Talk:Complex number/archive1

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Contents 1 Removed text 2 Spanish translation 3 Graphical explanation 4 De Moivre's formula 5 Was this copied? 6 Exercise Needed 7 Proper fields isomorophic 8 j or i - usage of the imag. unit in maths, physics, electrotechnology 9 The non-italic writing style of the imaginary unit 9.1 Can we agree? 10 Public voting table for (English) Wikipedia Notation of imaginary unit 11 Algebraic characterization 12 Elementary geometry 13 The imaginary part 14 Article name: shouldn't it be in the plural form? 15 Geometry? 16 Elementary geometry 17 ln(-1) 18 correct symbol? 19 Whats this? 20 Useless... 21 History 22 Conventions

[edit] Removed text

In the discussion of the argument of a complex number, I removed the following:

Note that this is exactly the same problem encountered when trying to define the inverse tangent as a function. The connection becomes more transparent when one considers that the formula for calculating arg(z) is arg(z) = arctan(b/a) if z = a + i b.

The values of the arctan function lie between -Pi/2 and +Pi/2, while arguments of complex numbers lie between -Pi and +Pi. The arctan formula for arg(z) is therefore incorrect. Most programming language have a function atan2(b, a) which returns the proper argument of a + i b by taking the signs of a and b into account. --AxelBoldt —The preceding comment was added on 04:27, 22 February 2002.

Are you sure about this? I've been calculating the angle on a complex number by arctan(y/x) for years in school. It's the way it is taught in all of my electrical engineering and applied mathematics courses. --virga —The preceding comment was added on 04:41, 24 January 2006.

Yes, this works, but φ one gets this way is up to a multiple of π, so the principal branch of the arctan function may not give the right answer. For example, arctan(1/1)=arctan(-1/-1)=&pi/4;, but the argument of -1-i is π+π/4. Oleg Alexandrov (talk) 05:01, 24 January 2006 (UTC)

In any case, something should be added to the article on symbolically calculating the angle on the complex exponential. It's a basic Cartesian --> Polar conversion neccesity. --virga —The preceding comment was added on 24 January 2006.

I believe it would be rather complicated to explain that thing fully at that place. Anybody willing to start a new article, argument of a complex number, to treat the issue fully? Or is circular coordinates a better place for doing that kind of thing? Oleg Alexandrov (talk) 19:32, 25 January 2006 (UTC)

[edit] Spanish translation

You can see the translation to spanish in http://es.wikipedia.com/wiki.cgi?Números_Complejos --Atlante —The preceding comment was added on 25 February 2002.

[edit] Graphical explanation

A graphical explanation is better than words. Should add images sometime. --Wshun —The preceding comment was added on 22:05, 25 February 2003.

[edit] De Moivre's formula

I could not find Demovire's theorem for complex numbers,should add this.-Raj.B (India-Karnataka-Mysore) —The preceding comment was added on 06:33, 13 December 2003.

It's at De Moivre's formula. I've put a link to it in the article. --Zundark 10:18, 13 Dec 2003 (UTC)

[edit] Was this copied?

Is this page originally copied from somewhere? -- Walt Pohl 18:04, 13 Mar 2004 (UTC)

Looks like it to me. Sounds rather archaic (1890-ish, perhaps). Probably public domain, but would be good to know where it comes from. Gwimpey 23:21, May 24, 2005 (UTC)

[edit] Exercise Needed

everything fine there in the article. as an encyclopeadian article. but shouldn't there be exercises? --jai 12:20, Aug 13, 2004 (UTC)

No. An encyclopedia isn't a problem-solving book. But wikibooks... That idea would fit right in. --Mecanismo 10:07, 16 September 2005 (UTC)

[edit] Proper fields isomorophic

Can someone give an example of a proper subfield of C isomorphic to C, or at least show one exists? I'm not saying I think it's not true; I just don't remember seeing this, so I'm interested how it's done. Revolver 13:49, 2 Nov 2004 (UTC)

There's an argument showing that these subfields exist at the end of the transcendence degree article. -- Fropuff 16:22, 2004 Nov 2 (UTC)

I get it. I think a comment saying that it's not constructive, that the axiom of choice is used, and so no such explicit subfield can be produced, would be good to say. Revolver 19:46, 2 Nov 2004 (UTC)

[edit] j or i - usage of the imag. unit in maths, physics, electrotechnology

According to the person sitting next to me, the root of negative one is represented by j not i now. Someone had better change it all. —The preceding unsigned comment was added by Borb (talk • contribs) 12:38, 24 May 2005.

Engineers tend to use j, and mathematicians i. Robinh 13:52, 24 May 2005 (UTC)

Yes, as far as I could observe, mathematicians and physicists prefer using i as the imaginary unit, while electrical engineering technicians prefer j as the character for the imaginary unit. For me (physicist) it does not matter, whether i or j is used in formulas. If they are written in non-italic characters, then a clear seperation from alternating current i and counting indexes like i or j is very easy. Originally, in electrical engineering the j was used to prevent confusion of imaginary unit i with the alternating current i. But I have not observed problems with this similarity, if italic and non-italic characters are used correctly. Nevertheless, on manually written pages, in any case a small legend of the used characters should be listed anyway, and there the non-italic symbol for the imaginary unit (i or j) can be clearly assigned. Personally, I prefer i as the imaginary unit, but because i = j = 'imaginary unit' both can be mixed/exchanged without problems. Enjoy working with the imaginary unit! -- Wurzel 10:00 UTC, 29 May 2005

[edit] The non-italic writing style of the imaginary unit

causes the lowest amount of trouble, if the imaginary unit i (mostly used in mathematics, physics), or j - frequently used in electrical engineering is written in non-italic writing style. I have observed this in the recent decade of my work in science, and a clear seperation of italic charactes for variables e.g. a,b, and non-italic characters for 'units' for which we also can count the 'imaginary unit' seems to be the best solution. This prevents mixing up the 'imaginary unit' i with e.g. the current i. —The preceding unsigned comment was added by Wurzel (talk • contribs) 09:42, 29 May 2005.

