Talk:Complex number/Archive 2

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Definition of a 'complex number'

According to WP:NOT we need to define the subject in the first sentence or two:

articles should begin with a definition and description of a subject

And Wikipedia:Wikipedia_is_not_a_dictionary:

Good definitions

"A definition aims to describe or delimit the meaning of some term (a word or a phrase) by giving a statement of essential properties or distinguishing characteristics of the concept, entity, or kind of entity, denoted by that term." (Definition)

A good definition is not circular, a one-word synonym or a near synonym, over broad or over narrow, ambiguous, figurative, or obscure. See also Fallacies of definition.

The current introduction doesn't quite do that. Is a complex number any pair of real numbers? No, it's a pair that are treated with particular set of operators (or something equivalent to that definition).

--- (User) WolfKeeper (Talk) 16:44, 20 February 2008 (UTC)

Don't you see the difference between:

"the complex numbers form an extension of the real numbers"

and

"a complex number is an extension of a real number"?

Yeah, but it's a very fine distinction that most readers probably won't get. Ultimately it begs the question: "what kind of extension?" rather. So you haven't really said anything important. Anybody reading that would just have to remember it, they would have nothing to fit it into.- (User) WolfKeeper (Talk) 17:03, 21 February 2008 (UTC)

Part of the purpose of the first formulation was to set the focus on the totality of complex numbers rather than, as is done currently, "a" complex number: after all, a complex number by itself is nothing; it is by virtue of being an element of a mathematical structure that they come to life. Since you reverted to a version that doesn't cut it either according to your own criteria (and I think the word "formally" is only confusing noise), do you have any practical suggestions?  --Lambiam 15:36, 21 February 2008 (UTC)

I suppose one problem is that the article is called 'Complex number' rather than 'Complex numbers'; but that's wikipedia policy.- (User) WolfKeeper (Talk) 17:03, 21 February 2008 (UTC)

I quite like:

"The complex numbers are the field C of numbers of the form x+iy, where x and y are real numbers and i is the imaginary unit equal to the square root of -1, sqrt(-1). When a single letter z=x+iy is used to denote a complex number, it is sometimes called an "affix." In component notation, z=x+iy can be written (x,y). The field of complex numbers includes the field of real numbers as a subfield."[1]

But is that style too complicated for the wikipedia?- (User) WolfKeeper (Talk) 17:03, 21 February 2008 (UTC)

I've never heard of "affix" with this meaning. I am a bit afraid of the use, early on in the introduction, of technical concepts (such as field) that are probably unknown to most people who have enough background to understand complex numbers. For that reason I wrote simply "extension" instead of "extension" (and, moreover, the complex numbers are also topologically an extension). What about the following?

In mathematics, the complex numbers are a number system that forms an extension of the familiar system of the real numbers. Certain algebraic equations that cannot be solved in the real numbers, such as x² + 1 = 0, have solutions in complex numbers.

A complex number can be thought of as consisting of two components: a real part and an imaginary part. These two components can be represented as an ordered pair of real numbers (a,b), and then the complex number can be written as:

$a+bi\,.$

The symbol i, called the imaginary unit, is itself a complex number, one of the two complex solutions of x² + 1 = 0, the other one being −i. The complex numbers for which b = 0 are identified with the usual real numbers; for example, 2 + 0i = 2. Thus, the real numbers form a subset of the complex numbers.

The system of complex numbers has addition, subtraction, multiplication, and division operations defined, with behaviours that are compatible with these operations on the real numbers.

 --Lambiam 18:03, 21 February 2008 (UTC)

I simply don't agree with your use of 'extension' there. It's a bit like 'An aeroplane is an extension of a car that can fly.' Well, both can go along the ground, but... It's not actually technically wrong, in a vague sort of way, but it will inevitably mislead, and I think the reader deserves better, and it doesn't define complex numbers well at all which is what this bit of the article is about.- (User) WolfKeeper (Talk) 18:39, 21 February 2008 (UTC)

Perhaps I am missing something here, but I had the opposite feeling, that extension is precisely the right word under the circumstances. Indeed, C is the unique proper algebraic extension of R. I objected to the mathworld definition you quoted because of the indefinite article "the" to refer both to the complex numbers themselves and to "the" square root of -1, which Lambiam's version clarifies rather nicely, I think. Silly rabbit (talk) 18:45, 21 February 2008 (UTC)

Yes, if you said that, that would be accurate, but removing 'extension' from the context doesn't seem helpful- the English meaning collides with the mathematical. But I'm not sure that saying ' C is the unique proper algebraic extension of R ' would be very helpful in other regards in the introduction; I think we want 15 year olds to mostly understand this article or at least the introduction.- (User) WolfKeeper (Talk) 18:55, 21 February 2008 (UTC)

Perhaps its the mathematician in me.  :-P Silly rabbit (talk) 19:04, 21 February 2008 (UTC)

I would like to reply to the edit summary: "The square root is a function." This seems to be moving further and further from an acceptable definition of the complex numbers. Complex numbers are needed to define a/the square root of -1. It would be better to say "where i² is defined to be -1." Also, the edit summary seems to suggest that each complex number has a unique square root. This is simply not the case. For positive reals, the square root is defined to be the positive root. However, in complex analysis the square root is always going to be a multiple valued function because of the branch point at 0. Silly rabbit (talk) 19:18, 21 February 2008 (UTC)

I agree with Silly rabbit here. What about an intro along the lines of:

In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying i² = −1. Every complex number can be written in the form x + iy, where x and y are real numbers called the real part and the imaginary part of the complex number, respectively. Pairs of complex numbers can be added, subtracted, multiplied, and divided in a manner similiar to that of real numbers. Formally, one says that the set of all complex numbers forms a field.

Just my 2 cents. -- Fropuff (talk) 19:25, 21 February 2008 (UTC)

But does this really follow the spirit of 'definition' above? Seems to me you're trying to define complex numbers by modifying reals, whereas a really good definition usually just says what it is.- (User) WolfKeeper (Talk) 23:29, 21 February 2008 (UTC)

I like it all, a lot, except the first sentence.- (User) WolfKeeper (Talk) 23:32, 21 February 2008 (UTC)

The complex numbers are defined by the properties they have, which are not something you can state all at once succinctly, clearly and unambiguously. Any definition of the form "A complex number is ..." will unduly favour one concrete representation over another and thereby miss the essence. Many articles on an abstract subject do not give a definition in the first sentence or so such that there is necessarily precisely one concept that fits the description; see for example Clifford algebra or Vector space.

Historically, the complex numbers did arise as an extension of the reals that made it possible to solve more equations. I am not convinced that the "English" meaning, as you call it, or at least one of the several meanings of extension, does not also cover the intention here. Perhaps you can accept:

In mathematics, the complex numbers are a number system that extends the familiar system of the real numbers.

Compare this dictionary meaning of to extend:

to enlarge the scope of, or make more comprehensive, as operations, influence, or meaning: The European powers extended their authority in Asia.[2]

 --Lambiam 15:54, 22 February 2008 (UTC)

A couple of comments; first, I know what extend means ;-) secondly, I don't have any problem if we only define it using one concrete description either, thirdly, that isn't significantly better.- (User) WolfKeeper (Talk) 18:01, 22 February 2008 (UTC)

Reading above, somebody points out that R*R and C are almost the same, but with arithmetic operators defined, so I'm thinking rearranging the current one slightly something like:

In mathematics, a complex number is a number which can be formally defined as an ordered pair of real numbers (a,b), with addition, subtraction, multiplication, and division operations defined. Complex numbers are often written:

$a+bi\,$

where $i^{2}=-1$.

Complex numbers have behaviours which are a strict superset of real numbers, as well as having other elegant and useful properties. Notably, the square roots of negative numbers can be calculated in terms of complex numbers.

- (User) WolfKeeper (Talk) 18:01, 22 February 2008 (UTC)

Some remarks:

1. You may have no issue with definition through "one concrete description", but I do, as explained above.
2. How does the word "formally" contribute to this definition? Does it make it in some way more clear or precise?
3. Stylistically the repetition of the word "defined" is ugly.
4. The algebraic operations listed are defined on the domain of the complex numbers, and not, as suggested by the formulation, on the individual complex numbers.
5. To a reader who doesn't already know what to expect, the part "where i² = −1" may appear nonsensical, or in any case in error.
6. I'd like to avoid ascribing "behaviours" to numbers, and the use of "strict superset" is pseudo mathematization that may not convey a meaning to the reader. One "behaviour" of the real numbers is that they are willing to divulge how they are ordered: x<y, x=y, or x>y. In that sense the complex numbers are less well-behaved, so strictly speaking we have a strict subset of a strict superset.

Unlike you, I think that the version I proposed is a significant improvement:

• it introduces the complex numbers as forming a system and establishes a connection with the reals;
• it introduces the motivating property right away (solving more algebraic equations);
• it introduces i as a new root of an equation;

 --Lambiam 15:12, 23 February 2008 (UTC)

Lambiam makes some excellent points. Rather than continuing this debate, I suggest that Wolfkeeper and Lambiam prepare drafts that can be compared side-by-side. (I guess Lambiam's is above. Does Wolfkeeper have a definite version he favors?) We can then have a strawpoll to determine what the consensus is. Silly rabbit (talk) 15:17, 23 February 2008 (UTC)

I happen to like the current definition a lot. Whatever you decide folks, please keep it very simple. Complex numbers are a rather complex subject, and better go for a sillier introduction that may not satisfy the mathematician inside of you but which is more accessible. Oleg Alexandrov (talk) 17:03, 23 February 2008 (UTC)

1. The policy says we have to give a definition; there's usually more than one equivalent definition
2. I agree
3. I agree
4. It is true for the domain as well as for the individual members in this particular case, I think adding domain would make it less clear for no advantage, although not formally incorrect; or are you actually saying that addition is not a defined operator for, say, 5+6i?
5. I think that misapprehension is cleared up in the article
6. that cuts both ways, that means that complex numbers aren't simply an extension to the reals

it's very, very probably a bad idea to try to motivate in the first paragraph, the motivation shouldn't be attempted in the definition in the first paragraph; we've got enough to deal with there; but it's important that it be mentioned in the introduction as a whole.- (User) WolfKeeper (Talk) 17:30, 23 February 2008 (UTC)

p.s. I also have another big issue with Fropuff's definition; the use of the word 'addition' is confusing (with adding), and in any case the imaginary unit is not sufficient to form a complex number, placing 'i' after a real forms an imaginary number, not a complex one.- (User) WolfKeeper (Talk) 17:30, 23 February 2008 (UTC)

I'm not sure what you are referring to with regard to my definition. I say that complex numbers can be added. What's confusing about that? Also inclusion of the imaginary unit is sufficient. The addition of a real number and a imaginary one is a complex number (I'm not sure why you would assume that you could multiply i by a real number but not add it to one). These points aside, I think Silly rabbit has a good point. We should stop arguing about the philosophy of definitions and put up 2 or 3 concrete proposals and go from there. -- Fropuff (talk) 17:49, 23 February 2008 (UTC)

Lambiam's proposed version

This version avoids using the word "extension" directly, and also the technical term "ordered pair" (since a reader might be like: "(1,2) is an ordered pair, but (2,1) is not: the elements are not properly ordered").  --Lambiam 17:51, 23 February 2008 (UTC)

I rather strongly disagree with the proposed version. It is unnecessarily complicated, and does not read well. The reader is bogged down into details. I very much prefer what is currently in the page (note: I did not write that). Oleg Alexandrov (talk) 22:33, 23 February 2008 (UTC)

Yeah, IMHO it's not looking good Lambian. I must admit, I was trying to improve on the current version (that I had a hand in but has been also worked on by others), but I've pretty much failed also. With introductions, it's more of a question of being destined to be imperfect, rather than truly achieving greatness, it's not very easy.- (User) WolfKeeper (Talk) 00:07, 24 February 2008 (UTC)

"Square Root"?

