Talk:Complex number/Archive 1

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 &phi; one gets this way is up to a multiple of &pi;, 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 &pi;+&pi;/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)

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.

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.

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)

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)

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)

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)

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

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. —Preceding unsigned comment 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):

[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 &#9742; 16:50, Jun 26, 2005 (UTC)
 * I've added them to the table above. Paul August &#9742; 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 &#9742; 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? $$e^{ix}\;dx \quad \mathrm{e}^{\mathrm{i}x}\;\mathrm{d}x$$ 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 &#9742; 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 [[Image:User_dmharvey_sig.png]] 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 &#9742;

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 [[Image:User_dmharvey_sig.png]] Talk 28 June 2005 19:08 (UTC)

Try using a different convention on this:

If e is a bivector, then we have $$e^{ix} = e^{-xi}\;$$ or equivalently $$e^{ij} = e^{-ji}\;$$. In quaternions much the same formula is true. We have $$\mathrm{e}^{\mathrm{i}\mathrm{j}} = \mathrm{e}^{-\mathrm{j}\mathrm{i}} = \mathrm{e}^\mathrm{k}\;$$, but $$\mathrm{e}^{\mathrm{i}x} = \mathrm{e}^{-x\mathrm{i}}\;$$ holds only if x is purely quaternionic: $$x = j\mathrm{j} + k\mathrm{k}\;$$ for real j and k.

Now let x, p, d and &mu; be real numbers and e a bivector on a compact 2-manifold R. Then we can calculate the integral $$\int_R expd\mu \in \mathbf{R}$$ and it is a real number. On the other hand it is much easier to simply calculate $$\int_\mathbf{R} \exp \; \mathrm{d\mu} = \infty $$, because it is infinity. Unfortunately Greek letters are autoitalicized.

This stuff is all pretty simple. Let's get into some deeper waters.

Let $$i := ({}^0_\mathrm{i} {}^{-\mathrm{i}}_0)\;$$. Then $$i^2 = 1\;$$ so we have $$\mathrm{e}^{ix} = \cosh x + i \sinh x\;$$. On the other hand $$\mathrm{e}^{\mathrm{i}x} = \cos x + \mathrm{i} \sin x\;$$.

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


 * 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.

Oleg Alexandrov 29 June 2005 16:16 (UTC) &mdash; trying to keep constructive. Oleg Alexandrov 29 June 2005 18:16 (UTC)
 * Some of the above are "actual opinions of mathematicians". Paul August &#9742; 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 &#9742; June 29, 2005 14:27 (UTC)

So far I've found : "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 : "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)

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:
 * 3) Revolver
 * 4) Paul August
 * 5) Dmharvey
 * 6) Oleg Alexandrov
 * 7) Pmanderson (signed as: Septentrionalis)
 * 8) linas
 * 9) Fropuff
 * 10) The following editors support the change to non-italic i:
 * 11) Wurzel
 * 12) MarSch
 * 13) Markus Schmaus
 * 14) 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 &#9742; 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 [[Image:User_dmharvey_sig.png]] 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 &#9742; 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)

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

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 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 $$\hbox{d}y/\hbox{d}x$$ 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. &mdash;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:


 * cos&theta; + i sin&theta;

or


 * $$\cos\theta + i \sin\theta,\,$$

or


 * $$\cos\theta + \imath\sin\theta.\,$$

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&mdash;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 &amp;CapitalDifferentialD; 	02145 	D for use in differentials, e.g. within integrals &amp;DifferentialD; 	02146 	d for use in differentials, e.g. within integrals &amp;ExponentialE; 	02147 	e for use for the exponential base of the natural logarithms &amp;ImaginaryI; 	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 &DifferentialD;, &ExponentialE;

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. 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)

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)

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)

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.
 * $$ F(F(x+iy))=F(y)=F(y+i0)=0 \ne y=F(x+iy)$$

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

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)

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)

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)

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(&minus;1) is a solution to ex = &minus;1. No real number x satisfy this equation. If x = it is an imaginary number, then ex = eit is a point on the unit circle. See Exponentiation. t is the length of the arc along the circle from point 1 ( = 1 + 0i) to eit. When you have walked the length &pi; along the unit circle starting at 1, then you have arrived at the point &minus;1 ( = &minus;1 + 0i). So ei&pi; = &minus;1. And so ln(&minus;1)=i&pi;. Keep asking ! Bo Jacoby 09:53, 3 November 2005 (UTC)

Except that ez = &minus;1 has more than one solution; i&pi; is not the only one. 3&pi;i is another. So log(&minus;1) (or ln(&minus;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)

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 $$\mathfrak{R I}$$. I'm thinking blackboard $$\mathbb{R I}$$ is appropriate or even calligraphy $$\mathcal{R I}$$. 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 $$\mathbb{R}$$ 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 $$\mathbb{R}$$ 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 $$\Re$$ and $$\Im$$, written as $$\Re$$ and $$\Im$$, 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 $$\mathcal{R}{e}(z) \mathcal{I}{m}(z)$$ 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 $$\operatorname{Re}(z)$$ typed as $$\operatorname{Re}(z)$$ . Very clean IMO. Plain text will be italicized otherwise and that is definitely wrong. Snafflekid 02:05, 11 November 2005 (UTC)

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. Musicians draw a circle of fifths where a note of frequency x is plotted on the point y=x2&pi;i/ln(2). Two notes differing by an octave, having frequencies x and 2x, are plotted on the same point on the circle because 22&pi;i/ln(2) = eln(2)2&pi;i/ln(2) = e2&pi;i = 1, so that (2x)2&pi;i/ln(2)=22&pi;i/ln(2)x2&pi;i/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. 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): eay+eby+ecy+1=0 where a=4+3i, b=3+i, c=i. This equation (2) has an infinity of roots. Substitute for exponential functions ex=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. 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 ex=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=&minus;4/(a+b+c)=&minus;4/(7+5i)=(&minus;14&minus;10i)/37. So x=ey=e&minus;14/37(cos(&minus;10/37)+i sin(&minus;10/37))=0,660-i0,183. Improved approximations give more roots and more precise roots. Bo Jacoby 08:24, 22 November 2005 (UTC)

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

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)

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. 2&pi;i is never written i&pi;2. 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 "cos&theta; + i sin&theta;"; 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 $$2+i\sqrt{3}$$ and $$cos\theta+i\sin\theta$$ 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) &minus; (c + di) = (a&minus;c) + (b&minus;d)i
 * (a + bi)(c + di) = ac + bci + adi + bd i 2 = (ac&minus;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)

Motivation for complex numbers
I think that the intro should spell out the basic motivation of introducing complex numbers: solving equations with real coeffs that don't have real solutions. It is true that the first sentence mentiones that i2=-1, but this is probably not enough. However, other people think that this mention encompasses the spirit of the basic motivation, so I might be wrong.

My point is that saying that the mention of i squared being equal to -1 suggests the equation solving motivation is like saying that merely defining complex numbers and their addition and multiplication suggests (by the fundamental theorem of algebra) the equation solving motivation (I know this is a gross exageration, but I think that it illustrates my point well) AdamSmithee 08:56, 9 January 2006 (UTC)


 * Originally the motivation was solving equations with real coefficient that DO have real solutions. In solving the cubic by radicals, when the solutions are real, one uses imaginary numbers along the way, and the imaginary parts cancel out.


 * And note that I said originally. Of course in the 19th and 20th centuries, they came to be used for many things. Michael Hardy 23:37, 9 January 2006 (UTC)


 * I pretty much meant basic motivation as in not advanced, not necessarily historical motivation (though I admit my wording doesn't make clear what I mean). And I still think that putting the info on solving some quadratic equations in the intro would be a good answer to "why should I care?". Besides, it links well to the remark in the intro about complex numbers being an algebraically closed field (a rather cryptic remark for a non-math IMHO).


