Talk:Complex number

For older discussion, including the discussion about whether to use italics in naming the imaginary number i, see Archive 1. Later archive: Archive 2.

Domain colouring graph

The caption incorrectly uses the word 'saturation' where it should say 'lightness' or 'value'. I changed it, but someone changed it back. What's the story with that? 83.70.252.146 (talk) 02:18, 15 January 2009 (UTC)

clarification

How about a layman's introduction to this article? Why do we need this? How is it done? Ect. To understand this article you need a degree or some very heavy study in mathematics, thus making this work unusable to the average person who just wants to know what the subject is about. —Preceding unsigned comment added by 65.23.116.46 (talk) 05:41, 7 March 2009 (UTC) (talk) 20:53, 11 August 2009 (UTC)

Confused

If a = 1 and b = 2 then what is the answer to a+bi? 95jb14 (talk) 19:41, 28 April 2009 (UTC)

1+2i. This cannot be simplified further, as i is just the placeholder (see section 1.2 of the article for details).— Kan8eDie (talk) 20:03, 28 April 2009 (UTC)

Suggestion

It should be noted that the complex numbers unlike the real numbers cannot be ordered by <, >, <=, >= since they are not ordered fields. Thus there is not linear relation that is applicable for the complex numbers. Given two different complex numbers it is not possible to say which of the two is greater. —Preceding unsigned comment added by 59.125.178.121 (talk • contribs) 21:54, 30 April 2009

It already does. See the Real vector space section. Oli Filth^{(talk|contribs)} 21:00, 30 April 2009 (UTC)

misleading description

The words "real" and "imaginary" were meaningful when complex numbers were used mainly as an aid in manipulating "real" numbers, with only the "real" part directly describing the world. Later applications, and especially the discovery of quantum mechanics, showed that nature has no preference for "real" numbers and its most real descriptions often require complex numbers, the "imaginary" part being just as physical as the "real" part.

This is terribly misleading. No scientist performs measurements of complex numbers. In fact, the example provided is incorrect-- quantum mechanics DOES provide a preference towards 'real' numbers, in that all measurements performed corresponding to physically observable quantities MUST be real quantities, not complex. For instance, the wavefunction is a quantity of complex magnitude, but one cannot measuring it--instead, we find that the magnitude squared of the wavefunction (a real valued function) corresponds to the probability of the particle in space.

As a result, this needs to be reworded. I suspect the author was attempting to describe the fact that complex numbers are commonplace in scientific analysis, which is correct, and an important point. However, it is misleading to suggest they are 'just as physical' as real numbers, when measurements that directly measure complex quantities are impossible. —Preceding unsigned comment added by 128.83.68.219 (talk) 20:08, 10 May 2009 (UTC)

arctan not correct for arg

In Complex number#Conversion from the Cartesian form to the polar form its say

$\varphi = \arg(z) = \pm\arctan\frac{y}{x}$ (taking the sign appropriately so that z = re^iφ)

This is not correct. arctan only gives results between -π/2 and π/2 so this cannot give all the values from -π to π. I tried just writing atan2 on the right in another context and somebody stongly objected on the grounds that atan2 was not mathematical. What are peoples feelings on A) leaving the arctan there which is wrong but lots of people do it with hand waving, B) removing the business entirely or C) putting in atan2? If hand waving is the option what would yyou put instead of the wrong statement here about changing sign? Dmcq (talk) 00:39, 18 May 2009 (UTC)

I would use a description of the atan2, and link to atan2. Something like this:

The argument arg(z) is the counterclockwise angle φ between the positive x axis and z. It can be computed as:

$\varphi = \arg(z) = \begin{cases} \arctan\displaystyle\frac{y}{x} &\mbox{ if } z \mbox{ is in the 1st or 4th quadrant}\\[9pt] \pi + \arctan\displaystyle\frac{y}{x} &\mbox{ if } z \mbox{ is in the 2nd or 3rd quadrant} \end{cases}$

Thus, in general, arg(z) = atan2(y,x).

— Carl (CBM · talk) 12:16, 18 May 2009 (UTC)

That looks much better, it doesn't actually give the same principal value in the 3rd quarter but it is correct modulo 2π. I like it because it isn't so long it obscures everything which is what the formulae in the atan2 article tend to do. I don't think one needs to really worry much about the the x=0 case. I had been thinking of removing the arctan and changing the comment to 'the value of φ in (−π,π] so z = re^iφ.' but that';s not straightforward. Dmcq (talk) 14:14, 18 May 2009 (UTC)

Formal development

in the Formal development section, what is the motivation for choosing (a·c − b·d, b·c + a·d) as the product? it seems rather arbitrary as currently written -- arbitrarily chosen such that i^2 = -1 that is. is there anything else that could be said to motivate that choice, without referring to the fact that it results in the square of i being -1?

User4096 (talk) 19:52, 15 June 2009 (UTC)

The better way to look at the complex numbers is as a field extension of the reals in which one adds a square root i of -1. Because of the way field extensions work, a+bi and c+di must multiply like polynomials. The formulas you asked about come from identifying these elements of the field extension with ordered pairs, writing down the formula one would get if you multiplied everything like polynomials, and then ignoring the fact that these ordered pairs originally came from a field extension. This approach allows our article (and other elementary presentations) to avoid using the term "field extension" early on, but has the cost of making the source of the formulas more obscure. A brief discussion of the field extension approach is in the section "Construction and algebraic characterization". — Carl (CBM · talk) 20:03, 15 June 2009 (UTC)

It's unclear to me why the choice "i^2 = -1" is considered arbitrary by User4096 in this context. That's what defines complex numbers, after all. As CBM notes, the formula for multiplying complex numbers is forced by requiring (1) the usual laws of algebra (more technically, the axioms for a commutative ring) and (2) that i^2 = -1. So yes, the formula is chosen so that i^2 = -1, and it is the only one that will work for that purpose. -- Spireguy (talk) 02:05, 16 June 2009 (UTC)

There are two roots of the equation x^2=-1. One of them is picked, arbitrarily, to be called i, and the other is called −i.—GraemeMcRae^talk 18:07, 16 June 2009 (UTC)

thank you carl. that answers my question precisely. User4096 (talk) 12:07, 17 June 2009 (UTC)

The square of complex numbers

In the lead, and elsewhere, it is stated that complex numbers give real numbers when squared. This is not true since (a + bi)^2 = a^2 +2abi + (bi)^2" ... the last term becoming -b^2. This result is also a complex number, not a real number. A simple imaginary number squared gives a real number but a complex number squared does not. Abtract (talk) 06:03, 23 June 2009 (UTC)

The lead says "negative real numbers can be obtained by squaring complex (imaginary) numbers" (emphasis mine), which is admittedly ambiguous. Perhaps you would be happier if the sentence were written "negative real numbers can be obtained by squaring some complex numbers (those that are purely imaginary)". I hesitate to make the change, though, for two reasons: it might not satisfy your concern, and it makes the sentence that much more wordy without improving it much at all.—GraemeMcRae^talk 10:50, 23 June 2009 (UTC)

I take your point but as currently worded it sounds as though this is a generic property of complex numbers - especially as this is in the lead. For my money I would remove it from the lead altogether. Abtract (talk) 11:23, 23 June 2009 (UTC)

Notation for exponentiation and laws of exponents

For a complex number z, a notation z^(1/3) could be interpreted as a binary operation, in which case it must either be undefined or produce a single result. Or it could be interpreted variously as meaning the set of third roots of z or "any one of the third roots of z". So it would useful if the article explained the conventions for exponential notation in the complex numbers and stated the properties of exponentiation as an operation.