I changed the opening section a little bit. It said that complex numbers include the "square roots" of negative numbers. Although I understand what was meant, the square root function is only defined for the positive reals, even on Wikipedia's own article about square roots, and so this might be confusing to a curious non-mathematician. I've changed it to something which keeps the meaning (I hope). Dissimul (talk) 09:32, 29 February 2008 (UTC)

Can we please say that there is multiplication before listing its properties?

I know there has now been considerable back-and-forth on the issue of how to define complex numbers. But I find the current version somewhat unsatisfactory. One should, I think, say that there is such a thing as multiplication defined on complex numbers before including an equation which explicitly uses multiplication. One possible change to the first paragraph could be something along the following lines:

In mathematics, a complex number is a number which can be formally defined as an ordered pair of real numbers (a,b), often written:

$a+bi.\,$

The quantity i is called the imaginary unit, and multiplication of complex numbers is defined so that $i^{2}=-1$.

However, I think this might impinge somewhat on the next paragraph, which mentions addition, multiplication, and division. My original idea, to simply put the $i^{2}=-1$ after multiplication has been mentioned, was reverted by Wolfkeeper. Silly rabbit (talk) 15:34, 29 February 2008 (UTC)

Mmm. I think that the distance on the page between the i and the multiplication is too small to worry about, but you want the a+ib and the definition of i together (I orginally removed it entirely, but I decided it was slightly better with it and eventually reverted myself).- (User) WolfKeeper (Talk) 15:53, 29 February 2008 (UTC)

You may be right about the distance not being enough to worry about. I still have reservations, though, that there doesn't seem to be enough tying the formula $i^{2}=-1$ to the multiplication. A reader who is not mathematically inclined may not make the connection, obvious to us, that the squaring here is the same squaring one gets from the multiplication defined on the complex numbers. Indeed, that this formula is true because of how that multiplication is defined, and vice versa. The order is not as important as making clear that there is a connection. Silly rabbit (talk) 16:05, 29 February 2008 (UTC)

There is another point here that i^2 = -1 is an axiom from which the arithmetic operators can be defined, and so it is more fundamental, so it probably should be first.- (User) WolfKeeper (Talk) 16:44, 29 February 2008 (UTC)

Again, I think the lead would be improved if there were a way to make it clear how this relates to multiplication. Here it isn't being stated as an axiom, merely a fact without being explicitly tied to the multiplication in any particular way. I would have no objection to introducing it as an axiom, but that doesn't seem to be what you are suggesting. Silly rabbit (talk) 16:57, 29 February 2008 (UTC)

I tried to do both here but Xantharius seemed to get ever-so upset, for what seemed to me to be no particularly good reason and he promptly reverted it. But I wondered what others thought.- (User) WolfKeeper (Talk) 02:18, 2 March 2008 (UTC)

I prefer silly rabbit's modification here. The expression $i^{2}=-1$ should be explicitly tied to multiplication. -- Fropuff (talk) 17:01, 29 February 2008 (UTC)

For the record, Xantharius is not upset. Xantharius (talk) 14:45, 4 March 2008 (UTC)

Shouldn't the lack of ordering be mentioned?

Quote: "Complex numbers have addition, subtraction, multiplication, and division operations defined, with behaviours which are a strict superset of real numbers, as well as having other elegant and useful properties. Notably, negative real numbers can be obtained by squaring complex numbers". The price to be paid is that, unlike real numbers, complex numbers are not ordered. You don't write a<b or a>b when a and b are complex numbers. This important fact is not mentioned in the entire article. Shouldn't it? Bo Jacoby (talk) 11:32, 2 March 2008 (UTC).

The Real vector space section mentions that the complex numbers can't be totally ordered in any reasonable way. --Zundark (talk) 12:21, 2 March 2008 (UTC)

They are orderable, in fact there's multiple ways to do it. They're just not totally or completely orderable.- (User) WolfKeeper (Talk) 13:30, 2 March 2008 (UTC)

There are ways to impose a total order on the complex numbers (see e.g. Wikipedia:Reference desk/Archives/Mathematics/2007 October 3#Comparing complex numbers); it just doesn't seem useful.  --Lambiam 06:42, 3 March 2008 (UTC)

No, there aren't, that's only a partial ordering. You've misunderstood.- (User) WolfKeeper (Talk) 07:07, 3 March 2008 (UTC)

The point Lambiam was referring to was that the complex numbers can be totally ordered. For instance, they are in 1-1 correspondence with a totally ordered ordinal, and so carry a total order themselves. However this total order has nothing to do with the field structure. Thus they can be given a total order, but not in a manner compatible with the field operations: they do not form an ordered field. Silly rabbit (talk) 12:38, 3 March 2008 (UTC)

I'm really unsure that complex numbers are actually orderable. Does this 1:1 correspondence include all the irrational complex numbers?- (User) WolfKeeper (Talk) 15:36, 3 March 2008 (UTC)

C is equinumerous with R, so it can obviously be totally ordered. (Besides, every set can be well-ordered.) --Zundark (talk) 23:23, 3 March 2008 (UTC)

The ordering defined in this decision table is a total (linear) order:

	Im u < Im v	Im u = Im v	Im u > Im v
Re u < Re v	u < v	u < v	u < v
Re u = Re v	u < v	u = v	u > v
Re u > Re v	u > v	u > v	u > v

Antisymmetry and totality can be read directly from the table; transitivity is easily checked, but is also an immediate consequence of the well-known fact that the lexicographical order on the Cartesian product of two totally ordered sets (in this case the reals) is again a total order. --Lambiam 10:45, 4 March 2008 (UTC)

Yeah, OK, I was a bit concerned with the table, since the obvious way to do the algorithm is to compare the numbers digit by digit; which for irrational numbers isn't guaranteed to terminate when the number is infinitely long for the Real part, so there's a real question as to whether you'd ever get to the complex part, but you can alternate digits between the two with the same cardinality as a real.- (User) WolfKeeper (Talk) 15:05, 4 March 2008 (UTC)

??? This is not an algorithm but a definition. Who is talking about digits? Alternating digits is not going to do you any good. Are you a mathematician?  --Lambiam 23:02, 4 March 2008 (UTC)

I'm not entirely sure that you really understand that table... but it doesn't matter.- (User) WolfKeeper (Talk) 00:44, 5 March 2008 (UTC)

To me it does matter if I don't understand that table. Do you understand it?  --Lambiam 19:20, 5 March 2008 (UTC)

Unlike the usual arithmetic of real numbers which is extended to complex numbers, the usual ordering of the real numbers is not, and cannot be, extended to the realm of complex numbers in the same way as the order of rational numbers is extended to the real numbers. The nonreal complex numbers are neither positive nor negative. This is important from an elementary point of view. Bo Jacoby (talk) 13:59, 5 March 2008 (UTC).

I've reverted as incorrect, Bo's recent attempt to add an explanation as to why there are no useful inequality relations for the complex numbers:

Unlike real numbers, which are either positive, negative or zero, the non-real complex numbers are neither positive, negative nor zero. So the inequality signs '<' and '>' are not used in the context of complex numbers.

Please see Total order#Orders on the Cartesian product of totally ordered sets for possibilities for total orders for C and Inequality#Complex numbers and inequalities for why none are useful.

Paul August ☎ 20:01, 10 March 2008 (UTC)

Hi Paul August. Which part of the edit do you consider incorrect ? The fact that total orders exists does not imply that they are used in the context of complex numbers, and in actually they are not. You state that the orders are not useful, and I wrote that it is not used. It seems as if we agree completely, so why revert? 11:14, 11 March 2008 (UTC).

Hi Bo. The reason you gave for why inequality relations are not useful, namely because " complex numbers are neither positive, negative nor zero", is not correct. The reason is given in the second link above. Paul August ☎ 17:16, 11 March 2008 (UTC)

It does not make any sense to mention order in any way in the context of complex numbers. I agree with Paul. Oleg Alexandrov (talk) 02:36, 12 March 2008 (UTC)

Paul, I wrote that "non-real complex numbers are neither positive, negative nor zero", and that is true. Bo Jacoby (talk) 13:44, 12 March 2008 (UTC).

Bo, the terms "positive" and "negative" are not ordinarily defined in the context of complex numbers. So the phrase "the non-real complex numbers are neither positive, negative nor zero" doesn't really mean anything. For the terms "positive" and "negative" to have any reasonable meaning you would have to define an order relation for the complex numbers. There are any number of ways such an order could be defined, in each of which you could define "positive" and "negative" complex numbers in the usual way. Some of these orders could be chosen so that every non-zero complex number was either "positive" or "negative". However none of these orders are particularly useful, as for why this is please read Inequality#Complex numbers and inequalities. Paul August ☎ 18:38, 12 March 2008 (UTC)

Oleg, let's share the fact that "it does not make any sense to mention order in any way in the context of complex numbers". Otherwise the reader assumes that complex numbers preserves all the nice properties of real numbers, and have additional benefits. The price to be paid should be mentioned. Bo Jacoby (talk) 13:44, 12 March 2008 (UTC).

That's a rather weak argument. Complex numbers are points in the plane. Not being all on the line is alone is sufficient for most readers to realize that order would be problematic. There's no compelling reason to state that explicitly. Oleg Alexandrov (talk) 05:15, 13 March 2008 (UTC)

Well, in fairness to Bo, including the content of Inequality#Complex numbers and inequalities in the article might be appropriate. Paul August ☎ 20:15, 13 March 2008 (UTC)

Complex problems

Is a complex problem a problem for which the underlying field of numbers is that of the complex numbers? And does "complex solution" mean that the solution has an imaginary part? That may happen, of course, but there are also complex problems involving quaternions, and there are some very complex problems in mathematical logic that have complex solutions not involving any field. Just like mathematical field may mean something like "graph theory" – not every instance of the word field refers to an algebraic structure – so the word complex can have its common dictionary meanings too. For that reason, I think the following statement in the article is potentially misleading, in particular for the non-mathematicians it is apparently aimed at:

In mathematics in particular, the adjective "complex" means that the underlying field of numbers is that of the complex numbers, for example [...].

I tried to address this by changing this into:

In mathematics in particular, the adjective "complex" often means that the underlying field of numbers is that of the complex numbers, for example [...]. The adjective may, however, also be used in its common, non-technical sense of "complicated" or "difficult", as in: "a complex problem".

This was, however, reverted with edit summary: not in article *about* complex numbers, presumably in response to my edit summary "complex" may also be used in a non-technical sense. If "article *about* complex numbers" refers to the present article, I think this is irrelevant, because the statement about the adjective complex referring to C is meant more generally. If it refers more in general to mathematical articles about complex numbers, I think it is even more irrelevant (and also almost certainly not true). What do others think?  --Lambiam 02:27, 9 March 2008 (UTC)

I am inclined to agree with the revert. I believe you are technically correct, but I do not think the original wording is sufficiently misleading to justify the extra sentence. It's a judgment call, of course. That's just my opinion.

--Bob K (talk) 05:54, 9 March 2008 (UTC)

In the article on Real numbers, would you also add a phrase like

The adjective real may, however, also be used in its common, non-technical sense of "genuine" or "not in disguise", as in: "a real threat".?

DVdm (talk) 09:54, 9 March 2008 (UTC)

If the article contained an absolute statement like In mathematics the adjective "real" means that the underlying field of numbers is that of the real numbers, I might feel a need to point out that this is not so absolute as it is phrased. However, as it is, the Real number article contains no such pronouncement, and the re is no need to avoid the reader being misled by it.  --Lambiam 21:40, 10 March 2008 (UTC)

These are examples of "overloaded" words (multiple meanings). Our language is full of examples, and readers already understand that. (Nevertheless, many editorial arguments have been caused by overloaded words.) Anyhow, DVdm's point is that if we stop and clarify every overloaded word, the articles will become too cluttered for enjoyable reading. So we have to let the readers help carry that load.

