# Talk:Complex number/Draft

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To learn how to fill out this checklist, please see CZ:The Article Checklist. To update this checklist edit the metadata template.
 Definition:  Numbers of the form a+bi, where a and b are real numbers and i denotes a number satisfying $i^2=-1$. [d] [e]

## APPROVED Version 1.0

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 "$\hbar$ is Planck's constant divided by $2\pi$" 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 |

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 |  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

$e^{z_1 z_2} = e^{z_1} e^{z_2} \$

halfway the section "The complex exponential" should read

$e^{z_1 + z_2} = e^{z_1} e^{z_2} \$

(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 $z_1/z_2 = z_1/|z_2|^2 \bar z_2$ then it does not have anything to do with the conjugate of $z_1 z_2$ which is $\bar z_1 \bar z_2$ -- so what does "conjugation" in this part of the article refer to? Compare this for example, with the discussion of $z_1 z_2$ 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 $x^3=15x+4$ 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)

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 $\frac 1 {\left|z\right|^2}$ (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 $1/(re^{a}) = e^{-a}/r$, which holds for all real and complex $a$. --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 $z = r\exp(i\theta)\;$
$c/z = c/(r\exp(i\theta)) = (c/r)\exp(-i\theta)= (c/r^2)\;r\exp(-i\theta)= (c/r^2)\;\overline{z}\;$,
--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)

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)

## 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  $r \ang \theta . \,$. Henry A. Padleckas 22:41, 28 February 2011 (UTC)