An amusing defect in presentation of the axioms for the real numbers in secondary school mathematics, is that texts and web pages often state that an operation like x^(2/3) is defined only for x > 0. They also state the law (x^a)^b = x^(a,b). Then they proceed to do examples like computing (-1)^(2/3) to be 1. This contradicts the law when x = -1, a = 2/3, b = 3/2. Attempts to straighten this out inevitably lead to a discussion of the complex numbers and multiple nth roots and so forth. That only leads to more confusion since it attempts to change the discussion from the properties of an operation to statements about sets of numbers. So clarifying the properties of the exponentiation operation would also help people understand exponentiation in the real number system.

Tashiro (talk) 17:00, 15 July 2009 (UTC)

Imaginary part

Recent edits (one of which I reverted) by two editors make me think I have misunderstood the phrase "imaginary part". I had assumed that it was the bi term but it seems to be just the real number b. Could someone else confirm this please. Abtract (talk) 22:49, 4 August 2009 (UTC)

Must admit it's not something I've actually thought about before, but Imaginary part says it is b, not bi. Dmcq (talk) 22:59, 4 August 2009 (UTC)

Yes I had already looked there but that is a completely unsourced "article". Abtract (talk) 23:01, 4 August 2009 (UTC)

I just put 'imaginary part of a complex number' into google books and I see your point, it gives either way. However Im(z) is pretty definitely b and it seems to be elementary texts when introducing complex numbers that say the imaginary part of a complex number is bi when given a+bi. Dmcq (talk) 07:46, 5 August 2009 (UTC)

Here is an interesting one from yourdictionary.com "the coefficient of the square root of negative one in a complex number as 5 in (3 + 5i): formerly, this coefficient multiplied by i was considered the imaginary part". Note my bolding on the word "formerly". If that is correct it would explain why there are two definitions around. It still seems odd to me; I would have thought that a real number could hardly be the imaginary part but it should rather be the coefficient of the imaginary part. I will leave it as it is but would be interested in any more informed views. Abtract (talk) 08:33, 5 August 2009 (UTC)

In Mathematics textbooks the imaginary part is almost always defined as the real coefficient - see for instance ([1], [2], [3], [4], etc...

In most standard dictionaries it sounds like the unit is included (for instance [5]), but I'm not sure that is what they really have in mind. In some "idiot's guide" [6]) the imaginary unit is included.

I propose we use the textbook approach :-) DVdm (talk) 12:05, 5 August 2009 (UTC)

Yes I am sure that is the correct approach but I remain sceptical so I will look into it in more depth for my own understanding (when term starts in October). Abtract (talk) 14:26, 5 August 2009 (UTC)

DVdm is correct, the textbook definition is that the real number b is the imaginary part, not bi. There are at least two reasons for this: (1) real numbers are simpler than pure imaginary numbers. Using b instead of bi when you are interested in the imaginary part reduces to a familiar context (with many many theorems available), namely the real numbers. (2) It's a special case of taking the components of a vector. If e1, e2 are basis vectors, then the e2 component of the vector a e1 + b e2 is simply b, not b e2. The reason is again number (1) above: the goal is to reduce to real numbers.

I could also add that the i is redundant: if I say "the imaginary part of z is 3" then I clearly mean z = a + 3i for some a, so I don't need to retain the i. -- Spireguy (talk) 20:39, 5 August 2009 (UTC)

Many sources do agree on the Projection (mathematics) interpretation of the phrase "imaginary part". However, then we are lead to conclude that this statment is false:

"A complex number is the sum of its real and imaginary parts."

The following two sources seem sensitive on this point and avoid the phrase "imaginary part" through alternative terminology:

• EJ Townsend (1915) Functions of a complex variable, page 6:

"axis of reals" and "axis of imaginaries"

• Philip Franklin (1958) Functions of complex variables, page 2:

"real component" and "imaginary component".

When proceeding to quaternions, the imaginary part contains direction information that cannot be simply dropped, so in that context the imaginary part of q is q deprived of its real part. Seemingly trivial matters as the one under discussion can sometimes block learning. The above false statement may be frequently repeated, quite innocently, due to the meaning of part in ordinary English.Rgdboer (talk) 22:37, 2 September 2009 (UTC)

I have removed the word "part" twice ... I hope this helps. Abtract (talk) 23:17, 2 September 2009 (UTC)

Yes, that's ok for the lead. Good idea. DVdm (talk) 08:42, 3 September 2009 (UTC)

This article is terrible

Do any of the contributors to this article actually think it is well-written? For starters the graph of the Mandelbrot set should be axed. No where in all of this discussion is there a discussion of the fact that |Z| = Sqrt(Z Z*) which is pretty fundamental (from my view anyway). The article should be rewritten from the ground up. Contributors should settle on an outline before writing. The lead in ought to be accessible to the proverbial intelligent layman. —Preceding unsigned comment added by 65.19.15.124 (talk) 15:06, 29 November 2009 (UTC)

So fix it. --M4gnum0n (talk) 15:24, 29 November 2009 (UTC)

Yes, be bold, but please follow the talk page guidelines, and don't forget to sign your messages here. Good luck. DVdm (talk) 15:56, 29 November 2009 (UTC)

I don't think it's too bad, at least not compared to some others. I agree that first diagram doesn't really help. However I see you have just started off on Wikipedia, I'd advise starting by having a look at WP:WPM and finding a low quality article with a reasonably high priority to practice on. That way your effort is more likely to contribute appreciably to Wikipedia. Starting off here there's been lots of other people done things and your effort will probably be highly diluted. Dmcq (talk) 16:57, 29 November 2009 (UTC)

What is this I dont even

This article is dense and incomprehensible to anyone who's not very familiar with the topic. If you want to know what a complex number is, and have some idea of their uses don't try reading this, it's just discouraging. Go to youtube, search 'complex number' and in 20 minutes you'll learn enough to make some sense of what's here. The Wikipedia sorely needs an article to explain this topic for those without advanced education in mathematics. 196.209.232.87 (talk) 13:46, 21 December 2009 (UTC)

Go ahead, make it better. DVdm (talk) 19:16, 21 December 2009 (UTC)

The introduction to this article is horribly written. A first glance at it does not even provide a remote definition of "complex". And the definition of "number" is also placed into question. It is inundated with a variety of links spinning elsewhere to other pages, seemingly comprehensible only to the mathematically oriented. Is there an easier-to-understand version of this article at all in it?71.108.26.48 (talk) 06:57, 22 February 2010 (UTC)

Some improvements to the article

Current version of the article is not good - cluttered and very confusing to beginners who are the primary audience as people familiar with advanced mathematics don't need to look up complex numbers on wikipedia.

I've tried to improve the article a bit by doing the following:

• I've reverted the lead picture to a previous version since the current one http://en.wikipedia.org/wiki/File:Complex_mandelbrot_illustration.png was extremely confusing (why did someone put a mandelbrot set there?!).
• I've rearranged basic properties a bit - restored section on equality, moved operations up, restored section on absolute value and conjugation and split them in two.
• Changed title of the first section to "Definitions and basic properties" and of section "Some properties" to "Some advanced properties"

As a result of the changes the article should be a bit more novice-friendly, advanced stuff is now a bit further down while basic stuff is near the top of the article, operations and properties of conjugation are now clearly explained near the top.

Some other changes which should be done IMO to further improve the article:

• Further separation of advanced topics like Formal development and Elementary functions from basic ones;
• The graphical interpretation of the operations is far from clear it should emphasize that those points are complex numbers and show which one is real and which imaginary part on the picture;
• Square roots of complex numbers should be better explained - especially that there are many of them since this is important and surprising to newcomers.
• Notation section should already explain that there are many forms in which complex numbers are written (a,b) a+bi |Z|exp(i*angle)

Sergiacid (talk) 08:41, 29 March 2010 (UTC)

K-algebra over a field?

The article says this:

Thus this is not an ad hoc construction, but can be applied to any K-algebra over a field.