--Bob K (talk) 03:00, 10 March 2008 (UTC)

I am not suggesting that we clarify every overloaded word. But suppose that the article Natural number stated that in mathematics "natural" means: related to the natural numbers. Wouldn't you feel a need to moderate that statement? I don't think there are many articles stating bluntly something of the form "In mathematics the word X means Y" when in fact in mathematical contexts the word X can also have different meanings.  --Lambiam 21:40, 10 March 2008 (UTC)

I agree with that too. Therefore perhaps we should simply reverse the offending sentence. Instead of saying the adjective complex means ..., we should say when the number field is complex, the adjective "complex" is prepended to the description. (I already took a shot at it.)

--Bob K (talk) 23:37, 10 March 2008 (UTC)

Heaviside

In the article on Oliver Heaviside it says that he applied complex numbers to equations in electrical engineering. Is this worth mentioning in the History section? Or is there any more detail that could be added to the Applications section. I was thinking it might be helpful to students of electrical engineering who come to this article. Itsmejudith (talk) 09:07, 2 April 2008 (UTC)

"formal definition" in lede

The first sentence of the current lede has some issues. I'm much happier calling the a + bi representation a representation, rather than a formal definition.— Carl (CBM · talk) 18:53, 9 April 2008 (UTC)

That's not what it says. It helps if you read it carefully enough to be able to describe it correctly.- (User) WolfKeeper (Talk) 19:37, 9 April 2008 (UTC)

I'm not happy with the phrasing in which it claims that the complex numbers are defined individually. Any definition defines the entire set of complex numbers at once, with the individual complex numbers then being the things that have been defined. What do people think about this very minor change [3]? — Carl (CBM · talk) 18:53, 9 April 2008 (UTC)

I'll point out that the second paragraph of the current lede already explains the manner in which the complex numbers extend the real numbers; the additional first sentence merely clarifies what's going on. — Carl (CBM · talk) 18:58, 9 April 2008 (UTC)

No, they continue to define complex numbers as best we can in the introduction by mentioning the operators that have been defined.- (User) WolfKeeper (Talk) 19:37, 9 April 2008 (UTC)

Also, the property "is a number which can be formally defined as an ordered pair of real numbers (a,b)" is true of many things. It's true of elements of the ring $\mathbb {R} \times \mathbb {R}$, for example, as well as for complex numbers. So this isn't much of a "definition" of a complex number. — Carl (CBM · talk) 19:30, 9 April 2008 (UTC)

Unfortunately we don't have space in the article lead to describe the operators in full. Hey, here's an idea, why don't you read the talk page as well as the article, and then come up with suggestions?- (User) WolfKeeper (Talk) 19:37, 9 April 2008 (UTC)

Lack of space isn't on its own a justification for anything. "Can be defined as an ordered pair of real numbers" is not a defining property of the complex numbers. On the other hand, "can be represented as an ordered" pair doesn't claim to be a definition. — Carl (CBM · talk) 19:49, 9 April 2008 (UTC)

I fully agree with Carl here – which shouldn't be a surprise, considering my earlier criticism of the lede on this talk page.  --Lambiam 20:46, 9 April 2008 (UTC)

It seems to me you're trying to, not describe what complex numbers are, but do a introductory tutorial on complex numbers. But the wikipedia isn't a how-to, it's an encyclopedia. The not a dictionary policy says that we should define the topic right at the beginning of the article. Complex numbers aren't sort of, kinda like reals, only different, they are a unique type of number from domain C that consists of an ordered pair of reals, which have certain operators defined for them. And that's pretty much what the lead says/said.- (User) WolfKeeper (Talk) 21:30, 9 April 2008 (UTC)

Incidentally, I also require a reference to your edit that claims that complex numbers existed before they were 'discovered', otherwise I will remove this latest edit also.- (User) WolfKeeper (Talk) 21:30, 9 April 2008 (UTC)

I do want to explain what complex numbers are. The complex numbers are the smallest extension field of the real numbers in which every nonconstant polynomial has a root. I would even be happy to see that in the lede. My concern is that I don't believe it is possible to define an individual "complex number" apart from the definition of the entire field of complex numbers.

What a complex number consists of is a subtle question. The complex numbers are the algebraic closure of the real numbers, for example; in that definition there is no mention of ordered pairs of reals whatsoever. There are several other definitions in the article that make no reference to ordered pairs. This is one reason why I prefer to say complex numbers can be represented as ordered pairs rather than saying they "are" ordered pairs.

I would be happy with neither "invented" nor "discovered" in the article, but I don't like "invented" in the case of something as concrete as the complex numbers because I think it goes too far. In any case, both "invented" and "discovered" appear all over the place in the mathematical literature. You can find them both on google books — Carl (CBM · talk) 21:54, 9 April 2008 (UTC)

For the record, I'm not saying the lede should be made longer or more detailed. I think this revision of the lede is generally OK. — Carl (CBM · talk) 22:00, 9 April 2008 (UTC)

Wolfkeeper, I'm hearing a lot of belligerence and threatening from you, but remember you don't own this article. So you don't get to decree the rules on how to edit this page, nor tell people they have to discuss something with you before they change it. --C S (talk) 01:52, 10 April 2008 (UTC)

We engineers were taught that the real numbers are merely degenerate complex numbers -- when b = 0 in the form a+bi. Thus the complex numbers are more "real" than the real numbers -- i.e. more fundamental -- certain physical phenomena require the existence of the complex numbers. Bill Wvbailey (talk) 22:06, 9 April 2008 (UTC)

Looking over the lede and the previous discussions, I think it's best to define the complex numbers, rather than set it up like we're defining a complex number. As has been pointed out, there's no such thing as defining just one complex number anyway. Even if we describe complex numbers as ordered pairs of reals with certain operations, those operations are defined on the whole set. So I recommend starting with something like, "Complex numbers are an extension of the real numbers obtained by adding in the solutions to x^2 + 1 - 0". --C S (talk) 01:48, 10 April 2008 (UTC)

For the record

Of all of the versions of the lede so far proposed, the current one with the tacked on "where i^2=-1" is perhaps the least suitable. May I again suggest that we look alternatives in which it is made explicit what this means? I think Fropuff's version way at the top is my favorite, modulo some possible adjustments of the wording here or there:

In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying i² = −1. Every complex number can be written in the form x + iy, where x and y are real numbers called the real part and the imaginary part of the complex number, respectively. Pairs of complex numbers can be added, subtracted, multiplied, and divided in a manner similiar to that of real numbers. Formally, one says that the set of all complex numbers forms a field.

Any thoughts? silly rabbit (talk) 02:20, 10 April 2008 (UTC)

I thought about that earlier, and decidied that maybe the average reader would be so naive about the nonunique choice of i that they wouldn't struggle too much. When I was editing the article earlier I tried to keep the changes minimal, based on Oleg's criticism of Lambiam's proposed lede above. Maybe there is some wording that is slightly less dense than yours but slightly more deep than what is currently in the article? I think we should expect the average reader here to be very untrained, perhaps in grade school, and so the introductory parts in particular should be kept to a high level of accessibility. — Carl (CBM · talk) 02:34, 10 April 2008 (UTC)

I actually find rabbit's introduction to be very nice (Lambiam's was sound too, but perhaps too technical). I just wonder if it is worth saying "pairs of complex numbers". Won't the wording "complex numbers can be added, etc" convey the same meaning? But this is a small point. Oleg Alexandrov (talk) 06:06, 10 April 2008 (UTC)

Too technical? It was formulated the way it was precisely to convey the motivation and underlying idea to non-mathematicians, while the lede it (only very briefly) replaced went straight to a "formal" definition of an "isolated" complex number and moreover required an understanding of notation (such as a standalone i) that had not been introduced.  --Lambiam 22:37, 10 April 2008 (UTC)

And if the current intro is kept, I think saying "can be represented" is better than "can be formally defined". Oleg Alexandrov (talk) 06:11, 10 April 2008 (UTC)

It's 'nice'. But the problem is that it's wrong though. By saying it can be 'represented as' something, you're implying that it isn't that thing, that this is only a representation. There's a difference between something and the representation of that thing. We're supposed to be saying what it is here not give a representation.- (User) WolfKeeper (Talk) 23:06, 10 April 2008 (UTC)

A complex number is not a pair of real numbers; that was the objection I had to the older lede to begin with. The plane is just one representation of the complex numbers. The construction of the complex numbers as the algebraic closure of the reals, for example, makes no reference to pairs of real numbers. In that construction, a complex number "is" constructed out of polynomials over the real numbers. — Carl (CBM · talk) 23:12, 10 April 2008 (UTC)

When you write a complex number as a + bi that is a representation and nothing but a representation. I could have picked other representations: for example, every complex number can be written uniquely as aj +bk where j = i + 1, and k = i-1, and a and b are real numbers. A more intriguing example, perhaps, is to use -i instead of i. Thinking of complex numbers as only being this pair of real coordinates with respect to the basis 1 and i is something artificial you are adding to what the complex numbers are. --C S (talk) 00:41, 11 April 2008 (UTC)

It would appear that more than one definition is possible. One text I have (Marsden 1973) defines the complex numbers as ordered pairs of integers and how they are manipulated i.e. how to calculate (a,b)+(c,d) and (a,b)*(c,d). Anther book (Cunningham 1965) defines them in terms of solutions to the "general quadratic equation", in particular what happens when the square root of minus one is necessary (Cunningham says that "The art of mathematics is largely concerned with symmetries and patterns" and therefore a need exists to treat those situations when one encouncounters the square root of a negative number). I like Cunningham's approach better because it says why the complex numbers are necessary, and where they come from.

So one could write that:

In mathematics, the complex numbers extend the real numbers to allow the solution of the quadratic equation when the square root of a minus number is encountered. In the most general form, the two roots of the equation can be written in the form x = a+bi and x = a-bi, where a and b are real numbers called the real part and the imaginary part of the complex number, respectively and i is the square root of minus 1. Pairs of complex numbers can be added, subtracted, multiplied, and divided in a manner similiar to that of real numbers. Formally, one says that the set of all complex numbers forms a field.

Bill Wvbailey (talk) 15:52, 10 April 2008 (UTC)

I would have issues with the above- as it introduces complex numbers, but the introduction of an encyclopedia article is supposed to be defining what a complex number is for the reader. It would be perfectly acceptable anywhere else in the article or even in the lead, but it's not very encyclopedic for the beginning of the lead of the article. You're not really supposed to be saying that something is similar to or an extension of something else, you're supposed to say what it is.- (User) WolfKeeper (Talk) 22:27, 10 April 2008 (UTC)

I have exactly the opposite feeling, that this actually tells you what the complex numbers are — the unique quadratic field extension of R. It also tells you that complex numbers are represented as expressions of the form x+iy. Again, apart from trivial changes in the wording (discussed elsewhere on this very talk page), this communicates precisely the same information as the current version and does it better, in my opinion. If you are unhappy that it is not enough of a definition, then perhaps the following form of the lead should be discussed:

In mathematics, the complex numbers, often denoted by C, are the unique quadratic field extension of the real numbers. The complex numbers are typically obtained by adjoining to the real numbers a generator i, called the imaginary unit, such that i²=-1. It is a theorem that every complex number can be represented in the form x+iy where x and y are two real numbers.

