Talk:Complex number/Archive 3

Archive 1

Archive 2

Archive 3

Archive 4

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 $z=re^{i\varphi }$)

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:

$|z_{1}+z_{2}|\leq |z_{1}|+|z_{2}|$

I think it should say:

$|z_{1}+z_{2}|\leq ||z_{1}|+|z_{2}||$

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:

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

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:

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

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)

Archive 1

Archive 2

Archive 3

Archive 4

Assessment comment

The comment(s) below were originally left at Talk:Complex number/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Needs a bit more detail on motivation in the introduction. Then consider nominating for Good Article status. Tompw 14:38, 7 October 2006 (UTC) Also needs more references. Geometry guy 20:44, 9 June 2007 (UTC) Please no references. It is completely unknown who first has given the formulas in this article. And any book on this subject has been copied from older books. But one can include the even-dimensional matrix representations of complex numbers because beginners don't understand what the complex unit i means, since it cannot be represented in our 3-dimensional environment. — Preceding unsigned comment added by 130.133.155.68 (talk) 10:53, 11 August 2015 (UTC)

Last edited at 10:54, 11 August 2015 (UTC). Substituted at 20:20, 2 May 2016 (UTC)

Metamath Proof Explorer

The recently added section on the Metamath Proof Explorer seems like undue weight. Also, it does not seem that even the (rather poor) references there actually address the subject of this article in a direct and substantive manner. Rather it seems more like an advertisement for Metamath and for the axiomatic method in a rather generic and non-specific way. Finally, the section is out of place. In a section about constructing models of the complex numbers, a generic section on the benefits of the axiomatic approach is off topic. This having been said, it would be interesting to see if there are axiomatizations of the complex numbers that are not reliant on the axioms of the reals. That would probably make a worthy addition to the article, unlike this section. But this will involve consulting multiple quality sources, rather than just flap for Metamath. Sławomir Biały (talk) 12:47, 1 February 2015 (UTC)

Agreed, this did not belong here. The addition seemed to be making a claim that would be a detour, even if it was valid and notable. On the question on sidestepping the axiomatization of the reals, the axioms listed actually seem (to my cursory reading) to include axiomatization of the reals essentially as a totally ordered field in which there is always an element between any distinct pair of elements. This seems to me to be incomplete as an axiomatization of the reals, and hence also of the complex numbers. —Quondum 15:24, 1 February 2015 (UTC)

It may also be equivalent to axiomatization of the real numbers (Dedekind-completeness); I have not checked properly. Anyhow, I see nothing interestingly new in this, in the sense of not relying on the axiomatization of the reals. —Quondum 16:11, 1 February 2015 (UTC)

Slawekb: The section I added isn't about the Metamath proof explorer; it's about the axiomatic definitions of complex numbers. I have added the example of Metamath set.mm database because it illustrates the axiomatic definition of the complex numbers. The featured article about groups mentions examples and applications, analogous to how this article already mentions example constructions of the complex numbers. My addition adds a reference to one example axiomatization. Axiomatizations of complex numbers are on-topic in an article about complex numbers. The examples are not taken to be "undue weight" or "out of place" in the aforesaid featured article and in this article (example constructions). Why this ad-hoc criterion against my edit?.

This having been said, it would be interesting to see if there are axiomatizations of the complex numbers that are not reliant on the axioms of the reals.

Free feel to contribute such an axiomatization.

In a section about constructing models of the complex numbers, a generic section on the benefits of the axiomatic approach is off topic.

This is a confusion. In my edit (that you undid and linked), the section I introduced about the axiomatic approach to define complex numbers is separate from Construction as ordered pairs of real numbers, under the section Formal definition, renamed from Formal construction. I was careful to make the titles in my revision correspond to the contents of the sections.

Also, it does not seem that even the (rather poor) references there actually address the subject of this article in a direct and substantive manner.

That would probably make a worthy addition to the article, unlike this section. But this will involve consulting multiple quality sources, rather than just flap for Metamath.

This sounds like you're taking your opinion of whether an addition is "worthy", whether a reference is "rather poor", and that it seems [to you] as an advertisement as something objective and as if it was the only criterion relevant; this is dangerously close to what is described in Ownership of articles. Bear in mind that editorial decisions are made by consensus, which in turn must be based upon arguments, not just stated opinions. I have stated the arguments for my posture referencing other articles and Wikipedia policy.

Before my revision, the article was missing a discussion about the axiomatic approach to defining complex numbers, and it's also missing it after you reverted it. I presume that your intention is to improve the article, so I make some suggestions:

• Revert your revert, and expand the section about axiomatic definitions of complex numbers. Don't demolish the house while it's still being built, instead make your contribution to build it.
• The whole section Formal construction is unreferenced. Since you are especially concerned about "rather poor" (in your opinion) references I guess that a whole unreferenced section is especially disturbing to you. You can turn your complaints into a contribution by adding references to this section.

Regards. Mario Castelán Castro (talk) 19:53, 1 February 2015 (UTC).

Quondum: Please specify which claim you're talking about when you wrote “The addition seemed to be making a claim that would be a detour, even if it was valid and notable.”. Regards. Mario Castelán Castro (talk) 20:36, 1 February 2015 (UTC).

At a first glance, the added section seemed to be to be claiming an alternate axiomatization that relied on potentially fairly deep aspects of set theory (through mention of ZFC and the like). To approach complex numbers through a new direction would be a detour: approaching a simply described mathematical object in a new way, the only purpose being to describe the new route.

However, looking at the axioms listed on the site, it seems not to be new at all: the axioms are essentially those of the real numbers with a few more to determine the complex numbers as containing them, and I've already indicated. The article already gives non-constructive approaches to reaching the complex numbers (e.g. algebraic closure). —Quondum 02:49, 2 February 2015 (UTC)

Yep, the source really doesn't inspire much confidence at all. The notion of an ordered pair is presented as though it is some hard problem in model theory. But all of the hard work is already done in the construction of the reals. In building axioms for the complex numbers, one is left with a number of rather trivial choices (a complex number can be an ordered pair of reals, or it can be any irreducible quadratic extension of the reals, etc.) all of which lead to structures that are rather trivially isomorphic modulo the Galois group. Designating any one of these as "the one true" axiomatic treatment seems to be missing the point. The conclusion, that some hard problem (e.g., ordered pairs—gasp!) has been solved by building a "portable" structure, does not hold up under scrutiny. Sławomir Biały (talk) 11:24, 2 February 2015 (UTC)

---

At a first glance, the added section seemed to be to be claiming (...) However, looking at the axioms listed on the site, it seems not to be new at all

The section I added never makes that claim, that's your interpretation, that, as you also notced, is invalid (why critize based a invalid point?). Maybe you were trying to read between lines and saw something that is simply not there. Free feel to make a quote from the aforesaid added section to support your claim. If you think that the section is unclearly worded (I disagree) free feel to suggest a better wording in order to materialize your complains into a contribution to Wikipedia.

To approach complex numbers through a new direction would be a detour: approaching a simply described mathematical object in a new way, the only purpose being to describe the new route.

If that argument were to hold, there would be place for only one approach to construct or describe the complex numbers. The article already mentions several ways to construct the complex numbers; how is it that the critique of “approaching a simply described mathematical object in a new way, the only purpose being to describe the new route” applies to the section I added and not to the current constructions and characterizations?. Of course, any approach to construct or define the complex numbers is equivalent to any other in the resulting properties, for that matter, all but one are redundant. Talking about an axiomatic construction of the complex numbers is on topic and not a detour on an article about complex numbers!. We can agree in that it wouldn't make sense to include as many axiomatizations as possible, but we can and should include a representative sampling (just as we can't include all properties of complex numbers, but we can and should include a representative sampling).