 * However, your info is extremelly interesting and I never knew any details about the original historic motivation. This brings about another point: the article should have a "who and what" history section. If you have some (preferably online) reference I'd like to contribute to that. AdamSmithee 07:59, 10 January 2006 (UTC) Oops, it seems that I just skipped the History section earlier, I don't know why, just didn't see it... AdamSmithee 13:55, 10 January 2006 (UTC)


 * The article does have a section on History, doesn't it? But I agree with Adam about including some motivation in the intro. However, I'm afraid I didn't like his original formulation very much. There are two things that could be mentioned. Firstly, every polynomial equation (of nonzero degree) has a solution if one allows complex numbers (by the way, I don't think we should use algebraically closed field without explanation in the intro). That's pretty neat, but rather theoretical: what use does a solution have if it doesn't have any sense? Secondly, what Michael says, complex numbers are useful as "bookkeeping device". As always, we should try to keep the intro short, preferably shorter than now. -- Jitse Niesen (talk) 12:22, 10 January 2006 (UTC)

I believe the intro was written by Bo Jacoby, when some anon complained that the previous intro (which I found fine) was incomprihensible. I like the new intro too, but indeed a bit shorter and a blurb about motivation would be appropriate. But I did not like AdamSithee's way of saying it. :) Oleg Alexandrov (talk) 20:17, 10 January 2006 (UTC)


 * Regarding the need for a shorter intro: I'm not sure that the exact rules for addition, subtraction and multiplication need to be in the intro. Maybe it's enough to just say that a form of addition...division exists and detail it in the main article. Regarding motivation: I didn't really like my formulation either :-), but at the moment I couldn't come up with anything better AdamSmithee 07:53, 11 January 2006 (UTC)
 * I agree, and have shortened by division. I know "polynomial algebraic" is redundant; but "polynomial" by itself is likely to be uninformative to anyone who doesn't know what a complex number is. Septentrionalis 22:52, 11 January 2006 (UTC)

Complex Field vs. R x R
Can someone explain the difference between the complex field and Euclidean 2-space? If a complex number is an ordered pair of real numbers, then are the elements of R x R such as (2, -4) complex numbers? This is something I have never been clear about.


 * Euclidean 2-space, R2 = R &times; R, is a vector space and has no (vector) multiplication defined on it. If you define a multiplication as given in this article you get the complex numbers C. In other words R2 and C both have the say underlying set of elements; C just has more algebraic operations defined on it. Therefore pairs such as (2, -4) can be thought of as elements of either. You can define other sorts of multiplication on R2 and get structures like the split-complex numbers. -- Fropuff 20:34, 12 February 2006 (UTC)


 * OK, thank you. I was thinking that it had something to do with the operations defined but wasn't sure.


 * omg wtf r^2 got dot product dude. product means multipication


 * The dot product is not an operation on R &times; R. The result is always from R. --MathMan64 01:34, 13 December 2006 (UTC)
 * To clarify, the range of the dot product is not R &times; R; but the range of complex multiplication is the complex plane. Septentrionalis PMAnderson 05:29, 13 December 2006 (UTC)

My reversion
My revert was caused by rather clumsy recent writing, and an unnecessary example of complex number multiplication. This is an introduction after all, not the whole article, so need to be kept short and consise. The issue of multiplication is dealt with very cleary right below the table of contents. Oleg Alexandrov (talk) 04:53, 21 February 2006 (UTC)

Clumsy
I rather prefer a clumsy but didactical and correct text, over a conc(!)ise, incorrect and less comprehensive one. Almost always in introductory texts on complex numbers somewhere the frase "square root" of -1 turn up. Is it meant to shock the reader and implicitly implying the author isn't? In introducing the complex numbers the square root is not yet defined.Nijdam 23:54, 22 February 2006 (UTC)
 * The issue of what is i is dealt with at length below. Oleg Alexandrov (talk) 03:40, 23 February 2006 (UTC)
 * Quite a lenghty discussion! And so it should be! Then what is i??Nijdam 15:57, 17 March 2006 (UTC)
 * Well, i is the number (0, 1) in the plane, which squared, by the rules of complex numbers, give (-1, 0), that is, -1. You are right that something more must be said about i, as it was mystifying people for a long time, but I don't find the intro appropriate for that. The text you want, about what is i, already exists at Imaginary unit, and that one is linked form the intro. Oleg Alexandrov (talk) 01:50, 18 March 2006 (UTC)
 * Well It wasn't a question from my side. I know what i is and what it is not. I would never say i is the point (0,1) ...It may be defined that way, with the complications of the embedding of the reals. It may be defined otherwise, like the historical way, or as a matrix.Nijdam 10:58, 30 April 2006 (UTC)

Complex line
There is a redirect from Complex line the this entry. But I cannot find it picked up somewhere. Is there space left to integrate this into this article? Hottiger 18:43, 21 March 2006 (UTC)


 * Well, complex line is described here. I don't think that should redirect to this article, for that reason I will now delete that redirect. Oleg Alexandrov (talk) 04:13, 22 March 2006 (UTC)
 * OK, one may think of the complex line as C viewed as vector space or manifold over itself. But I don't think that terminology is used that much. Oleg Alexandrov (talk) 04:27, 22 March 2006 (UTC)

Identities
I know these are easy to prove but as a reference it is easier to just look up these values then to have to prove it every time. These also provide and way to work backwards in proofs since it gives a hint to the answer.

$$\sqrt i={\frac{1+i}{\sqrt2}}$$,$$\mathbf{i^2}=-1$$,$$\mathbf{i^3}=-i$$,$$\mathbf{i^4}= 1$$Adhanali 03:04, 4 April 2006 (UTC)
 * Yeah, but this is a big article, and including a lot of various miscellaneous formulas make it overall harder to read. That is, at some point one needs to decide what to include and what to skip when writing something, and, at least in my view, these identities are not worth having in. Oleg Alexandrov (talk) 03:06, 4 April 2006 (UTC)
 * Is there any place where we can put $$\sqrt i={\frac{1+i}{\sqrt2}}$$. Even if it is not on this page. I am not sure about others but it is not a common identity I run into often and when I need to use it I can never remember it. I just thought it might be useful. But I understand the length versus content issue. Cheers Adhanali 03:28, 4 April 2006 (UTC)
 * Special case of Euler's formula for ei&pi;/4? Septentrionalis 05:46, 4 April 2006 (UTC)


 * puting root i in... lol u mean helping ppl do their homework?

The formula may belong in the article root of unity. Bo Jacoby 07:52, 3 May 2006 (UTC)

polar form
Why does the page polar form redirect here? The word polar isn't even in the article. Fresheneesz 05:06, 26 April 2006 (UTC)


 * Polar form is another way of writing complex numbers, when written in the form a + bi, the are said to be in rectangular form. When in the form of r cis $$\vartheta$$ it is said to be in polar form.--Phoenix715 08:59, 16 June 2006 (UTC)


 * Polar forms also exist e.g. for quaternions (e.g. K. Carmody, Circular and hyperbolic quaternions, octonions, and sedenions, Appl. Math. Comput. 28 (1988) 47–72. doi:10.1016/0096-3003(88)90133-6) and other numbers. Let me look around and make sure that "polar form" is defined as an expression using angles and one (absolute) length (distance to origin / zero). A quick internet search indeed returned only complex numbers expressed in polar form, so a new article here with more examples should be interesting, and I'd be glad to write it up. Through polar coordinates, it would give a geometrical viewpoint on multiplication on some hypercomplex numbers, next to the rectangular / Cartesian product. Any comments are always welcome. Thanks, Jens Koeplinger 13:46, 31 July 2006 (UTC)


 * I am puzzled by the fact that we seem to be using polar and geometrical synonymously, as if rectangular/Cartesian representation is not geometrical. Polar representation makes multiplication easier but addition much harder.  Are we implying that multiplication is more geometrical than addition?  --Bob K 19:56, 31 July 2006 (UTC)


 * Thanks for pointing out this wording inconsistency. Rectangular / Cartesian coordinates are just as "geometrical" as polar forms / coordinates. I should really have said an "additional geometrical viewpoint" or so. From my end, this mishap came from an abstract algebra mindset, creating algebras by demanding certain abstract properties (distributivity, associativity, commutativity, etc). But of course, this cannot be separated from geometry. I'll check my edits and make sure to eliminate this incorrect wording if I find it. Thanks, Jens Koeplinger 22:04, 31 July 2006 (UTC)

Passerby says, The sections, "Polar form" and "Conversion from..." are duplicated in the Polar Coordinates article, which is their natural place. They don't belong here. I was struck by this as a reader. 71.65.246.124 23:16, 14 March 2007 (UTC)

Organization or... not?
This page seems a bit disorganized to me. For example, all those things under the header "definitions" don't seem to be definitions. They're not definitions of "complex number" in any case. I think this page should have a header called "properties" that includes indenties, and other properties of complex numbers - instead of having the properties strewn all over the place.