This gives a completely rigorous definition of the complex number. Furthermore, unlike the current version, it defines the field structure as well (at least, for those who know how to read it). We could go down this sort of route, I suppose. I still think it is better to, as you say, introduce them before trying to give an abstract definition, though. silly rabbit (talk) 22:45, 10 April 2008 (UTC)

Who says we need to be completely rigorous in the lede? Not to mention, Bill Bailey's version was set in terms of solving the quadratic equation (which many high schoolers could understand) but your version now has technical jargon like "quadratic field extension". Of course, there is no reason we can't say that as an aside after a more informal introduction. I think we both agree that we don't want to go down "this sort of route". --C S (talk) 22:59, 10 April 2008 (UTC)

I don't agree that introductions must define a mathematical object. Sometimes the definition itself is too complicated to lay out in the lede. In the case of complex number, whether it is "too complicated" to explain at the beginning depends on the intended audience. Since I expect nobody that could easily understand the formal definitions would actually need them, I think it's best not to lay out a formal definition right away. Also, note I used the word "formal". There is another point of disagreement. I think Bailey's introduction does in fact define the complex numbers. Not formally but informally. You might feel it doesn't really explain anything, but for me, and I expect, a great deal many others, it does explain the gist of the concept of complex numbers. In addition, as Bill Bailey and Silly Rabbit have pointed out, it really gets at the heart of what the difference between complex and real numbers are. In fact, this leads very naturally into the next sentences on solving cubics and the fundamental theorem of algebra. --C S (talk) 22:53, 10 April 2008 (UTC)

I really like the version proposed by silly rabbit (with Oleg's improvements). It is likely to be better understood by the Lay person, yet is not inferior for the expert. We may either extend the Reals by the solution of the single equation $x^{2}=-1$, or by all quadratic equations, or by all polynomial equations. The two extremes are more natural to consider than the intermediate. (Though the result is the same, of course.) Oded (talk) 17:10, 10 April 2008 (UTC)

I put more some quotes from Cunningham 1965 re the origin of complex numbers; see Trovatore's talk page. Bill Wvbailey (talk) 17:47, 10 April 2008 (UTC)

I think that silly rabbit's version can rephrased slightly, so that it is more clear how the definition works, and also slightly more accessible:

In mathematics, the complex numbers are the extension of the real numbers obtained by adjoining an imaginary unit, i, which satisfies i² = −1. Every complex number can be written in the form x + iy, where x and y are real numbers called the real part and the imaginary part of the complex number, respectively. The complex numbers have addition, subtraction, multiplication, and division operations that extend the corresponding operations on real numbers.

— Carl (CBM · talk) 22:55, 10 April 2008 (UTC)

This version is fine with me. I'm not married to the idea of menitioning a field (mathematics) in the lead. silly rabbit (talk) 14:17, 11 April 2008 (UTC)

Note on a recent addition

I will argue that the recently added section Complex number#An immaterial notational ambiguity does not add value to the article, and is rather a long-winded discussion would be better omitted. Comments? Oleg Alexandrov (talk) 03:47, 11 April 2008 (UTC)

Agreed. Oded (talk) 05:49, 11 April 2008 (UTC)

Agreed as well. It might be useful though to add at some (earlier) point that the complex number represented by the pair of reals (a,b) is usually written $a+bi$ because the number happens to be the value of that expression. This kind of thing occurs often: 104 is pronounced "one hundred and four" because that happens to be a computation having 104 as result. If univariate polynomials are formally defined as sequences of coefficients, and X denotes (0,1,0,0,…) then the polynomial with sequence (-2,0,0,1,0,0,…) is written X³ − 2 because that expression happens to designate the polynomial in question. Marc van Leeuwen (talk) 10:35, 11 April 2008 (UTC)

Well, I'd argue that such observations confuse more than what they illuminate. While you are correct, the reader is better served by saying $(a,b)$ can be written as $a+bi,$ and then not worrying about the result of computing $a+bi$ brings us back to $(a,b).$ Complex numbers are already complicated enough for readers without the need to go to the very bottom of things. Oleg Alexandrov (talk) 15:12, 11 April 2008 (UTC)

But is it clear to the reader who doesn't know yet what complex numbers are and what i is, that "writing $(a,b)$ as $a+bi$" is not merely a notation, but that it involves an addition operator "+" and an (invisible) multiplication operator, both representing operations not yet introduced at that point, and that in fact a, b and i are all complex numbers here? If all this may be assumed to be clear to the non-mathematician reader, then we can simplify the article further by skipping all statements such as that a is the same as a + 0i — of course it is the same, since 0 × anything = 0 and a + 0 = a. But there is in fact nothing "of course" about this, and there is something absurd in defining complex addition by formulas requiring "+" in the definiens to be interoreted as complex addition, as if we have a recursive definition (which, however, is not grounded in this case). This can conceivable be salvaged by declaring a + bi to be a canonical representation, and to proclaim that we define the operations by rewriting to a canonical representation, but that may in fact require more explanation than pointing out that the notation a + bi is not ambiguous – at least not in any sense that is material. In helping students who felt hopelessly lost, I have often found that this was largely because the instructional material was taking things for granted that should not have been taken for granted in presenting the material to newbie mathematicians, but instead should have been justified.  --Lambiam 00:51, 12 April 2008 (UTC)

Some problems in this article

Reading through this article I noted some problems, unclear or incorrect statements. I corrected a few in passing, but do not have the time to do all, so I'll just signal them here.

• The section operations says all arithmetic operations are defined by applying the associative, commutative and distributive laws. This should not be presented this way, and is certainly false for division (anyone doubting this should do the exrecise of trying to extend the complex numbers again in the same way, introducing a new imaginary unit say l with $l^{2}=-1$, new numbers $a+bl$ with a and b complex numbers, and defining the arithmetic operations just like when building the complexes from the reals; the problem is this gives zero divisors, and division cannot be defined). The right way to go is say that addition and multiplication are defined by explicit formulas, and that inverse operations of subtraction and division turn out to exist.
• The section Geometric interpretation seems unclear about what it wants to assert. The set R² is a plane; does one want to interpret the already defined complex arithmetic operations geometrically, or does one want to construct the complex numbers independently, using only geometric constructions?
• the section polar form introduces the argument function without even vaguely stating what it is (unlike the absolute value which is already mentioned before), except that values are defined modulo 2π. In fact the reader has to deduce what is meant by "argument" from the formulae involving cosine/sine that follow somewhat later. Also the formulation is sloppy (are two real numbers differing by 2π considered equivalent just because they occur as argument of the same complex number?). The use of "atan2" for the argument function seems a bit too computer-oriented to be used as it is here.
• The section on matrix representation could do with a more factual tone. I would start with something like "Instead of defining arithmetic operations somewhat arbitrarily on R², the complex numbers could be defined as a subring of the ring of 2×2 real matrices that happens to be a field". Also one doesn't stretch or rotate individual points of the plane (at least it is hard to see the effect of doing so).
• The section on Characterization as a topological field strikes me as having a level of abstraction/mystery rather distinct from the rest of the article. Does it serve well here? Is there some better place for this infomation?

Marc van Leeuwen (talk) 11:53, 11 April 2008 (UTC)

You're right that many of these would benefit from copyediting or rewriting. Want to lend a hand? I noticed this the 87th most frequently viewed mathematics article on wikipedia. It only drew may attention recently, but I am hoping to participate here and help polish it for a couple weeks. — Carl (CBM · talk) 12:48, 11 April 2008 (UTC)

Sounds like a good idea to me also. By the way, thanks to Marc for writing down all of these points. silly rabbit (talk) 12:58, 11 April 2008 (UTC)

I would suggest no big rewrites of the entire thing though. Gradual changes while everybody else has a chance to follow and comment would work best I think. Oleg Alexandrov (talk) 15:14, 11 April 2008 (UTC)

It seems like there are enough interested parties that most if not all of the issues can be fixed incrementally, with no big rewrites. silly rabbit (talk) 15:21, 11 April 2008 (UTC)

• the article is far less than minimally referenced. The article is B-class, and if anything in my opinion, currently slipping downwards in quality (and I'm not pointing at anyone in particular when I say that). IMO people need to think about referencing FIRST, and then trivialities of phraseology second. It's not an exageration to say that every paragraph or so needs an in-line reference.- (User) WolfKeeper (Talk) 16:50, 12 April 2008 (UTC)

Another "formal" definition

We are back to having a statement about "formally" defining the complex numbers, in the section "The field of complex numbers":

"Formally, the complex numbers can be defined as ordered pairs of real numbers (a, b) together with the operations:

$(a,b)+(c,d)=(a+c,b+d)\,$

$(a,b)\cdot (c,d)=(ac-bd,bc+ad).\,$

So defined, the complex numbers form a field, ..."

My objections to this are twofold:

1. It suggests the preceding was informal. This, however, is not in any sense more formal than the earlier presentation; it just repeats the same in another notation.
2. It rather strongly suggests we have new definitions of addition and multiplication here that are needed to make the complex numbers into a field. This is unnecessarily confusing; they are as much a field with the definitions of these operations given in the immediately preceding section.

 --Lambiam 13:36, 12 April 2008 (UTC)

I simply reverted it because:

1. the notation was still in use in the article
2. the notation was not described anywhere
3. it is the way that complex numbers are defined
4. after the revert, the quality of the article increased

I'm not particularly wedded to any particular wording, but when people make the article internally inconsistent with their edits, I get out my revert pen.- (User) WolfKeeper (Talk) 13:54, 12 April 2008 (UTC)

I don't see any use of the ordered pair notation below that section. It may be that I missed something small, but it should be easy enough to fix one or two instances of different notation. — Carl (CBM · talk) 14:23, 12 April 2008 (UTC)

I tried rearranging it a different way. I think it makes sense to discuss the operations in the section on operations, and then discuss the field nature in a separate section. — Carl (CBM · talk) 14:36, 12 April 2008 (UTC)

Unreferenced claim in intro

In mathematics, the complex numbers are the extension of the real numbers obtained by adjoining an imaginary unit,

This requires referencing. - (User) WolfKeeper (Talk) 16:55, 12 April 2008 (UTC)

Which part exactly? That C = R[i] where i satisfies i^2 = -1? That's a completely standard fact; see google books. — Carl (CBM · talk) 17:17, 12 April 2008 (UTC) — Carl (CBM · talk) 17:15, 12 April 2008 (UTC)

The bit where there's no reference in the article text?- (User) WolfKeeper (Talk) 18:03, 12 April 2008 (UTC)

There's no requirement to have an inline reference, the requirement is only that the claim must be verifiable. If you look through the results of that google books search that I linked above, you'll be able to find numerous sources to back it up. If you really want to add an inline citation for it, please feel free.

But I'm confused by your request for a reference because this is such a completely standard fact about the complex numbers (that they are obtained as a field extension of the real numbers by a root of x^2 + 1). If you already know this fact, I don't know why you're asking someone else to give you a reference. If you don't already know it, it might be worthwhile for you to do some more background reading. — Carl (CBM · talk) 18:14, 12 April 2008 (UTC)

This isn't a trick question, nor because I don't know the answer. It's an actual request for you to actually reference that part of the introduction. I'm sorry if you find the concept confusing.- (User) WolfKeeper (Talk) 23:26, 12 April 2008 (UTC)

I have provided a reference below, proving the material is verifiable. Let's move on to to a more useful discussion. — Carl (CBM · talk) 23:37, 12 April 2008 (UTC)

No. Unless you add it to the article, I will remove your additions, and add my own, and reference them.- (User) WolfKeeper (Talk) 00:15, 13 April 2008 (UTC)

Everyone is free to edit the article of course, regardless whether there are references or aren't. In the discussion above, there was a lot of support for the wording that is now in the article. — Carl (CBM · talk) 00:23, 13 April 2008 (UTC)

In the end, the material in this article will be verifiable in numerous places, because it's so standard. That's why my concern is more with getting the right content in the article, and presenting things in a clear and correct way. — Carl (CBM · talk) 00:36, 13 April 2008 (UTC)

Is there a need to be so combative? Not to mention, since there is consensus for the additions (that you don't like) and are clearly reference-able, aren't you able to add the reference yourself, instead of threatening removal? Or do you not feel like helping out in referencing additions that haven't been approved by you? I hope you remember this is a collaborative encyclopedia. An attitude of "I don't like this, so I'm not going to help out" isn't going to get you very far. I will add the reference myself. --C S (talk) 02:33, 13 April 2008 (UTC)

Quite honestly, I think the particular way you're introducing complex numbers is over helpful, and it obscures or fails to make well many important points, but that aside, much worse there's a worrying tendency of the editors here to remove other ways of looking at complex numbers. That's not the wikipedian way. The article should be trying to encompass all ways of looking at them, all notations and all important connections, and to the extent that is possible that includes the introduction.- (User) WolfKeeper (Talk) 17:27, 15 April 2008 (UTC)

The ordered pair representation is discussed in the section on operations. The a+ bi representation is discussed right at the start of the introduction, and the ordered pair representation is just a different way of writing the same thing. I don't think anything has been removed from the article. — Carl (CBM · talk) 18:58, 15 April 2008 (UTC)

---

I cc'd this from Trovatore's talk page:

John Cunningham, 1965, Complex Variable Methods in Science and Technology, Van Nostrand, New York, no ISBN, card catalog number 65-20159.