The article already gives non-constructive approaches to reaching the complex numbers (e.g. algebraic closure).

That's one such possible axiomatic approach, though it's not mentioned as such. Why should any other axiomatization (and an explicit mention of the fact that the axiomatic approach is an alternative to the constructive approach) be excluded?. It's possible to merge those and my added content into a single section about axiomatic approaches.

Yep, the source really doesn't inspire much confidence at all. The notion of an ordered pair is presented as though it is some hard problem in model theory.

It doesn't inspires much confidence to you but you have no backed your claims, leaving them as just your opinion. You're entitled to draw any opinion or conclusion whatsoever by any text for yourself. However, the Wikipedia policy doesn't supports removal of content based on the mere opinion of an editor. None of your complaints about [12] are supported by the actual article. Specifically, I mean the following claims, quoting from your message above, interleaved with my annotations:

• The notion of an ordered pair is presented as though it is some hard problem in model theory.

  No, the page makes absolutely no mention of model theory, take note that it doesn't even includes the word “model”.

• Designating any one of these as "the one true" axiomatic treatment seems to be missing the point.

  The article never postulates that axiomatization as "the true one", there's even a reference to a similar but different axiomatization.

• The conclusion, that some hard problem (e.g., ordered pairs—gasp!) has been solved by building a "portable" structure, does not hold up under scrutiny.

  The article doesn't says this this, it even describes exactly what it means by portable:

  The construction is "portable" in the sense that the final axioms below hide how they are constructed, and another construction that develops the same axioms could be plugged in in place of it.

If you a find an actual factual error in my source (That is, it actually makes a claim that is objectively incorrect), free feel to bring it up and suggest another source without that mistake. If you have no objections to the section, restore it; free feel to suggest and work on improving it and the rest of the article; otherwise, please mention the reasons for the removal of the section I added based on policy and arguments.

I'm more than willing to work together with any editor interested in improving the article. It's for my desire of contributing that I donated my time adding a section on a topic not currently covered. However, I don't think that criticizing a source for saying something that it doesn't says will lead us anywhere, so I really recommend to avoid that.

We can merge the section Algebraic characterization and Characterization as a topological field with my added section into a single one (or subsections of a single (sub)section) about axiomatic descriptions of complex numbers.

Regards.

Mario Castelán Castro (talk) 17:57, 15 February 2015 (UTC).

Mario, just because you have a source for something doesn't mean that the something is more deserving of inclusion than the unsourced material. If this article were riddled with original research and unverified claims, then you might have a point. But, as far as I am aware, virtually all of the article is of such a basic nature as to be covered in many many very reliable sources. There are almost too many to pick from. Now, more inline citations are usually a good thing, but two wrongs definitely do not make a right. In this case, the material you added, and the source that it was attributed to, were clearly inappropriate for inclusion in a general-purpose article on complex numbers. People have been studying complex numbers for hundreds of years. There are reliable and very distinguished books written on the subject. Axiomatics, too, has been around for a very long time. If there are notable perspectives on the axiomatizations of the complex numbers, they would be the sort of thing routinely lurking in the usual places like analysis textbooks. But of the sources that we actually do use in writing this article, there should be no place for using random websites as sources. Moreover, we should be very leery of material that is not readily supportable by an overwhelming preponderance of sources. This is the policy. Sławomir Biały (talk) 20:59, 1 February 2015 (UTC)

Slawekb:

Mario, just because you have a source for something doesn't mean that the something is more deserving of inclusion than the unsourced material.

The policy does not talks about content being “deserving” of inclusion. Maybe you're talking about it being notable. or verifiable and confused the terms. Please don't distort the policy.

In this case, the material you added, and the source that it was attributed to, were clearly inappropriate for inclusion in a general-purpose article on complex numbers.

This is again your opinion, to which I disagree. Using the word “clearly” doesn't turns opinions into facts, but by stating our arguments we can reach a consensus. When you state your opinion, please also state your arguments.

(...) If there are notable perspectives on the axiomatizations of the complex numbers, they would be the sort of thing routinely lurking in the usual places like analysis textbooks. But of the sources that we actually do use in writing this article, there should be no place for using random websites as sources.

I suggest that you take a look at the wiktionary entry for random. The Metamath website is not random in any sense, it's not a probability distribution or characterized by or often saying random things; all on the contrary. Metamath and it's website is written mostly by Normal Megill, a professional mathematician, who has published papers in mathematical journals, and several professional mathematicians contribute to its set.mm database. The page I used as source is itself a secondary source which I used as an example of the axiomatization of complex numbers. The set.mm database and the prose pages accompanying the proof explorer contain references to treatises by other professional mathematicians, including books (anybody can verify this claims by browsing the Metamath website and seeing for himself).

People have been studying complex numbers for hundreds of years. There are reliable and very distinguished books written on the subject. Axiomatics, too, has been around for a very long time

That's right.

If there are notable perspectives on the axiomatizations of the complex numbers, they would be the sort of thing routinely lurking in the usual places like analysis textbooks.

It's not your article to design a place from which all the citations must come (analysis textbook and other "usual" places) from and exclude all other contributions such as mine that don't use references from those places (anyway, by the same logic, since Complex_number#Formal_construction contains no such citations, then it needs to be removed as well, but I disagree with that). I did add a citation to my section to comply with the verifiability policy already, which I must note, doesn't forbids using websites as citations. If you would like to add a particular citation from an analysis textbook so as to improve my addition (Wikipedia is a collaborative work), free feel to materialize your complaints into such a contributions by restoring my edition and adding such a citation. Likewise, if you think that Metamath being the sole example that I added about axiomatic descriptions from the

Moreover, we should be very leery of material that is not readily supportable by an overwhelming preponderance of sources.

Hence my suggestion that you add citations so that we don't have to be be leery of any of the unsourced claims of this article, including the whole unreferenced section Formal construction. (I disagree that I have to be leery of any of that, but materializing your complaints into an action would be a contribution in this case, hence my encouragement).

Regards. Mario Castelán Castro (talk) 22:34, 1 February 2015 (UTC).

Mario, there are real problems with the source you want to cite, as well as the content in question. If you have some relevant high-quality sources concerning the axiomatization of complex numbers, you are certainly welcome to present them. But further argument, and petty sniping, apparently for its own sake, does not seem to me to be a very constructive use of time. Sławomir Biały (talk) 23:34, 1 February 2015 (UTC)

I have addressed your concerns above (I.e: the real problems which are claims completely unsupported by the criticized source). See my coment above. — Preceding unsigned comment added by Mario Castelán Castro (talk • contribs) 17:57, 15 February 2015 (UTC)

This argumentation is pointless. Also, the recent refactoring of this discussion, with the same points brought in, is entirely unconstructive. The source you proposed is not good. And the content you tried to include in the article is thoroughly idiotic. Go find a real mathematics textbook or peer reviewed secondary source, per our guidelines. It's not really that hard to find such sources on complex numbers, and I'm astonished that someone editing scientific content on an encyclopedia would think that this is an unreasonable standard of reliability and weight for inclusion. If you don't like this, start a formal RfC process. We're done here otherwise. Sławomir Biały (talk) 19:25, 15 February 2015 (UTC)

Helpful?

