Talk:Complex number/Archive 1

Complex number

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Archive 1

Definition

I reworked the text a bit. So this is why.

• I think ${\sqrt {-1}}$ 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 pairs 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 $a+bi$, 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 $\mathbb {C}$ is the splitting field of $x^{2}+1$ over $\mathbb {R}$. 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 $\mathbb {Q} [{\sqrt {-1}}]$ or $\mathbb {Q} [{\sqrt {2}}]$. But I suppose that's a topic for another article. Greg Woodhouse 06:21, 2 April 2007 (CDT)

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. --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. Greg Woodhouse 16:27, 3 April 2007 (CDT)

sketch of a plan

The status of the ${\sqrt {-1}}$ 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 $x$ we call ${\sqrt {x}}$ (the positive one), but there is not such a canonical way to choose amongst the two square roots of -1. Once $\mathbb {C}$ 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 ${\sqrt {-1}}$ 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.

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 $x^{3}=15x+4$ 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! At this stage, one can denote the mysterious number by ${\sqrt {-1}}$, 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.

Next we need a model to legitimate this mysterious number, and then, Alek Stos's suggestion is best : considering that $\mathbb {C}$ is $\mathbb {R} ^{2}$ 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.

Then, another section may deal with a more abstract point of view, that is $\mathbb {C} =\mathbb {R} [X]/\left(X^{2}+1\right)$, and more generally, introduce the notions of splitting fields, 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.

Finally, some applications of complex numbers must be cited : a few words about complex analysis and holomorphic functions, 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.

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!

--Sébastien Moulin (talk me) 11:21, 2 April 2007 (CDT)

I guess I like the idea! --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. Greg Woodhouse 19:34, 2 April 2007 (CDT)

I just added an aside on mathematical notation that I hope will address some of the concerns raised here. Greg Woodhouse 23:07, 2 April 2007 (CDT)

You made good work. I wrote the introductory example about the equation $x^{3}=15x+4$. 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. --Sébastien Moulin (talk me) 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. Greg Woodhouse 11:34, 4 April 2007 (CDT)

Entertainingly written, good job so far --Larry Sanger 11:38, 4 April 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. --Sébastien Moulin (talk me) 11:41, 4 April 2007 (CDT)

Closing the loop (pun intended)

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). Greg Woodhouse 14:38, 6 April 2007 (CDT)

Developing or Developed?

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? Greg Woodhouse 14:41, 6 April 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). --AlekStos 11:53, 11 April 2007 (CDT)

Comments in footnotes

I think the use of footnotes is preferable to the "sidebar" comments I used originally. Greg Woodhouse 15:11, 11 April 2007 (CDT)

What now?

It seems to me like we've pretty much covered Sébastien Moulin's proposed outline. What's the next step? Greg Woodhouse 11:53, 4 April 2007 (CDT)

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).

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 $z^{n}=a,\quad a\in \mathbb {C}$. 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.

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. --AlekStos 16:16, 11 April 2007 (CDT)

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 ($z^{n}=1$) 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. Greg Woodhouse 19:25, 11 April 2007 (CDT)

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 :-). --AlekStos 03:08, 12 April 2007 (CDT)

Extension and slight reorganization began... Meanwhile, I realized that we need also

• perhaps a minor remark on equality of two complex numbers
• 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. --AlekStos 07:26, 12 April 2007 (CDT)

Complex numbers in physics

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 $\psi$ 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. Greg Woodhouse 22:58, 16 April 2007 (CDT)

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. --AlekStos 02:59, 17 April 2007 (CDT)

Well, in a naïve way, there is the obvious fact that $e^{it}$ 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. Greg Woodhouse 07:09, 17 April 2007 (CDT)

New section

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. Greg Woodhouse 23:19, 17 April 2007 (CDT)

Request for approval

Editors: Could you take a look at this article and, if you think it's ready, initiate the approval process? Greg Woodhouse 13:24, 22 April 2007 (CDT)

Comments

I'm not an editor, but: excellent article! I 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.

Thank you. The historical motivation (cubic equations) was contributed by Sebastién Moulin, and is most appreciated. Greg Woodhouse 21:02, 22 April 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 $a+bi$?