Cunningham starts off his Chapter 2 Complex Numbers like this:

"The concept of an 'imaginary number' such as the square root of minus one was first introduced, with great scepticism, late in the sixteenth centruy. Mathematicians and physicists soon discovered that the device led to many simplifications in difficult problems, and having proved its worth in practical situations the imaginary number became a reputable and powerful mathematical tool...

" 2.1 The Square Root of Minus One The art of mathematics is largely concerned with symmetries and patterns. Let us pick up any school textbook on the solution of quadratic equations. we are likely to find the following type of statement 'The equation x^2 - 9 = 0 has two roots x = +/-3, but the eqution x^2+9 has no roots'. The mathematician finds this sort of assymetry rather unpalatable ... this can in fact be achieved by adding to the real number system the imaginary number which is the square root of minus one [etc -- he goes on to discuss the solutions to ax^2 + bx + c = 0].... In this way the original notion of an imaginary number gives way to the concept of a complex number.(p. 27)

The only difference that I can see here is the notion of "adding" as opposed to "extension of". Bill Wvbailey (talk) 18:50, 12 April 2008 (UTC)

Another reference from google books states:

For example, the classic view of complex numbers is that they are obtained by 'adjoining' to the real number system, the 'imaginary' square root of — 1. (Foundations of Discrete Mathematics by K. D. Joshi - Mathematics - 1989 - Page 398)

One reason the word "adjoin" or "extend" is used is because there is an addition operation already hanging around, so if you say you are "adding" i to the real numbers it can be confusing. — Carl (CBM · talk) 18:58, 12 April 2008 (UTC)

Agreed. Bill Wvbailey (talk) 22:01, 12 April 2008 (UTC)

That's not a definition though; that's a note as to how they were historically arrived at, the introduction is supposed to contain a definition.- (User) WolfKeeper (Talk) 17:33, 15 April 2008 (UTC)

More often than not a definition is "operational" in the sense that the definiendum's behavior, or how it is used, or how it is built, defines the object. For example: "A human is a mammal which most commonly in its unimpaired adult phase, talks, walks on two legs, drinks beer or coffee or both, and laughs at jokes, uses hammers and lives in houses." "A hammer is a tool used to pound nails." "A house is a structure built by humans wielding hammers." In fact there may not be any other way of forming the definition after the object has been generally classified (e.g. a human as a mammal of a certain genomic-type, a hammer as a tool, a house as a "structure"). My trusty Webster's 9th New Collegiate dictionary says that a complex number is "a number of the form a + b*sqrt(-1) where a and b are real numbers." This does not tell me what it "is" just how it is built, and what its components are. Bill Wvbailey (talk) 19:20, 15 April 2008 (UTC)

We shouldn't turn to dictionaries for definitions of mathematical concepts. The problem with that "definition" is that it ignores the key properties of complex numbers - the field and topological structure - that make them what they are. In short, it doesn't really define a complex number at all. — Carl (CBM · talk) 19:27, 15 April 2008 (UTC)

Doesn't In mathematics, the complex numbers are the extension of the real numbers obtained by adjoining an imaginary unit, i, refer to imaginary numbers, not complex numbers???- (User) WolfKeeper (Talk) 03:05, 19 April 2008 (UTC)

The two terms refer to the same set of numbers. Unless I misunderstand what your asking. Thenub314 (talk) 11:51, 19 April 2008 (UTC)

"Imaginary number" can be either "complex but not real" or "real multiple of i" depending on context. Either way, the real numbers are complex but not imaginary. — Carl (CBM · talk) 12:57, 19 April 2008 (UTC)

Well I like this usage of the terms better. I was under the impression that high school teachers (and text books) had a habit of using ther terms interchangebly. In which case I would point to the term "extension" as the word that should make it clear the real numbers are contained in the complex numbers.

The fact that is being referred to is that the complex numbers are a field extension of the real numbers obtained by adjoining a root of -1. This is not to say that every complex number can be obtained by multiplying a real number by i; it's saying that if you start with the reals, throw in a new object that squares to -1, and close the structure under addition and multiplication, you obtain a field, and this field is the field of complex numbers. — Carl (CBM · talk) 12:57, 19 April 2008 (UTC)

And that's what the introduction says is it?- (User) WolfKeeper (Talk) 18:16, 19 April 2008 (UTC)

I'm not sure what you're asking; you must be thinking of something that I'm not. In the previous question, why did you think the introduction might be only covering the imaginary numbers? Maybe that will help me see what you're saying. — Carl (CBM · talk) 18:19, 19 April 2008 (UTC)

<unindent>I think Wolfkeeper's concern is that in that sentence "adjoin" could be interpreted by someone to mean only multiples of i. But I think any merit in that is negated by the very next sentence of the lede which clarifies that this is not what it means. I think as fruitful as this discussion over the first sentence has been, it is time to move on. --C S (talk) 13:00, 20 April 2008 (UTC)

It's simply a heap of junk. It's totally bad; the two first sentences say different things- they pointedly contradict each other. The first says, you just adjoin/add a real to an i, and out pop complex numbers. The next says, oh, by the way, one real isn't enough, you need two reals, and they're called x and y.

And the first sentence implies that there's only one way to add an i to a real, so imaginary numbers don't exist then? So you sort of have to guess what the heck they mean. And the whole notation is a historical accident anyway.- (User) WolfKeeper (Talk) 04:19, 21 May 2008 (UTC)

If you at least defined an imaginary number, and then defined a complex number, I could get onboard, but this is simply just disaster; bad and wrong.- (User) WolfKeeper (Talk) 04:19, 21 May 2008 (UTC)

I don't follow what you are saying here. What do you mean by "And the first sentence implies that there's only one way to add an i to a real,... "? The term adjoin in the first sentence is used in the sense of Adjunction (field theory) and has a specific meaning. It does not mean that one adds i to a real number in the sense of addition. Are you familiar with the abstract algebra underlying field extensions? — Carl (CBM · talk) 10:48, 21 May 2008 (UTC)

references

Perhaps I'm being too obtuse. It is not nearly sufficient to have a very few references more or less as a bibliography on the end of the article- those are deprecated in the wikipedia. The article needs in-line references scattered throughout, if it stands any chance of getting anywhere above 'b' grade.- (User) WolfKeeper (Talk) 23:26, 12 April 2008 (UTC)

I can sympathize with your comments on references. A carefully selected bibliography is likely to be more useful than a collection of more-or-less random references. A few of the parts could use inline references, but carefully selected to be particularly useful to the reader. If you have some ideas about which references would be good, I'm sure everyone would be glad to hear them. — Carl (CBM · talk) 23:36, 12 April 2008 (UTC)

The article does need some more references, but you should be aware that the scientific citation guidelines apply here. They may be somewhat different than what you are accustomed to. --C S (talk) 02:33, 13 April 2008 (UTC)

There is no requirement that every sentence, or indeed every paragraph, be referenced (per the guideline you cite). Material that is potentially in any standard textbook does not need a specific reference. It is more important that the reader should be able to verify the information. For that reason, I suggest supplying some bibliographical notes; e.g., indicating which are "standard references" for the material presented in the article. The material in the article is relatively uncontroversial, with the possible exception of the history section, which I think could use some inline citations. silly rabbit (talk) 13:04, 16 April 2008 (UTC)

Second Paragraph

I suggest removing the sentance "The complex numbers form an algebraically closed field, unlike the real numbers, which are not algebraically closed." from the second paragraph. If you understand what it is ment it relates to the paragraph it is in. If your not familiar with algebraic closer it seems a bit out of place. It is also explained a bit better later in the article which makes it seem a little too soon to mention it here. Thenub314 (talk) 17:05, 15 April 2008 (UTC)

So that would make the paragraph say something like:

Complex numbers were discovered when attempts to find solutions to some cubic equations required intermediate calculations containing the square roots of negative numbers, even when the final solutions were real numbers. Research in this area led to the fundamental theorem of algebra. This theorem states that when complex numbers are employed, it is always possible to find solutions to polynomial equations of degree one or higher, unlike when real numbers are employed.

That seems fine to me. It makes the lede more accessible, and we already mention algebraic closure lower in the article. — Carl (CBM · talk) 18:55, 15 April 2008 (UTC)

Division of Complex Numbers

Would it be useful to mention that complex numbers can be divided by multiplying the nominator and denominator by the conjugate of the denominator, rather than just listing an equation? —Preceding unsigned comment added by 82.108.65.117 (talk) 21:41, 5 May 2008 (UTC)

The relationship is given in the subsection "Absolute value, conjugation and distance", where the concept of conjugate is introduced.  --Lambiam 08:24, 6 May 2008 (UTC)

j is the new i

As of 19XX - 200X some people changed i to j.... I'll (or someone) look into this. —Preceding unsigned comment added by Ovarninehundred (talk • contribs) 15:12, 1 June 2008 (UTC)

In virtually every textbook on mathematics, i is used for the imaginary unit. I know that electrical engineers sometimes prefer j, and that is mentioned in the article on the imaginary unit. However, the article should stick to the more common notation, which is to use i. siℓℓy rabbit (talk) 15:19, 1 June 2008 (UTC)

We electrical engineers use both indiscriminantly and without hesitation, but not together in the same derivation (to avoid confusing ourselves let alone anyone else). But really there's no reason for confusion: we never use the symbols i and j for anything else but the imaginary unit. Bill Wvbailey (talk) 01:43, 2 June 2008 (UTC)

According to the article, EEs use i for current. (v=i·r)

--Bob K (talk) 04:06, 2 June 2008 (UTC)

The wily and cautious ones avoid "i" for current and use i(t) or I(t) or I, as in v(t) = i(t)*r(t), for example. But you have a point, there are those who are sloppy and/or incautious. I suggest they use V = I*R. Bill Wvbailey (talk) 04:40, 3 June 2008 (UTC)

Very often, I is used to denote large-signal effects (bias currents, etc.), whilst i is used to denote the small-signal current. It's nothing to do with being "sloppy" or "incautious"! Oli Filth^(talk) 09:12, 3 June 2008 (UTC)

Au contraire. Otherwise we wouldn't be having this dialog. I frequently lapse into sloppy-hood and write a script-"i" myelf for "current". In those cases when i (imaginary number written as "i" in script) has to live with i (current written in Roman lower-case i), I also use j. I come from the same generation as Bob K, below. Does anyone know why "I" aka "i" appeared in the first place to reperesent "current"? Bob K hit it on the head ... editorial thrashing. Bill Wvbailey (talk) 14:34, 3 June 2008 (UTC)

It's hardly the first incidence of two variables having the same symbol! (Incidentally, if you want further worry about the overlap in physics/electronics, e is often used to denote electromotive force (voltage).) Using the standard symbol for a quantity can hardly be described as "sloppy" or "incautious". Oli Filth^(talk) 14:46, 3 June 2008 (UTC)

Yes, i(t) and I(t) are what I remember from my 2 circuits courses, circa 1966-67. Obviously that doesn't make me a reliable "witness", but I think the context usually sorts things out nicely. Personally, I use j for the imaginary unit, but I think the article has it right. i is "more standard", and j is acknowledged as a common alternative. Both are found throughout Wikipedia, which is not all bad or good. The good thing is that it accurately reflects the reality of no universal standard. The bad thing is that it leads to editorial thrashing... the Achilles Heel of Wikipedia.