What does the "K" mean? I've seen the term

k-algebra

meaning an algebra over the field k. But no particular algebra seems to have been given the name K in this context, and if it had, the phrasing "a field" would not make sense; it's not just "a" field; it's this particular field. The phrase "algebra over the field K would make sense.

So what does this mean?

possible todo's

I'm planning to work on this article. Here are a list of suggestions that I hope to carry out soon. Of course, help is very much welcome. Jakob.scholbach (talk) 21:07, 1 November 2010 (UTC)

• cx numbers in computing: briefly discuss cx arithmetic in programming languages (GNU Scientific Library, Atlas Autocode, Ruby, computer algebra system), Complex Number Calculator, C standard library
• completeness of C: complete field, giving rise to notions such as Banach space, Hilbert space, C* algebra,
• e^πi=-1
• complex conjugation: Hermitian form, complex conjugate vector space, unitary operator, unitary matrix, conjugate transpose, inversive geometry
• Abel-Ruffini theorem. real closed fields, F/E finite extension, F algebraically closed --> dim_E F=2). Matrix decomposition methods?, axiom of choice, model theory: transfer principle, Lefschetz principle
• complex plane: de Moivre's formula, rotation in R^2, complex projective line, Siegel's upper half-space
• elementary functions: exp, log (--> amoeba (mathematics), sine, cosine,
• Applications:
  • quantum mechanics. wave function
  • complex wavelet transform
  • Fast Fourier transform
  • transcendence theory??
  • signal processing, electronics. reflection coefficient
  • Eisenstein integers, Kummer rings
  • Galois theory: Dessin d'enfant
  • analytic number theory: special functions (Gamma(s), polylog, Dirichlet series)
  • real analysis: methods of contour integration
  • complex analysis: Cauchy-Riemann equations, Riemann mapping theorem, Cauchy integral theorem, several complex variables
• odds and ends: group character, Pontryagin duality, complex base systems

Jakob.scholbach (talk) 21:07, 1 November 2010 (UTC)

Brushing over

Per wp:RETAIN I undid some of the HTML-rendering by Jakob.scholbach (talk · contribs), first in this edit, and now with an undo. I also left a message on talk page. I propose we keep the math-rendeing of the article. DVdm (talk) 21:14, 12 November 2010 (UTC)

Ah, sorry, I just see this post. Well, see below for the reply. Please note that WP:RETAIN applies to British vs. American English, so is clearly unapplicable here. Jakob.scholbach (talk) 21:19, 12 November 2010 (UTC)

DVdm, thanks for your caring about complex numbers. However, you reverted me thrice, which you should not. Working on an article that is notationally unbalanced as this one does not require to ask for consensus. Beyond notation, your removal of the instructional example explaining the notation at the beginning of the article is not very helpful. I ask you to please undo your revert of my edit. Jakob.scholbach (talk) 21:25, 12 November 2010 (UTC)

Jacob, I know that wp:RETAIN is about British/American but I assumed that as a regular editor you would understand what I meant. I also know the 3RR rule, but again I assumed that you were arare of wp:BRD (-and yesyes, it's just an essay, not a policy -) and frankly I was a bit amazed over your revert, so I reverted back. Anyway, I guess you reverted before you saw my message at your talk page, so I understand. I really don't mind the remainder of your edit, on the contrary. I took the trouble in my first edit of manually re-math-rendering, so I guess it's your turn now. In other words, feel free to undo my revert, but please leave the rendering as it was? Thanks. DVdm (talk) 22:17, 12 November 2010 (UTC)

I'm with DVDm on the formatting issue. One of the overriding principles in maths articles is consistency, so readers can move backwards and forwards between sections and not be surprised or made to pause because it looks different, and formatting is a key if not the most important part of this. And in a substantial article TeX formatting is almost always preferred as there are many things only possible with it, as seen later on this article. Unless there are technical reasons, such as inline formulae breaking line width, TeX should be used. This is also explicitly allowed by the math manual of style: "Changing to make an entire article consistent is acceptable", and one of the key principles of the general manual of style.--JohnBlackburne^words_deeds 21:49, 12 November 2010 (UTC)

I hope we all agree that the article is in no way consistent. (And was not before I started editing it.) Since, as I emphasize again, will overhaul the article, which in particular includes a lot of new material. Given the unconsistent formatting, I am therefore free to choose the formatting I want in these additions and rewrite efforts. Right? Unless somebody, maybe you?, is willing to make the article consistent, there is no point in using this argument.

Even if there was a consistent formatting, as you point out, "Changing to make an entire article consistent is acceptable". I think, this guideline has to be applied with common sense, i.e., "entire" article means as consistent as possible. Other articles I worked on, including FA group (mathematics), GA's matrix (mathematics) and vector space and GA nominee Logarithm all follow this pattern and no MOS expert objected. I guess because this is coherent with MOS.

On the practical side, edits like the ones of DVdm simply hinder other editors in working on this. I cannot edit the section one in question for a while since I have to wait for his/her reaction. His/her (maybe accidental) removal of content I just added is also just a hindrance to work on this. Jakob.scholbach (talk) 22:07, 12 November 2010 (UTC)

Jacob, I have just seen what you have been doing with formatting at Logaritm with for instance this edit. You left the article in this state, with a mix of rendering. Even now, the current version, for example, the first section has two equations in HTML, and two in math. I think that this is particularly ugly. This article (Complex number) had full math-rendering, so please let's keep it. Thanks. DVdm (talk) 22:39, 12 November 2010 (UTC)

Yes, logarithm is a good example of how not to do it. Inconsistent, difficult to read and then with colour added, presumably to help readers pick out unnecessarily small expressions differing only by superscripts (which themselves are subscripted), in violation of WP:COLOR: "Especially, do not use colored text unless its status is also indicated using another method such as italic emphasis or footnote labels." But more generally we don't use brightly coloured text for the same reason the BBC and NYT, say, don't use it for articles: it is ugly and makes text much less readable.--JohnBlackburne^words_deeds 22:54, 12 November 2010 (UTC)

No, you are not free to choose the formatting. If the article is in one style you should not change it to another just because you prefer it. That paragraph from MOS:MATH#Very_simple_formulae in full:

"Either form is acceptable, but do not change one form to the other in other people's writing. They are likely to get annoyed since this seems to be a highly emotional issue. Changing to make an entire article consistent is acceptable."

So you can't change it just because you prefer one style over another, if another style is already established. You can change it for technical reasons, such as if you move a formula from being inline to its own paragraph or vice versa, or for consistency, a guiding principle of the general MOS. But as noted above in substantial articles that almost always means TeX because as here there are some formulae only possible with TeX.--JohnBlackburne^words_deeds 22:44, 12 November 2010 (UTC)

Work in progress

As outlined in the previous post, I'm going to overhaul this article. As a minor part, this includes a unified choice of notation. I propose (and will start doing so) the following: use <math> markup only where necessary. This worked in other articles. For the MOS-lovers: The relevant guideline is MOS:MATH#Typesetting_of_mathematical_formulae. Please don't revert my changes with the argument that the article is notationally inconsistent. I am working on the whole article (for more important reasons than notation) and kindly ask anyone to either help me or wait until a uniform appearance is reached. Jakob.scholbach (talk) 21:17, 12 November 2010 (UTC)

As I said before, go ahead, but please keep math-rendering for every standalone equation, and use HTML at your discretion for small inline equations. Thanks. DVdm (talk) 22:28, 12 November 2010 (UTC)

Jacob, again you made an edit ([7]), leaving a section with mixed rendering. We really don't like your HTML-rendering, and we certainly cannot accept mixed rendering. In your edit summary you say: "This is not affecting the discussion about reformatting existing text discussed at talk." I think this is affecting that discussion, since the formatting is the only discussin going on. Therefore, please stop mixing? Thank you. DVdm (talk) 23:07, 12 November 2010 (UTC)

OK I'm fading away now and this discussion is tiring me, too. So just briefly: what guideline prevents me from writing new (as opposed to reformatting existing) material in whatever math-markup I want? (my edit summary simply meant I did not revamp existing material). IMO, the consistency argument mentioned above by John does not apply to this article, because the article is not consistently formatted.

