Talk:Complex plane

Comment by Karl

I believe the following passage is incorrect: "The unit circle itself (|z| = 1) will be mapped onto the equator, and the exterior of the unit circle (|z| > 1) will be mapped onto the northern hemisphere,..."

Actually, I believe the equator is mapped onto |z| = 2. — Preceding unsigned comment added by 178.190.43.45 (talk) 20:20, 27 December 2012 (UTC)

Comment by User:The Anome

Whoops, confused Bode plots with Nyquist plots there for a moment. Fixed now. -- The Anome 22:17, 2 October 2005 (UTC)

The explications here do exceed the ones in complex numbers only in the relation given to control theory. I propose putting that part somewhere to control theory and replacing the rest by a redirect to complex numbers. D'accord? Hottiger 18:26, 21 March 2006 (UTC)

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Since when has the y-axis been imaginary and the x real? --HantaVirus 13:44, 28 July 2006 (UTC)

I suppose you're right, I'll change that back to how it originally was. --Neko18 00:32, 11 November 2006 (UTC)

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Just a little question, has it ever been postulated to have a 3-dimensional system wherein the x-axis represented the input to a function, the y-axis represented the real output, and the z-axis represented the imaginary output? -- ThatOneGuy

Wouldn't that just be a complex function of a real variable? They certainly have been considered and studied. Madmath789 18:24, 9 September 2006 (UTC)

I've actually thought about that idea before as well, though I don't know enough about mathematics to form any sort of conclusion on that yet. --Neko18 00:34, 11 November 2006 (UTC)

Magnitude of complex number

Does one really say magnitude of a complex number. I thought, modulus or absolute value is the correct term? Well, I am not a native.

Disambiguation needed

The complex field C is not the only complex plane. Split-complex numbers and dual numbers are also used in the same sense, a plane of numbers equipped with a product. Rgdboer 21:11, 12 October 2006 (UTC)

Furthermore, Peter J. Olver notes another ambiguity:

...as a complex vector space, C is one-dimensional and will therefore be referred to as the "complex line", with C² being the "genuine complex plane".To minimize misunderstanding, we shall try to avoid using the term "complex plane" in this book.

Classical Invariant Theory', p. 18

Thus we have four different meanings for this term in general use, so some disambiguation is required.Rgdboer 19:32, 16 October 2006 (UTC)

I do understand what you're driving at. The cardinality of C is clearly the same as the cardinality of R. So each of those sets is just an example of a "number line", since their elements can be placed in 1-1 correspondence.

But I've also read quite a bit of stuff from complex analysis, and I've looked at several Wikipedia articles. Historically the term "complex plane" as a synonym for C is quite old and, unfortunately, replacing "complex plane" with "Argand diagram" or "Argand plane" would be quite a lot of work. Would complex² plane as a synonym for C×C work for you? DavidCBryant 19:09, 10 December 2006 (UTC)

New content

I've been adding quite a bit of new content to this article recently, in response to the "stub" tags. My general plan to complete the article looks like this.

1. Explain the connection between the Cartesian plane and the (traditional) complex plane. (done)
2. Explain the representation of the extended complex plane as a sphere. (done)
3. Explain various motivations for "cutting" the plane. (done)
4. Explain the process of "gluing" cut planes (sheets) together to form Riemann surfaces. (pending)
5. Allude to alternative "complex planes", specifically C×C. (pending)

Can you think of anything important that I've left out? I've put in quite a few examples, because I think people who are curious but not really mathematicians might read this article. Are there too many examples? My aim was to introduce infinite products, infinite sums, and continued fractions within this one article, just to pique the reader's curiosity. Is that a good strategy? Or not?

Your feedback is welcome, either on this talk page or on my user talk page. DavidCBryant 16:02, 4 January 2007 (UTC)

In response to your note on my talk page, and to the above, I do appreciate the new content. As for the relative prominence of the various planes, one might note from Euclidean geometry#Treatment using analytic geometry that the ordinary complex plane has great utility in geometry. Such utility for the other planes exists but needs communcation to the wider community of users of quatitative methods. The lack of departmental boundaries in WP gives me hope, for instance, that the split-complex number plane will become well enough known that the obscurantism practiced in the name of spacetime physical science can be overcome.Rgdboer 21:49, 10 January 2007 (UTC)

The first two sentences in the article are excellent for motivating the alternative complex planes: the paragraph could be extended to say "If i represents the imaginary unit, then i² ∈ {−1, 0, +1}. This article treats the case ii = −1 ; see dual numbers for ii = 0 and split-complex numbers for ii = +1."Rgdboer 04:59, 12 January 2007 (UTC)

Real Coffee

Where does real Coffee fit in an Argand plane? —The preceding unsigned comment was added by 82.211.73.34 (talk) 12:23, 1 February 2007 (UTC).