--Bob K (talk) 11:30, 3 June 2008 (UTC)

I was curious to see what my own usage is, so I went back to a recent notebook full of derivations I did before this "dialog." Both are of a "differential amplifier" with input resistors to protect the amplifier against high common-mode voltages (for example, Analog Devices makes a couple such parts e.g. the AD628 and in particular the AD629). Anyway, the common text-book formula (without V_os in it ... or i_b(T) there's the clue...) can be found in an online TI guide called "Op Amps for Everyman" or something like that. This derivation requires two current loops. But it turns out there's another derivation that involves only one current loop -- this is the one that occurs when your input is truly floating and has no common-mode connection -- and the results are different. (Thus the value of cranking the formulas...). So anyway what did I do in my own derivations?:

(1) I used i₁ and i₂ for the two loop currents (here's is how the (handwritten) formulas look; note I used capitals for everything except i and the subscripts):

-V_in- + i_a(R₁ + R₂) + V_o = 0

-V_in+ + i_bR₁ + (V₊ + e) = 0

(2) I used i, drawn over an arrow with a loopy current. This single i does appear in the following loop equation as I wrote it:

-V_r + i(R₁ + R₂ + R₃ + R₄) + V_in + V_o = 0

The upshot of this second equation is stuff that looks like:

V₊ = iR₂ + V_o

This of course has one meaning in the context of my hand-drawn diagram (shown with a current loop without a label). But if a mathematician encountered this equation as written, by itself, on the page without context and knowledge of EE's, they would interpret this as if i were an imaginary number. So, by definition of the word precision (having to do with L. prae + caedere to cut, and L. distinguere, as in the word di-sting-guish to separate by "picking apart"), this usage across technical fields is less precise. Another trick I've learned over the years is to put a * between e.g. i₁*R so there's no ambiguity about the nature of what the symbol "i₁R" means; then if there's even more possibility of confusion, I use a "key" to define the symbols. (We've done this alot in wikipedia, it's almost a requirement for a lot of us). Usually, also, in data sheets and texts you see i_b, or i_bias, or i_os or whatever. This BTW is my last entry on this; I've got better things to do with my time. Bill Wvbailey (talk) 15:17, 4 June 2008 (UTC)

External Links

There's no shortage of online complex calculators and tutorials in the world; it seems crass to allow some arbitrary subset of those -- especially when they are a gateway to money making ('buy amazon stuff through me', 'click on these google ads here'). I propose we remove the links to all calculators and tutorials that feature google ads and/or amazon click throughs. If there is no objection before 25.june.2008, i'll do such. Quaeler (talk) 09:10, 21 June 2008 (UTC)

2nd sentence -- a run-on sentence?

The 2nd sentence: "Every complex number can be written in the form a + bi, where a and b are real numbers called the real part and the imaginary part of the complex number, respectively."

Forgive the intrusion; I'm an 'outsider'. I got stuck on the 2nd sentence. It seems like some sort of run-on sentence needing a simple fix; maybe it's just not clear. Perhaps ya'll haven't noticed it because you're so familiar with the article ;-) or maybe I'm crazy! Delete this new section if appropriate; no offense taken. —Preceding unsigned comment added by Gloucks (talk • contribs) 17:33, 14 July 2008 (UTC)

Would this make more sense?

Every complex number can be written in the form a + bi, where a and b are real numbers that are called the real part and the imaginary part of the complex number, respectively.

— Carl (CBM · talk) 18:59, 14 July 2008 (UTC)

Or perhaps, "... are real numbers, which are called ..."? After all, their being called thus is not restrictive. I'd suggest:

Every complex number can be written in the form a + bi, in which a and b stand for real numbers, called the real part and the imaginary part of the complex number, respectively.

 --Lambiam 00:05, 20 July 2008 (UTC)

an/the in lede

I noticed the changes in the first paragraph. I don't think the intended reader is worried much about uniqueness, so I don't mind if it says 'an'. On the other hand, the comment about quaternions isn't apropos, since the quaternions aren't a field extension of the reals and can't be obtained by adjoining roots of polynomials to the reals. In any case, I don't think it's worth spending too much time on that particular word choice. On reflection, I think the quaternions concern may be more common among the intended readership, who won't really know about field extensions, so the 'an' might be less confusing. — Carl (CBM · talk) 00:07, 20 July 2008 (UTC)

If it just says "extension", then it seems that the quaternions (or even octonions) are reasonable candidates, so "an" is more appropriate. However somewhere the article should mention that the complex field is the unique (up to isomorphism) as a field extension. Since this is a little technical, maybe the lede is not the right place to put it.

On another note, I don't think that the statement (made later in the article) that the complex numbers are the topological closure of the algebraic numbers makes much sense as a stand-alone characterization of C. Closure in what? -- Spireguy (talk) 14:15, 20 July 2008 (UTC)

The statement should probably be that it is the the completion of the metric space of algebraic numbers. This is obviously what was intended, and does not depend on any ambient embedding of the algebraic numbers. siℓℓy rabbit (talk) 15:14, 20 July 2008 (UTC)

Historical focus of the lead

I'm wondering if the current historical focus of the lead is appropriate. I think it might be stylistically better for the lead to state as concisely as possible the main points of the article without trying to contextualize it historically, and then to have an "Introduction" section in which the historical motivations for studying complex numbers and their generalizations can be examined in a more leisurely manner. siℓℓy rabbit (talk) 12:53, 25 July 2008 (UTC)

I also think an "introduction" or "history" section would be nice. Also, the history in the lede is a little incoherent. It says one person discovered complex numbers, and another discovered the arithmetical operations. The way it is written, it will probably appear contradictory to some readers. — Carl (CBM · talk) 13:00, 25 July 2008 (UTC)

At least a part of the lead has to be moved into the History section. I'd like to tag the article with the 'longintro' template to attract other editors' attention on this issue. --M4gnum0n (talk) 13:56, 25 July 2008 (UTC)

You have our attention - it's what silly rabbit and I are discussing. You could just move the content yourself. The people who are likely to comment here will have the page on their watchlist; the tag isn't going to get their attention more than this discussion will. — Carl (CBM · talk) 13:59, 25 July 2008 (UTC)

Inverse

Inverse really is not a big deal when it comes to mathematical operations that can be done with complex numbers. However, I believe it should be added to the article as a mathematical operation. What does anybody else think? --Jparrish88 (talk) 07:33, 4 September 2008 (UTC)

The two forms of inverse -z and 1/z are discussed in Complex number#The field of complex numbers.--Salix alba (talk) 08:08, 4 September 2008 (UTC)

No commercial external links in Wikipedia!

I inserted following very valuable link to a free(!) ressource and canceled commercial(!) links to Wolfram.com ressources (which is not allowed in Wikipedia):

• Dimensions: a math film. Embedding Complex Numbers in a demonstration of higher dimensions (also Chaostheory) in particular chapter 5 and 6

User Quaeler (talk) pressed the undo-button without to think it over. And so he/she deleted the free ressouce too. What to do?

--92.227.30.255 (talk) 11:15, 7 September 2008 (UTC)

As i stated the first time on your warning (which i'm about to re-warn on), Wolfram links are widely accepted; the irony of your complaint is that you're linking to a site that offers DVDs for sale of the content. Quaeler (talk) 11:39, 7 September 2008 (UTC)

No. You misunderstand. Everything on that Site is for FREE download. It is all under the terms of Creative Commons license. The DVD has 10 Euro for shipment only not for commercial profit! You also have to pay for the shipment of Linux-DVDs. So please accept inserting that FREE Ressource. Wolfram.com links are profitoriented and commercial only like Microsoft stuff. --92.227.30.255 (talk) 11:52, 7 September 2008 (UTC)

I misunderstand because it says "Price: 10 E[uros], shipping cost included!" The site is likely written by people whose first language is not English (they use the French-ism of "We propose" on the page in question) - but translation issues aside, a quoted price is a quoted price, and any one of adult age realizes that people roll all sorts of "shipping & handling" fees together to allow commercial profit.

That being said, i'm totally off on the wrong foot with you, so let's leave it up to a community consensus on this page as to what to do with your link, and the Mathworld encyclopedic references. Thanks. Quaeler (talk) 12:00, 7 September 2008 (UTC)

Links to Mathworld are widely accepted on Wikipedia. It is a "commercial" site in the sense that it advertises Mathematica, but it is also an extensive, free, accurate resource for brief summaries of mathematical topics. I don't see any reason to remove them.

On the other hand, it may be worthwhile to add the link to the movie as well, I'm going to try to watch chapter 5 of the film, so I can comment on that more sensibly. I need some time to download it first. — Carl (CBM · talk) 12:02, 7 September 2008 (UTC)

TradeOff: If Wolfram.com is accepted, then Dimensions.org (***.org!) should be accepted too. @Carl: Thank you. @Quaeler: Think about what Creative Commons license means. An "Price : 10 €, shipping cost included!" means: profit free printingcosts + profit free DVD-burningcosts + shipment. A Non-profit organization does not have money to afford such costs. --92.227.30.255 (talk) 12:12, 7 September 2008 (UTC)

Re 92.227.30.255: as you would expect, we can't include every possible external link. The quality of each link is important, in addition to free access.

I downloaded and watched chapter 5 of the film. It is a well-produced animation of an idealized blackboard presentation introducing the complex arithmetic, argument and modulus, and stereographic projection. I think it would be useful for a reader who is trying to learn the basics of complex numbers but wants a multimedia presentation. — Carl (CBM · talk) 12:41, 7 September 2008 (UTC)

Once I download chapter 6 (so I can write an accurate summary), I'm going to reinsert the link. I'm planning to do some other link maintenance at the same time: removing "John and Betty's Journey Through Complex Numbers", and merging the Mathworld bullets into one. — Carl (CBM · talk) 12:50, 7 September 2008 (UTC)

@Carl: I agree the merging. But the "John and Betty's Journey Through Complex Numbers" is an approach for little children, I like. It doesn't has to be pretty to me but for those youngster it could be nice. --92.227.30.255 (talk) 13:00, 7 September 2008 (UTC)

It might be, again, your poor grasp of the English language however it is not a "Trade Off" to graciously allow the original link which you tried to delete (and which everyone agreed with from the start) in addition to the link you're attempting to sell (any more than a bully who takes one dollar and gives you 50 cents back is making a compromise). Making this situation worse, there have been notes placed on both my talk page and your talk page, stating that you've been warned about shilling this link on other language wikipedias.

To that extent, i am nominating that your link be removed as not only is your behaviour in bad faith, but i see nothing unique that wouldn't be found in the dozens of hits returned by a brief google search (and so fails notability). Barring any evidence of notability prior to 12.00 UTC 8.September, 2008, i'll remove the link. Quaeler (talk) 19:58, 7 September 2008 (UTC)

I think it's worthwhile to actually download chapter 5 and watch it to get a sense of the movie. There's no requirement for external links to be notable. This one is free content that is also provided with no cost or registration, and unlike many of the other external links in this article it doesn't violate the "Any site that does not provide a unique resource beyond what the article would contain if it became a Featured article." bullet here. — Carl (CBM · talk) 20:13, 7 September 2008 (UTC)

It is indeed painful for me to admit, but the chapter 5 video is a worthwhile asset (after the narrator finishes stroking himself for the first 1/13th of it); it's always unsettling to reward what seems like bad behaviour, but i suppose the content makes the link worth keeping.... Quaeler (talk) 05:25, 8 September 2008 (UTC)

Definition

I'm sure people are aware that it is possible to define the set of complex numbers C without appeal to phrases such as "structure" or "number system" or "let's introduce a new object called i so that i^2=-1". Even the most anti-formal person would admit this. Even if we want to start off with a little bit of friendly hand waving, shouldn't a precise definition be given at some point down the page? Maybe in a section entitled "formal definitions of C"? It seems ridiculous that some sort of OTT fear of being seen to prefer one possible definition ("Nazis!, NAZIS!") over another leads to no definition being given at all.