In Complex number#Conjugation there is the sentence: "In the network analysis of electrical circuits, the complex conjugate is used in finding the equivalent impedance when the maximum power transfer theorem is used." Is this perhaps burdening the reader a bit too much with a detail of a specific application in a section that is introducing a very basic abstract concept? --Stfg (talk) 17:33, 31 May 2015 (UTC)

Yes, I think it would be better to find a place for this in the Complex number#Applications section. - DVdm (talk) 17:42, 31 May 2015 (UTC)

Vector sum notation ambiguity

The vector-sum diagram at the right of the section on "Addition and Subtraction" of complex numbers should use identification characters other than "a" and "b" (e.g. use "p" and "q" instead) because current usage creates an ambiguous use of the characters "a" and "b": for the entire complex numbers being added in the diagram vs. real and imaginary parts of complex numbers in the explanation text for addition and subtraction of complex numbers, at the left.

Though mathematically savvy readers will likely not be confused by that ambiguous usage, after noting it, novices may be very confused by it. — Preceding unsigned comment added by 173.164.80.150 (talk) 02:23, 14 December 2015 (UTC)

We can't just change the image without possibly creating a similar "problem" in other articles that include this image. On the other hand, novice readers could actually benefit from this, by being forced to realise that symbols depend on context. Note the preceding image, where the symbols x and y are used instead of a and b. - DVdm (talk) 09:35, 14 December 2015 (UTC)

Shouldn't this be 'b < 0' and not 'b > 0'?

Section 'Definition' reads Moreover, when the imaginary part is negative, it is common to write a − bi with b > 0 instead of a + (−b)i, for example 3 − 4i instead of 3 + (−4)i.. Shouldn't it be less than 0? Or am I reading it wrong?2001:981:9B5E:1:88F4:B16F:4BD3:87E2 (talk) 04:00, 18 January 2016 (UTC)

I'm afraid that you are reading it wrong. If b was itself negative (i.e., b < 0) then a - bi would have a positive imaginary part. What is being talked about is the common way of writing a complex number with a negative imaginary part without the use of unnecessary parentheses, that is, by making the "negativeness" explicit in the notation. To put it another way, if -b is negative, then b must be positive. Bill Cherowitzo (talk) 05:16, 18 January 2016 (UTC)

Seems like being dead tired and learning about complex numbers at 5 AM isn't a good idea. I read it once and it makes complete sense now. Feel free to remove this section. I'm not an wikipedia etiquette expert. 2001:981:9B5E:1:88F4:B16F:4BD3:87E2 (talk) 12:44, 18 January 2016 (UTC)

"Complex numbers are used in many scientific and engineering fields, including ...statistics, as well as in mathematics"

Qn: Should maths be included in the list of 'scientific and engineering fields' in 2nd paragraph?

According to the Wikipedia page on scientific fields. Obviously, maths is much more than just science, but that doesn't mean it isn't science. Also, maths is more a part of science than economics will ever be, and no one wants to remove that? Thanks. Mmitchell10 (talk) 22:35, 7 June 2013 (UTC)

I don't have an opinion about either edit. I reverted a similar edit a little while ago that removed all mention of mathematics and statistics, which is clearly rather absurd. I'm happy as long as mathematics is mentioned, but it does seem a little one-sided that mathematics is not considered a science in this article while statistics is. But I don't think any of these questions are particularly worth arguing over. The reference in the edit summary to WP:BRD seems backwards. JamesBWatson was the one who made the bold edit and was reverted, or have I missed something? Sławomir Biały (talk) 00:36, 8 June 2013 (UTC)

I agree. — Carl (CBM · talk) 01:38, 8 June 2013 (UTC)

Ah yes indeed, the WP:BRD was somewhat misplaced, sorry for which, but we're discussion now, so that's a good thing.

Something our old professor used to throw at us every now and then during his lectures on complex analysis:

Philosophy is the mother of science.
Physics is the queen of science.
Mathematics is the whore of science.

Quite true i.m.o. :-)

Seriously, our Science article says: "In modern use, "science" more often refers to a way of pursuing knowledge, not only the knowledge itself. It is often treated as synonymous with 'natural and physical science', and thus restricted to those branches of study that relate to the phenomena of the material universe and their laws, sometimes with implied exclusion of pure mathematics. This is now the dominant sense in ordinary use"

Let's be modern... - DVdm (talk) 08:19, 8 June 2013 (UTC)

My opinion on this is not strong either. JamesBWatson has made a suggestion to me for how to rephrase this which I think is excellent, so I've suggested he make the change. :-) Mmitchell10 (talk) 11:00, 8 June 2013 (UTC)

Actually, mathematics is the language of science.173.75.21.87 (talk) 02:57, 3 March 2016 (UTC)

• Note: The following message was edit conflicted by Mmitchell10, but I still feel it is valid. JamesBWatson (talk) 11:16, 8 June 2013 (UTC)

I don't see it as worth arguing about whether mathematics is a science or not: usage varies. However, to me the essential meaning of the sentence is that complex numbers have applications, and are not just an abstract piece of pure mathematics. Of course complex numbers are used in mathematics, because they are part of mathematics, but the sentence is telling us that they also have uses in other fields. That is why I rephrased the sentence to give "mathematics" a different status in the sentence. I confess that my edit summary was not well thought out, and did not adequately express my purpose. However, I didn't give much thought to the edit summary because I thought that the minor change of wording that I made would be uncontroversial, and it never occurred to me that anyone would object to it. What I really meant was something like "use of complex numbers withing pure mathematics is not application of them in the same sense as applications of them to other fields". In answer to "it does seem a little one-sided that mathematics is not considered a science in this article while statistics is", the point is not really whether mathematics or statistics or economics or anything else "is a science", but rather that mathematics does not have the same status as the other fields in this context, because use of complex numbers within mathematics is not an external application, as it is with the other fields. To me, the essential message of the sentence is that, as well as appearing in pure mathematics, which is obvious, complex numbers also have practical applications in many fields, which is less obvious, and therefore worth mentioning. The fact that there are external applications is the new information that is introduced by the sentence, while the fact that complex numbers are used internally in mathematics is part of the background, which must already be clear to anyone who has reached that part of the article. It therefore seems to me natural to separate the mention of mathematics from the list of applications.

Since all this has led me to think about the whole sentence, another point has been brought to my attention. The sentence says that "complex numbers are used in many scientific and engineering fields". At best, specifying "scientific and engineering" is redundant, as the list of examples clearly contains scientific and engineering fields. Arguably it is worse than redundant, because many people do not regard economics as scientific or engineering. It therefore seems to me that nothing would be lost, and something might even be gained, by missing out those words. "Complex numbers have practical applications in many fields, including..." followed by the list of examples, would convey the same meaning, and also emphasise the practical nature of applications. Any opinions? JamesBWatson (talk) 11:16, 8 June 2013 (UTC)

• I initially misread Mmitchell10's comment above, which I thought said "so I've made the change", not "so I've suggested he make the change". For what it's worth, my suggestion is: As well as their use within mathematics, complex numbers have practical applications in many fields, including physics, chemistry, biology, economics, electrical engineering, and statistics. I will wait and see if anyone else has any opinion on the matter before deciding whether to go ahead with it. JamesBWatson (talk) 11:24, 8 June 2013 (UTC)

Perfect! - DVdm (talk) 12:01, 8 June 2013 (UTC)

OK, that looks like consensus to me, so I have made the change. JamesBWatson (talk) 13:59, 8 June 2013 (UTC)

Complex numbers

I am an 11 class student.i want to know how to prove de movires theorem and all questions related to that. Rupak kafle (talk) 13:24, 26 January 2017 (UTC)

This is not the place for that—see wp:Talk page guidelines. You can take this to the wp:Reference desk/Mathematics. - DVdm (talk) 15:40, 26 January 2017 (UTC)

Good example of poor complexity

The overview reveals how poorly complex numbers are taught and thought.