Aside question: should an edit, which improves the content (a lot, say), but decreases the formatting quality of an article (a little bit, say), be reverted? I think no, because otherwise we may end up having super-formatted but content-wise crappy articles (this one is not even the former). Jakob.scholbach (talk) 23:47, 12 November 2010 (UTC)

The guideline you're looking for is Wikipedia:MOS#Internal consistency. If the article were inconsistent then it would be good to fix, but the version before you started editing was consistent. The format used for non-inline formulae was TeX so that should be what's used in any new math added, i.e. you should fix any math you've added since to make it match, otherwise someone else might fix it for you, or revert it if it consists largely of inappropriate formatting changes.--JohnBlackburne^words_deeds 00:04, 13 November 2010 (UTC)

Jacob, you ask "Should an edit, which improves the content (a lot, say), but decreases the formatting quality of an article (a little bit, say), be reverted?". No, indeed normally it should not, but it depends on the circumstances and on the work needed to repair the formatting quality. I had just reformatted one edit and left you a message. I noticed that you ignored both the edit summary and the message on the talk page. So, assuming that you were not interested in any of this, I undid, you undid and I undid. In my view you had decreased the formatting quality a lot.

Regarding your other question: yes, there is another guideline that can prevent you from writing new material in whatever math-markup you want -- see wp:consensus, and that is not just a guideline, but a fundamental policy. Ignoring other contributors' wishes and remarks is not constructive, and it obviously tends to waste a lot of tiresome discussion on talk pages. Perhaps a smoother way to rebrush (or "overhaul", or rewrite) an article, would be to do it in user space, and when it is finished, propose and discuss on the article talk page, to finally put an agreed upon version in place when a consensus is reached. DVdm (talk) 10:48, 13 November 2010 (UTC)

I disagree in a number of points with you, but for the sake of the time (and sanity) of all of us, I decided to follow your wish concerning the math markup in standalone formulas. Jakob.scholbach (talk) 23:33, 13 November 2010 (UTC)

Thanks. And keep up the good work! - DVdm (talk) 11:06, 14 November 2010 (UTC)

Isn't Wikipedia just great! Even when people initially don;t agree we all get together and work stuff out! Why is there an image of the construction ofa pentagon on this page? It is not referenced at all in the text. I suggest it be removed. 192.16.184.140 (talk) 13:54, 16 November 2010 (UTC)

The picture was added with this edit, probably because the text contains the phrase "...it can be shown that it is not possible to construct a regular 9-gon using only compass and straightedge...". So, while the pentagon is not mentioned in the text, the text does talk about constructing polygons with compass and straightedge, so I.m.o. the pic has its place in the article. I made a little change to the caption. DVdm (talk) 14:10, 16 November 2010 (UTC)

I see. Thanks for pointing that out, it makes more sense now. Although it does raise the point whether that section is really that relevant, especially as its link to complex numbers is barely explained. 192.16.184.140 (talk) 10:56, 22 November 2010 (UTC)

Hm, I think I agree with you. This constructibility question is probably not important enough (to complex numbers) to justify more than one or two lines in this article. If the picture is there, it has to be integrated more smoothly. I hope to keep working on the article this week. Why don't you join in editing? Upstairs there is a long list of possible things to include etc. etc. We don't always discuss formatting :) Jakob.scholbach (talk) 12:10, 22 November 2010 (UTC)

Edit request from 208.65.73.102, 3 December 2010

Resolved

{{edit semi-protected}} typo: one instance of "octionions" should be "octonions" - notice extra "i" after the "t".

208.65.73.102 (talk) 21:27, 3 December 2010 (UTC)

Well spotted, done.--Salix (talk): 22:27, 3 December 2010 (UTC)

Edit request from 93.103.110.15, 4 December 2010

Cut and paste of article removed. Rather than just pasting in the whole article it will be easier to see what needs to be changed if you describe what needs to be changed with a shorter context.--Salix (talk): 10:14, 4 December 2010 (UTC)

Triangle inequality

Under the section "Complex exponential and related functions", the triangle inequality is stated incorrectly I think. It says:

$|z_1 + z_2| \le |z_1| + |z_2|$

I think it should say:

$|z_1 + z_2| \le ||z_1| + |z_2||$

Agreed? --logixoul 20:17, 4 December 2010 (UTC)

I don't think that's needed. |z₁| is the magnitude of the complex number z₁, so is always a non-negative real number. The same is true for |z₂|. So |z₁| + |z₂| is already non-negative and equals ||z₁| + |z₂||, and the extra '|'s do nothing.--JohnBlackburne^words_deeds 21:13, 4 December 2010 (UTC)

Edit request from Kourzanov, 21 December 2010

{{edit semi-protected}} $\varphi = \arctan\frac{y}{x}$ for x ≥ 0 is clearly wrong, must be
$\varphi = \arctan\frac{y}{x}$ for x > 0 and, for x = 0, y > 0, $\varphi = \frac{\pi}{2}$ and, for x = 0, y < 0, $\varphi = -\frac{\pi}{2}$ and otherwise undefined.

Kourzanov (talk) 14:08, 21 December 2010 (UTC)

It's slightly more complicated: see our article Polar coordinate system where x=y=0 results in φ=0. I propose we take this over over from there:

$\varphi = \begin{cases} \arctan(\frac{y}{x}) & \mbox{if } x > 0\\ \arctan(\frac{y}{x}) + \pi & \mbox{if } x < 0 \mbox{ and } y \ge 0\\ \arctan(\frac{y}{x}) - \pi & \mbox{if } x < 0 \mbox{ and } y < 0\\ \frac{\pi}{2} & \mbox{if } x = 0 \mbox{ and } y > 0\\ -\frac{\pi}{2} & \mbox{if } x = 0 \mbox{ and } y < 0\\ 0 & \mbox{if } x = 0 \mbox{ and } y = 0 \end{cases}$

and cite the same source (<ref>{{Cite book|first=Bruce Follett|last=Torrence|coauthors=Eve Torrence|title=The Student's Introduction to Mathematica|year=1999|publisher=Cambridge University Press|isbn=0521594618}}</ref>). Is that OK for everyone? DVdm (talk) 15:13, 21 December 2010 (UTC)

Why use specifically the arctangent to express the angle? Using arcsine or arccosine would reduce the number of cases. Tkuvho (talk) 15:21, 21 December 2010 (UTC)

Sure, no problem. Afaiac, anything goes, provided a proper source is cited. There's way to much of this unsourced stuff in the math articles. - DVdm (talk) 15:28, 21 December 2010 (UTC)

I don't think the relation between the angle and x,y requires any more sourcing than .999...=1. Tkuvho (talk) 15:32, 21 December 2010 (UTC)

I hope you're joking :-) - DVdm (talk) 15:45, 21 December 2010 (UTC)

The last case though is a convention of polar coordinates ("One must also choose a unique azimuth for the pole, e.g., θ = 0."), clearly needed for a coordinate system. Here it's surely undefined if and only if x = y = 0.--JohnBlackburne^words_deeds 15:35, 21 December 2010 (UTC)

Yes, that could be right, but then we need another source. Do we have one? DVdm (talk) 15:45, 21 December 2010 (UTC)

Eh, maybe this one http://mathworld.wolfram.com/ComplexArgument.html Kourzanov (talk) 22:47, 21 December 2010 (UTC)

A bit sloppy... and let's avoid over-advertising Mathematica here. I have taken this source and made the edit:

$\varphi = \begin{cases} \arctan(\frac{y}{x}) & \mbox{if } x > 0 \mbox{ and } y \ge 0 \\ \arctan(\frac{y}{x}) + \pi & \mbox{if } x < 0 \\ \arctan(\frac{y}{x}) + 2\pi & \mbox{if } x > 0 \mbox{ and } y < 0\\ \frac{\pi}{2} & \mbox{if } x = 0 \mbox{ and } y > 0\\ \frac{3\pi}{2} & \mbox{if } x = 0 \mbox{ and } y < 0\\ \mbox{undefined } & \mbox{if } x = 0 \mbox{ and } y = 0 \end{cases}$

DVdm (talk) 08:20, 22 December 2010 (UTC)

User Dmcq, I notice that you have been tweaking and modifying the entry. Please note that the version with φ in [0,2π[ is properly sourced in the article. DVdm (talk) 09:49, 22 December 2010 (UTC)

That publication dates to 1956 and -pi to pi is the normal principal value nowadays. I'll look around for a good source. I didn't actually see the formula given when I looked at the page cited. Dmcq (talk) 09:59, 22 December 2010 (UTC)

Ok. Happy hunting! DVdm (talk) 10:10, 22 December 2010 (UTC)

Wasn't as easy as I though, most just gave a definition rather than an algorithm. I guess people just use atan2 nowadays or press appropriate buttons on a calculator. Anyway I've found something that is practically exactly what's there so that'll saves me some work :) Dmcq (talk) 10:48, 22 December 2010 (UTC)

Excellent, we're in 2005 now. I made another tweak to the ref. DVdm (talk) 11:42, 22 December 2010 (UTC)

┌────────────────────────────────────────────────────────────────────────────────────────────────────┘ seeing as this appears to have gotten attention from people who can edit the article, I've untranscluded the edit request. Cheers. sonia♫ 05:47, 23 December 2010 (UTC)

Copyedit Polar coordinate system

The section Complex number#Polar form just needs a little copyedit from someone who has a better feel for that sort of thing thanks.

The modulus is given by |z| rather than r, but the argument is given as φ rather than arg(z). Both refer to figure two. r is used later on with only the reference to the figure defining it. I was thinking of having r=|z|= and φ=arg(z)= and removing the references to the figures but it just didn't look nice to me. Dmcq (talk) 11:52, 23 December 2010 (UTC)

Yes, I think that r=|z|= ... and φ=arg(z)= ... is much better. I boldly changed this and removed the refs to the figure, as this speaks for itself now. Feel free to hone. DVdm (talk) 14:28, 23 December 2010 (UTC)

Um yes, it doesn't look bad when someone else does it. I think I must be a bit hard on myself ;-) Dmcq (talk) 17:54, 23 December 2010 (UTC)

Cfr tickling :-) - DVdm (talk) 19:24, 23 December 2010 (UTC)

Addition diagram

The diagram showing addition beside the section 'Addition and subtraction' should really have the green line going from O to X I believe. I couldn't find a suitable diagram easily on commons. I really must get myself some nice tools for this sort of thing. Dmcq (talk) 20:05, 30 December 2010 (UTC)

I should have looked under vector addition! I'll stick in File:Vector Addition.svg Dmcq (talk) 20:09, 30 December 2010 (UTC)

The original diagram was horror. Excellent job! DVdm (talk) 20:14, 30 December 2010 (UTC)

Signal processing caveats

Complex numbers are used as the article notes in manipulating sinusoids in signal processing applications. There are a few caveats to this that the article would be best to note: first it should be more strongly stated that when this is done, the complex plane can only represent a set of sinusoids that share the same frequency. Second and more importantly, that the algebra of the complex plane is closed while the algebra of the set of all sinusoids of the same frequency is not closed. While summing works, multiplication does not: e.g. sin(t) * sin(t) == 1/2 - cos(2t)/2 and cos(2t) is decidedly *not* in the set of sinusoids represented by the complex plane, nor are constant offsets. As such it needs to be pointed out that this representation is valid only if operations are restricted to summation and scaling by a real number. (140.232.0.70 (talk) 19:18, 14 January 2011 (UTC))

Not sure what you mean by all that. Fourier analysis can be applied to a discrete or continuous spectrum of frequencies. Impedance for a circuit of resistors capacitors and inductors encapsulates information about behaviour for all frequencies. Dmcq (talk) 20:43, 14 January 2011 (UTC)

Descartes' quotation

About the citation nedeed at the line: The term "imaginary" for these quantities was coined by René Descartes in 1637 and was meant to be derogatory[citation needed]

I've found the following quotation in "E. Hairer G. Wanner - Analysis by Its History - Springer, 2008" page 57

Neither the true nor the false roots are always real; sometimes they are imaginary; that is, while we can always imagine as many roots for each equation as I have assigned, yet there is not always a definite quantity corresponding to each root we have imagined. (Descartes 1637)

Extracted from: R. Descartes (1637): La Geometrie, Appendix to the Discours de la methode, Paris 1637 English translation, with a facsimile of the first edition, D. E. Smith & M. L. Latham, The Oper Court Publishing Comp., 1925, reprinted 1954 by Dover.

--Nchiriano (talk) 17:24, 21 February 2011 (UTC)

I've incorporated this in an edit but kept the French because I'm not sure about the translation. FightingMac (talk) 05:33, 20 April 2011 (UTC)

Recent formatting changes

I just undid a large number of formatting changes done by two anon IPs, but possibly the same person, yesterday, changing C to $\mathbb{C}$. As described at MOS:MATH#Blackboard bold either format is allowed but the second format has more issues and an editor should not change the format from one to another without good reason. As the IP editor(s) did not finish the task, or at least missed some items, restoring the non blackboard-bold formatting makes the text more consistent too.--JohnBlackburne^words_deeds 21:49, 14 April 2011 (UTC)

History

The paragraph "Wessel's memoir appeared ... with a success that is well known" is copy-paste from Project Gutenberg's History of Modern Mathematics, by David Eugene Smith and really the section needs a rewrite if only to get rid of its somewhat archaic language which is a dead give away. Also a certain Bernhard Riemann really ought to get a mention ... :-) (but hold a while on that outraged blanking button - I'll do it myself a fortnight or so hence if no one else offers) FightingMac (talk) 05:15, 18 April 2011 (UTC)

Actually on reflection I'll back out of the offer of a rewrite, sorry. There's already a 'Brief history' section I've noticed since which seems to me adequate enough. FightingMac (talk) 13:40, 18 April 2011 (UTC)

That is dated 1905 so there is no real problem provided we say where it came from. It is still better to phrase it in own words. Dmcq (talk) 19:41, 18 April 2011 (UTC)

Hi Dmcq. Beg to differ but no matter and of course I value your own contributions very highly. I've now provided a small edit of what I consider the worst offending material. There's still stuff there I'm unhappy about and I expect I shall make small additions bye and bye. Meanwhile any contributor who wants to entirely rewrite or add in any way is very welcome to. Just doing a first-aid job here. FightingMac (talk) 17:41, 19 April 2011 (UTC)

The edits are improvements, but please be careful with dates. Obviously the 1897 date for Gauss’s work is spurious. References would be helpful. Strebe (talk) 18:34, 19 April 2011 (UTC)

Whoops, sorry about that and thank you for pointing it out. I'll look around for a good reference to add. Normally I would add them into the article on a sentence by sentence basis but I noticed that this article has a different structure for its references. FightingMac (talk) 19:51, 19 April 2011 (UTC)

Adding references now. Thank you. FightingMac (talk) 04:57, 20 April 2011 (UTC)

Any apologists (they had better be good) for the penultimate paragraph commencing "A complex ring or field is a set of complex numbers ..."? I am especially curious about Felix Klein's geometrical basis for Kummer's ideals, new to me. Failing enlightenment I suggest blanking it. Agree? FightingMac (talk) 20:56, 19 April 2011 (UTC)

Sorry to be a bore possibly but most of the first four paragraphs (less an edit provided by me concerning Descartes and another on the cubic equation x^3 - x = 0 - but surely Reneissance mathematicians would have reduced to x = 0 or x^2 - 1 = 0 [added: but a very good illustration, understood now - apologies]) appears to be a plagiarism of material in Together with Mathematics published Rachna Sagyar which is certainly in copyright even if Google Books have helped themselves to an extract notwithstanding. What is the Wikipedia policy here? Does anyone have a good source for the remark about Heron? I suggest keeping the reference to Euler but striking the stuff about the identity sqrt(a) * sqrt(b) = sqrt(ab) interesting though that is? FightingMac (talk) 04:55, 20 April 2011 (UTC)

Regarding Together with Mathematics published Rachna Sagyar, the plagiarism goes the other way. The author of that book just copy/pasted the history section of the Wikipedia article Number into his book.