Books with a high scientific level and a high acceptance prefer using this way. To them belong (upper part of the table):

Used imag. units in several scientific works

References using non-italic i	imag. unit	comment
Haken, Wolf; Molecule- and Quantum Physics, 1.-8. ed.	i
Haken, Wolf; Atomics- and Quantum Chemistry, 1.-8. ed.	i
Slichter; Principles of Magnetic Resonance 1.-3. ed.	i
A.C.S. van Heel; Advanced Optical Techniques, 1967, Delft, Holland, 1st ed.	i
Nightingale; A short course in General Relativity, 1986, 2nd ed.	i
Milnor and Stasheff, Characteristic Classes, 1974	i
R Menzel, Photonics, Springer, 2001	i
HW Harris (Yale Univ.), H Stocker, Handbook of Mathematics and Computational Science, Springer, 1998	i
S Feferman (Stanford Univ.), The number systems, Foundations of Algebra and Analysis, Addison-Wesley, 1964	i
I Steward, D Toll, Complex Analysis, Cambridge University Press, 1983	i
GM Henkin, J Leiterer, Theory of Functions on Complex Manifolds, 1983	i
Mathematics texts using italic i	imag. unit	Comment
Schechter, Eric, Handbook of Analysis and Its Foundations, Academic Press (1997)	i
Ahlfors, Lars V., Complex Variables, McGraw-Hill (1953)	i
Townsend, E. J., Functions of a Complex Variable (1942)	i
Knopp, Konrad, Theory of Functions Dover Publications (1947)	i
Brownwell, Arthur, Advanced Mathematics in Physics and Engineering McGraw-Hill (1953)	i
Spiegel, Murray R., Complex Variables, McGraw-Hill (1964)	i
Spiegel, Murray R., Advanced Mathematics for Engineers and Scientists, McGraw-Hill (1971)	i
Web references using italic i	imag. unit	Comment
Britannica	i
Mathworld	i
PlanetMath.org	i
Web references using upright i, e, and d	imag. unit	Comment
W3 Consortium - MathML specifications	i, e, d	Also see below.
Web references using blackboard bold i, e, and d	imag. unit	Comment
W3 Consortium - MathML specifications	i, e, d	Search for "imaginary". Also see above.
Physics texts using italic i	imag. unit	Comment
N. Minorsky; Nonlinear Oscillators, 1962, New Jersey, 1st ed.	i	i is also used for indexes and as alternating current i see p.174
L.B. Felsen, N. Marcuvitz; Radiation and Scattering of Waves, 1973, 1st ed.	i,j	both are used dependent on chapter
Itzakason and Zuber, Quantum Field Theory	i
Jackson, Classical Electrodynamics	i
Griffiths, Introduction to Electrodynamics	i
Sakurai, Modern Quantum Mechanics	i
Peskin and Schroeder, An Introduction to Quantum Field Theory	i
Weinberg, The Quantum Theory of Fields	i
José and Saletan, Classical Dynamics	i
Wald, General Relativity	i
Hassani, Mathematical Physics	i
Eisberg and Resnick, Quantum Physics	i
Linas Pauling and E. Bright Wilson Jr., Introduction to Quantum Mechanics, Dover (1985)	i
Feynman, Leighton, and Sands, The Feynman Lectures on Physics, Addison-Wesley (1963)	i
P. C. W. Davies, Quantum Mechanics, Routledge & Kegan Paul, (1984)	i
J. P. Elliot and P. G. Dawber, Symmetry in Physics, Oxford University Press, (1984)	i
Other usage	imag. unit	Comment
www.wikipedia.fr : 'Nombre complexe', on 9.3.2005 14:45 CET		Hmm, is also an interesting alternative
Griffiths-Harris: Principles of Algebraic Geometry		makes the formulas quite cumbersome
Lorrain and Corson, Electromagnetic Fields and Waves	j

[I will update this list, as soon as I have newer data.]

If somebody has other/better proposals then he can offer them. Wurzel

I have a solution: change it back to italics. You are proposing a radical change in notation. Virtually all math textbooks use italicized i for imaginary unit. There are other variables to use for an index besides i. Like n. Or k. If you need i for current, use j then. But don't pretend non-italicized i is such that

"Books with a high scientific level and a high acceptance prefer using this way."

Give me a break. Are you saying that all the graduate and research math textbooks and papers I have aren't of a high scientific level or high acceptance?? I noticed all your "high level books" are of a particular area or field, namely physics. Physicists aren't the only people to use i. BTW, I think the algebraic geometry Griffiths-Harris example may not be appropriate, as in abstract algebra one often uses sqrt(-1) as opposed to i, because one is working in fields other than the complex numbers, where square roots of -1 exist, but are not "the imaginary unit" because we're not in the complex field. Revolver 09:04, 8 Jun 2005 (UTC)

The first seven books I've just pulled from off my shelves, all use an italic i, for the complex unit. Paul August ☎ 16:50, Jun 26, 2005 (UTC)

I've added them to the table above. Paul August ☎ 18:36, Jun 26, 2005 (UTC)

I too have tried seven random books from my shelf. Five use italics. One sticks to sqrt(-1), but uses italic i when discussing quarternions. One uses non-italics: Characteristic Classes, Milnor + Stasheff, 1974, but in my opinion the typesetting in this book is not that great overall. Dmharvey Talk 18:07, 26 Jun 2005 (UTC)

Dmharvey: Would you mind adding your books to the table above? Paul August ☎ 18:36, Jun 26, 2005 (UTC)

I like more the italic style. Oleg Alexandrov 21:08, 26 Jun 2005 (UTC)

The table above suggests a hypothesis: Mathematicians use i consistently; physicists and chemists are divided, but more use "i" than i. (Counterexamples welcome.) This may imply that i is the natural usage, avoided chiefly when the author is accustomed to discussing current.

Looking at my own post, I think that italicized i makes the valuble point: this is not an English word, in the usual sense. I would use it, except in articles which use complex numbers and use i for current; and I would suggest that all articles which discuss both imaginary numbers and electric current state explicitly what they are doing. (But then, as a mathematician, I expect to see i except for texts still in (mechanical) typescript.) Septentrionalis 23:03, 26 Jun 2005 (UTC)

I believe it is a fundamental error to try to go around and edit a mass of WP articles to change the notation of something like i. These articles were written the way they were written because the writers employ a common convention that all agree to. This convention is widely used in mathematics and physics. To single-handedly try to convince hundreds or thousands of people to change their notation, especially on something trivial like this, is an error of judgement. The changes should not be made; the changed articles should be reverted, and this whole argument is a rather poor rat-hole to get drawn into. linas 04:25, 27 Jun 2005 (UTC)

There are two conventions. I don't believe it is clear that one is much more widespread. But these conventions are not really only conventions, because theya re not equivalent. One convention uses generic variable symbols for standard objects and is thus ambiguous and confusing. The other convention is a solution to this problem. There is a good reason why we use R instead of R for the reals. The same goes for exp(1) and the exterior derivative. What does latex do when you type \cos or some other standard function? People ignore good style because they are lazy and not because it is a different equivalent convention or perhaps because they are ignorant of good style. Since this is about style (or lack thereof) and not convention we can and should take a stand. --MarSch 28 June 2005 12:31 (UTC)

MarSch, I'm not sure I undersatand what you are proposing. Could you please elaborate? Paul August ☎ June 28, 2005 14:10 (UTC)

MarSch, I absolutly agree. Markus Schmaus

The above table is very unhelpful. There are literally thousands of "scientific and technical books" out there to choose from. No brief sample like this can prove anything.