(x+1)^2 = -9 has an obvious solution: (x+1) = -3 and 3, once each. At a glance (2+1) and (-2-1). Why not follow this obvious route? Because formalisms were drummed into us by rote?

i(x+1) = (x+1)(-x-1))

(x+1)(-x-1) = -9

-(x^2) - 2x - 1 = -9

-(x^2) - 2x = -8

x = 2 and -2.

(2+1)(-2-1) = -9

Is this really so horrible?

But because a complex number has a real and 'imaginary component,' we have to add lots of complexity. — Preceding unsigned comment added by 208.80.117.214 (talk) 20:37, 15 May 2014 (UTC)

Please put new talk page messages at the bottom of talk pages and sign your messages with four tildes (~~~~). Thanks.

(x+1) = -3 and 3 are not solutions of (x+1)^2 = -9. They are solutions of (x+1)^2 = 9. - DVdm (talk) 20:51, 15 May 2014 (UTC)

The roots of 9 are EITHER -3 OR 3. The roots of -9 are BOTH -3 AND 3. This can't be wished away by a principal branch assumption. i is a function which produces the BOTH...AND root. Brian Coyle — Preceding unsigned comment added by 208.80.117.214 (talk • contribs) 02:07, 16 May 2014 (UTC)

Would you care to define a "root"? I am unable to determine a framework in which to interpret what you're trying to say. Please sign your posts using four tildes as indicated by DVdm. —Quondum 03:40, 16 May 2014 (UTC)

Thank you for your response.

Root is often defined as a number that when multiplied by itself gives a given number. But the reverse is equally true. A given number has roots that generate it. If the number is negative, and the square root is sought, then a complex number is needed: a two dimensional mathematical object. Roots as often defined generate two dimensional objects. But a complex root is a two dimensional object already. The usual projection of a + bi on the x,y axis is an approximation; i is not represented by a single point. i is both positive and negative. Thus 3i is -3 and 3. This isn't a weird conclusion, it generates -9. Imaginaries are not imaginary; they're a different way solving the always present issue of negative and positive roots. Again, I appreciate your interest. Brian Coyle 208.80.117.214 (talk) 07:19, 16 May 2014 (UTC)

Brian, note that this is not the place where we teach elementary algebra to our readers. This is where we discuss proposed changes to the article. We are not allowed to discuss the subject here—see wp:talk page guidelines. So if you have a proposal to make a specific change to the article, and you have a reliable source to back it up (see wp:reliable sources), you are free to do so here. Otherwise, I'm afraid this is off-topic on this article talk page. - DVdm (talk) 09:01, 16 May 2014 (UTC)

I think that roots of -9 are -3i and 3i. Only -3 and 3 cant be roots of -9 because -3*-3 is 9 not -9. — Preceding unsigned comment added by 178.149.167.66 (talk) 15:43, 12 February 2017 (UTC)

ALL "NON-REAL" NUMBERS ARE IMAGINARY--a + bi as well as bi--SO SAYS THE DICTIONARY!

EVERY MATH TEXTBOOK I've ever read has said that "imaginary numbers" are complex numbers a + bi such that b is not zero. That is, all complex numbers other than real numbers (a) are imaginary--not just bi, which is called pure imaginary. This is also what Merriam Webster's Collegiate Dictionary, Eleventh Edition (published 2014!) says--and this is a 1,600+-page dictionary with terms ranging from tech-math like Fourier series to dirty words like fuck. Definition of imaginary number as follows (page 620):

"A complex number (as 2 + 3i) in which the coefficient of the imaginary unit is not zero--called also imaginary; compare PURE IMAGINARY"

This is a dictionary that has a lot of stuff not found in most dictionaries--and very technical-minded to boot. Besides, the idea that, within the set of complex numbers, the set of imaginary numbers represents the full complement of the set of real numbers is consistent with the other English meanings of the word "imaginary"--anything not real. If real numbers are a and imaginary numbers are only bi, what the hell are a + bi numbers called? Think about it.

I recommend that this article be rewritten, and a category/article created for "pure imaginary numbers".RobertGustafson (talk) 06:31, 15 April 2017 (UTC)

Proposal: multi-way split

This article is being used at User talk:Jimbo Wales as an illustration of math vs. readability. My opinion was as follows:

The problem is simply too few articles

... We have too many people thinking that things don't "deserve" separate articles, who leave a mountain of stuff crammed into a sardine tin, and then wonder why people can't follow the details. Things we should have for complex numbers include:

• History of complex numbers. Incredibly, the article actually applies WP:summary style with a history section in brief --- that links to a longer history section in the same article! No, no, no, no, that's not how you do it, guys.

• Operations with complex numbers. We should do this in summary style in the main article, giving examples and relevant high-level information, but this should be pursued at length in a proper sub-article.

• Polar form of complex numbers. Note that a number of the complex operations need to refer to polar form to make sense. In the main article, this section follows the other, but once properly split, it can be referred to more conveniently as needed. We may also want a separate article on the complex plane.

• Complex field is a different, more specialized topic.

• We should also have a separate article encompassing the various ways at points in the text in which the complex numbers are treated as a mathematical set.

• Matrix representation of complex numbers, with some more informative examples to show why this is useful.

• Complex analysis is, thankfully, a separate article, but the summary style relationship (what is covered and how) might use a reexamination.

• Physical applications of complex numbers are described in a grab-bag format. There is some risk of it being called a "coatrack". If split as a separate article, this would help to emphasize that we should have secondary sources that enumerate these applications. Each application typically has its own article, but there is definitely room for a high-level article considering them all.

• Generalizations of complex numbers should work the same way - we ought to have sources beyond Wikipedia that make the statement of what is considered one of these, which we can use to navigate the sub-articles.

Once we split the complex numbers article into these ten components, we will have room to both give a rigorous mathematical statement and to explain what that statement means. Wnt (talk) 11:52, 23 October 2017 (UTC)

Anyway, I was encouraged by one reader and challenged to do this by another, so I'm thinking to try taking some of these steps shortly -- starting with the history and its ridiculous internal summary style, and proceeding to a few of the other sections. I can't split any totally unreferenced sections to new articles. My feeling is that with more precise topics and more room to work with, people can give ample explanations and examples, making it possible to be both clear and correct. Wnt (talk) 22:52, 23 October 2017 (UTC)

• Oppose. It sounds like what is being proposed is that the better bits of the article be farmed off to separate articles, with a half-baked attempt to fix things here after the decimation has taken place. The remaining sections of this article are in such bad shape that this will leave a useless non-article behind in its wake. Readers are almost certain not to be served by this, and editor time is much better spent trying to fix this one very flawed article, than eight really bad ones. Get this article up to scratch. Then we can talk about what should be cut out to some other one. Finally, does User:Wnt have any experience writing formally correct mathematical prose? If not, who is going to do all the work to execute this proposal without making a massive cockup? Sławomir Biały (talk) 23:35, 23 October 2017 (UTC)

• Holy Crap No – No no no no no no no. --Deacon Vorbis (talk) 01:02, 24 October 2017 (UTC)

  You might be right, but it would be nice if you stated your reasons. Retimuko (talk) 01:20, 24 October 2017 (UTC)

  Maybe this clip would explain it better? But seriously, this is a single topic. Splitting it up artificially into 10 (yes, ten) articles would serve absolutely no purpose except to scatter information, making it harder to find and maintain. It's a lot of work, and the editor suggesting it doesn't seem to have the requisite background to do so. For example, what in the world is the suggestion that "Complex field is a different, more specialized topic."? What's a complex field? Does he mean the field of complex numbers? If not, then what? If so, then how is that even remotely deserving of a separate article? --Deacon Vorbis (talk) 02:09, 24 October 2017 (UTC)

By complex field I mean complex field, an existing redirect, or "complex number field", as presently described in the article text, or the field of complex numbers if you will.