Weird redirects

Argand diagram redirects here, but Argand Diagram redirects to Complex Number. Surely they should both go to the same place?--Physics is all gnomes (talk) 14:14, 31 December 2010 (UTC)

Other complex planes

The issue of other complex planes arose when Hartenberg and Denavit were preparing appendix A-1 (pp 370–9) of Kinematic Synthesis of Linkages. In the footnote on page 372 one reads

Rather, c = a + b i should be called the ordinary, or usual, complex number. There are others, for example c = a + ω b, where it is agreed that ω² = 0. These are not our concern at this time.

The most immediate context in which the triple of canonical complex planes is necessary for comprehension is in the algebra of 2 × 2 real matrices. This reference has been posted here to backstop the footnote to the lead.Rgdboer (talk) 02:59, 14 April 2011 (UTC)

See also Anthony A. Harkin & Joseph B. Harkin (2004) Geometry of Generalized Complex Numbers, Mathematics Magazine 77(2):118–29 for more details on the alternative complex planes.Rgdboer (talk) 22:38, 16 February 2015 (UTC)

Argand diagrams

This article mentions Argand diagrams (and indeed, it is the redirect for it) but it doesn't explain what they are or provide a labeled example. I can see how that might appear trivial but there should at least be a full definition or sample construction. 76.181.233.121 (talk) 18:37, 17 June 2016 (UTC)

Edit war about an image

3D complex plane model to visualize usefulness for translation of domains

TimsDesigns added repeatedly (3 times) an image to the article, which I have reverted three times because the image appears to be not convenient for this article. The reasons of my revert follow. IMO, each of the following issues is sufficient for justifying my reverts.

• The choice of the colors is very poor and makes almost impossible to understand what is represented, even for an expert of the subject.
• The image is confusing, and even misleading, since the complex plane is a 2D space and this is a 3D figure
• The image is out of the scope of this article, since it seems to be about the use of complex numbers for modeling sinusoidal wawes.

D.Lazard (talk) 13:45, 16 October 2023 (UTC)

I deeply disagree with the assessment. And thus (now a third time) I revert the removal.

• The representation of a 2D object in a 3D space is the most intuitive way to understand it.

• For anyone new to the subject who is not a mathematician, this is much easier to grasp than expressions.

• The image is uploaded in full HD and therefore it is incredibly easy to zoom in to get the details.

• The colors are oriented on the Euclidean coordinates represented in most CAD Programs.

• A hint of scope allows for a context to be applied to the rest of the theory. TimsDesigns (talk) 19:10, 26 October 2023 (UTC)

I find this image a bit out of place seeming as a basic representation of "complex numbers".

To begin with, I have serious issues with low-contrast colored labels that seem a bit hard to read, oriented in various 3d directions, and partially obscured by other elements, and with somewhat non-obvious/potentially confusing labels like "real (complex) cosine wave". I'm also concerned about low-contrast line elements whose spatial positions is a bit hard to figure out due to lack of context, and not immediately obvious spatial relationships between the different parts of the chart.

But leaving that aside, this image isn't trying to demonstrate the most basic features of complex number arithmetic (interpreted geometrically in a 2d plane), which is what I think we should be depicting in the top half or so of this page. Instead, the subject depicted is three orthogonal projections of a helix. This can be interpreted as a graph of the complex exponential of an imaginary parameter, but that interpretation is not at all self evident, and explaining the details to readers who have never heard of complex numbers before (and therefore are also unlikely to have thought deeply about the significant number of nontrivial prerequisites) is a pretty demanding task that our current page doesn't try to do and I don't think would be reasonable to do in the available space adjacent to the image. –jacobolus (t) 21:59, 26 October 2023 (UTC)

I share these concerns. Furthermore, I am puzzled by statements like The representation of a 2D object in a 3D space is the most intuitive way to understand it. Shouldn't a 2D object be represented in 2D? XOR'easter (talk) 17:27, 27 October 2023 (UTC)

Aside: even ignoring the color contrast issue for readers with typical color vision, the use of green and red to indicate meaningful distinctions is a very bad idea in diagrams intended for a general audience, because a very significant part of the human population (especially among men) is red–green colorblind and will not be able to distinguish them. –jacobolus (t) 00:26, 28 October 2023 (UTC)

I have been trying to figure out how this image is helpful to that section or the article, and honestly I don't see it. The image, or a variation of it, might be helpful to some other article, though. So I would tend to side with D Lazard on this one. Dhrm77 (talk) 20:38, 27 October 2023 (UTC)