Regarding Heron I find this on an internet search here

Imaginary numbers almost appeared in the geometry of Heron of Alexandria in the first century A.D. Attempting to compute the volume of a truncated pyramid, he came across the expression √(81-144), which produces the square root of a negative number, √-63. Without explaining his logic or identifying his dilemma, Heron bypassed the negation and wrote √63. His mistake might seem sloppy, but negative numbers themselves were regarded warily—if known at all—in his time. No wonder he ignored their imaginary square roots, which must have seemed doubly absurd.

which doesn't seem worth perpetuating so I will strike unless smacked. Worrying of course about all those high school projects citing the Greeks as inventors of complex numbers ... :-) FightingMac (talk) 05:23, 20 April 2011 (UTC)

actually I've found a good reference and I've retained it with a small expansion FightingMac (talk) 20:08, 20 April 2011 (UTC)

{re: high school students] No really. Here's a quote from a student paper on the internet

The very first mention of people trying to use imaginary numbers dates all the way back to the 1st century. In 50 A.D., Heron of Alexandria studied the volume of an impossible section of a pyramid. What made it impossible was when he had to take √81-114 (sic). However, he deemed this impossible, and soon gave up. For a very long time, no one tried to manipulate imaginary numbers. Although, it wasn’t for a lack of trying. ...

His named sources included Wikipedia... FightingMac (talk) 11:52, 20 April 2011 (UTC)

Algebraic number theory

Bit naff this section I fear. Some expert attention needed? Nice graphic though. Perhaps better shunted somewhere else? FightingMac (talk) 05:47, 18 April 2011 (UTC)

Likewise the section on 'Analytic number theory' could do with a makeover from someone punching within their weight :-) FightingMac (talk) 05:52, 18 April 2011 (UTC)

Matrix Representation of Complex Numbers

Just a suggestion. Not faulting the final answer, but when you made the statement the square of the absolute value of a complex number expressed as a matrix you should follow with something written "in terms of the absolute value squared of a complex number expressed as a matrix." I see the modulus squared |z|^2, but it is set equal to the what looks like the absolute value of an 'matrix-like' array of numbers (using the || meaning 'absolute value') which numbers look like the a's and b's that made up the earlier matrix representation of a complex number (using the () to represent a matrix). I.e., I don't see any |()|^2 in the example.Langing (talk) 19:18, 24 May 2011 (UTC)

Definition

From a mathematical viewpoint, the Definition section is not complete. It cannot be understood without reference to the rest of the article. There is no mention, for example, of the fact that i² = -1, only that i is a "mathematical symbol". As such, the definition simply defines pairs of numbers with some notation. Please either include i² = -1, or else include a statement along the lines of something like "such that the operations described in the section below hold". --seberle (talk) 03:42, 11 September 2011 (UTC)

Although the definition explicitly says that i is a mathematical symbol, which is called the imaginary unit, indeed I think it's better to say immediately that i is the imaginary unit, satisfying i²=-1. There's no need to confuse the symbol with the object, and it certainly does no harm to mention the defining property of the object. DVdm (talk) 08:30, 11 September 2011 (UTC)

Dot and Cross product

I've removed an addition even though cited to Schaum's series which defined a dot and cross product for complex numbers. |They certainly shouldn't be stuck with the normal basic operations as like conjugation they are not holomorphic. I guess there could be a section down somewhere but I'm not sure where, it looks to me like the authors just stuck in vector operations into the book to pad it out but it might be of interest. There might be some interest to some people if there is some source showing how these operations can eb done in terms of the usual operations and conjugation so someone could calculate the usual vector results when using complex numbers to represent 2D vectors. Dmcq (talk) 15:28, 13 September 2011 (UTC)

Here is the edit if you want to look at it [8] Dmcq (talk) 15:31, 13 September 2011 (UTC)

I saw it before you removed it and it was very non-standard. The dot product is pretty standard in 2D, but the cross product is usually replaced with the perp dot product. None of it has much to do with complex numbers. --JohnBlackburne^words_deeds 15:58, 13 September 2011 (UTC)

Fine, but if its non-standard why do you think the author's included it? What do you mean pad out? Maschen (talk) 16:46, 13 September 2011 (UTC)

As in:

pad something out

Fig. to make something appear to be larger or longer by adding unnecessary material. lf we pad the costume out here, it will make the person who wears it look much plumper. Let's pad out this paragraph a little.

Dmcq (talk) 18:29, 13 September 2011 (UTC)

Ok... I'm sorry if i'm not understanding you two completley, but what exactly do you want to do about this section I added? Above you mentioned it might be of interest and to look for a better source using the common complex number operations, though unless that happens the section is not to be included in the article. I'm happy with that. Now I know what you mean about the book I cited, they probably inlcuded it for sake of analogy and as you say its non-standard content. Maschen (talk) 19:39, 13 September 2011 (UTC)

Definition of term "imaginary number"

The math textbook in which I learned the most about complex numbers defined an "imaginary number" as any non-real complex number--that is, any number a + bi where b, the imaginary part, is non-zero. Numbers in the form bi--the kind referred to in this article as "imaginary"--were called pure imaginary numbers. This nomenclature, unlike what's given in this article, gives a name to numbers that are not a or bi but a + bi. If the naming convention's been changed, then what is the term for the specific latter form of complex number? (According to my textbook, the set of complex numbers is the union of the sets of real numbers and imaginary numbers; according to this article, it's the union of real numbers, "imaginary numbers", and what other kind of numbers?) This article should give a name to numbers in the a + bi form, where neither a nor b is zero. RobertGustafson (talk) 04:41, 11 November 2011 (UTC)

Nowadays the most common nomenclature seems to be one we have —with sources— in the articles. If you look into modern books, you'll notice that imaginary numbers are, so to speak, like bi and complex numbers like a+bi. Pick a few from this Google books search,, for instance this or this etc...