As I suggested above, the split would not all be at once since some parts are in too poor a shape to split. The first order of business is something so mind-numbingly obvious it seems hard to imagine anyone disputing -- to get rid of the "detailed" history section that is first introduced in the "brief introduction". And all of this should be in WP:summary style, which is to say, you get to keep the biggest points about each sub-article in this article, but can leave out the less critical parts here. Wnt (talk) 02:30, 24 October 2017 (UTC)

The field of complex numbers is not a number field, contrary to the suggestion here. Sławomir Biały (talk) 02:32, 24 October 2017 (UTC)

Yes, and complex number fields are different things than the field of complex numbers, which is a different thing than the field of complex number theory (which is not about the theory of complex numbers). Fortunately we don't have to talk about all of those things here. —David Eppstein (talk) 03:07, 24 October 2017 (UTC)

@David Eppstein: If we don't have to talk about these things here, and we can't split them to another article either, then I take it Wikipedia is not supposed to talk about them at all. Can we have a notice on this page saying "You can't possibly be told the proper distinctions about complex numbers without buying a fancy textbook, because free encyclopedias are just a myth"? Wnt (talk) 11:50, 24 October 2017 (UTC)

A complex number field is a number field that is complex. No portion of this article is about number fields, so there is nothing to discuss. If you feel that our treatment of complex number fields is lacking, your best bet would be to post a request at Talk:Algebraic number field, but ideally after you've read the article number field so that you know what it is that you're actually requesting. Sławomir Biały (talk) 12:52, 24 October 2017 (UTC)

(edit conflict) Sure, the way the history is presented is pretty screwy, but seems fixable. There are lots of other potential tweaks that could improve this (or any) article as well. But there isn't really anything here that would benefit from being spun off into a separate article. --Deacon Vorbis (talk) 03:11, 24 October 2017 (UTC)

• Oppose. This seems likely only to make this material more messy, unwieldy, and unsearchable, rather than the intended goal of making things easier for readers. —David Eppstein (talk) 03:03, 24 October 2017 (UTC)
• Oppose. You're right about the history summary. I'm not opposed to a separate history article. I'm not opposed to shortening the generalizations and applications, making the reader go more quickly to the linked articles. But the rest of the proposal makes no sense to me. The "operations" and field structure are too central and fundamental not to remain in this article. If they left, then what would remain? The set structure of C does not deserve its own article. Why should polar form should be split off if rectangular form is not? Why should matrix form should be split off if vector form is not? Mgnbar (talk) 03:07, 24 October 2017 (UTC)
• Oppose. However the complex analysis part here is I think too long - and the complex analysis article is too short! What's here is certainly not a summary of the article. I guess the history section is big enough to start an article and it might encourage people to improve the history. Dmcq (talk) 08:32, 24 October 2017 (UTC)
• Oppose. The proposed change would make impossible for readers to quickly find what they need, and for editors to provide convenient links. For example, the complex field is nothing else than set of all complex numbers, equipped with its operations and the basic properties of these operations. It is thus logically impossible to have 3 separate articles for complex numbers, for their operations, and the properties they have (that they form a field). Another example, is the "polar form representation", which would be better named "imaginary exponential representation". One can hardly find any text that uses the exponential form $\rho \exp ^{i\theta }$ without using the rectangular form $\rho (\cos \theta +i\sin \theta )$ several lines before or after. This makes unrealistic to have a specific article for the polar form.

The "matrix representation of complex numbers" is nothing else than the fact that the multiplication by a complex number is a linear operation in the complex plane. Thus the corresponding section of the article should be moved to the section on complex plane.

I agree that the history section could be made a separate article, and that the "application section" is not useful as it is, as it does not contain any other useful information than "Complex numbers are used in almost every area of mathematics, science and engineering". An exception: An article Complex numbers in electrical engineering could be useful, as this is an application that may appear as counter-intuitive.

On the other hand, this article deserves to be restructured for making clearer the global coherency of its subject, which is somehow hidden by the present presentation (for example, the operations are defined long after the definition, and one must read carefully the article for learning that the operation rules are simply the application of associativity and commutativity. Also, strangely, a + bi and a + ib are presented as different notations (why not also bi + a and ib + a?) D.Lazard (talk) 08:39, 24 October 2017 (UTC)

• Oppose. Holy content fork, Batman! The goal of our article should be to have a single reference for complex numbers - not to farm out all the information to numerous separate pages. There's no reason I can see that "polar form of a complex number" or "operations with complex numbers" or "matrix representation" requires an entire article - these are all topics that we should summarize in just a paragraph or two in this article. Remember that our goal is to present a coherent reference on a topic, not to present a textbook-length exposition on it. — Carl (CBM · talk) 10:25, 24 October 2017 (UTC)

@CBM: By many accounts this is not viewed as a coherent reference, at least by non-mathematicians. The standard way of arguing about whether a topic "deserves" an entire article is WP:GNG, and I would say that all these things very readily pass WP:GNG. As a rule, a long article/section is better than a short one. Wnt (talk) 11:53, 24 October 2017 (UTC)

Your proposed solution to the incoherence of this reference is a splitting of the article? That does not seem sensible. A coherent account would surely not be spread across ten different articles. See dict:coherent. What you're apparently arguing for is the opposite of coherence. And no one is talking about whether a subject "deserves" a separate article, rather what is better as a coherent reference. Complex multiplication is important, but it is better to treat it in this article rather than a separate article, because it's a defining property of the complex number system: you cannot have an article about complex numbers without discussing their algebraic properties. To do so would be omitting important details (WP:NPOV) that are part of the discussion of complex numbers found in all reliable sources. GNG is not the only deciding factor in whether it is appropriate to split. Sławomir Biały (talk) 13:07, 24 October 2017 (UTC)

Moreover, it would be silly to have an article just about complex multiplication, because there isn't enough to say even if we said everything that could be said. In the end there is multiplication in the rectangular form and in the polar form - what else would be in that article? Making articles with artificially narrow topics only encourages textbook-like writing rather than reference writing. It also encourages violations of NPOV, as writing about an artificially narrow topic encourages editors to editorialize. — Carl (CBM · talk) 14:32, 24 October 2017 (UTC)

@Wnt: I don't see much argument that the article is incoherent - what I saw was that people wanted to rephrase some sentences in various ways, or include examples. But that has little to do with coherence, which I understand as the organization of the article, its flow from one subtopic to another, the choice of subtopics, etc. — Carl (CBM · talk) 14:35, 24 October 2017 (UTC)