From the discussion happening here, it sounds like people working in different areas use different notation. Mathematicians tend to use italics (this is certainly my experience), and perhaps other people like physicists use non-italics (my experience is too limited to say).

An analogy: in Australia, people spell differently to people in the U.S. Neither side is "right" or "wrong". However, when in Australia, you should spell the australian way. When in the U.S., you should spell the american way. When at the U.N., you can do either. (Although I bet at the U.N. they do it the american way :-) ). When an american comes to Australia, maybe they find it a little disorienting, but they easily get used to it and can understand everything that's going on. Similarly: articles primarily mathematical should be done using the mathematicians' notation; articles primarily in other areas can happily use other notation. Articles on the boundary... well maybe that can be discussed. Dmharvey Talk 28 June 2005 14:14 (UTC)

Using "i" is more common but using "i" would be better. "i" is a symbol, not a variable or an index, it's more similar to "1" or "sin" than to "x". Variables and indices are placeholders and, unlike symbols, they can be substituted. Distinguishing between symbols and variables improves readability, but many math books do not. I think wikipedia can and should deviate from popular notations if this increases readability and comprehensibility. Markus Schmaus 28 June 2005 15:18 (UTC)

I disagree. (1) I don't agree that using non-italic "i" increases readability and/or comprehensibility. (2) But even if it did, it's not a good argument, because the role of wikipedia is not to change the way people write things. For example, I think "elliptic curve" is very poor term for the object that it describes (see elliptic curve). Nevertheless, the term has stuck, and WP doesn't go around changing the name of "elliptic curve" to something more meaningful, on the grounds of increased comprehensibility. Dmharvey Talk 28 June 2005 16:19 (UTC)

For reference, I've added Britannica, mathworld and PlanetMath.org to the table above, all of which use italic i. Paul August ☎

Paul: I am saying that this is not about conventions. Using italics for standard objects is confusing/bad/lazy/ignorant style. Just the same as using non-italics for variables. This is not like the difference between American and Australian and British english spelling, this is like spelling errors, orput tingth espaci ngint he wrongpla ce. This is not about conventions this is about doing it right.--MarSch 28 June 2005 17:46 (UTC)

I disagree. I would certainly not characterise prolific mathematical authors such as Jean-Pierre Serre, John H. Conway, John Milnor and Richard P. Stanley, all of whom have been awarded the Leroy P Steele Prize for Mathematical Exposition by the American Mathematical Society, and all of whom use italicised "i" and "e" in their work, as ignorant or lazy. I think they know exactly what they are doing, and it's not at all like placing spaces in the wrong positions. Dmharvey Talk 28 June 2005 19:08 (UTC)

Try using a different convention on this:

If e is a bivector, then we have or equivalently . In quaternions much the same formula is true. We have , but holds only if x is purely quaternionic: for real j and k. Now let x, p, d and μ be real numbers and e a bivector on a compact 2-manifold R. Then we can calculate the integral and it is a real number. On the other hand it is much easier to simply calculate , because it is infinity. Unfortunately Greek letters are autoitalicized. This stuff is all pretty simple. Let's get into some deeper waters. Let . Then so we have . On the other hand .

enjoy. --MarSch 28 June 2005 18:55 (UTC)

The above examples are highly misleading. You have deliberately chosen your variable names to conflict with standard symbols. Anyone seriously using the above expressions would, in the interests of clarity and good style, refrain from using the symbols "i", "j", etc as variables, if they were using the standard meanings of those symbols in the same piece of writing. Dmharvey Talk 28 June 2005 19:14 (UTC)

Metaquestion: Should the original usage of this article be changed while this discussion is ongoing? especially since it appears to be majority opinion to retain i? Justify, or this will be sufficient grounds to revert. Septentrionalis 28 June 2005 21:12 (UTC)

I say yes.

I also propose that one of the articles Complex number, Imaginary unit or something similar, include a sentence similar to the following:

The symbol i is usually written in italics in mathematical writing; however in other scientific contexts, including physics and engineering, italics are often not used.

Dmharvey Talk 28 June 2005 21:52 (UTC)

I strongly disagree with MarSch that using italic vs. non-italics is a matter of correctness. All mathematical formatting is a matter of convention, even using non-italics for function names like sin. I have never heard of a mathematical formatting Bible in which are written a divine set of rules never to be broken. That being said, I believe this article should revert to the italic formatting of imaginary unit, as that appears to be the dominant convention. -- Fropuff 28 June 2005 22:38 (UTC)

The math conventions are non-italic for "sin" and "cos", italic for "i" and "e", and italic for "dx" in the integral. MarSch seems to be trying to say why one better use nonitalic "i" and nonitalic "dx". But that "why" is irrelevant here, we don't discuss what things should be or why they should be that way or another, we are discussing what the math convention is. Oleg Alexandrov 29 June 2005 02:55 (UTC)

I concur. Dmharvey Talk 29 June 2005 11:35 (UTC)

I completely agree with Oleg. Also, this is not just a mathematics text issue. The first ten physics books I pulled off my shelf all use italics for the imaginary unit. These are

• Jackson, Classical Electrodynamics
• Griffiths, Introduction to Electrodynamics
• Lorrain and Corson, Electromagnetic Fields and Waves (uses j instead of i)
• Sakurai, Modern Quantum Mechanics
• Peskin and Schroeder, An Introduction to Quantum Field Theory
• Weinberg, The Quantum Theory of Fields
• José and Saletan, Classical Dynamics
• Wald, General Relativity
• Hassani, Mathematical Physics
• Eisberg and Resnick, Quantum Physics

The above list includes such reputable publishers as John Wiley, Cambridge, Addison-Wesley, Springer, and Prentice Hall. -- Fropuff 29 June 2005 04:32 (UTC)

Perhaps this is an emerging style. We're not that far from typewritten mathematics books with penned in greek letters and such. Our library has quite a bit of those and also books that don't bother to deitalicize anything (det, tr, SU(N), etc.) I would expect mathematics gods to be lazy, after all they expect things to be effortless ;) It would be interesting to get some actual opinions from mathematicians or publishers on this, I'll try and see if I can google something up. --MarSch.

::: I think this is getting silly. Oleg Alexandrov 29 June 2005 16:16 (UTC) — trying to keep constructive. Oleg Alexandrov 29 June 2005 18:16 (UTC)

Some of the above are "actual opinions of mathematicians". Paul August ☎ June 29, 2005 14:27 (UTC)

Inspired by Fropuff, I have looked at my physics text and all of them I've consulted so far use italic i also. I've added them (along with Fropuff's examples) to the table above, I've also reorganized the table a bit. Paul August ☎ June 29, 2005 14:27 (UTC)

So far I've found [1]: "The numbers themselves do not act as containers for other values and so are set to upright. The complex number "i" we consider to be a creature of the same genus as the numbers and so we set it upright. Often it is represented as "j." Upright presentation is also a great aid in differentiating the complex number "i" from "i" used as a running index. The mixed use of "i" is very common."--MarSch 30 June 2005 10:56 (UTC)

On the other hand [2]: "In mathematical equations, use italics for all letter symbols (caps, lowercase, superscripts, and subscripts). Use regular type for all numbers.

