Talk:Complex number

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Mathematics grading:	B+ Class	Top Importance	Field: Number theory
A vital article
Needs a bit more detail on motivation in the introduction. Then consider nominating for Good Article status. Tompw 14:38, 7 October 2006 (UTC)

This page has been selected for the release version of Wikipedia and rated B-Class on the assessment scale. It is in the category Math. It has been rated High-Importance on the importance scale.

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

Contents 1 Motivation for complex numbers 2 Complex Field vs. R x R 3 My reversion 4 Clumsy 5 Complex line 6 Identities 7 polar form 8 Organization or... not? 9 First sentence 10 EGAD geometric interpretation of complex number multiplication MISSING 11 History 12 Formula for complex argument 13 When Im(z)=0 14 Polar form 15 Department of redundancy department 16 Continuous operations? 17 Direction 18 Notation 18.1 Electrical Engineering

[edit] 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)

[edit] 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 × 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 × 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 × R; but the range of complex multiplication is the complex plane. Septentrionalis PMAnderson 05:29, 13 December 2006 (UTC)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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.

,,,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 . 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 e^iπ/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)

[edit] 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 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)

[edit] 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)

[edit] 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

where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i ² = −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

where a and b are real numbers and i is the imaginary number defined so that i ² = −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

where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i ² = −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)

[edit] 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...   --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! . Madmath789 07:50, 30 July 2006 (UTC)

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

Congratulations.   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 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(a_i)+b = lim(a_i+b)

lim(a_i)b = lim(a_ib)

lim(a_i)* = lim(a_i*)

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)

[edit] History

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

[edit] 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 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

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 π/2. If x is zero and y is negative, use -π/2. If x is negative and y is nonnegative, use arctan (y/x) + π. If x is negative and y is negative, use arctan (y/x) - π. 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:

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:

How about that? --IP 23:28, 27 December 2006 (UTC)

To 194.97.124.49 (talk • contribs): 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)

[edit] 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)

[edit] 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#Converting_between_polar_and_Cartesian_coordinates 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).

[edit] 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)

[edit] 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)

[edit] Direction

I inserted the following sentence in the subsection on absolute value: "The other factor e^iφ = 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 ≥ 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 e^iφ = 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=ae^iφ does not imply that a and φ are real numbers, because ae^iφ is defined also for complex values of a and φ. 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)·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 e^iφ. 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).

[edit] 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

instead of using the standard (universally accepted, I think) notation

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)

[edit] 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).