• Oppose. I want to support the given arguments against the proposed multi-way-split, also perhaps setting aside the treatment of the History-section. Maybe, there could be some general "History of Numbers", covering the different notions of numbers and their treatments, and the variance in the meaning of these notions over time. Btw, I am lucky, too, that we don't have to discuss "(complex) number field" here. On the reason of this discussion, I strongly oppose to the hope that an article covering an abstract topic gets more readable by adjusting its wording to semi-laymen's expectations, like, e.g., sqrt(-1) effectively defining i. (see JW-talk, triggered by M. Byrne). Purgy (talk) 13:55, 24 October 2017 (UTC)

• Oppose I don't really see the benefits here, at best some topics may in addition be treated in more detail separate "main article", but not by removing content from this article.--Kmhkmh (talk) 21:51, 24 October 2017 (UTC)

Examples

I see an example

For example, the number 2+3i is a complex number, with real part 2 and imaginary part 3.

was removed from the lead citing WP:NOTTEXTBOOK. How do people feel about that? I'd just put in an example to another article saying 'For example, 1.6 would be rounded to 1 with probability 0.4 and to 2 with probability 0.6'. It wasn't in the lead but wouldn't the same argument apply everywhere? I though my example would illustrate the meaning of a formula well. Dmcq (talk) 15:11, 24 October 2017 (UTC)

There is (still) a similar example here in the Overview→Definition section, of course. There is always an issue with examples and NOTTEXTBOOK, but I think the lede is a particularly constrained part of the article, which is meant to summarize the entire article vaguely like the way an abstract summarizes a printed article. Using that rough rule of thumb, I thought the example in the lede here didn't fit. — Carl (CBM · talk) 15:16, 24 October 2017 (UTC)

Being told of a, b being the real- and imaginary parts of a+ib, I consider an example with a=2 and b=3, telling me that 2 and 3 are the respective real- and imaginary parts as an overkill. And, honestly, I would object to naming distinct probabilities when rounding, but I also complained about WP:NOTTEXTBOOK being a weasely guideline, already. Purgy (talk) 17:17, 24 October 2017 (UTC)

WP:NOTTEXTBOOK does not appear to be relevant. If anyone thinks otherwise, please copy to here the relevant excerpt from that policy section. --Bob K31416 (talk) 18:57, 24 October 2017 (UTC)

Indeed, the only thing it speaks against is "It is not appropriate to create or edit articles that read as textbooks, with leading questions and systematic problem solutions as examples." It specifically says "Some kinds of examples, specifically those intended to inform rather than to instruct, may be appropriate for inclusion in a Wikipedia article." That is a policy practically tailor-made for this instance. Wnt (talk) 19:20, 24 October 2017 (UTC)

I think an issue is that many editors want the lead to do possibly too much. Carl typically advocates very minimal leads. In this case, the additional example is not informative beyond what already appears in the first paragraph, and can be completely omitted without sacrificing meaning and readability. Sławomir Biały (talk) 20:02, 24 October 2017 (UTC)

I think omitting it would sacrifice meaning and readability for some readers who aren't very familiar with the terms real, imaginary, and complex numbers. Now it may be that such readers are too small a minority of the people who come to this article for information, but I don't know. There does seem to be criticism of Wikipedia math articles for not being concerned with such readers, as evidenced by the discussion at Jimbo's talk page that most of us were aware of. --Bob K31416 (talk) 20:59, 24 October 2017 (UTC)

Content added to the lead must be judged by both a standard of prominence (WP:WEIGHT), and other policy requirements (notably, WP:V, WP:NOR). Sławomir Biały (talk) 22:25, 24 October 2017 (UTC)

It seems reasonable to include it to me. A nice simple example which doesn't involve variables will be helpful to lay readers. -mattbuck (Talk) 21:38, 24 October 2017 (UTC)

Misuse of the English word "real"

Recently a sentence was deleted on the spurious grounds that it is a misuse of the English word "real". It seems to me that this is a vital sentence for the lead, intended to convey that the is nothing imaginary about imaginary numbers, to a lay audience. I don't quite understand the objection to the use of the English word "real". Real numbers are a thing, and imaginary numbers are too. But imaginary numbers aren't less real, in the colloquial sense of the word, than real numbers. (One could argue that real numbers are just as imaginary as imaginary nbers, but this fails to convey the point as clearly). If there is a different, better way to say the same thing, please propose it. But I don't see how getting rid of the word "real" (or "inaginary") can convey the same message to a lay reader. Sławomir Biały (talk) 19:28, 26 October 2017 (UTC)

In the first paragraph of the lead is,

"Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of our description of the natural world."

To begin with, what is the page number of the source? --Bob K31416 (talk) 19:29, 26 October 2017 (UTC)

Apart from being one of the central themes of the entire book, in particular pages 72 and 73. Undoubtedly other sources can be found to satisfy your objections. But I still don't clearly understand the nature of your objection to the sentence. In your edit summary, you appeared to object to the word choice "real", but declined to offer details. Here you are asking for a page number (which should be done with one of the standard citation templates, not by deletion). Would you like some other sources? Sławomir Biały (talk) 20:14, 26 October 2017 (UTC)

I looked on 72 and 73 and I didn't see the idea that complex numbers are as "real" as the real numbers. Please provide an excerpt from that source that you think expresses that idea. --Bob K31416 (talk) 20:35, 26 October 2017 (UTC)

Quoting that source at length: "Presumably this suspicion arose because people could not ‘see’ the complex numbers as being presented to them in any obvious way by the physical world. In the case of the real numbers, it had seemed that distances, times, and other physical quantities were providing the reality that such numbers required; yet the complex numbers had appeared to be merely invented entities, called forth from the imaginations of mathematicians who desired numbers with a greater scope than the ones that they had known before. But we should recall from §3.3 that the connection the mathematical real numbers have with those physical concepts of length or time is not as clear as we had imagined it to be. We cannot directly see the minute details of a Dedekind cut, nor is it clear that arbitrarily great or arbitrarily tiny times or lengths actually exist in nature. One could say that the so-called ‘real numbers’ are as much a product of mathematicians’ imaginations as are the complex numbers. Yet we shall Wnd that complex numbers, as much as reals, and perhaps even more, Wnd a unity with nature that is truly remarkable. It is as though Nature herself is as impressed by the scope and consistency of the complex-number system as we are ourselves, and has entrusted to these numbers the precise operations of her world at its minutest scales. In Chapters 21–23, we shall be seeing, in detail, how this works." :::: Sławomir Biały (talk) 20:44, 26 October 2017 (UTC)

(edit conflict) This is what Penrose says on page 73: "One could say that the so-called 'real numbers' are as much a product of mathematician's imaginations as are the complex numbers."

So in fact Penrose says that "the real numbers are as "imaginary" as the complex numbers". Apparently this was re-interpreted here as that "the complex numbers are as "real" as the real numbers." So I can more or less agree with Bob's remark. This could be a slight case of wp:synthesis. I propose we turn it around again, in order to stay closer to the source. A good compromise i.m.o. - DVdm (talk) 20:52, 26 October 2017 (UTC)

No, this is a counterpoint. He is saying that someone could say this, but in fact: "complex numbers, as much as reals, and perhaps even more, find a unity with nature that is truly remarkable. It is as though Nature herself is as impressed by the scope and consistency of the complex-number system as we are ourselves, and has entrusted to these numbers the precise operations of her world at its minutest scales." That seems pretty clear cut as it is, but for added context see where he blathers on in the introduction about being a strict Platonist (pinging @Trovatore:, our friendly neighborhood fundamentalist), making the question of the reality of the real numbers a something of a settled thing for him. I think a better compromise would be simply to quote Roger Penrose in the first paragraph of the lead. A bit unconventional, but he expresses it far better than any of us will. Sławomir Biały (talk) 21:06, 26 October 2017 (UTC)

I'm okay with that. - DVdm (talk) 21:09, 26 October 2017 (UTC)

I don't see the excerpt supporting the statement which is apparently OR. Also, regarding another part of the statement, "complex numbers are regarded in the mathematical sciences as" – it appears that you are concluding that it is generally accepted in the mathematical sciences, which also isn't mentioned in the excerpt you gave from the source. Also, you use scare quotes for "real" because you have no clear definition of what you mean by that term. In any case, the sentence is poor writing that isn't informative, and certainly shouldn't be in the lead, or anywhere in the article.