Print chemical symbols, units of measurement, and abbreviations such as log, max, exp, tan, cos, lim, etc., in regular type. "--MarSch 30 June 2005 11:19 (UTC)

[edit] Can we agree?

Consider:

1. Until a few days a go the consistent practice on Wikipedia was two use an italic i to represent the imaginary unit.
2. The following editors oppose the change to non-italic i on Wikipedia:
   1. Revolver
   2. Paul August
   3. Dmharvey
   4. Oleg Alexandrov
   5. Pmanderson (signed as: Septentrionalis)
   6. linas
   7. Fropuff
3. The following editors support the change to non-italic i:
   1. Wurzel
   2. MarSch
   3. Markus Schmaus
   4. PizzaMargherita

I conclude from the above that, so far, there is no consensus for changing the usage on Wikipedia to a non-italic i.

Regarding common practice:

1. The web sources Britannica , Mathworld and PlanetMath.org all use italic i.
2. Of the first seven mathematics texts from my library and the first seven mathematics texts from DMharvey's library, 12 use italic i, one uses non-italic i, one uses sqrt(-1).
3. For the physics texts in the above table, 16 use italic i, 5 use non-italic i, one uses italic j.
4. Two of the three supporters of the change (Wurzel hasn't yet said) concede that italic i usage is more common.

I conclude from the above that common practice, (particularly in mathematics) is probably italic i, and that also seems to be the consensus view. In addition several editors have expressed, and I agree, that Wikipedia should follow common practice. It is no part of Wikipedia's mission to create or promote usage conventions. Because of all the above, I think it would be a good idea to reinstate usage of the italic i, in this and other articles. Later we can change it again, if a consensus for change is reached. Can we agree on this?

Paul August ☎ June 29, 2005 17:59 (UTC)

Support: This article is a really bad place for the non-italic experiment. It does not now discuss current flow; although maybe it ought to. More importantly, consider the first sentence: (In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying i² = -1.) The first "i" should be italicised or bolded anyway, because it is being defined. Septentrionalis 29 June 2005 18:12 (UTC)

You can't go italicizing or bolding symbols simply because they are being defined. It changes the symbol.--MarSch 30 June 2005 10:45 (UTC)

Support. Oleg Alexandrov 29 June 2005 18:16 (UTC)

Support. Fropuff 29 June 2005 18:34 (UTC)

Support. But still think the article could briefly mention the different formatting usage in different fields. This could be near the discussion of "i" vs "j" in different fields. Dmharvey Talk 29 June 2005 18:48 (UTC)

Support, although if these are just two conventions then we shouldn't be changing either way. Just as pages using different english shouldn't be changed, unless they are inconsistent.--MarSch 30 June 2005 10:45 (UTC)

Support. Gandalf61 June 30, 2005 13:51 (UTC)

Support. linas 1 July 2005 00:17 (UTC)

Ok I am going to start changing back to italic i, in this atricle and others, feel free to help. Paul August ☎ June 30, 2005 14:51 (UTC)

Accepted (temporarily). Hi all, soory, I haven't seen earlier this discussion going on here. Recently, I have checked again my books (math + phys) and compared with earlier verions, and most of them have changed from i (older books) to i for imag. unit in newer revisions, but it is my personal collection of books and may not represent the average. Currently, I still do not see a conflict if the definition of imag. unit is done by i² = -1, compared to i² = -1, because it is a definition. Thus, I agree at least to the updates in my math + phys books. Regards, Wurzel July 01, 2005 15:16 (UTC)

Summary of reasons

Reasons for pro italic usage of imaginary unit in Wikipedia: i
most mathematics books/papers use italic notation of imag. unit
italic notation of imag. unit looks better Oleg Alexandrov
is a conceptual case of definition, italic i is needed Septentrionalis
Reasons for pro non-italic usage of imaginary unit in Wikipedia: i
Better semantics. This has several beneficial implications. PizzaMargherita
prevents confusion with running index i, electr. current, etc. Wurzel
offers electrical engineering technicians an imaginary unit notation which has no interference with neither (Maxwell's) current density j nor with electr. current i Wurzel
allows parallel usage with running indexes i,j Wurzel
improves readability of formulas containing the imag. unit i because of no overlapping definitions Wurzel
i is easily acessible on many computers/text systems / fonts Wurzel
Reasons for usage of \imath
Is an alternative offered by TeX Michael Hardy

[Please add entries if something is missing - or revise if needed.]

[edit] Public voting table for (English) Wikipedia Notation of imaginary unit

To find some more public data, I have added here a voting table. This may help (from the side of vote counts) to find a useful decision for all involved people in the various scientific fields. I hope that this table will not collapse here. I would propose to stop counting at around 100 votes. Nevertheless, the content based discussion has to be continued and should be used for decision. Please notify, that Wikipedia should be as precise as possible and simultaneously, as compatible as possible to all the different (e.g. scientific) fields, because many people are using it. (Wurzel 01 July 2005 16:30 UTC)
Meanwhile, we need some rough overview of the public meaning. If you, dear reader, are very interested for one of the decision possibilities, and if you think to give a vote here, then

Please vote here and use the format:
[Name]/[Priority(L[ow]/M[edium]/H[high])],

Voting: Pro italic usage of imaginary unit in Wikipedia: i ; Sum of votes: 9:
Revolver/H, Paul August/H, Dmharvey/H, Oleg Alexandrov/H, Pmanderson/H, linas/H, Fropuff/H, Blotwell/L, Flex/L
Voting: Pro non-italic usage of imaginary unit in Wikipedia: i; Sum of votes: 3:
Wurzel/H, MarSch/H, PizzaMargherita/H
Voting: None of the above; Sum of votes: 1:
Jitse Niesen/ε (see explanation below)
Voting: Usage of \imath; Sum of votes: 1:
How about ? Michael Hardy 00:06, 3 August 2005 (UTC)

Please do not forget to add a character (L/M/H) for the priority, so that we have an intension how deep is your interest for your vote. [[The list was started with the article editors from the table in section 'Can we agree?'.