Perhaps a "representation of complex numbers" section would be good to place the matrix representation and vector representation, along with the complex plane and the complex number field. As it stands, the TOC is long and this page is hard to sift through to find what you need. Fresheneesz 04:07, 19 May 2006 (UTC)


 * Agreed, organization was the main reason I rated the article as B-class. I'll take a look and see if we can get the ToC trimmed a bit. --JaimeLesMaths 03:37, 6 October 2006 (UTC)

First sentence
Currently, the first paragraph reads:


 * In mathematics, a complex number is a number in that field of numbers which includes the real numbers and the imaginary numbers, of the form
 * $$ a + bi \,$$
 * where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = &minus;1. The real number a is called the real part of the complex number, and the real number b is the imaginary part. When the imaginary part b is 0, the complex number is just the real number a.

The previous version read:


 * In mathematics, a complex number is an expression of the form
 * $$a + bi\,$$
 * where a and b are real numbers and i is the imaginary number defined so that i 2 = &minus;1. When the imaginary part $$b=0$$, the complex number is just the real number a.

The version before that read:


 * In mathematics, a complex number is an expression of the form
 * $$ a + bi \,$$
 * where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = &minus;1. The real number a is called the real part of the complex number, and the real number b is the imaginary part. When the imaginary part b is 0, the complex number is just the real number a.

I like the version mentioned last better. The version mentioned first has the disadvantage that it uses "field" (difficult) and "imaginary number" in the very beginning, and I don't like the construction of the first sentence. The second version is a bit imprecise in that there are two numbers satisfying x^2 = -1.

I disagree with the remark "a number is not an expression". -- Jitse Niesen (talk) 12:40, 13 June 2006 (UTC)


 * First, a number is absolutely not an expression. A number can be expressed by an expression, which is the very meaning of the term expression, or it can be represented by an expression, but the number itself is not an expression. In the same way, the numeral "4" is not a number, but a representation of that number. That number can also be expressed by "IV", "four", "1+2+1", or by four stones in a square, or infinitely many other representations, but there is only one abstract number. It is the difference between the form of language and its meaning.
 * Second, it is essential to the meaning to define "complex number" in terms of the imaginary unit and imaginary numbers. Moving the word "imaginary number" below the MATH equation does nothing to actually simplify the description. A person reading the first part is simply not told what a complex number is; the expression a + bi is meaningless without defining what a, b, and especially i are. There is no reason to shunt "imaginary number" down, it would only obscure the definition unnecessarily, whereas a reader can get an accurate impression that the set of complex numbers can be thought of as two sets together, the real numbers and the imaginary numbers.
 * Third, it may be desirable to exclude "field" from the initial description. However, it must be stated that it is some set, some collection of numbers, and "field", while also being the accurate mathematical term, is not obscure: a reader can very well adduce from common English that it is some area, some region with things in it. —Centrx→talk 14:00, 13 June 2006 (UTC)
 * I would agree that the first sentence needs improving, and we need to strike a fine balance between mathematical correctness and the danger of confusing less experienced readers with excessive mathematical precision at the very beginning of the article. I would prefer not to mention field or imaginary number quite so early in the page. It might also be argued that it is not essential to define complex number in terms of imaginary numbers at all: the rigorous process is to define complex numbers as ordered pairs (not that I am suggesting we should do that here!), and then define i as an abbreviating for the ordered pair (0,1). Madmath789 14:47, 13 June 2006 (UTC)
 * Yes, but if you don't use one of these, there is no definition at all. It is impossible to define "complex number" without defining i, and it serves no purpose to simply move it down a bit. The reader is not going to have any better idea of what "a + bi" means without it. —Centrx→Talk 23:42, 13 June 2006 (UTC)

I reverted the "In mathematics, a complex number is a number in that field of numbers". That adds no mathematical rigour, and is clumsy and confusing. I would say the intro better be kept informal and simple, we arrived at this intro after long arguments that it was too complicated. The rigurous definition of complex numbers is not simple, and can't be summarized in a sentence. Also, most people couldn't care less about it, as long as the properties work right. I think the current intro is better. Oleg Alexandrov (talk) 16:02, 13 June 2006 (UTC)


 * Replying to Centrx: First, I see what you mean with "a number is not an expression". I think "expression" in mathematics is a more abstract concept. Both "1 + 1" (usual notation) and "+ 1 1" (Polish notation) represent the same expression, which differs from the number 2. I think complex numbers can be defined as being expressions of the form a + bi, though this definition is hard to formalize. Whether complex numbers exist separately from their definition depends on your philosophy, I guess. But it may indeed be better to write "A complex number is a number of the form a + bi" instead of "an expression of the form a + bi."
 * Second, moving "imaginary number" under the formula does have a purpose: it improves the flow of the text.
 * Third, why "must be stated that it is some set"? Of course, the complex numbers form a set, but that adds (almost) no information. -- Jitse Niesen (talk) 03:22, 14 June 2006 (UTC)
 * Yes, I agree that those represent the same expression, still they both represent "one thing" "put together" with "one thing", and any other operator or number used would make it different. So, I think you are correct that it is not the same distinction as simply a "representation", and that multiple representations can represent the same expression. Still, multiple expressions represent the same number, they are still representations, but at another layer. I don't think it is so much a matter of philosophy, the fact is that if both "1+1" and "3-1" express the number 2, they cannot all be the number two; I am not asserting that there must exist ideal objects dancing around in some other realm. As noted above, there is also the ordered-pair definition, and they
 * By "set" I do not mean it so much in its mathematical definition, but simply the common meaning of a "collection" or "group", that there are a bunch of numbers, and all the real numbers and all the imaginary numbers are in there. "Set" serves this purpose, it is a normal English word, which is also mathematically accurate. —Centrx→Talk 03:00, 15 June 2006 (UTC)

EGAD geometric interpretation of complex number multiplication MISSING
The _MOST_ important fact of complex numbers that is the basis of the entire goddamn complex number and complex functions and complex analysis and functional analysis is not mentioned in this article!!

that is, the essence of complex numbers is the vectors with rotation as algebraic operation. i.e. the geometric interpretation of complex number multiplication.

Jesus.

Xah Lee 14:06, 28 July 2006 (UTC)


 * I would argue strongly about it not being the most important feature of complex numbers, but it IS important - feel free to add something about it to the article! The whole point of Wikipedia is that when we se something missing, we add it ourselves, rather than complain about it being missing. Madmath789 14:10, 28 July 2006 (UTC)


 * Ok. The top two most important aspect of complex numbers are: (in no particular order) • rotation as a operation of multiplication. • Sqrt[i] == -1. Xah Lee 23:38, 28 July 2006 (UTC)


 * EGAD... $$\sqrt{i} = \frac{1 + i}{\sqrt{2}} \ne -1\,$$   --Bob K 01:01, 29 July 2006 (UTC)


 * oops, of course i meant Sqrt[-i]==i. Xah Lee 07:28, 30 July 2006 (UTC)


 * Nope - don't think you meant that either! $$\sqrt{-i} = \pm\frac{1 - i}{\sqrt{2}} \ne i\,$$. Madmath789 07:50, 30 July 2006 (UTC)


 * Sqrt[-1]==i. (^_^) Xah Lee 08:04, 30 July 2006 (UTC)


 * Congratulations.  $$i\cdot i = -1 \,$$ is of course just a special case of the more general multiplicative property, which brings us right back to your original point.  --Bob K 13:19, 30 July 2006 (UTC)


 * Hello Xah Lee - you said : "rotation as a operation of multiplication". I agree that this may be pointed out more, or maybe just adjusted in the existing text; and I really encourage you to do some work to the page, e.g. to the existing sections "Notation and operations" and/or "The complex plane", that multiplication in complex numbers can be interpreted in two ways: In the "rectangular" (is this a real term?) execution (a + ib)(c + id) and in the geometric execution by means of arguments and angles. The section "The complex plane" mentions just this, but I see now where you're going and agree that this is notable on a higher level (i.e. at the beginning, without going into details yet) and should not be separate from the "Notation and operations" section: When multiplying two numbers, you multiply their absolutes ("lengths") and add their angles ("directions"). I hope you can find the time to put this into nice wording and update the page accordingly. Again, I think all the material you're writing about is there, but in different sections, and possibly with not enough wording, or wording that goes to quickly into some detail while bringing other high-level notables at a later point. - Many people have their eyes on this page, so I wouldn't worry about typos, grammar, etc ... someone will come by and straighten it out for sure. There are a few other curiosities which I might want to add at a later point, e.g. the infinite amount of solutions of a logarithm in the complex number plane, or the infinite amount of solutions to $$~i^i$$ which curiously are all real numbers. Thanks, Jens Koeplinger 01:39, 29 July 2006 (UTC)


