off topic: I have corrected the typo in Literature/Draft. There should be now problem to correct it on the Approved page. And the other open cases should also be possible, even if Editors are needed. There are History and Biology Editors active.
Peter Schmitt 00:20, 23 December 2009 (UTC)
I took care of the spelling error on Literature/Draft, thanks! My understanding is that any corrections to content on this page would have to be made by Paul (including image captions) in order for the single editor process to remain a viable option. D. Matt Innis 15:11, 23 December 2009 (UTC)

Nomination for reapproval

Thanks, Paul, for making the corrections! I have now nominated the draft for reapproval. I have set the date to today, so that this step can be performed whenever a Constable feels ready. (For the reasons for this reapproval see above.)

For more information see also my talk page: Complex numbers again. I hope that no further problems will surface in this article. However, as discussed there, Paul and I think that eventually this article should be replaced by a more encyclopedic one, while this colloquial introduction should find another home.

Peter Schmitt 12:46, 25 December 2009 (UTC)

APPROVED Version 2.1

Phasors

I think there should be a section in this article about complex numbers expressed as phasors, which are commonly used in electrical engineering calculations involving alternating current voltages, currents, and impedance. Phasors are complex numbers expressed in terms of magnitude and phase angle. Henry A. Padleckas 11:29, 28 February 2011 (UTC)

You are talking about the polar form, aren't you? Calling this "phasor" in the context of the complex number would be confusing.
It is rather the other way: Phasors are described with the help of complex numbers (and not: complex numbers are expressed as phasors). --Peter Schmitt 22:24, 28 February 2011 (UTC)
Yes, the polar form, as in  . Henry A. Padleckas 22:41, 28 February 2011 (UTC)