I don't see why we should have a standard on this. Just use the convention of whoever started the article. In my experience as a mathematician, Americans almost always use italic i, d and e, while both italic and upright are in use in Europe (if you want to know, I prefer upright i, d and e). -- Jitse Niesen (talk) 4 July 2005 13:32 (UTC)

I could believe that. In my experience as a mathematician, my (British) high school text contained upright i, d and e and I've rarely seen them since. My personal opinion is that I like upright d (the differential operator, as in dy / dx versus dy / dx) a lot and don't care much either way for e and i the constants. But as a Wikipedian I think that the consensus in mathematics (don't know about physics or engineering) is italics for all and we should go with that. Otherwise, every single contribution will have to be picked through by a pedant to change the typesetting. —Blotwell 7 July 2005 05:17 (UTC)

FWIW, Abramowitz and Stegun use an italic i (and this IMHO is the ultimate arbiter of mathematical notation). The Mathematica website uses something that looks like a Blackboard bold lower-case i. Robinh 7 July 2005 08:10 (UTC)

why is it your ultimate arbiter of math notation? Are we talking about the 1964 version? --MarSch 15:33, 8 September 2005 (UTC)

I don't know how I created a second column in the voting table above; maybe someone could fix it?

Table fixed - created extra voting entry. Wurzel 9 August 2005 22:40 (UTC)

I have a slight preference for the italic i, thus:

cosθ + i sinθ

or

or

The last is created in TeX by \imath. Michael Hardy 00:06, 3 August 2005 (UTC)

I think \imath is there to let you put any kind of vector arrow, circumflex or double dot on top of an i if you would want -- it's not for using a dotless i really. So I think that would be a severe misunderstanding on our part if we start to use an i that's not an i -- for i. ;-P ❝Sverdrup❞ 00:57, 11 January 2006 (UTC)

Since this discussion is relatively recent, I'll add my support for the "upright" convention, for the reasons already stated. My theory (which I read somewhere authoritative but I can't remember where) is that the "upright" convention was the original (and correct) one, and then a lazy convention started to spread and became the most widely used, but (as many things that "most" people do) still wrong. It's not a matter of arbitrarily spelling the British or the American way, the upright convention is objectively superior, for exactly the same reasons why "sin" and "cos" should be upright (or in a world where variables and generic functions are upright, italicised). I'll keep looking for some official references to support this theory. Also this is not only about "i", it's also about "e", "pi" and any other constants, plus the "d" for total differential, which when found in integrals should be "\,\mathrm{d}". The Italians seem to have got it right somehow—look at the formula at the bottom of this article. PizzaMargherita 00:59, 21 November 2005 (UTC)

I found something interesting in the MathML specifications. I think WP has made a long-term committment towards MathML support, so I think we should read this carefully.

They clearly agree with the points made above, regarding better semantics. "e" and "i" are not variables, and they deserve special entities.

To begin we list separately a few of the special characters which MathML has introduced. These now have Unicode values. Rather like the non-marking characters above, they provide very useful capabilities in the context of machinable mathematics.

Entity name        Unicode         Description
ⅅ      02145   D for use in differentials, e.g. within integrals
ⅆ     02146   d for use in differentials, e.g. within integrals
ⅇ      02147   e for use for the exponential base of the natural logarithms
ⅈ        02148   i for use as a square root of -1

As for rendering, I don't think they make it quite clear. Compare this example (clearly supporting upright rendering) with this description, which seems to imply that "traditionally" (whatever that may mean, possibly "by a lot of people [fools! :-)], but not by us") these entities are rendered in italics

Certain MathML characters are used to name operators or identifiers that in traditional notation render the same as other symbols, such as ⅆ, ⅇ

But that's not the end of the story. To make things even more confusing, look at the rederings used for those four entities (search for "Imaginary" within that page).

Also, this highlights that out-of-the-box (La)TeX is semantically less powerful than MathML, and therefore information is lost, which is a shame. This was probably well-known by most, but having no MathML background, I just discovered/thought about it. PizzaMargherita 12:34, 25 November 2005 (UTC)

As you say, it is not yet clear how e and i will be rendered in mathml. And I said it before, mathml is not here yet, and will not be here soon. Oleg Alexandrov (talk) 16:08, 25 November 2005 (UTC)

I just want to mention the following passage from Springer-Verlags instructions to their authors, "The Differential d, exponential e and imaginary i should be set upright in Springer books." Springer Author instructions, pdf (User:Berland) 30 November 2005

Yes, Springer wants upright d. In general, in Europe people like upright things more. But it has not been the rule on Wikipedia, and I see no reasons for change. See also Wikipedia talk:WikiProject Mathematics/Archive7#straight_or_italic_d?. Oleg Alexandrov (talk) 01:07, 1 December 2005 (UTC)

Reasons for change are above. You can't deny the semantic superiority of the upright convention. You can't justify "sin" and "cos" being upright and "differential d" being italicised. PizzaMargherita 07:20, 1 December 2005 (UTC)

Actually, yes, I can; it visually distinguishes the derivative, which is a functional, from the functions. That way you don't need parentheses to divide the three parts of dsinz. But that misses the real point: it is not Wikipedia's business to attempt to impose a standard on the world, however "logical" it may be. Septentrionalis 01:17, 11 January 2006 (UTC)

Right, wikipedia should not impose a standard on the world, I agree. But it should give best recommendations for a good usage of symbols, concepts, etc. For me it would be absolutely "logical" to use e.g. an upright differential symbol d or i as the imaginary unit. Wurzel 22:52, 21 January 2006 (UTC)

Over my dead body. :) Oleg Alexandrov (talk) 23:48, 21 January 2006 (UTC)

I would also like to object to some of the "reasons" given in the table.

• "italic notation of imag. unit looks better" This is POV. I can say the same about upright "i".
• "is a conceptual case of definition, italic i is needed" Ok, so can we change all the other instances in all the other articles? Very poor argument.
• "[upright] i is easily acessible on many computers/text systems / fonts" Another very poor argument. Shall we start writing everything upright then? Including integral symbols and the like? Come on...

"[upright] i is easily acessible on many computers" - sure, it is not a 'strong' reason. It just means that this is the simpliest notation alternative to an italic i. Blackboard i or special TeX symbols for i are tendentially more problematic for the usage in standard texts. Wurzel 23:05, 21 January 2006 (UTC)

So anyway, the only real reason for not changing to upright is "go with the flow", which I don't buy for a minute because the flow is sometimes wrong. PizzaMargherita 07:46, 1 December 2005 (UTC)

[edit] Algebraic characterization

The article says:

The field C is (up to field isomorphism) characterized by the following three facts:

• its characteristic is 0
• its transcendence degree over the prime field is the cardinality of the continuum
• it is algebraically closed

OK, so one adds a set of transcendental elements of cardinality c. If one adds only a proper subset of those, with the same cardinality, does one get a proper subfield? Does C therefore have proper subfields isomorphic to C? If so, one gets an infinite descending chain of proper subfields isomorphic to C, and their intersection is also a subfield; what does it look like? Michael Hardy 00:14, 3 August 2005 (UTC)

If you add a proper subset the algebra might change such that you end up with C no matter what. --MarSch 15:44, 8 September 2005 (UTC)

What is the reference for this algebraic characterization? - Gauge 05:59, 30 January 2006 (UTC)

[edit] Elementary geometry

Alas the subject of 'complex numbers' is made advanced from the beginning. Consider the geometric plane of points. Choose a zero point, 0, and a unit point, 1. If the triangle (0,A,X) is similar congruent to triangle (X,B,0), then X=A+B. If the triangle (0,1,A) is similar to triangle (0,B,X), then X=AB. This definition of addition and multiplication of points in the plane is nice to the beginner who only needs to know the geometric concept of similar triangles, but no knowledge of real numbers is needed. Complex numbers are easier than real numbers.

