Talk:Complex number/Draft

Article Checklist for "Complex number/Draft"

Workgroup category or categories	Mathematics Workgroup [Categories OK]
Article status	Developed article: complete or nearly so
Underlinked article?	No
Basic cleanup done?	Yes
Checklist last edited by	Greg Woodhouse 19:26, 11 April 2007 (CDT)

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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.

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 $a+bi$?

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.

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 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)