 * looks like Oleg Alexandrov is taking over. He probably knows more deep theories about complex numbers than I. Xah Lee 08:04, 30 July 2006 (UTC)


 * Xah Lee, thank you for your contribution, I hope that our discussion doesn't get sour. I do believe that the article needs updating, in the direction you are proposing, and I'm glad to run this by this forum. At the moment, we have three sections "Notation and operations", "The complex plane", and "Geometric interpretation of the operations on complex numbers". The last section doesn't mention anything about multiplication. In "Notation and operations" we have a forward-reference "Division of complex numbers can also be defined (see below).", the section "Complex fractions" comes two sections after the multiplicative inverse is demonstrated, and the geometric interpretaion (in polar coordinates) of multiplication as multiplying arguments and adding angles comes after division is defined, in "The complex plane" section. I understand that in the current form, the article first defines multiplication and then shows later the geometric interpretation of multiplication in polar coordinates. For an encyclopedic entry, however, I don't see the need of keeping an order of deductions, instead we should be free to put key properties first. Right now, the order of complex number facts appears confusing to me. Jens Koeplinger 15:10, 30 July 2006 (UTC)


 * "rectangular" is a system of vector coordinates, applicable to 2 or more dimensions. Complex numbers are the special case of 2 dimensions.  I believe the most common terms are Cartesian and polar coordinates.  --Bob K 11:48, 29 July 2006 (UTC)


 * Thanks for clarifying the terminology, I appreciate this. Jens Koeplinger 15:10, 30 July 2006 (UTC)

I have changed to figure text to simplify and clarify. There are 3 figures: 1: addition, 2: multiplication, 3:conjugation. Nothing is missing. Bo Jacoby 12:00, 31 July 2006 (UTC)


 * Thanks for your update, I did overlook the pictures before. As far as the rest of the article, maybe my (and Xah Lee's) opinion is just personal preference. I noticed through the discussion above that polar form redirects here; let me pick-up from there. Maybe a modification there will accomodate my concern. Thanks, Jens Koeplinger 13:41, 31 July 2006 (UTC)


 * Dear Koeplinger and others:


 * I'm forfeiting further edits on this page. However, I think there are few items that needs to be addressed. (in no particular order below)


 * • there needs to be a explanation of the geometric interpretation of addition, negation, multiplication, multiplicative inverse (complex inversion), and conjugation. (Please look at my last edit on how I think this should be done.) And, the article should somehow discuss the much neglected fact that it is the definition of complex multiplication that sprang the entire field of complex analysis and attempted generation into higher dimensions. (i.e. otherwise it is just vectors.)


 * I think your point is that there is no vector equivalent of (A+iB)·(C+iD) = (AC-BD)+i(AD+BC). That does seem to be a neglected fact.  And no doubt that is not just an arbitrary definition.  It would be nice to explain how it was chosen, and who actually "discovered" its utility.  What problem was he trying to solve?  etc.  --Bob K 02:19, 1 August 2006 (UTC)


 * As far as the historical reference is concerned, I'm tempted to try to pull in others. Maybe we should ask (beg, bribe, ...) "hyperjeff" (from http://history.hyperjeff.net/hypercomplex ) whether he can provide us with some historical help? I've already sent him an e-mail last week about the term "hypercomplex number", so I'll hold-off with bothering him until I hear back. From his account, the 2D plane representation goes back to Argant and later Gauss. His opinion may help us sort out the line of discovery, and subsequently the order in which one may want to present complex numbers here. Thanks, Jens Koeplinger 02:59, 1 August 2006 (UTC)


 * • the current section on geometric interpretation is rather quite inane. It seems to have a fixation on similar triangles, which results in (1) mathematical ambiguity, (2) vague in explanatory power.


 * • there is a technical error on explaining 1/z. It says it is just reflections. It should at least say it is reflections of a line thru the origin. (with dilation) Or, better, it is a rotation and dilation, followed by a reflection around the real-axis. (See my past edit.) The explanation of 1/z should be part of the geometric interpretation on complex inversion.


 * the complex inversion 1/z should mention that it is circle inversion with a reflection around the real-axis. If I recall correctly, the article makes no mention of circle inversion.


 * Xah Lee 20:45, 31 July 2006 (UTC)


 * Xah Lee, thank you for detailing and itemizing your concerns; I appreciate your legwork which is tedious, time-consuming, and often not gratifying - but it is needed. I do like the current "geometric interpretation" section that uses triangles, because the polar form of complex numbers are really more a coordinate representation issue than a geometric issue (thank you, Bob K, for pointing this out). So here I disagree with you. Of course, coordinates cannot be separated from geometry, so I also understand your concern. As far as 1/z, do you think that if we get your concern about multiplication and polar form sorted out, this would also address the inverse operation here? Thanks, Jens Koeplinger 02:59, 1 August 2006 (UTC)

Similar triangles is elementary geometry. Vector and rotation is not elementary geometry. The definition of 1/a, which follows from that of multiplication, is:
 * if Δ(0,1,a)~Δ(0,x,1) then x=1/a.

The proportional sides of similar triangles are generalized to complex numbers:
 * If Δ(0,a,b)~Δ(0,c,d) then a/b=c/d.

So the definitions of complex operations are elementary and has a lot of explanatory power. There is no ambiguity in the definition, but if the 3 points in question are on a straight line then no triangle is defined. This insufficiency can be fixed by demanding continuity:
 * lim(ai)+b = lim(ai+b)
 * lim(ai)b = lim(aib)
 * lim(ai)* = lim(ai*)

Historically, the geometric interpretation of complex multiplication is due to Caspar Wessel.

By the way, what is the meaning of 'EGAD' ?

Bo Jacoby 07:19, 1 August 2006 (UTC)


 * definition of egad   --Bob K 11:37, 1 August 2006 (UTC)


 * A quick comment on Bo Jacoby's reply above.


 * The geometric explanation of complex numbers using similar triangles you gave above, is week in explanatory power in comparison to what i was saying, which is also utilized in almost every text on complex analysis. In particular, that the multiplication of two complex numbers is the rotation of one by the other (with scaling). One easy way to see this is that with the similarity triangle explanation, one cannot fathom in term of geometry of any given polynomial or rational function of complex numbers. For example, say a*x^2+b*x+c, one can see that it is a number rotated, scaled, rotated and scaled again by a, and moved by the same number rotated and scaled by x, and then translated by c. With similarity of triangles interpretation, it can't explain expressions with such procedural visualization. Xah Lee 02:19, 28 August 2006 (UTC)


 * also, how's the similarity triangle thing explain addictive inverse and multiplicative inverse? i.e. -x and 1/x. With the geometric interpertation i gave, it is simply the reverse of the geometric operations. The geometric interpretation i gave is simply from a transformational geometry point of view.


 * Besides this, yeah, sure, one may say that “vectors” or “rotation” isn't elementary concepts (give me a break!!), but nor is similarity of triangles unless you are talking about the Era before non-Euclidean geometry. Xah Lee 02:04, 28 August 2006 (UTC)


 * Btw, where did you get these from? Is it from some text on Euclidean geometry with complex numbers? It is interesting, but i'm not sure what is the utility. Xah Lee 02:07, 28 August 2006 (UTC)

Hi Xah Lee. The polynomial ax2+bx+c, or (ax+b)x+c using the Horner scheme, has the geometrical interpretation that triangle(0,1,x) is similar to triangle(0,a,ax), short: (0,1,x)~(0,a,ax); that (0,ax,b)~(ax+b,b,ax); that (0,1,ax+b)~(0,x,(ax+b)x); and that (0,(ax+b)x,c)~((ax+b)x+c,c,(ax+b)x). Here 0, 1, x, a, b, c, ax, ax+b, (ax+b)x and (ax+b)x+c are points in the plane, and so the explanation is in terms of elementary geometry, (Compass and straightedge), without requiring knowledge of rotation or scaling. A child can do it. The utility is that a geometrical problem is translated into the language of algebra, where it is easier to study. For example, the construction of the regular 17-gon is reduced to studying the equation x17 = 1. Bo Jacoby 23:33, 7 December 2006 (UTC)

History
Isn't history directly copied from source 1? Is this illegal or unethical?

Formula for complex argument

 * I moved the following comment, which was prompted by this revert by me, here from my talk page. -- Jitse Niesen (talk) 02:23, 7 December 2006 (UTC)

Unfortunatelly the one of the most important formulas of the complex argument $$\varphi = \tan^{-1}(b/a) $$ doesn't appear in the atricle. If you think I added it in the wrong place, just copy it to more convinient place, but why do you just delete it ? —The preceding unsigned comment was added by Dima373 (talk • contribs) 20:59, 5 December 2006 (UTC).