About your question, I have never seen a specific name in the literature for a+bi numbers, where neither a nor b is zero. I guess one could call them "complex off-the-axes numbers" or something. But of course it's not for us to invent a name and inject it into the WIkipedia. I you can find a name and a wp:reliable source, we can do so of course. But I think that question was more or less tongue-in-cheek, right? :-) - DVdm (talk) 10:58, 11 November 2011 (UTC)

There's no point naming things when there's ...... no point naming them. Dmcq (talk) 11:22, 11 November 2011 (UTC)

Perplexing non-identity" (a^b)^c≠a^(bc)

The recent addition [9] gives some food for thought. I think it highlights that the notation $e^z\,$ for $\exp z\,$ is not equivalent to $a^z\,$ with $a=e\,$, and this distinction should be highlighted in the article. In particular, regarding $e\,$ as a constant (and not as a notation for $\exp\,$), we get $e^z=\exp (z\log e)=\exp (z(1+2\pi in))\,$, which is multivalued for non-integer $z\,$, unlike the $\exp z\,$ function. The recent edit must then be corrected with this in mind. Quondum^talk_contr 10:11, 30 November 2011 (UTC)

Done. Shoot me if I got it wrong. Quondum^talk_contr 14:42, 30 November 2011 (UTC)

e^z is normally considered as being a positive real number to a complex power which is single valued. The example there exploited a gotcha because the result of such an exponentiation should be treated as a complex number but the result was supposedly the real number e. If one had to adhere to the strict typing of a formal proof system it would flag the problem immediately but us being humans practically immediately treat e+0i as the real number e without thinking. Dmcq (talk) 15:58, 30 November 2011 (UTC)

Tagging "real e" and "pure real complex e" as different "types" is not the way I'd do it; one should rather think in terms of what function is one using (and hence "principal value" or "multivalued"); it'd be nice knowing what the "standard" way of treating this is. If changes I've made (labelling e^z as a convention for exp z) are not mainstream, feel free to adjust that. There are places one needs to be a bit more explicit about exactly what the notation means. Quondum^talk_contr 17:31, 30 November 2011 (UTC)

First sentence

While I appreciate the effort to keep formulae out of the first sentence, I don't think it's strictly speaking correct as it is. Firstly (due to mathematicians' rather confusing terminology) the "imaginary part" of a+bi is not bi, but b, so the complex number doesn't really consist of its real and imaginary "parts". Also the imaginary part is optional (unless we count 0 as an imaginary number). Any way we can rephrase this to make it accurate but not offputting?--Kotniski (talk) 17:18, 13 December 2011 (UTC)

Fixed. Duoduoduo (talk) 17:52, 13 December 2011 (UTC)

That's better, thanks.--Kotniski (talk) 08:55, 14 December 2011 (UTC)

Meger complete

The page real and imaginary partshas been merged to here. Please don't revert the change - I tried to slot content together as carefully as I could. If the order of content is not correct (hardly see why not) or anything else wrong then please just edit those bits.-- F = q(E + v × B) 10:13, 19 December 2011 (UTC)

First sentence is inconsistent with Wiki def of imaginary number

In the article on Imaginary numbers, an imaginary number is specifically defined to be nonzero. This conflicts with the first sentence of the article on complex numbers, which says that a complex number is a real plus an imaginary, either of which can be zero.

I do see that you've all been struggling mightily with all this. The difficulty is to find a one or two sentence description that summarizes this subject for casual users with limited math background.

Still, some effort should be made to make the page on complex numbers and the page on imaginary numbers consistent.

My own preference is to allow 0 to be the only number that is both real and imaginary. But on the imaginary numbers page, people have been arguing over that for months.

76.102.69.21 (talk) 06:50, 29 December 2011 (UTC) stevelimages@your-mailbox.com

I've changed it from '0' to 'omitted', which is effectively of the same (zero of something means "none of that" so it's omitted) but doesn't have the problem you've identified.--JohnBlackburne^words_deeds 09:18, 29 December 2011 (UTC)

Nifty solution :-) - DVdm (talk) 12:40, 29 December 2011 (UTC)

LOL. Cute. 76.102.69.21 (talk) 02:29, 30 December 2011 (UTC) stevelimages@your-mailbox.com

Wiki definition of imaginary number didn't follow the citation given in the article. Fixed, imaginary number can have coefficient zero. The source said it was a number such that the square was equal to the negative of a real squared. It never said the real had to not be zero. Dmcq (talk) 12:01, 30 December 2011 (UTC)

Look guys, authors define imaginary number in different ways, some including zero in the definition some not, we need to mention and source both and not pick just one. Paul August ☎ 13:14, 30 December 2011 (UTC)

I'd say, come on John, do it again! - DVdm (talk) 15:37, 30 December 2011 (UTC)

Remove mention of imaginary numbers in the lead and definition

I believe we should just refer to the imaginary unit and remove most mentions of imaginary numbers in the lead and definition. Imaginary numbers are not referred to often and I think the only mention we should keep is to that 0+bi is called an imaginary number. Complex numbers form a field. Imaginary numbers are just not very useful for anything. Dmcq (talk) 17:54, 30 December 2011 (UTC)

Suggestions for the introduction and definition of complex numbers can be found here:

• Anton, "Elementary Linear Algebra", 10th ed., p. 521 [10]
• Poole, "Linear Algebra: A Modern Introduction", 3rd ed., p. 664 [11]

I think what you suggest is fine, however in the overview we should avoid discussing field extensions, since a typical reader who wants to know what a complex number is probably does not know what a field is. Isheden (talk) 21:32, 30 December 2011 (UTC)

Let's not confuse the reader

IMHO the current use of the word "optionally' in the lead is more likely to confuse than elucidate. If it is really felt necessary in a purist way then it should surely be confined to the body of the article. Abtract (talk) 23:29, 30 December 2011 (UTC)

The imaginary part is not optional. A real number is not a complex number, it is still a real number. This business about optional is confusing how complex numbers are written and what they are. Complex numbers with a zero imaginary part are written without the complex part but they are still complex numbers, it is just that such complex numbers can be treated the same as reals in most circumstances. Dmcq (talk) 23:47, 30 December 2011 (UTC)

Dmcq, also referring to this edit summary: "Imaginary part is not optional. A real number is not a complex number see talk": We say that the set of all complex numbers is denoted by C or $\mathbb{C}$. Surely you don't claim that $5 \notin \mathbb{C}$ and that $\mathbb{R} \nsubseteq \mathbb{C}$? Of course every real number is also complex.

Anyway, the result of your edit is good now, but only provided that the definition of and at imaginary number doesn't change anymore. It is defined now as allowed to be zero, so the word "optional" is indeed not needed anymore in our lead here. As soon as the definition overthere changes again (by forbidding zero as an imaginary number) we need to come back here and add "(optionally)" for the imaginary number bit again.

But again... a real number is also a complex number: $\mathbb{R} \subset \mathbb{C}$. Get Real, please (pun intended). - Cheers and happy 2012! - DVdm (talk) 10:24, 31 December 2011 (UTC)

Yes of course I claim that a real number is not a complex number. A complex number is a pair of real numbers. Complex numbers can also be viewed as a field extension of reals but that is still different from the reals, what one can do is identify elements of one set with the other and one normally does that. People automatically use order relationships only with real numbers and a^z is a well defined single valued function if a is a real number, complex numbers aren't ordered and the exponentiation would have multiple possible values if a was complex - even if the imaginary part was zero. Have a look at Construction of the real numbers and tell me which of those is the same as any definition here of a complex number. Saying a real number is a complex number is like saying the counting numbers are the same as your fingers. Dmcq (talk) 11:25, 31 December 2011 (UTC)

So you agree that $\mathbb{R} \subset \mathbb{C}$ but a real number is not a complex number???

Look: $\mathbb{R} \subset \mathbb{C}$ is short for $\forall x: x \in \mathbb{R} \implies x \in \mathbb{C}$, in words, if x is a real number then x is a complex number.