I am only familiar with one (in terms of ordered pairs of real numbers), and assumed that was *the* rigorous definition, up to exchanging the real and imaginary axes. This would be in contrast to Cauchy sequences and Dedekind cuts and so forth for real numbers. But however many possible definitions there are, could we not just list a few and point out that there are structure preserving maps between the sets given by the various definitions?

Dan, 22:12, 9 September 2008

The formal definitions you are looking for are in the "Algebraic characterization" and "Characterization as a topological field" sections. The definition sketched in the lede is one of the formal definitions of the complex numbers: they are the unique field extension of the reals with a root of x^2+1.

There's no way, in the end, to define the complex numbers without referring to their structure. An ordered pair of real numbers stands for an element of R², an element of C, and an element of RP¹ among other things. What makes the complex numbers different than those other structures is its algebraic structure, not the fact that complex numbers can be represented as pairs of real numbers. The "friendly hand waving" is to not link field extension to the word extension in the first paragraph. — Carl (CBM · talk) 21:47, 9 September 2008 (UTC)

I don't think the characterizations you point out really answer the question. Yes they characterize C. (Well, I assume they do, I have never seen these particular characterizations and no reference in this section is given.) But at some point one must answer the question "Why does there exists a set satisfying these properties?" In just the way that showing the axioms for the reals makes them unique does not substitute for a construction for the reals. I think it would be good to have a section with some formal definition. I also object to the term unique above, it's only unique up to field isomorphism. Thenub314 (talk) 08:22, 10 September 2008 (UTC)

What more can you expect of a construction in the category of fields? I see what you mean though. I edited the algebraic characterization section to describe the construction. I doubt that many people spend too long wondering whether the complex numbers actually exist, since naive readers won't think of that at all and more experienced people will be familiar enough with abstract algebra to know that C is the algebraic closure of the real numbers. — Carl (CBM · talk) 12:48, 10 September 2008 (UTC)

I just expect something like: "Formally C can be defined as ordered pairs of real numbers with the following algebraic operations (x,y)+(u,v)=..." etc. This construction always seemed to me to carry the least background necessary. That is, it would make sense even if you hadn't taken an abstract algebra course. But I think your expansion of the algebraic characterization is quite nice. Would two constructions be over kill in the article? Thenub314 (talk) 14:17, 10 September 2008 (UTC)

They're literally the same construction, just expressed in different words. I'd rather expand the text currently in the article to explain in more detail, rather than have the same construction twice. One reason I tend to avoid the "ordered pairs" language up front is that it emphasizes the coordinates very early in the construction, when in fact the coordinates (in particular, the choice of i vs.-i) are somewhat arbitrary.

The article does give a very long description of the arithmetical operations on the complex numbers, of course, in the first two sections, using the normal a+bi notation. — Carl (CBM · talk) 14:29, 10 September 2008 (UTC)

Fair enough, they are essentially the same construction. Except to understand the algebraic construction I need to know (or read about): fields, field extensions, irreducibility, and quotient rings, before I would have a sense of what C was. I think ordered pairs are a bit more accessible. I agree one should make the point that i and -i are in some sense indistinguishable. But I think it is a point that could easily be explained. Thenub314 (talk) 15:02, 10 September 2008 (UTC)

A reader who wants to get a sense of what C is can do so by reading the first several sections of the article. Those explain complex numbers in great detail, and already include the formulas for ordered pairs in the "Operations" section, as a lead-in to discussion of the complex plane... — Carl (CBM · talk) 16:58, 10 September 2008 (UTC)

Sure. It is a good article. The only question that wasn't answered before this discussion is what if someone came to the page looking for the answer to "What is the formal definition of the complex numbers?". Now there is an answer to this question in the algebraic characterization section. I just think there is some value off setting this a bit more so the information can be more clearly seen as a definition. I also think there is some value by taking the more pedestrian approach of ordered pairs. I agree they may i seem a bit too unique. But I think the benefit of being more accessible out weighs this problem. Thenub314 (talk) 06:50, 11 September 2008 (UTC)

Topological characterization

I put a fact tag on this only because it's sufficiently esoteric that someone might be interested in looking it up but unable to find it in elementary texts. The explanation below the characterization is perfectly reasonable.

Also, it seems to me that the line:

• If S is any nonempty subset of P, then S+P=x+P for some x in C.

is more clear as

• If S is any nonempty subset of P, then S+P=x+P for some x in P.

They're equivalent, because of the other assumptions on P, but the second one expresses the idea better I think. — Carl (CBM · talk) 13:17, 10 September 2008 (UTC)

Somehow Dedekind completeness of the real subfield seems to have been left out of the definition. Otherwise, doesn't Q[i] satisfy these properties? Or am I just not seeing something. siℓℓy rabbit (talk) 23:52, 23 November 2008 (UTC)

This is the part that achieves that:

"If S is any nonempty subset of P, then S+P=x+P for some x in P."

Intuitively, for example, let S be the set of elements z of P such that z^2 < 5 z^2 > 5. Then the x from the quoted statement must be the square root of 5. The same is true when S is any other Dedekind cut of P. — Carl (CBM · talk) 03:07, 24 November 2008 (UTC)

I see. Yes, that makes sense. Thanks, siℓℓy rabbit (talk) 03:14, 24 November 2008 (UTC)

I need to change the example to say S is the set of z in P such that z^2 > 5. The point is that since inf(P) = 0, inf(P+S) = sqrt(5), which is why x has to be sqrt(5). Sorry for the typo, — Carl (CBM · talk) 04:39, 24 November 2008 (UTC)

(unindent) I hadn't even looked closely enough at your example to notice. Anyway, I have a second criticism to float: this is not really a characterization of C as a topological field. The definition given in this section boils down to saying that C is a field containing R as a subfield, together with a nontrivial involution whose set of fixed points is exactly R. Now for it:

• Why doesn't the article just come out and say it? We know what R is, whether it is constructed by Dedekind cuts or Cauchy sequences. There seems to be no advantage in trying to reinvent the wheel by effectively saying that R must be a subfield of C, but in much more obscure and unfamiliar way. Instead this definition seems more likely to lead to confusion, and should be replaced with a version which comes right out and says what it is all about.
• This isn't really a characterization of C as a topological field. Such a characterization should, I think, begin: "Let F be a topological field such that..." How this sentence is completed may depend on what reliable sources tell us. I can think of a few plausible ways to do it. For instance, "...the topological closure ${\overline {E}}$ of the prime subfield E of F is a linear continuum, and $F\backslash {\overline {E}}$ has two connected components."

-- siℓℓy rabbit (talk) 13:24, 24 November 2008 (UTC)

The first bullet is fine with me. The second seems to be just semantics - whether you want to say "C is the only field such that..." or "Any field E such that ... must be C" is just a matter of taste. — Carl (CBM · talk) 13:51, 24 November 2008 (UTC)

Sorry if I gave the impression that it was just semantics. What I mean to say is that there should be conditions on the toplogy of a field that one can impose which will ensure that the field is isomorphic as a topological field to C with its usual topology. Currently, the "characterization" section gives a perfectly fine characterization of C (modulo my first bullet point), but one which seems to have very little to do with topological fields in general. What I more envision is something along the lines of, e.g., in the theory of Lie groups one knows that a compact Hausdorff topological group with no small subgroups is in fact a Lie group. This is what I would call a characterization of a compact Lie group as a topological group. Presumably some analogous statement exists for C over R, but the current statement (or even the statement modified to accord with my first point) seems to fall short of the mark. siℓℓy rabbit (talk) 20:18, 24 November 2008 (UTC)

More: The current logical structure of the section is: here is this field with these (not-so-very-topological) properties. Then the field turns out to be C, and we can write down a topology. This puts the cart before the horse. It's like having a section entitled "Topological characterization of normed spaces" in which you define a normed space to be a space with a norm. Taulogically correct. But not topological. A topological characterization of Banach spaces requires some analytic conditions on the Minkowski functionals of a convex balanced absorbing open set (in particular, such a thing needs to exist in the first place). siℓℓy rabbit (talk) 21:28, 24 November 2008 (UTC)

So it's a characterization as a topologizable field instead of a topological field? I didn't add the characterization, but I don't think it's so off topic that it needs to be removed. I don't know whether we can find any better characterizations that are actually sourced somewhere. — Carl (CBM · talk) 23:57, 24 November 2008 (UTC)

Comments.

In the section on "Geometric interpretation of the operations on complex numbers" there is the sentence:

For example, the problem of the geometric construction of the 17-gon is thus translated into the analysis of the algebraic equation x¹⁷ = 1.

It is not clear to me what is supposed to be meant by this. Specifically what makes 17 special in this context, if there is something that is different from x¹⁷ = 1 as opposed to xⁿ = 1 we should say what it is and include a reference.

In the section on "Multiplication, division, exponentiation, and root extraction in the polar form" there is the phrase:

The addition of two complex numbers is just the vector addition of two vectors, …

But this has nothing to do with the operations listed in this section, nor does it have much to do with polar form, and really belongs elsewhere. Perhaps in either in the section on Geometric interpretation. Thenub314 (talk) 13:34, 9 October 2008 (UTC)

ordered field

Someone asked why C cannot be an ordered field. This is really a basic fact of abstract algebra. As the article on ordered fields points out, -1 is always negative in an ordered field. If i were positive, then i^2 = -1 would show that two positives multiply to make a negative, a contradiction. On the other hand, if i were negative, then -i is positive, and so (-i)^2 =-1 shows there are still two negatives that multiply to make a positive. So there's no way to assign i as either negative or positive that is consistent with the requirements of an ordered field. Indeed, the article ordered field already included a justification why C can't be ordered. h— Carl (CBM · talk) 12:41, 24 October 2008 (UTC)

Is conjugation commutative or distributive

~(a+b) = ~a + ~b seems like a distributive law to me, but the article implies the commutative law ("That conjugation commutes with all the algebraic operations ..."). —Preceding unsigned comment added by 194.67.106.32 (talk) 13:49, 26 November 2008 (UTC)

Yes, two operators P and Q commute, when [P,Q] = 0, i.o.w. when for all x: P(Q(x)) - Q(P(x)) = 0, i.o.w. when for all x, P(Q(x)) = Q(P(x)). In order for the phrase to make sense, we need P as addition and Q as conjugation, but in one case with conjugation of one element, and the other case of two elements, which is a bit silly.

So I guess we need to write "... conjugation distributes over all the algebraic operations". I'll make the change. - DVdm (talk) 18:23, 26 November 2008 (UTC)

Let's not confuse the reader

IMHO the current use of the word "optionally' in the lead is more likely to confuse than elucidate. If it is really felt necessary in a purist way then it should surely be confined to the body of the article. Abtract (talk) 23:29, 30 December 2011 (UTC)

The imaginary part is not optional. A real number is not a complex number, it is still a real number. This business about optional is confusing how complex numbers are written and what they are. Complex numbers with a zero imaginary part are written without the complex part but they are still complex numbers, it is just that such complex numbers can be treated the same as reals in most circumstances. Dmcq (talk) 23:47, 30 December 2011 (UTC)

Dmcq, also referring to this edit summary: "Imaginary part is not optional. A real number is not a complex number see talk": We say that the set of all complex numbers is denoted by C or $\mathbb {C}$. Surely you don't claim that $5\notin \mathbb {C}$ and that $\mathbb {R} \nsubseteq \mathbb {C}$? Of course every real number is also complex.

Anyway, the result of your edit is good now, but only provided that the definition of and at imaginary number doesn't change anymore. It is defined now as allowed to be zero, so the word "optional" is indeed not needed anymore in our lead here. As soon as the definition overthere changes again (by forbidding zero as an imaginary number) we need to come back here and add "(optionally)" for the imaginary number bit again.