Please do. Greg Woodhouse 21:02, 22 April 2007 (CDT)

I added some math html tags near the beginning of the article. I also added "scriptstyle" — 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. --Catherine Woodgold 19:05, 24 April 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

1. no math tags, e.g., of the form a+bi obtained by which renders as

   … of the form a+bi obtained by …

2. emulate maths formatting in HTML, e.g., of the form ''a'' + ''bi'' obtained by which renders as

   … of the form a + bi obtained by …

3. use math tags, e.g., of the form <math>a+bi</math> obtained by which renders as

   … of the form $a+bi$ obtained by …

4. use math tags and textstyle, e.g., of the form <math>\textstyle a+bi</math> obtained by which renders as

   … of the form $\textstyle a+bi$ obtained by …

5. use math tags and scriptstyle, e.g., of the form <math>\scriptstyle a+bi</math> obtained by which renders as

   … of the form $\scriptstyle a+bi$ obtained by …

Options 1 and 2 are not always possible. For instance, $x_{1}^{2}$ cannot be displayed in HTML; the best you can do is x₁². 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. -- 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 post what is written above. Hope to see you there. --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 both. I 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. Greg Woodhouse 21:02, 22 April 2007 (CDT)

Thanks. So, is that the way Gauss defined them? --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. 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.

I've no idea what Gauss did in 1831. Perhaps whoever wrote this can clarify? -- Jitse Niesen 02:08, 24 April 2007 (CDT)

Reading the forums, I see that Catherine has a much better idea of what the approval process entails. -- Jitse Niesen 21:41, 24 April 2007 (CDT)

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. --Catherine Woodgold 08:00, 25 April 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.

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.

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 $1+i^{2}$" or "modulo $1+i^{2}$".

End of first paragraph of "geometric interpretation": "both ... but also" doesn't sound right to me. I would delete "both" or change "but" to "and".

" 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".

"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.

"But, by the triangle inequality, we know that outside a neighborhood of the origin..." OK, maybe I should have understood this. But I didn't. I 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".

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 $x^{2}+1$, " inserting "(i.e. the set of polynomials with real coefficients modulo $x^{2}+1$)", 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 $K/\mathbb {C}$ is a finite normal extension" I don't understand the notation "$K/\mathbb {C}$". "...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 $\mathbb {R}$ 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.

I included this proof to illustrate how a very different(?) sort of argument could be used to show that $\mathbb {C}$ 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 $2^{n}$ 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. Greg Woodhouse 21:02, 22 April 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".)

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.

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.

"This seemingly innocuous difference actually has far reaching implications." I would hyphenate "far-reaching" when used as an adjective phrase.

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".

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 waves. Could 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?

Thanks for an enjoyable read. --Catherine Woodgold 19:01, 22 April 2007 (CDT)

Well, I suppose what I had in mind was that if $\psi$ is an eigenstate corresponding to a real eigenvalue (for $\Delta$, 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. Greg Woodhouse 21:09, 22 April 2007 (CDT)

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)?

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. -- Jitse Niesen 21:08, 24 April 2007 (CDT)

Quantum Mechanics

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 $-i\hbar \Delta$ 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. Greg Woodhouse 22:42, 24 April 2007 (CDT)

I found this exposition by Scott Aaronson enlightening when it made rounds on the interweb a few weeks ago (scroll down to "Real vs. Complex Numbers"). Fredrik Johansson 02:15, 25 April 2007 (CDT)

What a thought provoking article! Unfortunately, I barely have time to glance at it right now, but I shall return to it soon. Greg Woodhouse 06:49, 25 April 2007 (CDT)

Can we make it even better?

Great work, everyone! Certainly a great article on this important topic is a significant plus for the Mathematics section of CZ.

I'm proposing a list of modifications below. All of them are (of course) debatable.

Move the formal definitions (the sections "Formal definition" and "Beyond the notation") much later in the article.

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).

Move the basic operations and polar coordinate operations very close to the top of the article.

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 - Greg W.'s suggestion to keep the Cardano's method example first seems reasonable; the basic operations could come right after. - Greg Martin 03:43, 26 April 2007 (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! - Greg Martin 00:08, 25 April 2007 (CDT)

I am no mathematician, and you may use the above to improve this article, but - as the famous surgeon 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. Nancy Sculerati 00:17, 25 April 2007 (CDT)

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.

However, Nancy also has a point. Looking at the 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:

1. "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.
2. "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?
3. "this is just the condition for the existence of a scalar potential" (in the complex integration section): needs some elaboration.
4. 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? -- 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. Nancy Sculerati 09:31, 25 April 2007 (CDT)

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. Greg Woodhouse 10:14, 25 April 2007 (CDT)

I've found these comments very thoughtful and helpful - thank you! In particular, Nancy and Jitse put in focus for me that the following question is really what's under discussion here: what does approving an article signify?

