Talk:Complex number Information & Talk:Complex number Links at HealthHaven.com

Mathematics portal

This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.

Mathematics rating:

B Class

Top Priority

Field: Analysis

A vital article.

One of the 500 most frequently viewed mathematics articles.

Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments.
Please add to or update the comments to suggest improvements to the article.
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)

Version 1.0 Editorial Team (Rated B-Class)

This article has been reviewed by the Version 1.0 Editorial Team.

This article has been selected for Version 0.7 and subsequent release versions of Wikipedia.

Additional information:
B	This article has been rated as B-Class on the assessment scale.
???	This article has not yet received an importance rating on the assessment scale.
	This article is a vital article.

WikiProject Citizendium Porting (Last updated: Never)

	v • d • e This article is within the scope of WikiProject Citizendium Porting, a collaborative effort to improve Wikipedia articles by working in any useful content from their Citizendium counterparts. If you would like to participate, you can visit the project page, where you can join the project and see a list of open tasks.
	This article is out of date with its Citizendium counterpart and needs to be updated.
	Current with its Citizendium counterpart article as of: Never

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

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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 $z=r e^{i \varphi}$ )

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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

Contents