But hey, this is Wikipedia and any kind of crappy writing can get into articles if there are editors determined to do it. So this is my last message about the subject, and have fun! --Bob K31416 (talk) 22:03, 26 October 2017 (UTC)

Which is why we have now compromised, and included an exact quotation from the source. I still feel that the original phrasing was a reasonable summary of that source. I even supplied a quotation that supported what I still feel is reasonable. You haven't proposed any constructive alternative. Instead, the entire premise of this thread seems to be the WP:FRINGE belief that there is some controversy over the existence of complex numbers. To dispel the false notion that there is any such controversy in mathematics, it is clearly necessary to point this out, especially as certain editors seem keen to banish such important details in a thinly disguised attempt to subvert the neutral point of view policy. You're now explicitly questioning whether the existence of the complex numbers is generally accepted in mathematics. Clearly you are edging closer to the brink in this discussion. Sławomir Biały (talk) 22:22, 26 October 2017 (UTC)

While I'm sure I have seen the statement about complexes being as real as reals in print, I can't find a reference. However, an alternative to Penrose might be an 1831 passage by Gauss where he talks about a mysterious obscurity attached to imaginary numbers due to a poor choice of notation (i.e., "imaginary"), which would vanish if better choices had been made. Gauss also originated the term "complex number" so there might be a possibility of leading into this quote from that point. --Bill Cherowitzo (talk) 23:44, 26 October 2017 (UTC)

Why imaginary "unit"?

Since this article garnered a lot of attention because of some traditional, and imho naive, but not without reason, discussion about math articles being only for the elite, I use the chance to ask for some motivation calling i a 'unit'. I did not find an answer within this article and also not within the article "Imaginary unit". Perhaps some illustrating words could be added here.

Sure, i is a unit in the sense of 'being invertible', but this notion of 'unit' is not quite easy to find within WP. I found it hidden in an article on rings. Certainly, i is no 'unit' in the sense of '(multiplicative) identity' or 'unity', but it has a magnitude of ' 1 ', a property shared with many other complex numbers, similar to being invertible. Associating 'imaginary unit' with something like 'meter' being the 'unit' belonging to a dimension of 'length' is absolutely alien to me. I think 'unit' is too laden with associations to leave it simply "unrefined". Purgy (talk) 08:36, 26 October 2017 (UTC)

IMO, this is simply a traditional terminology. Nevertheless, this is a unit (ring theory) of every subring of the complexes that contains it. D.Lazard (talk) 10:02, 26 October 2017 (UTC)

I suppose imaginary root of unity might be more technically correct. Sławomir Biały (talk) 10:37, 26 October 2017 (UTC)

I do think it's just traditional terminology. Rather than meaning "unit" in the sense of "invertible", it seems to mean "unit" as in "basic unit of measurement", as purely imaginary numbers are "measured" in multiples of i. This make some sense because |i| is 1. But I think that in many cases we can avoid the term "unit" here and use more direct wording. — Carl (CBM · talk) 14:55, 26 October 2017 (UTC)

I don't think I've ever said 'imaginary unit' in my life so it shouldn't be too hard to avoid! Dmcq (talk) 14:58, 26 October 2017 (UTC)

I agree with D.Lazard's comment; I could imagine that the name is related to units as in Dirichlet's unit theorem. However, unless there is a clear source backing up this idea or any other idea, I would not write anything in the article itself. Jakob.scholbach (talk) 15:10, 26 October 2017 (UTC)

I was surprised to find imaginary unit the other day. — Carl (CBM · talk) 15:29, 26 October 2017 (UTC)

Is there a more standard way to refer to i? Sławomir Biały (talk)

A little history may help here. The term comes from being invertible, a unit in the ring sense. Gauss introduced it when he talked about the four "units" in the Gaussian integers, 1, −1, i and −i. As with several other terms, once Gauss used it, it became standard to copy him. --Bill Cherowitzo (talk) 23:32, 26 October 2017 (UTC)

Thanks for all the thoughts. Please check my attempt on implementation in the lede. Purgy (talk) 09:28, 27 October 2017 (UTC)

I think this will simply make things more confusing for likely readers. If there is a place to discuss the meaning of the word "unit" in this context, it is surely the article imaginary unit, rather than the first paragraph of the lead. The meaning of the word "unit" is almost irrelevant to the subject of this article. I think we just need to establish that it is a standard way to refer to "i" in prose, but not to belabor the point. Sławomir Biały (talk) 10:43, 27 October 2017 (UTC)

I certainly would have used a "standard way to refer to "i" in prose", if such would have been mentioned above. But, as articulated in my thread starter, I perceived misguidance by the word "unit" in "imaginary unit", which is used throughout the whole article and also in the linked one with this very title, without being sourced at all, so I tried to establish this term at the beginning of this article in a most simple and plausible manner. Omitting the list of 4 values would have taken away the plausibility. Getting rid of the term appears to me not sooo easy as Dmcq seems to think.

Commenting further on the lede, I think the citation of Penrose in its whole length is only justified to refute the, imho, unsubstantiated objection by Bob31416 to "real". I perceive it a bit, say, flowery. I do hope that my changes are not considered rubbish, wholesale, but certainly they are improvable. Purgy (talk) 16:47, 27 October 2017 (UTC)

I actually think a better solution would be to try to remove "imaginary unit" from the lead. It can be introduced in context later in the article. I've taken a swing at this. If we can agree that the original wording of complex numbers being just as real as real numbers is an adequate summary of the scientific literature, I'm certainly open to saying things in that much shorter way. The only reason I included the quotation at length was, of course, that a certain editor seemed to be pushing a WP:FRINGEPOV. Sławomir Biały (talk) 16:59, 27 October 2017 (UTC)

I agree, I don't think that the term adds anything here and hinting at its origins will definitely raise more questions than it answers. I mentioned it above to give a little historical background, but never intended it to be used in the article. It provides the answer to why these and not the other roots of unity are called units, but the context is Gaussian integers and that is not something that should be brought up in the lead of this article. While I'm at it, I also dislike the use of "imagined solution" in the first line, even with the scare quotes. This is repeating (at least according to Gauss) the mistake that Descartes made in terminology. I would prefer all mentions of imaginary to be dropped into the second paragraph where the issue can be dealt with. And, referring to the existence of imaginaries as a "settled question" seems to me to be too blunt a statement, making it sound like the issue was dealt with head-on. A better phrasing would be something like, "Mathematicians' unease with the concept was gradually dispelled by ...". --Bill Cherowitzo (talk) 16:58, 27 October 2017 (UTC)

Any objections to just this? Significantly shorter and easier to read, IMO. Sławomir Biały (talk) 17:03, 27 October 2017 (UTC)