Bo Jacoby 07:40, 8 September 2005 (UTC)

this is a nice alternative definition, but whether it is simpler is debatable. --MarSch 15:49, 8 September 2005 (UTC)

Feynman used it in QED, and it should be at least included. Septentrionalis 17:01, 9 September 2005 (UTC)

1. If (0,A,X) and (X,B,0) are similar, then they are congruent too, so the requirement that they be congruent is not necessary.
2. Construction of similar triangles is done by compass and ruler. Multiplication of reals requires some theory of continuity. Bo Jacoby 13:52, 19 September 2005 (UTC)

   So does the use of compass and straightedge, to establish the existence of the point of intersection. Septentrionalis 17:34, 19 September 2005 (UTC)

OK. I've split the section geometry into two called geometry and coordinates. Bo Jacoby 12:49, 20 September 2005 (UTC)

[edit] The imaginary part

If z=x+iy where x and y are real, then x is the real part and iy is the imaginary part. y is neither imaginary nor a part. Why do you call y the imaginary part ? Bo Jacoby 12:49, 20 September 2005 (UTC)

This is the standard def for the imaginary part. Im(x+iy):=y. It is projection onto the second coordinate. --MarSch 17:40, 25 September 2005 (UTC)

Sorry, but the projection of x+iy onto the second axis is iy, not y. Surely the confusing definition is widespread. Does that mean that it should be promoted ? Bo Jacoby 09:41, 26 September 2005 (UTC)

No, C is R^2 with the product (a, b)(c, d) = (ac - bd, ac + bd). Then you can make the definition i := (0, 1). Thus x + iy = (x, y) and if you project to the second coordinate you get y. i is just a basis vector. --MarSch 13:07, 28 September 2005 (UTC)

The mapping F(x+iy)=y is not a projection. The mapping P(x+iy)=iy is a projection. See the article projection operator. The point is that a projection P is idempotent, PP=P.

while P(P(x+iy))=P(iy)=iy=P(x+iy). So while PP=P. F is not idempotent. P is idempotent. Bo Jacoby 08:17, 29 September 2005 (UTC)

[edit] Article name: shouldn't it be in the plural form?

• The article reffers to the number set and therefore it's name should be in the plural form. Why is it in the singular form? --Mecanismo 10:13, 16 September 2005 (UTC)

Well, in links the mention of complex numbers often comes as

Let z be a complex number...

There is also a Wikipedia convention, that whenever possible, article titles should be singular not plural.

Let me try a different explanation. This article is as much about the set as it is about its individual elements. So, ultimately, to call it singular or plural is a matter of convention, and I would prefer singular for the reasons in the paragraph above. Oleg Alexandrov 16:15, 16 September 2005 (UTC)

It disturbs me too, Mecanismo. The object of interest is the "set of complex numbers", together with its topology, algebra structure, involution and what have you. A complex number in isolation doesn't have these things. --MarSch 17:48, 25 September 2005 (UTC)

This would need a wider discussion, at Wikipedia talk:WikiProject Mathematics if anybody feels it is worth it. I think it is not. Oleg Alexandrov 22:52, 25 September 2005 (UTC)

It does sound pretty awkward tho, to me. Fresheneesz 05:07, 26 April 2006 (UTC)

[edit] Geometry?

The Geometry section should be deleted, or at least rewritten so that it makes some kind of sense. The article begins with a nice definition of complex numbers and continues with discussions of complex numbers as coordinates, which are arguably the two most common definitions but stuck in the middle is this incoherent rigamarole discussing similar triangles. The article is intended to serve as an encyclopedia article, not a pedagogical tool (again, assuming the Geometry section is teaching anyone anything).

The section is so egregiously bad I was tempted to just delete it without consultation, but there seems to be some who think it has some value. Please show me what that value might be.--andersonpd 01:28, 21 October 2005 (UTC)

I don't much like that section either, it is rather badly written and does not seem to be extremely relevant. There is some text above it explaining the complex plane and polar coordinates, that should be enough. If somebody has the energy to write a nice Geometry of complex numbers article to explain in more coherent way the stuff in that section, it would be good. Otherwise, bring the axe brother. Oleg Alexandrov (talk) 04:43, 21 October 2005 (UTC)

The geometry section was intended to be the first one, because it is elementary, but was moved to the middle. The geometry section does not depend on knowledge on real numbers or coordinates, but only on elementary geometry. So it can be read by non-mathematicians. It would profit by some drawings. Actually complex number multiplication is simpler than real number multiplication, and should not rely on that. Most of the article on complex numbers brings the impression that understanding real numbers is a prerequisite for understanding complex numbers, and that is not true. This point, of cause, should be made more clear in the section of geometry. Bo Jacoby 08:10, 21 October 2005 (UTC)

Would you be willing to move the "Geometry" section to a new article? That material is of course related to complex numbers, but it is more like an application (and not a really important one). As such, it was not right to put it before other, more immeadiate properties of complex numbers, like division, absolute values, etc. I think that section is not even as relevant as the sections now below it, which are "Solutions of polynomials equations", "Algebraic characterizations", etc. And by the way, the "Geometry" section is indeed not well written. Oleg Alexandrov (talk) 08:48, 21 October 2005 (UTC)

I made some clarifications. The fusion of geometry and algebra by the geometrical interpretation of complex numbers is very important. Historically Descartes preceded Gauss, and so the use of real coordinates in geometry came before the use of complex numbers. In teaching, the historical road from Euclid via Descartes to Gauss is usually followed. Logically, however, a shortcut can be made, from Euclid directly to Gauss. This is what I did. To define the arithmetic of points in the plane you don't need coordinates and you don't need real numbers. If you already know about coordinates and real numbers, then you do not need this shortcut of cause, but some other Wikipedia readers don't, and they would prefer the direct road to 'complex numbers', without the detour to coordinates and real numbers. So I think that an elementary geometrical explanation should precede the advanced stuff. Bo Jacoby 09:29, 21 October 2005 (UTC)

Geometry means pictures, sir. I found the formulas in there hard to follow. Several pictures, and replacing all those triangle symbols plain words, say caption to the pictures, would go a long way towards improving that section. Also, it needs to be made shorter. There is no need to rederive again the complex numbers, with X^2+1 and all that. What is needed is a rather short section explaining how addition, multiplication and conjugation would look like in geometric terms. Oleg Alexandrov (talk) 09:43, 21 October 2005 (UTC)