 * That's a good point. I didn't realize the formula wasn't in the article at all. I put it back in, but I moved it a bit higher up, where the argument is actually introduced. -- Jitse Niesen (talk) 02:23, 7 December 2006 (UTC)


 * Thank you. Dima373 20:24, 9 December 2006 (UTC)


 * While we're at it, I should also thank JRSpriggs for noting that the formula works only in half the cases and fixing it (diff). That's especially embarrassing for me since I actually taught this a couple of months ago! However, I question the usefulness of the new formula
 * $$\varphi = 2 \arctan \frac{y}{r+x} \mbox{ when } r+x \neq 0 \mbox{, otherwise } \varphi = \pi.$$
 * A correct formula is obviously better than an incorrect formula, but my first reaction when I see the formula is more of "that's a nice trick" and less of "that's what I'm going to use next time I need to compute the argument". -- Jitse Niesen (talk) 03:05, 10 December 2006 (UTC)
 * The other alternative is to give several different cases. If x is positive, use arctan (y/x). If x is zero and y is positive, use &pi;/2. If x is zero and y is negative, use -&pi;/2. If x is negative and y is nonnegative, use arctan (y/x) + &pi;. If x is negative and y is negative, use arctan (y/x) - &pi;. Of course, if both x and y are zero, then it is undefined (which I did not bother to mention in my version). I doubt that these cases are easier to remember than the half-angle formula. If neither is useful, perhaps that explains why there was no formula at all before. JRSpriggs 11:29, 10 December 2006 (UTC)
 * You're talking about this formula:
 * $$\varphi =

\begin{cases} \arctan(\frac{y}{x}) & \mbox{if } x > 0\\ \arctan(\frac{y}{x}) + \pi& \mbox{if } x < 0 \mbox{ and } y \ge 0\\ \arctan(\frac{y}{x}) - \pi& \mbox{if } x < 0 \mbox{ and } y < 0\\ +\frac{\pi}{2} & \mbox{if }x=0 \mbox{ and } y > 0\\ -\frac{\pi}{2} & \mbox{if }x=0 \mbox{ and } y < 0\\ 0 & \mbox{if } x = 0 \mbox{ and } y = 0 \end{cases}$$
 * I think that this is the standard formula and should be mentioned in the article. However, often not all cases are considered, sometimes even only the first case. Admittedly, in the complete form it's not easy to remember, but in principle it's easy to understand. Furthermore, many programming languages have a variant of the arctan-function which is called "atan2" and has the case differentiations build-in. The formula with "2 arctan" is really interesting in my opinion, because it avoids the many cases in the formula above, but it's not easy to understand. Perhaps we could add it and give a reference for a proof. Then there is another interesting formula that has only few cases and is easy to understand and remember in my opinion:
 * $$\varphi =

\begin{cases} +\arccos\frac{x}{r} & \mbox{if } y \geq 0 \mbox{ and } r \ne 0\\ -\arccos\frac{x}{r} & \mbox{if } y < 0\\ 0 & \mbox{if } r = 0 \end{cases}$$
 * How about that? --IP 23:28, 27 December 2006 (UTC)
 * To : That is a pretty formula. Mention it in the article, if you like. JRSpriggs 08:30, 28 December 2006 (UTC)
 * I have added both variants. Amendment: For the case y = 0 in the arccos formula it is required that r ≠ 0. --IP 12:18, 29 December 2006 (UTC)

When Im(z)=0
I'm not sure how to edit this, but perhaps there needs to be revision of the sentence "when b=0, then this is just the real number a" (not verbatim)

The real number a is actually a complex number a+0i. For non mathematicians, this is not just being petty this is involved with a fundamental definition in algebra that the complex numbers provide the solutions to all equations etc. I can't think how better to word it though, so perhaps someone with better linguistic skills could give it a go? Triangl 12:38, 19 December 2006 (UTC)


 * I changed the sentence to "We usually identify the real number a with the complex number a+0i.". Does that satisfy you? JRSpriggs 08:22, 20 December 2006 (UTC)

There is no need to change it. R is a substructre of the complex number field. Indeed, there is no difference in saying that we may identify R with the line Im(z) = 0 in C and saying that it is the line. Why? Topologically, algebraicly, etc, they are isomorphic. Phoenix1177 (talk) 05:34, 2 January 2008 (UTC)

Polar form
I was wondering if perhaps the "Polar form" section could be made shorter, and with most of the material moved to its own article, called Polar form. Then, at the top of the current section "Polar form" in this article we could point to the article Polar form for more details. Would that be a good idea? Oleg Alexandrov (talk) 17:00, 29 December 2006 (UTC)
 * Yes it is a good idea. But I also think that the link to the 17-gon is a good idea. Gauss' use of complex numbers for analyzing the 17-gon was a historic breakthrough, and your edit comment: "This section is about geometric interpretation of algebraic stuff, and not the other way around", is a misunderstanding. Bo Jacoby 12:46, 30 December 2006 (UTC).


 * I don't think that this is a good idea, because the polar form seems to be a basic of the complex number article. However, perhaps some details on calculating the argument when converting from Cartesian to polar form could be moved to Polar coordinate system, but that article is already a bit too long in my opinion. Perhaps the section on complex numbers in that article could be merged-in here. --IP 18:56, 30 December 2006 (UTC)


 * The article on polar coordinates contains the information that now pollutes the article on complex number. Bo Jacoby 16:40, 1 January 2007 (UTC).
 * Sorry, but that's currently not correct. The article on polar coordinates does not contain a formula to calcuate the angular coordinate in the interval (-π, π] as it is usual for complex numbers. I had added this, but it was deleted again. --IP 02:34, 2 January 2007 (UTC)
 * Polar_coordinate_system contains formulae for the requested calculation. I think we should link to that article and not include the formulae here.Bo Jacoby 23:20, 11 January 2007 (UTC).

Department of redundancy department
I've been re-reading this article, looking for ways it might be improved, and one thing keeps grabbing my attention. The two sections "Notation and operations" and "The complex number field" are very repetitive. I think we should either separate some of these ideas and only state them once, in one section or the other, or else we should merge these two sections into a single section, perhaps with a new title. What do you think? DavidCBryant 23:22, 24 January 2007 (UTC)

Here's another thing I noticed -- there's no mention of the fact that C has the same cardinality as R. Cantor produced a very cute argument showing how to place the complex numbers in 1-1 correspondence with the real numbers. Does that idea deserve a mention here? DavidCBryant 17:59, 25 January 2007 (UTC)


 * That is more about set theory than complex numbers per se. It should be mentioned, but in a different article. JRSpriggs 07:41, 26 January 2007 (UTC)

Continuous operations?
The section "Absolute value, conjugation, and distance" contains the sentence, "The addition, subtraction, multiplication and division of complex numbers are then continuous operations." First, I've never heard of a "continuous operation" (though I think I understand what the sentence was trying to get at), and second, the sentence does not seem to add anything to the paragraph it's in. I'm removing it for now, but, if you think it should be retained, a rephrasing is probably in order first. I'm happy to talk through it with anyone who wants to take a crack at it. --JaimeLesMaths (talk!edits) 10:41, 26 January 2007 (UTC)


 * These "operations" are just functions of two variables. Of course, continuity makes sense for them. JRSpriggs 11:38, 26 January 2007 (UTC)


 * OK, well, yes, the operations are continuous functions from C x C to C, not from C to C (which is what I thought was meant and was why I was confused). Thus, I don't think the sentence is relevant to the paragraph and will only add confusion. --JaimeLesMaths (talk!edits) 04:36, 27 January 2007 (UTC)

Direction
I inserted the following sentence in the subsection on absolute value: "The other factor eiφ = z / |z| is the direction of z. The length of a direction is one, and the direction of a length is one". However DavidCBryant removed it immediately. Please notice that you are not supposed to revert edits made in good faith. Make forward steps rather that backwards steps. If you feel that the information should be placed somewhere else, then place it somewhere else rather than remove it. The concept of direction is historically and conceptually important, and the very word 'direction' is in the title of Caspar Wessel's paper from 1799. The unique factoring of a nonzero complex number into lenght and direction is as useful as the decomposition into real and imaginary parts. Please behave wikipedialike rather than commit vandalism. Bo Jacoby 11:42, 5 February 2007 (UTC).


