Talk:Complex plane

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This article is within the field of Control theory.

Comment by Karl[edit]

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 (talk) 20:20, 27 December 2012 (UTC)

Comment by User:The Anome[edit]

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)

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)

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[edit]

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[edit]

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: 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[edit]

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[edit]

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

Weird redirects[edit]

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[edit]

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)