The geometric view is just another way of defining the complex numbers and as such should be in this article. Pictures would definitely improve it a lot, but that is no reason to delete what's there now. What I don't understand yet is how the plane is defined if you don't use real numbers. I mean you can't say, the plane is R^2, so what do you say? --MarSch 11:08, 21 October 2005 (UTC)

[edit] Elementary geometry

(The subsection grew, so I make a new header here.) I agree completely that pictures will improve it, but alas I'm no good at drawing. We need (1) an addition picture showing by similar triangles (0,1,1+i) and (2+i,i,1) that (1+(1+i)=2+i); (2) a multiplication picture showing by similar triangles (0,1,1+i) and (0,2i,-2+2i) that (1+i)(2i)=-2+2i; (3) a conjugation picture showing by mirror triangles (0,1,2+i) and (0,1,2-i) that (2+i)*=(2-i), and (4) an i picture showing by similar triangles (0,1,i) and (0,i,-1) that ii=-1. The plane was defined by Euclid many hundred years before real numbers were invented. Bo Jacoby 12:47, 21 October 2005 (UTC)

I can take care of the pictures, in the next several days. Bo, one important thing which I also said earlier is I believe you got a bit carried away in that section. There is no need to talk about the factorization of X^2+1 and all that stuff below it. This is one of those cases in which putting more information does not help, but rather confuses the reader. If you do want to elaborate, you should I think start a new article, and keep here only the important points. Oleg Alexandrov (talk)13:49, 21 October 2005 (UTC)

I believe that moving the geometry info to its own section has improved the overall flow of the article. I concur that adding pictures would be a great help in making the geometric definition clear. And I agree that a separate article would be a good way to cover the information in depth. In fact, I think that's probably the basis for my original objection -- the subject was presented without any transition or explanation of its purpose or goal. It is an interesting sidelight, but it is, IMHO, only that -- a sidelight. (unsigned post by Paul D. Anderson, 10:35, 21 October 2005).

Thank you, gentlemen. Your objections has so far lead to a substantial improvement, which is what this is all about. I wondered why elementary 'complex' number theory assume advanced stuff like trigonometry, exponentials and vectors, and what is the square root of minus one? The geometrical approach shows why ii=-1. No big deal, just similar triangles. One guy's sidelight is another guy's mainlight. As to the factorization of polynomials: it would be nice to have a bridge between geometry and algebra instead of separate islands. The points of intersection between circles and lines are the roots of a polynomial, leading to the factorization of the polynomial. This insight motivate the study of factorizations. Bo Jacoby 16:55, 23 October 2005 (UTC)

[edit] ln(-1)

I am a high school student who excels in math. However I do not completely understand the meaning behind complex numbers. Why is it that ln(-1)=(pi)x(i) What is its application or real world significance?-nick

Hi Nick ! ln(−1) is a solution to e^x = −1. No real number x satisfy this equation. If x = it is an imaginary number, then e^x = e^it is a point on the unit circle. See Exponentiation#Powers_of_one. t is the length of the arc along the circle from point 1 ( = 1 + 0i) to e^it. When you have walked the length π along the unit circle starting at 1, then you have arrived at the point −1 ( = −1 + 0i). So e^iπ = −1. And so ln(−1)=iπ. Keep asking ! Bo Jacoby 09:53, 3 November 2005 (UTC)

Except that e^z = −1 has more than one solution; iπ is not the only one. 3πi is another. So log(−1) (or ln(−1) if you like writing "ln" instead of "log", either of which means the natural or base-e logarithm) is "multiple-valued". Michael Hardy 19:17, 3 November 2005 (UTC)

Hi Nick, I recommend you check out a text or course on complex analysis. If you like math, then you'll probably enjoy the opportunity to re-derive many of the rules you know from real numbers but with complex numbers. I took complex analysis in college and see no reason sharp high school students couldn't manage the material (with the possible exception of some of the calculus that only a few high school students have learned). I used an out of print book from the seventies and Schaum's. I recommend the latter. Cheers - --rs2 17:17, 5 November 2005 (UTC)

[edit] correct symbol?

I need input on what is or has a consensus for being the correct symbol for the real part of a number and the imaginary part. I have seen someplaces using Black Forest . I'm thinking blackboard is appropriate or even calligraphy . I don't want to wreck a few pages and then find out I was wrong. Snafflekid 06:19, 10 November 2005 (UTC)

I don't think there is any universal consensus on font, but blackboard is often used to signify the set of real numbers. The real part function is not used as much as the complex conjugate function. Many authors do without it, and so the problem is solved. Bo Jacoby 07:29, 10 November 2005 (UTC)

I agree that is not suitable, but I think that the functions can be useful. As notations simply Re(z) and Im(z) are fine too.--Patrick 12:24, 10 November 2005 (UTC)

Agree with Patrick that Re and Im are good enough. Some people use and , written as <math>\Re</math> and <math>\Im</math>, but I am not sure it is worth it. Oleg Alexandrov (talk) 19:29, 10 November 2005 (UTC)

I have reviewed my books and textbooks by various lettered and sundry authors. There seems to be a 3:1 ratio of to Re(z) Im(z). I recall all my professors writing the function using script as well. But I'm an electrical engineer and this function comes up a lot in discussing phasor notation. Selection bias? Snafflekid 19:37, 10 November 2005 (UTC)

Put me in the Re(z) and Im(z) camp. Maybe my age is showing, but I've never seen either the script or Black Forest versions.--andersonpd 20:26, 10 November 2005 (UTC)

One reason to stick with the simpler Re and Im instead of fancy fonts is that fancy fonts will become images, and will look out of proportion when embedded in text. Plain text is preferrable to PNG images, as per the math style manual. Books are written on paper and don't have this issue. Oleg Alexandrov (talk) 01:04, 11 November 2005 (UTC)

Plain text looks fine, I think, but that doesn't help if LATeX is being used. I found a page using typed as <math>\operatorname{Re}(z)</math>. Very clean IMO. Plain text will be italicized otherwise and that is definitely wrong. Snafflekid 02:05, 11 November 2005 (UTC)

[edit] Whats this?

My question is what do you do if your faced with x^(4+3i)+x^(3+1i)+x^i+1=0? How do you solve it exactly? Is there even a solution to complex polynomials? --anon

This is not a polynomial equation. For a polynomial, the powers must be positive integers.