I don't think 'indeterminate' is a good word, it is correct as Indeterminate (variable) but it has too much other baggage and is used more often in indeterminate form, otherwise I think your changes were good. Dmcq (talk) 17:24, 27 October 2017 (UTC)

The question whether mathematical objects are discovered, invented, created, etc. is a deep issue in mathematical philosophy, and there is no reason our article here should weigh in on it. So IMO we should avoid claiming that complex numbers were "invented", "created", etc. There is almost always a more neutral way to discuss the situation without opening that can of worms. — Carl (CBM · talk) 17:29, 27 October 2017 (UTC)

My issues with the lead have been dealt with, so I am okay with it. I totally agree with Carl, but would like to point out that the notation associated with a concept is always "invented" (whether or not we can trace to its origins) and that might be a way to avoid stepping in that can. --Bill Cherowitzo (talk) 17:41, 27 October 2017 (UTC)

The literature has no problem with the term "imaginary unit"—see Google Books—so I don't see any reason why Wikipedia should shun it. - DVdm (talk) 19:49, 29 October 2017 (UTC)

Obviously, it is not WP's but my problem to associate too many possible meanings with the word 'unit', and being unwilling to assume that the context suffices for most readers to lightheartedly brush over different technical meanings of 'unit'. In no way I want WP to shun the term "imaginary unit", but I want to see it, say, introduced, be it as a sourced traditional, with some associated historical meaning, or as cited definition taken from some handbook.

Meanwhile, I am inclined to take the "imaginary unit" as baptized, neither for Gaussian integers, nor for ring units, but for its valuation and for being in an "imaginary dimension", like j and k. Nothing to shun, but worth an encyclopedic info, imho. Purgy (talk) 08:10, 30 October 2017 (UTC)

(a+bi)^n

can we add the formula for (a+bi)^n? Jackzhp (talk) 08:55, 11 January 2018 (UTC)

Look here, please. Purgy (talk) 09:10, 11 January 2018 (UTC)

However, it should be probably better to put the simpler and important case of integer exponents before the general case. D.Lazard (talk) 09:42, 11 January 2018 (UTC)

Definition of the argument

Section § Conjugate, contains the formula

$\quad \operatorname {arg} ({\overline {z}})\equiv -\operatorname {arg} (z){\pmod {2\pi }}.$

HaydenWong changed $\equiv$ into =, and this has been reverted by Deacon Vorbis. IMO, both versions are wrong, and the formula should be simply

$\quad \operatorname {arg} ({\overline {z}})=-\operatorname {arg} (z).$

However, this depends on the chosen definitions for an angle and for the argument of a complex number. The argument of a complex number is defined, later, in § Polar form, but this definition is a mess, and, in particular, suggests that 0 and 2π are different angles. Also, in this section, the formula with multiple cases should better be replaced by the equivalent formula:

The argument of z = x + iy is $\arctan {\frac {y}{x+|z|}}$ except if z is a negative real number, where arg z = π, and |z| = –x.

Thus, for fixing this particular formula, one has to define the argument of a complex number before the conjugate and to cleanup the section § Polar form. I'll not restructuring this article myself for lack of time. Someone is willing? D.Lazard (talk) 14:01, 11 December 2018 (UTC)

A while ago I attempted to improve on the elementary operations, but shied away from touching the polar form (a small caveat was reverted), because I am not aware what might be the nowadays most appropriate form to touch this subtle matter. Furthermore, I had a very disappointing encounter while trying to bring some consistency to the atan2 article, so I won't touch anything arctanny. As an aside, I stranded in my efforts to find sources for a consistent denotation and an agreed upon selection of principal branches in the inverse trigs, e.g., I found the use of Sin and Arcsin as opposed to sin and arcsin rather arcane in the superficially scanned literature. Sadly, I am not sure about the reasons why the congruence is considered wrong, I certainly lack a rigorous definition of arg.

As a result, I don't feel capable to touch the addressed problem. Purgy (talk) 16:32, 11 December 2018 (UTC)

I'm confused. The range of $\arctan$ is $(-\pi /2,\pi /2)$, so how can $\arctan(y/(x+|z|))$ give the correct values for $\arg$ on the left side of the complex plane?

I agree that the polar form should be introduced before the Conjugate subsection mentions facts about it. Mgnbar (talk) 01:41, 12 December 2018 (UTC)

- As a proxy: Simply, a factor of 2 is missing (the derivation might be based on the half-angle).

- I am not absolutely sure about the sequence. Personally, I prefer to look at the complex numbers as entities on their own, primarily neither as a sum of real and i×imaginary part, nor as modulus×exp(i×arg). Complex numbers may be added, subtracted, multiplied and divided by non-zeros AND -as a new feature- conjugated. This conjugation, by relying on its properties, allows for defining real- and imaginary parts, modulus and, via the multivalued arctan- the argument, which is not unique. Of course, the conjugation can be mirrored also in the newly introduced real/imaginary and in the modulus/argument scenario. This reverses the long standing, probably more common route, but I think it gained increasing acceptance over time. Just a personal remark. Purgy (talk) 08:20, 12 December 2018 (UTC)

Yes, I forgot the factor 2.

I agree that complex numbers should be viewed as entities on their own. A way of reaching this goal would be to rename as "Basic operations" the section § Elementary operations, and to include at its beginning subsections on the real-valued unary operations (real and imaginary part, modulus and argument). D.Lazard (talk) 10:22, 12 December 2018 (UTC)

Not the same as a vector in 2 dimensions as multiplication defined differently

The article now uses the word vector a few times, implying that a complex number is the same as a vector in two dimensions. I would question this identification, on the ground that the product of two complex numbers is defined differently from both the scalar product and the vector product of two vectors in vector multiplication. In fact the product of two complex numbers is a complex number in the same complex plane as the factors, whereas the scalar product of two 2-D vectors is of course a scalar, and the vector product is a vector perpendicular to the plane of the vectors being multiplied.

Would it be better to remove all mention of vectors from this article? Or perhaps to add a section explaining why a complex number is an object different from a vector in two dimensions? Dirac66 (talk) 22:34, 14 February 2019 (UTC)

But $\mathbb {C}$ is a (two-dimensional) vector space over $\mathbb {R} .$ Moreover, addition of complex numbers and multiplication by a real number correspond to the normal vector space operations of addition and scalar multiplication. This is a standard notion and shouldn't really be removed. I'm not sure much extra explanation is needed. The dot product is a bi- (or sesqui-)linear form, not a normal multiplication; and the cross product is something a bit special in 3 dimensions, not 2. Maybe just a quick note could be added, but not much. –Deacon Vorbis (carbon • videos) 22:58, 14 February 2019 (UTC)

Perhaps at the end of the section Multiplication, we could add a sentence such as: This product is different from both the scalar product and the vector product of simple vectors in 3-dimensional space. This would help readers who have learned about scalar and vector products in first-year physics but not in math, by clarifying that we are now not talking about the same thing. Dirac66 (talk) 01:36, 15 February 2019 (UTC)

My edit is reasoned by: I think the vector space property is quite elementary, is used in the top pic, and should therefore be addressed already in the lead, and not only way down in the article. I prefer to keep the binary operation "complex multiplication" free of both the form and the quick and dirty cross-product. Purgy (talk) 09:19, 15 February 2019 (UTC)

Yes, I think the paragraph you have added explains the point quite well. Thank you. Dirac66 (talk) 02:26, 16 February 2019 (UTC)