 * Bo, there's no need to argue in two different places. Here's a part of what has already been written on my talk page.


 * You actually undid my edit, exactly as you wrote in the edit comment. Now that we agree that this is bad behaviour, you are requested to reinstall my edit and return to the discussion if you disagree, or to improve if you have a contribution to make. Bo Jacoby 19:59, 6 February 2007 (UTC).


 * You think it's bad behavior. I don't. So there is no agreement on that score. I think I did a good thing by improving the article, and I also think you did a bad thing by scribbling nonsense.
 * Here's the edit comment I made. "(→Absolute value, conjugation and distance - Removed extraneous material that does not belong in this section of the article. Also removed redundant information.)" For the record, I left part of your previous edit intact. The History page does not lie.
 * On to specifics. I removed the phrase "the nonnegative real number" because it is redundant. The immediately preceding discussion of polar coordinates makes it abundantly clear that r &ge; 0. Saying the same thing over and over again annoys the reader.
 * I also removed a phrase "or 'length'" because it is extraneous (and, in fact, misleading). Complex numbers do not have lengths. Vectors have lengths. Line segments have lengths. While a line segment or a vector can be represented by a complex number, and vice versa, the three things are not identical. This article is about complex numbers. Information about geometric representations of complex numbers ought mostly to go in the article complex plane.
 * I also removed two entire sentences – "The other factor eiφ = z / |z| is the direction of z. The length of a direction is one, and the direction of a length is one." This material is not only extraneous (it deals with neither absolute value, nor conjugation, nor distance); it is patent nonsense to boot. I have read at least a hundred books about complex analysis, and I have never before encountered a statement like "The length of a direction is one, and the direction of a length is one." That isn't even a mathematical concept. It sounds like liturgical dogma, and it has absolutely no place in this article.   DavidCBryant 20:51, 6 February 2007 (UTC)


 * I intend to do my best to improve this article. If and when you insert more patent nonsense into Wikipedia I will not hesitate to remove it. Protestations of innocence do not behoove you, Bo. Your reputation among the community of mathematicians on Wikipedia is well-deserved. DavidCBryant 21:12, 6 February 2007 (UTC)

Two different edits are here confused by DavidCBryant. See User talk:DavidCBryant. Bo Jacoby 22:08, 6 February 2007 (UTC).


 * 1) The equation z=aei&phi; does not imply that a and &phi; are real numbers, because aei&phi; is defined also for complex values of a and &phi;. So it is not redundant to be explicite on this point.
 * 2) I don't mind using the word magnitude instead of length, like in in the article on vectors.

The two idempotent mappings Realpart(a+ib)=a and Imaginarypart(a+ib)=ib lead to the 'liturgical dogma': The Realpart of an Imaginarypart is zero, and the Imaginarypart of a Realpart is zero, (because Realpart(Imaginarypart(z))=Imaginarypart(Realpart(z))=0), and to the additive splitting: z=Realpart(z)+Imaginarypart(z). (Conventionally the 'imaginary part' of z is b, which is neither imaginary nor a part, but ib is needed here.)

The two idempotent mappings Magnitude(z)=|z| and Direction(z)=z/|z|, lead to the 'liturgical dogma': The Magnitude of a Direction is one, and the Direction of a Magnitude is one, (because Magnitude(Direction(z))=Direction(Magnitude(z))=1), and to the multiplicative splitting: z=Magnitude(z)&middot;Direction(z).

One may link to vector, but note that complex number multiplication generalizes the concept of multiplying a vector by a scalar. (The direction z/|z| is a vector while the magnitude |z| is a scalar, in vector lingo).

When reading an article on complex numbers, one should not be supposed to understand the complex exponential function eiφ. Elementary stuff is not explained in terms of advanced stuff. The concepts of magnitude and direction are explainable to the beginner.

Bo Jacoby 11:26, 7 February 2007 (UTC).

Notation
I noticed that someone has added "root extraction" to the list of operations "addition, multiplication, and exponentiation in polar form". While I was doing some minor mopping up after that, I also noticed that somebody, I don't know who, has very carefully miscoded the exponential function as
 * $$\mathrm{e}^z\,$$

instead of using the standard (universally accepted, I think) notation
 * $$e^z.\,$$

So now I'm reverting that to standard notation, and I'm leaving a note here just in case someone objects. DavidCBryant 16:43, 26 February 2007 (UTC)

Electrical Engineering
In EE the notation preferred is A+jB as opposed to A+Bj. —The preceding unsigned comment was added by Streyeder (talk • contribs) 15:18, 2 March 2007 (UTC).

wow
ok look, im a high schooler doing some research on various topics concerning complex numbers and im having a really hard time trying to understand most of the information in this article. Anyone know any place where complex numbers may be easily comprehended? thanks.76.18.202.211 18:48, 22 April 2007 (UTC)


 * If you could give some examples of things in this article that aren't easy to understand, then maybe someone could improve the explanations. --Zundark 19:07, 22 April 2007 (UTC)

square of a negative number
isn't the square root of a negative number -|x²| —The preceding unsigned comment was added by 24.187.129.93 (talk) 21:27, 25 April 2007 (UTC).


 * No, it isn't. Every real number, when squared, gives a positive result, because (&minus;1)2 = +1. That is why every positive real number has two square roots (a positive root, and a negative root). To represent the square root of a negative real number we must use the imaginary unit i. DavidCBryant 21:49, 25 April 2007 (UTC)

thank you for removing my additions
Would you like me to cite the source -- that is, Birkhoff and MacLaine? --VKokielov 12:57, 22 May 2007 (UTC)

I have not added them back. Please suggest a better place for them. --VKokielov 13:01, 22 May 2007 (UTC)


 * Hi, VKokielov.
 * I don't suppose you and I are going to agree on this. I really don't see any reason to explain a definition like "a + bi = c +di if and only if a = c and b = d." If you feel such an explanation is necessary, then I'd suggest it be presented after the definition is given, as the "motivation" for the definition. That would flow better. But since the definition as given is equivalent to saying that two vectors (x, y) and (u, v) are equal if and only if they are equal componentwise, and since the article goes on to give a formal definition of complex numbers as ordered pairs of real numbers, I honestly don't think the explanation adds anything to the article.
 * Anyway, I don't want to argue. If you want to add the justification for the definition to the article again, please do so. I'll let somebody else decide what to do about it. DavidCBryant 13:34, 22 May 2007 (UTC)

More operators to Notation and operations
I've already taken the liberty (hopefully correctly) to adding the division operator to complete the basic /*+- set in the "Notation and operations" section. I was wondering whether it might be advisable to add simple exponent, log or root operators too? Also, how about saying something along the lines that arithmetic with complex/imaginary numbers is somewhat counter intuitive compared to the calculation of real numbers. - Dan - 04:02 02/07/2007 GMT

The section on definitions should mainly contain definitions.
The present table of contents of section 1 is:

1 Definitions
 * 1.1 Equality
 * 1.2 Notation and operations
 * 1.3 The field of complex numbers
 * 1.4 The complex plane
 * 1.4.1 Polar form
 * 1.4.2 Conversion from the polar form to the Cartesian form
 * 1.4.3 Conversion from the Cartesian form to the polar form
 * 1.4.4 Notation of the polar form
 * 1.4.5 Multiplication, division, exponentiation, and root extraction in the polar form
 * 1.5 Absolute value, conjugation and distance
 * 1.6 Complex fractions
 * 1.7 Matrix representation of complex numbers

I suggest that sections 1.4 and 1.7 be moved elsewhere, as they go beyond basic definitions.

Bo Jacoby 15:39, 20 July 2007 (UTC).