Surely you must be joking??? - DVdm (talk) 11:53, 31 December 2011 (UTC)

No I am not joking. There is a subset of the complex numbers that can be treated like the reals but that's it. The complex numbers are pairs of real numbers. If the reals werte complex numbers then complex numbers would be pairs of complex number for instance. Dmcq (talk) 12:35, 31 December 2011 (UTC)

Your "pair of real numbers" (x,y) is an element of the product set $\mathbb{R}^2$ and $\mathbb{R}^2 \neq \mathbb{C}$. There is an isomorphism (check the article) between these sets ($\mathbb{C} \cong \mathbb{R}\cdot 1 \oplus \mathbb{R} \cdot i = \mathbb{R}^2$), but they are not equal, so a pair of real numbers is not a complex number. It is a pair of real numbers. - DVdm (talk) 12:53, 31 December 2011 (UTC)

Surely there is no one right or wrong answer to this - it depends on the particular formalization being used. We shouldn't nail the article to one particular approach when we know that different authors present these things in (formally) different ways (that all essentially come to the same thing).--Kotniski (talk) 14:06, 31 December 2011 (UTC)

See Complex number#Formal construction where they are defined as pairs of real numbers plus rules for how they are combined. There's other representations too as matrices for instance. I guess each definition defines something different. If we have a categorical definition of them and just say these represent it if they are isomorphic then we could have the reals as a subset of the complex numbers after leaving out exponentiation I guess, I'm not sure how all that works but it strikes me as probably a viable top down way of doing things rather than going from the axioms up. Personally I think of it as a programmer and say double or complex or whatever and type them all differently. Dmcq (talk) 14:12, 31 December 2011 (UTC)

I also notice that Kotniski added the "(either of which may be 0)" phrase again. It is of course correct (at least with the current status at imaginary number), but not needed anymore.

So... riding my hobby horse again... will this ever end? No, not until Imaginary number is merged into Complex number i.m.o... :-) - DVdm (talk) 10:39, 31 December 2011 (UTC)

I don't see any particular problem with having two separate articles. However I would move all the detailed maths either to the complex number article or the imaginary unit article, and leave the imaginary number article as a friendly article explaning the concept in fairly simple terms (and pointing out the slight discrepancies between definitions, without getting bogged down). I would also prefer at least the first sentence of the complex number article to be free of algebra, as it was when I first saw it - giving an intuitive (but not inaccurate) description of what a complex number is.--Kotniski (talk) 12:24, 31 December 2011 (UTC)

I've got rid of the imaginary number business and just use i just like most books do as per what I was suggesting in the previous section. Dmcq (talk) 12:35, 31 December 2011 (UTC)

I think I'm going to stop looking at (or at least editing and commenting to) this article (and cousins) for a while. When I notice on my watchlist that they have been stable for a week or so, I'll come and check again. Cheers to all and don't forget to enyoy the end of the year! - DVdm (talk) 13:04, 31 December 2011 (UTC)

I like the current version

A complex number is a number which can be put in the form a + bi where a and b are real numbers and i is the imaginary unit such that i2 = − 1.[1]

That is nice as it deftly avoids the contentious question of whether it has to be in that form or whether it can optionally be put in one or another abbreviated form. Duoduoduo (talk) 19:14, 31 December 2011 (UTC)

I like it too. Abtract (talk) 19:29, 31 December 2011 (UTC)

I like it in terms of accuracy, but not in terms of user-friendliness - the first sentence ought to give some kind of general indication, in terms that a relatively mathematically unsophisticated reader will readily relate to, what it is that these complex numbers are. I would prefer to postpone any algebra at least until the second sentence, after we've said something about it being a sum of a real number and an imaginary number (with such provisos as to make the statement not mathematically wrong).--Kotniski (talk) 08:37, 2 January 2012 (UTC)

The imaginary unit is the basic thing people have to work their heads around when starting with complex numbers. Imaginary numbers are a practically useless idea and confusing as the term has often been used to refer to complex numbers. Dmcq (talk) 13:01, 2 January 2012 (UTC)

The newest version starts out with

A complex number is an ordered pair (a, b) of real numbers a and b, referred to as the real part and the imaginary part, respectively.

I don't think this is at all user-friendly for someone who knows little or nothing about complex numbers other than that they look like 3+2i. An ordered pair? I think that starting out like that, with a concept the typical reader has never heard of, is likely to discourage the non-mathematician from reading further. I think it should start out with the familiar. Maybe start out with an example like 3+2i rather than a+bi? Duoduoduo (talk) 17:17, 2 January 2012 (UTC)

Yes, this is getting worse and worse. I always used to wonder how the maths articles on Wikipedia got to be in such an unhelpful state - now I can see the process in action, I can understand it. People are so hooked on making everything rigorously correct (which, for mathematicians, is understandable) that they forget their audience; formal errors are vigorously corrected, while matters of comprehensibility take a lower priority, and we have a gradual drift away from something readers might understand to something that can only be appreciated by people who, basically, already know it.--Kotniski (talk) 17:34, 2 January 2012 (UTC)

I'm not a mathematician; my intention was simply to go straight to the geometrical interpretation. A complex number is just a pair of coordinates in the complex plane. I thought it was clear what a pair of real numbers is, whereas starting with a formula containing roots of negative numbers might not be very user-friendly according to Kotniski above. However, if people are uncomfortable with this we can go back to the description as a number on the form a+bi. Isheden (talk) 19:38, 2 January 2012 (UTC)

The definition as a pair is correct and is used in formal definitions but it is not the usual definition used in introductory texts. I believe it should be left till later in the article like it is in the formal definition section. The a+bi definition is I believe the most common simple definition. Dmcq (talk) 19:42, 2 January 2012 (UTC)

Gets my thumbs up with both definitions/forms there in the lead and it shows where the accompanying Argand diagram comes from. Dmcq (talk) 21:52, 2 January 2012 (UTC)

Subfield of ℝ isomorphic to ℝ?

Really? (relinked to page from history) --COVIZAPIBETEFOKY (talk) 15:30, 15 January 2012 (UTC)

Contradiction. --COVIZAPIBETEFOKY (talk) 15:56, 15 January 2012 (UTC)

Damn, this is some old shit. I shall now remove the line entirely. --COVIZAPIBETEFOKY (talk) 16:21, 15 January 2012 (UTC)

I realize now that the pdf I linked to doesn't quite say what I thought it said. If anyone can find a citation stating that there exists a proper subfield of $\mathbb{R}$ isomorphic to itself, I would be very interested in seeing it (and we could reinclude the clause with the citation). --COVIZAPIBETEFOKY (talk) 16:34, 15 January 2012 (UTC)

I came across this page, apparently authored by Lounesto, that suggests that there is only one subfield of ℝ isomorphic to ℝ: "In contrast, the real field R has only one automorphism, the identity." I do not know whether this excludes injective automorphisms, so I do not know whether this would settle the question. I find it fascinating that he claims the converse holds for ℂ as a field, though not for a proper subfield. — Quondum^☏✎ 18:26, 31 January 2012 (UTC)

Yeah, I knew about that result. But that doesn't outrule an isomorphism with a proper subfield, ie, an injective endomorphism that's not onto.

Just for the record, I also put this in a discussion here, which seemed to be pretty conclusive. --COVIZAPIBETEFOKY (talk) 02:39, 4 February 2012 (UTC)

i=/=sqrt-(1)

Is it? I don't think it is necessarily. Cause if i=sqrt-(1) then i*i=sqrt-(1)*sqrt-(1)=sqrt(-1*-1)=sqrt(1)=1

Which we know isn't true since i^2=-1 not 1. — Preceding unsigned comment added by Fipplet (talk • contribs)

You have made a common mistake. See Imaginary unit#Proper use for one explanation. --JohnBlackburne^words_deeds 13:41, 31 January 2012 (UTC)

The principal value of sqrt(-1) is defied to be i rather than −i so yes they are the same. There is no requirement though that the principal value of square root the product be equal to the product of the principal values of the square roots, there are two possible square roots and it so happens that you should have taken the negative square root in the last step above. And in fact if a second square root of -1 occurs there is no guarantee without extra checks that it will be i if i already occurs. This is a bit like if you have x²=1 and y²=1 the possible values of x+y are −2, 0, and 2. Dmcq (talk) 15:56, 31 January 2012 (UTC)

Thanks guys I think I understand. I'm sorry you must get this question alot. I just started linear algebra so that's why. Fipplet أهلا و سهلا 14:28, 4 February 2012 (UTC)