I agree that a read through 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 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.

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.

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. - Greg Martin 13:31, 25 April 2007 (CDT)

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. :) Nancy Sculerati 20:50, 25 April 2007 (CDT)

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? - Greg Martin 03:43, 26 April 2007 (CDT)

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 sentences. It 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? --Matt Innis (Talk) 21:59, 25 April 2007 (CDT)

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" - Greg Martin 03:43, 26 April 2007 (CDT)

It's been awhile, but I believe I started out by saying outright that complex numbers are formal sums of the form $a+bi$ where $a$ and $b$ 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 $\mathbb {C}$ as the splitting field of $x^{2}+1$ 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. Greg Woodhouse 22:42, 25 April 2007 (CDT)

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. - Greg Martin 03:43, 26 April 2007 (CDT)

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 older version and the top of this talk page). At the same time the "reviewer" used the formal sum instead (see 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[1], 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? :-) --AlekStos 10:07, 26 April 2007 (CDT)

Complex analysis section

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. Greg Woodhouse 23:05, 25 April 2007 (CDT)

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. - Greg Martin 03:47, 26 April 2007 (CDT)

Approval

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. Robert Tito |  Talk  23:42, 25 April 2007 (CDT)

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? Nancy Sculerati 00:42, 26 April 2007 (CDT)

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. Robert Tito |  Talk 

graphic for geometrical/polar interpretations?

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. - Greg Martin 03:53, 26 April 2007 (CDT)

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. Greg Woodhouse 09:12, 26 April 2007 (CDT)

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.--AlekStos 14:10, 26 April 2007 (CDT)

I think it looks great. Greg Woodhouse 17:04, 26 April 2007 (CDT)

Looks good, but the text should be readable in the thumbnail. Fredrik Johansson 17:14, 26 April 2007 (CDT)

Right. I'll fix proportions. --AlekStos 17:41, 26 April 2007 (CDT)

Roots of polynomials?

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

It's fixed now. Thanks. Greg Woodhouse 09:09, 26 April 2007 (CDT)

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. --AlekStos 12:28, 26 April 2007 (CDT)

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. Greg Woodhouse 17:58, 26 April 2007 (CDT)

examples

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. Robert Tito |  Talk 

Ugh!

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. Greg Woodhouse 18:12, 26 April 2007 (CDT)

bad \scriptstyles, bad

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. - Greg Martin 19:31, 29 April 2007 (CDT)

I agree. I was going to say something like that. We should just say "math", and the browsers should figure out how to display it. --Catherine Woodgold 19:37, 29 April 2007 (CDT)

The first sentence

Here's a try at defining them completely in one sentence as Greg Martin suggests:

The complex numbers $\mathbb {C}$ are numbers of the form $a+bi\scriptstyle$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, which is a solution of the equation $x^{2}+1=0$.

--Catherine Woodgold 19:35, 29 April 2007 (CDT)

less notation?

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

... if $\alpha \in \mathbb {R}$ is real ...

let's prefer to write

... if $\alpha$ is a real number ...

In fact, I bet we could compose this entire article without even using the symbols $\mathbb {C}$ and $\mathbb {R}$! 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. - Greg Martin 19:55, 29 April 2007 (CDT)

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. Nancy Sculerati 20:00, 29 April 2007 (CDT)

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 article. Or whereever some particularly unusual notation is used there could be such a footnote. I'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 question. Perhaps there exists a math editor who will comment on these ideas?  :-) --Catherine Woodgold 20:11, 3 May 2007 (CDT)

Comments II

In "algebraic closure" it says "We offer two separate proofs," but I think the proofs are no longer included in the article.

Fixed. Greg Woodhouse 08:14, 30 April 2007 (CDT)

In "historical example" where it says "Now we choose the second condition on $u$ and $v$, namely $3uv-15=0$, or $uv=5$. " I don't like "namely" here because it suggests something that had previously been defined (or decided). I would replace "namely" with "as".

I'm not yet sure how to rephrase this. Greg Woodhouse 08:14, 30 April 2007 (CDT)

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). --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.--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 $x=u+v$, which leaves us free to choose another condition on $u$ and $v$ later; for example, we could assign any value to $u$ and then $v$ would be determined. ... Now we choose the second condition on $u$ and $v$ as $3uv-15=0$, or $uv=5$. 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 $u$ which we will choose later, and letting $v$ be $x-u$. ... Now we choose $u$ to be the number such that $3uv-15=0$, or $uv=5$.