Oleg's revert
Hi Oleg. You removed my clarifications:
 * $$   \triangle (0,A,B) \sim \triangle (X,B,A) \implies X=A+B.\, $$
 * $$   \triangle (0,1,A) \sim \triangle (0,B,X) \implies X=AB.\, $$
 * $$   \triangle (0,1,A) \sim \triangle (0,1,X) \implies X=A^*.\, $$

I intend to clarify why the implication sign goes only one way and how to mend it, but you seem to disagree. What is the problem? Why not improve rather than remove? Bo Jacoby 10:26, 28 July 2007 (UTC).
 * Such notation is unfamiliar to most Wikipedia readers, and it adds very little value in my opinion (in 95% of the cases people will just gloss over it rather than understand the finer details you mean to tell by this formula). Having this said in words, and then in pictures, as things are now, should be enough, no need to say the same thing in symbols. Let us see what others say. Oleg Alexandrov (talk) 15:03, 28 July 2007 (UTC)
 * i concur that for the majority of readers of the article, it will be non-beneficial to them to have this included. That being said, i think the relevance of that is for a discussion over the general purpose of a Wikipedia page, and in this specific instance i believe we should serve the 5% by including the symbolic write-up. Quaeler 15:23, 28 July 2007 (UTC)
 * But what's the gain? The same thing is said twice before, once in words, once in the picture. See Complex_number. This symbolism adds nothing new, it just distracts the reader's attention. Complex notation should be avoided, unless really necessary. Oleg Alexandrov (talk) 15:29, 28 July 2007 (UTC)
 * Some people operate much better on reading symbolic representation than on reading english (or whichever language) sentences describing it. i'm not hell-bent on this, i just believe that some people can better grok a concept when they have it presented to them via symbology. Quaeler 07:06, 29 July 2007 (UTC)

Clarifying the direction of the implication sign. $$   X=A+B \, $$ does not imply $$    \triangle (0,A,B) \sim \triangle (X,B,A) \, $$ because if $$   0,A,B \,$$ are collinear then the triangles are undefined and the geometrical construction is a little different. The symbolism is the one used in Similarity (geometry). Anyway, observe WP:DR. Bo Jacoby 16:14, 28 July 2007 (UTC).
 * The clarification can be done in words. I don't see your point about WP:DR. Oleg Alexandrov (talk) 02:05, 29 July 2007 (UTC)

As to WP:DR you removed rather that improve, and you didn't comment on the talk page. Bo Jacoby 07:11, 29 July 2007 (UTC).
 * There was no dispute. You used your Jimbo-given right to add content, I used my own editorial taste to delete it, and explained in the edit summary. Once we have a dispute, we can surely use the talk page, that's what we are doing apparently. Oleg Alexandrov (talk) 15:19, 29 July 2007 (UTC)


 * Saying its just a one way implication only gives half the story. Don't we mean
 * $$   \triangle (0,A,B) \sim \triangle (X,B,A) \Leftrightarrow X=A+B,\, $$ provided A, B not co-linear.
 * Indeed with a suitable definition of similarity for degenerate triangles we can get rid of the caveat altogether.
 * I don't think the symbol help much, as is not immediately obvious what the symbols mean and in this case they add more complication than clarification. --Salix alba (talk) 16:03, 29 July 2007 (UTC)

Hi Oleg. WP:DR says: "Be respectful to others and their points of view. This means primarily: Do not simply revert changes in a dispute. When someone makes an edit you consider biased or inaccurate, improve the edit, rather than reverting it". When you used your editorial taste to simply revert my added content, WP:DR is right that it feels disrespectful. I do not know the meaning of "Jimbo-given".

Hi Salix Alba. Can a suitable (non-circular) definition of similarity for degenerate triangles be given?

Bo Jacoby 20:09, 29 July 2007 (UTC).


 * There was no dispute. I did not show disrespect. There was no edit war. You added some content, I took it out. You started the discussion, I joined in. What's the problem? How about discussing the things on its merits? Oleg Alexandrov (talk) 20:22, 29 July 2007 (UTC)


 * On its merits, I think the symbolic expression detracts from the article, and makes it more difficult for the typical reader to understand. See this discussion for a reaction from a typical reader. Why introduce symbolic logic when it's unnecessary? DavidCBryant 20:51, 29 July 2007 (UTC)


 * On degenerate triangles. From Similarity (geometry) we have for two triangles ABC and DEF are similar if
 * AB|/|AC| = |DE|/|DF|, |AB|/|BC| = |DE|/|EF|, |BC|/|AC| = |EF|/|DF|
 * Corresponding angles are equal
 * take the above as a definition of similarity. Now allow for degenerate triangles where the points are allowed to be co-linear and use the above formula as a definition. Its quite simple to show that if X=A+B then triangles OAB and XBA obey the above, this even works if A=B, or A=O or B=0 or A=B=O. There is a curious case if A=B: take the triangle OAA this will be similar to a triangle YAA for any value of Y. Extending def of similarity to degenerate triangles is fairly standard see for example Alignments of random points where statistical analysis of triangles was used examine the probability of three points being co-linear. --Salix alba (talk) 07:45, 30 July 2007 (UTC)

Ok friends, I hear what you are saying. My feeling about the article is that it has too much emphasize on real coordinates, (either real cartesian coordinates, such as a and b in a+ib, or real polar coordinates,) and too little emphasize on the coordinate-independent explanation. In Mathematics, the complex numbers are considered a peak of sofistication. You walk a long way, from positive integers via negatives and rationals through limits to the real numbers, and then at last to the complex numbers. Isn't there a shortcut? Yes, there is! The geometric understanding of complex numbers is easy. Easier than real numbers. The construction of the product AB by similar triangles is elementary when 0,1,A are not colinear, and it is a straight-forward generalization of the idea that similar triangles have proportionate sides. On the other hand, the product of points on a line cannot be constructed without auxillary points outside the line. The hellenistic geometers, having the ingredients at hand, were only a tiny step away from inventing addition and multiplication of points, but they didn't do it. So the 5-gon was constructed by Euclid, while the 17-gon was not constructed until Gauss made those inventions. The purpose of an encyclopedic article is to offer an elementary introduction rather than to repeat the tiresome detours of history. Summarizing my point: Complex numbers are easier than real numbers. The article should reflect that insight, but it doesn't. Let's accept the challenge to make the article easier to the beginner. Bo Jacoby 08:43, 30 July 2007 (UTC). PS. Thank you, Salix alba. Our edits crossed. Bo Jacoby 08:43, 30 July 2007 (UTC).

Salix alba. I like your definition, but alas I am afraid it is circular. We define similar triangles: they have equal corresponding angles. We define proportionality: similar triangles have proportional sides. This definition matches the arithmetic definition of proportionality of rational numbers, but for irrational proportions the old definition does not work and the similar triangles serve as the definition of proportionality. When the points are colinear, then there is no similar trianges, and then there is no definition of proportionality. Of course there is not really a problem in defining (complex) multiplication and addition using similar triangles. You can for example use continuity to approximate colinear point by non-colinear points. You can use the associative rule: if 0,1,A are colinear, then chose X such that 0,1,X is not colinear and set AB = (A+x-x)B = (A+x)B-xB. We just need elegance. Bo Jacoby 09:17, 30 July 2007 (UTC).


 * A better def of similarity is that two objects P,Q, (in the real plane) are similar if there exists a 1 to 1 mapping, f, from P to Q such that f is a Dilation (mathematics), i.e. f has the property than
 * $$d(f(x),f(y))=rd(x,y)$$
 * where d is the distance metric, and r is some constant..The def I gave above is a direct consequence of this, indeed equivilent. The fact that angles remain equal is a consequence of this def not the other way round. --Salix alba (talk) 18:09, 30 July 2007 (UTC)

You multiply by r. You cannot use multiplication to define multiplication. Bo Jacoby 18:26, 30 July 2007 (UTC).
 * It seems like we are trying to do two different things here. I'm just trying to show a simple lemma for non-degenarate triangles extends in a natural manner for degenerate triangles, using standard definitions for similarity and complex multiplication. The multiplication above is multiplication for reals not complex numbers. You seem to be trying to do something much grander and non-standard, by actually defining multiplication via a geometric constructs. --Salix alba (talk) 19:05, 30 July 2007 (UTC)

Yes. We can define generalized similar triangles having corresponding sides proportional. If the 'triangles' in question are not degenerate, then this slightly generalized definition matches the original definition of similar triangles, having corresponding angles equal. Then we can use these generalized similar triangles to define complex number operations. But this approach assumes knowledge of real number arithmetic, (which is tedious, involving arithmetic of integers, fractions, and convergent sequences), so not much is gained pedagogically. I want to assume that the reader know not about real numbers, but only about constructions with compass and ruler. In that case (complex) arithmetic can be partly defined by nondegenerate similar triangles. 2&middot;i is not immediately defined, because the points (0,1,2) are colinear, but i&middot;2 is defined, because the points (0,1,i) define a triangle, and the similar triangle (0,2,X) can be constructed, and then i&middot;2 = X. The rule a&middot;b = b&middot;a can be proved geometrically when both sides are defined, and when only one side is defined, it can be used to supplement the definition. There are several cases to consider. I might not be as nice as could be wished, but it is not as difficult as real number arithmetic. Bo Jacoby 20:00, 30 July 2007 (UTC).
 * I don't think anything like this is appropriate in the article. Bo, we had this conversation many times before at many articles. Wikipedia is not the place for your constructions. Oleg Alexandrov (talk) 02:58, 31 July 2007 (UTC)