This equation is not well-defined. One cannot easily and uniquely define the concept of complex number raised to the power of another complex number. Oleg Alexandrov (talk) 01:26, 19 November 2005 (UTC)

That complex polynomials have roots is the so-called fundamental theorem of algebra (a misnomer, really) proved by Carl Gauss in (or about?) 1799. And Oleg is right: what you've written doesn't look like a polynomial. Michael Hardy 03:00, 19 November 2005 (UTC)

Complex exponents sometimes make sense. See Exponentiation#Arbitrary_real_and_complex_exponents. Musicians draw a circle of fifths where a note of frequency x is plotted on the point y=x^2πi/ln(2). Two notes differing by an octave, having frequencies x and 2x, are plotted on the same point on the circle because 2^2πi/ln(2) = e^{ln(2)2πi/ln(2)} = e^2πi = 1, so that (2x)^2πi/ln(2)=2^2πi/ln(2)x^2πi/ln(2)=1y. This is convenient because such notes are equivalent from a musical point of view. However, if you ment (4+3i)x³+(3+i)x²+ix+1=0, then see Root-finding_algorithm#How_to_solve_an_algebraic_equation. Bo Jacoby 09:53, 21 November 2005 (UTC)

Complex exponents make sense if you decide which cut in the plane you are going to use, and which branch of the logarithm you are going to use.

Otherwise, in that equation on the top of the section we are dealing with a sum of three-multivalued functions which gives us a lot of combinations for what the sum may equal to. A pain for sure. So I would agree with Bo that the anon most likely made a typo in there. Oleg Alexandrov (talk) 12:44, 21 November 2005 (UTC)

It need not be all that bad. Substitute y=ln(x) in the equation and get (2): e^ay+e^by+e^cy+1=0 where a=4+3i, b=3+i, c=i. This equation (2) has an infinity of roots. Substitute for exponential functions e^x=1+x, truncating the power series to degree 1. Solve the resulting equation of degree 1. The root is likely to be an approximate solution to (2). Include terms of degree 2 and repeat the process, using Root-finding_algorithm#How_to_solve_an_algebraic_equation. Continue with higher degrees until you are happy or tired, (whichever occurs first). Bo Jacoby 13:35, 21 November 2005 (UTC)

Substituting y=ln (x) forces you to choose the branch of the log, so you miss some solutions. Substituting e^x=1+x is bad, because you are back to an equation with complex powers, the think you started with. :( Oleg Alexandrov (talk) 17:51, 21 November 2005 (UTC)

Hello Oleg ! I'm sorry I used letter x in two meanings. The degree 1 approximation to (2) is (1+ay)+(1+by)+(1+cy)+1=0, having the solution y=−4/(a+b+c)=−4/(7+5i)=(−14−10i)/37. So x=e^y=e^−14/37(cos(−10/37)+i sin(−10/37))=0,660-i0,183. Improved approximations give more roots and more precise roots. Bo Jacoby 08:24, 22 November 2005 (UTC)

[edit] Useless...

Still doesn't tell me what the **** a complex number is.

Are you saying that the introduction is not idiot-proof? I'd agree that "closing under addition and multiplication" in the first sentence sounds a bit daunting. Can't we move it down where it says that it's a field? PizzaMargherita 23:44, 4 December 2005 (UTC)

First sentence in the article is a bit complicated, but it is fine. The second sentence in the article does a good job though. Either the anon did not reach to that second sentence, or the anon did not understand that second sentence. Either way, whatever we do will not be helpful for this particular person. Oleg Alexandrov (talk) 00:04, 5 December 2005 (UTC)

Our anonymous friend has a point. I'll make the introduction a little bit more elementary. Bo Jacoby 08:14, 5 December 2005 (UTC)

I think your edit went a little overboard in that direction. Fiedorow 09:19, 5 December 2005 (UTC)

Thank you. Wikipedia needs to be... more layman-friendly.

I agree. Although some topics do require a degree of mathematical sophistocation, many -- if not most -- physics and math articles seem unapproachable for a layperson. This is frustrating . . . maybe there could be a policy of having a general introduction for non-specialists, then a more detailed and techincial intro for students or advanced users. Arundhati bakshi 22:53, 13 February 2006 (UTC)

id say he doesnt know wat it is cuz its 'complex'. lol

[edit] History

Fiedorow's large section on history is well written! But is it well placed too? Are there general WP articles on the history of mathematics to refer to or to be referred to? Bo Jacoby 08:00, 7 December 2005 (UTC)

[edit] Conventions

The discussion on whether the imaginary unit should be written i or i is unsettled, but the same formula should stick to the same convention.

Do we write a+ib or a+bi ? 3+i2 or 3+2i ? This is a matter of convention.

In the expression ax² the convention tells that x is variable and a is constant. You rarely see x²a. Accordingly to this the constant i should be written first in the expression ix but last in expression 2i; '2' is even more constant than 'i'.

In the Einstein formula E=mc² the constant is c², and the variable is m. You never see E=c²m. I wonder why.

The expression cm² means squarecentimeter, not centisquaremeter, so the general rule ax²=a(x²) is violated. This leaves us with insufficient units for area and volume. (1km²=1000000m². There is no SI-name for 1000m²). I think the rules ought to overrule the conventions. Bo Jacoby 08:00, 7 December 2005 (UTC)

I never saw a complex number written as 1+i2. Did you? :) Oleg Alexandrov (talk) 17:54, 7 December 2005 (UTC)

Certainly not, but nor did I ever see x+yi, but only x+iy. It is strange that substituting x=1 and y=2 into x+iy cannot give 1+i2. 2πi is never written iπ2. We teach that ab=ba, but apparantly we are not quite to be trusted. :-) Bo Jacoby 10:03, 8 December 2005 (UTC)

OK, I would suggest we go with the flow rather than invent new conventions. That means, as far as I am concerned, x+iy, 3+2i, and 1-i/2. No? Oleg Alexandrov (talk) 18:25, 8 December 2005 (UTC)

I've seldom if ever seen "2 + i5" or the like, but obviously one sees "cosθ + i sinθ"; that's where the "cis" abbreviation comes from (not that "cis" is used all that often, though). Michael Hardy 01:05, 13 December 2005 (UTC)

Whether one writes x+yi or x+iy is purely a matter of style. While the latter convention is more common, there are notable texts, e.g. van der Waerden's Modern Algebra and Birkhoff's and MacLane's A Survey of Modern Algebra which use the former convention. On the other hand quaternions are generally written as x+yi+zj+wk rather than x+iy+jz+kw. One thinks of i, j, k as vectors and y, z, w as scalars, and the usual convention is to write the scalar first. On the other hand, I would tend to write and cosθ + isinθ to avoid any possible confusion that i might be interpreted as part of the argument to the square root or sin.Fiedorow 15:46, 14 December 2005 (UTC)

Would the author of the following formulae

(a + bi) + (c + di) = (a+c) + (b+d)i

(a + bi) − (c + di) = (a−c) + (b−d)i

(a + bi)(c + di) = ac + bci + adi + bd i ² = (ac−bd) + (bc+ad)i

please make up his mind as to whether the imaginary unit should be written i or i ? Bo Jacoby 09:59, 14 December 2005 (UTC)

Sorry, that was an oversight. I've fixed it.Fiedorow 15:46, 14 December 2005 (UTC)