Or yet another try:

Cardano's method for solving it suggests looking for a solution by writing it as a sum $x=u+v$, 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 $3uv-15=0$, or $uv=5$.

--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 $u$ and $v$; we only required that $x=u+v$. Hence, we can choose another condition on $u$ and $v$. We pick this condition to be...

--Catherine Woodgold 08:06, 5 May 2007 (CDT)

In "formal definition", "It follows that ${\sqrt {-1}}$, 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 ${\sqrt {-1}}$, the hypothetical number whose square root gives -1, is well-defined as (0,1)" (or maybe has been constructed instead of is well-defined.)

That should just be "defined". (I think thi is a remnant of previous edits.) Greg Woodhouse 08:14, 30 April 2007 (CDT)

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. --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". --Catherine Woodgold 17:37, 1 May 2007 (CDT)

Re multiline equations: The following syntax works on Wikipedia but does not work here: <math> \begin{align} f(x) & = (a+b)^2 \\ & = a^2+2ab+b^2 \\ \end{align} </math> --Catherine Woodgold 07:58, 30 April 2007 (CDT)

I reported this issue on the forums. -- Jitse Niesen 08:33, 3 May 2007 (CDT)

General structure revisited

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.

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.

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'. --AlekStos 09:55, 30 April 2007 (CDT)

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 $x+iy$ corresponds to the the ordered pair $(x,y)$, 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. Greg Woodhouse 18:04, 30 April 2007 (CDT)

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. --Aleksander Stos 02:43, 1 May 2007 (CDT)

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.

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. Greg Woodhouse 13:50, 1 May 2007 (CDT)

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).

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.

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. --Aleksander Stos 16:46, 1 May 2007 (CDT)

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.

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. -- Jitse Niesen 09:34, 3 May 2007 (CDT)

Approval discussion

I'm not sure I understand the approval process correctly, but I believe that if nothing happens the 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.

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 $\mathbb {Z} [i]$ 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. Greg Woodhouse 10:44, 3 May 2007 (CDT)

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.

It seems that Sébastien Moulin might know more on this, so I tried to e-mail him to ask for clarification. -- Jitse Niesen 10:48, 5 May 2007 (CDT)

Greg Martin, if you change the approval template to mention 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. -- Jitse Niesen 09:22, 3 May 2007 (CDT)

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. - Greg Martin 00:26, 5 May 2007 (CDT)

discovered algebraically closed?

In the current article: (As a final comment in this analysis, we discover that $\mathbb {C}$ has no finite extension and must therefore be algebraically closed.) The fact itself is true of course, but I don't see how the preceding discussion allows us to conclude this "discovery". - Greg Martin 00:30, 5 May 2007 (CDT)

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? - 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" ${\sqrt {-1}}$.

with this:

In the remainder of the article, we will use the letter $i$ to denote a solution to the equation $i^{2}=-1$.

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 ${\sqrt {-1}}$ 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 $-i$ 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" ${\sqrt {-1}}$ 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"? --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 $i$ in that section is the same as the symbol ${\sqrt {-1}}$ in the previous section. -- 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 $0$ are of the form $a+0i$. In this way, the real number $a$ is considered as the complex number $a+0i$ whose imaginary part is zero.

to this:

The real number $a$ is considered to be the complex number $a+0i$ 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 $a+0i$ whose imaginary part is zero is considered to be the same thing as the real number $a$.

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. $m$ and $\hbar$ should be defined, for example. Is $\hbar$ Planck's constant? (Or maybe a reduced Planck's constant, divided by 2 pi?) Presumably $m$ is mass, e.g. the mass of the electron in the example stated. I would like to see $\psi$ defined earlier in the section: it could be introduced by saying that the equation is an equation for the "probability amplitude" $\psi$ (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 $\psi$ before giving the equation for $\psi$. Maybe more could be said about why it would be hard to do this without complex numbers, e.g. "In this way, using $\psi$ 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." --Catherine Woodgold 12:35, 5 May 2007 (CDT)

• I think that the following in "formal definition" is misleading:

Positive real numbers satisfy the identity

${\sqrt {a}}\times {\sqrt {b}}={\sqrt {a\times b}},$

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 ${\sqrt {a}}$ for example is taken to mean a solution of $x^{2}=a$ 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 $x^{2}=a$, 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 $i$, the number satisfying $i^{2}=-1$, 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 $i$, 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 $i^{2}=-1$; the other is now denoted (0,-1)." --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. Nancy Sculerati 16:22, 6 May 2007 (CDT)

I think we are there Nancy. Robert Tito |  Talk