Dead references
References section contains dead link: http://people.bath.ac.uk/aab20/complexnumbers.html Dan Kruchinin 22:41, 23 October 2007 (UTC)

Recent changes to the intro
Looking at this edit, I think that the intro before Wolfkeeper is more elementary and easier to understand (if more informal). Comments? Oleg Alexandrov (talk) 00:39, 17 November 2007 (UTC)


 * I find the informality surrounding complex numbers is not particularly helpful, but the informality is still there, I've just moved it into the body of the article, it feels more of a better fit there. FWIW I think that the understanding of complex numbers as pairs is deeper, and avoids issues with 'i' being a non existent number that many people seem to handle poorly (e.g. Schroedinger has said he would never have written his quantum wave equation if he had known it would be using complex numbers!), but I'm not about to make any large scale changes to the article, the pair notation is notable, but the a + bi is more notable.WolfKeeper (talk) 01:33, 17 November 2007 (UTC)
 * The informality is very helpful to people who are not mathematically sophisticated, who are most of the wikipedia readers. People would be having a much easier time with a+ib where by some kind of magic i*i=-1, than with ordered pairs of reals. One's got to choose between informality and mathematical correctness, and for this general purpose intro to complex numbers I prefer the former. Oleg Alexandrov (talk) 02:20, 17 November 2007 (UTC)
 * I don't see that this topic is for the mathematically unsophisticated. It seems to me that most people interested in this would be doing relatively advanced maths. The 'magic' i*i = -1 is right there in the very next section, this is only the introduction. I also basically disagree that examples should be in the introduction, I think we're trying to give just a very brief summary of the whole article, the meaning of the notation can wait till the next section.WolfKeeper (talk) 06:54, 19 November 2007 (UTC)
 * Also, WP:LEAD says that we should have up to 4 paragraphs, which is what there currently is.WolfKeeper (talk) 06:54, 19 November 2007 (UTC)


 * It's getting worse and worse, with one editor even claiming that some way is the only way to formally define complex numbers . What happened to the quotient ring of the monomials over one variable X "divided" by the ideal generated by X2 + 1 – which miraculously turns out to be a field? The representation as a pair of reals is just that – a representation from among several provably equivalent ones.
 * To stay in line with the general approach to start simple, then move toward more abstract and technical statements as the article proceeds, as recommended in the Manual of style for mathematics, we should not set out with a "formal" definition (and, in addition, I don't think the definitions offered are really formal). I emphatically disagree with the opinion that this topic is not for "the mathematically unsophisticated". A reader with a general working knowledge of elementary algebra at the level of real numbers ought to be able to get a good idea from reading at least the beginning of the article and not be deterred already in the first sentence. --Lambiam 00:56, 20 November 2007 (UTC)
 * P.S. I see I misunderstood the edit summary. But the actual edit makes the sentences even more complicated. --Lambiam 01:09, 20 November 2007 (UTC)


 * I realised that my edit made the sentence more complicated. But it was necessary in order to make the sentence less misleading, because, as you point out, this is far from being the only way to define complex numbers. I would have preferred to remove all mention of pairs of reals from the opening sentence, but I considered that it would be pointless for me to do this, as Wolfkeeper would probably just add it back again. I agree that the beginning of this article ought to be suitable for the mathematically unsophisticated - this is, after all, a topic that is often taught to schoolchildren (and the mathematically sophisticated will already know much about complex numbers anyway). --Zundark (talk) 10:58, 21 November 2007 (UTC)

We could do with a decent diagram in the intro. It gives people a warm fuzzy feeling.WolfKeeper (talk) 17:14, 21 November 2007 (UTC)

Article needs an introduction
The article fails to explain in modern terms why having complex numbers is a good idea. It has a history section, but that's at the end, of the article, and I'm not sure that the history is optimal for describing why we have complex numbers.

I think it would be desirable to go through the different sorts of numbers and explain why negative numbers, fractions and the complex numbers are needed in a simple way (and refer out to the main article number for more information.)WolfKeeper (talk) 02:02, 21 November 2007 (UTC)


 * I don't think that guided tour through the garden of numbers should be done here. --Lambiam 10:11, 21 November 2007 (UTC)


 * I agree that a full description of the garden would be inappropriate. However, the unit of the wikipedia is the article. To the extent that complex numbers need to be put in the context of the other numbers, this is the correct place to do so.


 * I just think it needs a paragraph or two.WolfKeeper (talk) 17:10, 21 November 2007 (UTC)

Subfield of ℝ isomorphic to ℝ?
Really? (relinked to page from history) --COVIZAPIBETEFOKY (talk) 15:30, 15 January 2012 (UTC)
 * Contradiction. --COVIZAPIBETEFOKY (talk) 15:56, 15 January 2012 (UTC)
 * Damn, this is some old shit. I shall now remove the line entirely. --COVIZAPIBETEFOKY (talk) 16:21, 15 January 2012 (UTC)
 * I realize now that the pdf I linked to doesn't quite say what I thought it said. If anyone can find a citation stating that there exists a proper subfield of $$\mathbb{R}$$ isomorphic to itself, I would be very interested in seeing it (and we could reinclude the clause with the citation). --COVIZAPIBETEFOKY (talk) 16:34, 15 January 2012 (UTC)
 * I came across this page, apparently authored by Lounesto, that suggests that there is only one subfield of ℝ isomorphic to ℝ: "In contrast, the real field R has only one automorphism, the identity." I do not know whether this excludes injective automorphisms, so I do not know whether this would settle the question.  I find it fascinating that he claims the converse holds for ℂ as a field, though not for a proper subfield. — Quondum☏✎ 18:26, 31 January 2012 (UTC)
 * Yeah, I knew about that result. But that doesn't outrule an isomorphism with a proper subfield, ie, an injective endomorphism that's not onto.
 * Just for the record, I also put this in a discussion here, which seemed to be pretty conclusive. --COVIZAPIBETEFOKY (talk) 02:39, 4 February 2012 (UTC)

Very awkward discussion of polar coordinates
Polar interpretations of complex numbers and their mathematics can be very intuitive. Why is the discussion of complex math in polar coordinates separated by section from the rectangular coordinate interpretation? Also, the discussion of polar vs. Cartesian coordinates happens early, but is not expanded until deep in the article.

Awesome results like "multiplication corresponds to multiplying their magnitudes and adding their arguments" are buried at the end of a discussion of the coordinate system, but do not appear in the multiplication section. Finally, an interpretation of sinusoids with complex numbers in polar form is not covered in anyway but the driest mathematical description of Euler's formula. The interpretation of sinusoids as projections of circles onto lines should make it's way into this article somehow. — Preceding unsigned comment added by OceanEngineerRI (talk • contribs) 18:55, 12 September 2013 (UTC)

i=/=sqrt-(1)
Is it? I don't think it is necessarily. Cause if i=sqrt-(1) then i*i=sqrt-(1)*sqrt-(1)=sqrt(-1*-1)=sqrt(1)=1

Which we know isn't true since i^2=-1 not 1. — Preceding unsigned comment added by Fipplet (talk • contribs)
 * You have made a common mistake. See Imaginary unit for one explanation. -- JohnBlackburne wordsdeeds 13:41, 31 January 2012 (UTC)


 * The principal value of sqrt(-1) is defied to be i rather than −i so yes they are the same. There is no requirement though that the principal value of square root the product be equal to the product of the principal values of the square roots, there are two possible square roots and it so happens that you should have taken the negative square root in the last step above. And in fact if a second square root of -1 occurs there is no guarantee without extra checks that it will be i if i already occurs. This is a bit like if you have x2=1 and y2=1 the possible values of x+y are −2, 0, and 2. Dmcq (talk) 15:56, 31 January 2012 (UTC)

Thanks guys I think I understand. I'm sorry you must get this question alot. I just started linear algebra so that's why. Fipplet أهلا و سهلا 14:28, 4 February 2012 (UTC)


 * Good explanation. Themekenter (talk) 02:52, 1 December 2013 (UTC)

Is ordering considered to be an algebraic property?
This edit makes the implication that ordering is an algebraic property. It has the feel of WP:OR about it, but I do not have the knowledge to be sure. I'll leave it to others to decide and possibly fix this. —Quondum 13:56, 13 March 2014 (UTC)


 * Replacing "the algebraic properties" by "the properties" fixes the problem. MvH (talk) 13:46, 10 April 2014 (UTC)MvH