But again... a real number is also a complex number: $\mathbb {R} \subset \mathbb {C}$. Get Real, please (pun intended). - Cheers and happy 2012! - DVdm (talk) 10:24, 31 December 2011 (UTC)

Yes of course I claim that a real number is not a complex number. A complex number is a pair of real numbers. Complex numbers can also be viewed as a field extension of reals but that is still different from the reals, what one can do is identify elements of one set with the other and one normally does that. People automatically use order relationships only with real numbers and a^z is a well defined single valued function if a is a real number, complex numbers aren't ordered and the exponentiation would have multiple possible values if a was complex - even if the imaginary part was zero. Have a look at Construction of the real numbers and tell me which of those is the same as any definition here of a complex number. Saying a real number is a complex number is like saying the counting numbers are the same as your fingers. Dmcq (talk) 11:25, 31 December 2011 (UTC)

So you agree that $\mathbb {R} \subset \mathbb {C}$ but a real number is not a complex number???

Look: $\mathbb {R} \subset \mathbb {C}$ is short for $\forall x:x\in \mathbb {R} \implies x\in \mathbb {C}$, in words, if x is a real number then x is a complex number.

Surely you must be joking??? - DVdm (talk) 11:53, 31 December 2011 (UTC)

No I am not joking. There is a subset of the complex numbers that can be treated like the reals but that's it. The complex numbers are pairs of real numbers. If the reals were complex numbers then complex numbers would be pairs of complex number for instance. Dmcq (talk) 12:35, 31 December 2011 (UTC)

Your "pair of real numbers" (x,y) is an element of the product set $\mathbb {R} ^{2}$ and $\mathbb {R} ^{2}\neq \mathbb {C}$. There is an isomorphism (check the article) between these sets ($\mathbb {C} \cong \mathbb {R} \cdot 1\oplus \mathbb {R} \cdot i=\mathbb {R} ^{2}$), but they are not equal, so a pair of real numbers is not a complex number. It is a pair of real numbers. - DVdm (talk) 12:53, 31 December 2011 (UTC)

Surely there is no one right or wrong answer to this - it depends on the particular formalization being used. We shouldn't nail the article to one particular approach when we know that different authors present these things in (formally) different ways (that all essentially come to the same thing).--Kotniski (talk) 14:06, 31 December 2011 (UTC)

See Complex number#Formal construction where they are defined as pairs of real numbers plus rules for how they are combined. There's other representations too as matrices for instance. I guess each definition defines something different. If we have a categorical definition of them and just say these represent it if they are isomorphic then we could have the reals as a subset of the complex numbers after leaving out exponentiation I guess, I'm not sure how all that works but it strikes me as probably a viable top down way of doing things rather than going from the axioms up. Personally I think of it as a programmer and say double or complex or whatever and type them all differently. Dmcq (talk) 14:12, 31 December 2011 (UTC)

I also notice that Kotniski added the "(either of which may be 0)" phrase again. It is of course correct (at least with the current status at imaginary number), but not needed anymore.

So... riding my hobby horse again... will this ever end? No, not until Imaginary number is merged into Complex number i.m.o... :-) - DVdm (talk) 10:39, 31 December 2011 (UTC)

I don't see any particular problem with having two separate articles. However I would move all the detailed maths either to the complex number article or the imaginary unit article, and leave the imaginary number article as a friendly article explaning the concept in fairly simple terms (and pointing out the slight discrepancies between definitions, without getting bogged down). I would also prefer at least the first sentence of the complex number article to be free of algebra, as it was when I first saw it - giving an intuitive (but not inaccurate) description of what a complex number is.--Kotniski (talk) 12:24, 31 December 2011 (UTC)

I've got rid of the imaginary number business and just use i just like most books do as per what I was suggesting in the previous section. Dmcq (talk) 12:35, 31 December 2011 (UTC)

I think I'm going to stop looking at (or at least editing and commenting to) this article (and cousins) for a while. When I notice on my watchlist that they have been stable for a week or so, I'll come and check again. Cheers to all and don't forget to enyoy the end of the year! - DVdm (talk) 13:04, 31 December 2011 (UTC)

I like the current version

A complex number is a number which can be put in the form a + bi where a and b are real numbers and i is the imaginary unit such that i2 = − 1.[1]

That is nice as it deftly avoids the contentious question of whether it has to be in that form or whether it can optionally be put in one or another abbreviated form. Duoduoduo (talk) 19:14, 31 December 2011 (UTC)

I like it too. Abtract (talk) 19:29, 31 December 2011 (UTC)

I like it in terms of accuracy, but not in terms of user-friendliness - the first sentence ought to give some kind of general indication, in terms that a relatively mathematically unsophisticated reader will readily relate to, what it is that these complex numbers are. I would prefer to postpone any algebra at least until the second sentence, after we've said something about it being a sum of a real number and an imaginary number (with such provisos as to make the statement not mathematically wrong).--Kotniski (talk) 08:37, 2 January 2012 (UTC)

The imaginary unit is the basic thing people have to work their heads around when starting with complex numbers. Imaginary numbers are a practically useless idea and confusing as the term has often been used to refer to complex numbers. Dmcq (talk) 13:01, 2 January 2012 (UTC)

The newest version starts out with

A complex number is an ordered pair (a, b) of real numbers a and b, referred to as the real part and the imaginary part, respectively.

I don't think this is at all user-friendly for someone who knows little or nothing about complex numbers other than that they look like 3+2i. An ordered pair? I think that starting out like that, with a concept the typical reader has never heard of, is likely to discourage the non-mathematician from reading further. I think it should start out with the familiar. Maybe start out with an example like 3+2i rather than a+bi? Duoduoduo (talk) 17:17, 2 January 2012 (UTC)

Yes, this is getting worse and worse. I always used to wonder how the maths articles on Wikipedia got to be in such an unhelpful state - now I can see the process in action, I can understand it. People are so hooked on making everything rigorously correct (which, for mathematicians, is understandable) that they forget their audience; formal errors are vigorously corrected, while matters of comprehensibility take a lower priority, and we have a gradual drift away from something readers might understand to something that can only be appreciated by people who, basically, already know it.--Kotniski (talk) 17:34, 2 January 2012 (UTC)

I'm not a mathematician; my intention was simply to go straight to the geometrical interpretation. A complex number is just a pair of coordinates in the complex plane. I thought it was clear what a pair of real numbers is, whereas starting with a formula containing roots of negative numbers might not be very user-friendly according to Kotniski above. However, if people are uncomfortable with this we can go back to the description as a number on the form a+bi. Isheden (talk) 19:38, 2 January 2012 (UTC)

The definition as a pair is correct and is used in formal definitions but it is not the usual definition used in introductory texts. I believe it should be left till later in the article like it is in the formal definition section. The a+bi definition is I believe the most common simple definition. Dmcq (talk) 19:42, 2 January 2012 (UTC)

Gets my thumbs up with both definitions/forms there in the lead and it shows where the accompanying Argand diagram comes from. Dmcq (talk) 21:52, 2 January 2012 (UTC)

a + bi (or ai + b) is necessary so the two-dimensional complex plane supports addition, multiplication, and division -- so it's a full-fledged dimension. The structure of i, or j, k, or sqrt -1, operates through multiplication. Merely adding i (essentially the sequence of -1, 1) to a number has no impact. Only multiplication changes the sign, i's key function. To have the complex plane behave normally, i is incorporated, via multiplication, in a linear function that uses addition: a + bi.

Brian Coyle — Preceding unsigned comment added by 208.80.117.214 (talk) 03:43, 16 May 2014 (UTC)

Use of a+bi in first diagram

The normal form of presenting a complex number, today, is to write a + ib and NOT a+bi.--Михал Орела 14:46, 29 September 2012 (UTC) — Preceding unsigned comment added by MihalOrela (talk • contribs)

Do you have any sources for this? I've always been taught that it's simply a matter of choice, nothing more, and that both forms are used. 91.156.57.136 (talk) 15:56, 17 December 2012 (UTC)

In all my time, at school, at 3 universities, and in books I've read, the form used by pretty much everybody has been a+bi, not a+ib - in my experience this is rare Mmitchell10 (talk) 17:00, 17 December 2012 (UTC)

Since b and i commute, order does not matter. However, b may be a long expression, in which case writing i out front makes clear that a complex number is being specified, not simply the sum of two real numbers.Rgdboer (talk) 21:39, 25 April 2013 (UTC)

• My experience is that a+bi is usual. A Google search for "a+bi" gave me 56,600,000 hits, and "a+ib" only 2,650,000. The editor who uses the pseudonym "JamesBWatson" (talk) 10:53, 16 May 2014 (UTC)

a + bi ?

We use the conventions a + bi and x + iy in this article to refer to general complex numbers. Would it not be best to stick to one convention? Martin Hogbin (talk) 14:55, 9 June 2014 (UTC)

It seems pretty consistently to be the a + bi ordering. The exceptions are where the other convention is being mentioned, and when a function is involved, when the function follows the i. Perhaps I missed some? —Quondum 16:02, 9 June 2014 (UTC)

It is not just the ordering but the letters used. Martin Hogbin (talk) 16:04, 9 June 2014 (UTC)

Square of the absolute value

What's the square of the absolute value called? Is it the same as |x|, which is used this in article but not explained? Olli Niemitalo (talk) 17:43, 27 July 2014 (UTC)

The absolute value, or magnitude, of a complex number is its distance from the origin, written as $|x|$. The square of the absolute value is $|x|^{2}$. Does that answer your question? --gdfusion (talk|contrib) 21:04, 27 July 2014 (UTC)

It does not seem to have a common name, other than "the squared magnitude". —Quondum 22:02, 27 July 2014 (UTC)

If we need one word, I propose we vote between "sqagnitude", "squodulus", and "sqorm" :-) - DVdm (talk) 07:52, 28 July 2014 (UTC)

Thanks all. I put it (but not those funky names) in the article . Olli Niemitalo (talk) 10:27, 28 July 2014 (UTC)

Weasel words

User:Wcherowi. According to WP:WEASEL, the section Notation consist of weasel words and it may be tagged with cleanup teaplates. Verbatim quote: “Articles including weasel words should ideally be rewritten such that they are supported by reliable sources, or they may be tagged with the {{weasel}} or {{by whom}} or similar templates so as to identify the problem to future readers (who may elect to fix the issue).“. Please note that your opinion stated in Wikipedia isn't a reliable source. According to the policy on verifiability “all quotations and any material challenged or likely to be challenged must be attributed to a reliable, published source using an inline citation.”. Per this policy, I'm restarting the cleanup tag. Free feel to remove it once reliable sources have been provided. Given that according to you “Just about everyone uses that form” it seems that you won't have trouble finding reliable sources to support this claim. Until this is done, please don't remove the cleanup tag, again, per the quoted policy. I have searched a couple of books to which I have access, to see if I could fix these weasel words, but they all use the notation a+bi; for instance Tom M. Apostol Mathematical Analysis. Regards. Mario Castelán Castro (talk) 14:41, 6 October 2014 (UTC).

It is trivial to find examples of authors using a+ib, so I fail to see what use the template has. The weasel word guideline (not policy) is not a moratorium on the use of certain words, but rather an injunction against using those words to create a false impression of authority. In this case, there is no such false impression: a great many authors do indeed use this alternative convention. In any event, I have referred to the standard classic "Complex Analysis" by Lars Ahlfors. Sławomir Biały (talk) 15:13, 6 October 2014 (UTC)

Thanks for providing a reference. I have made it link specifically to the full reference in the article References section so that it's clear. Mario Castelán Castro (talk) 16:50, 6 October 2014 (UTC)

Perhaps I was a bit "flip" in my edit summary, for which I apologize, but I believe that this highly non-controversial statement does not require a source. The article itself uses both forms (especially the diagrams), and the references, that I checked, uniformly used z = x + iy. The predominance of this form is likely due to the fact that when the imaginary part of a complex number is given by a real valued function (such as a trig function) putting the "i" first cuts down on the likelihood of misinterpreting the expression. However, due to the commutivity of multiplication, it doesn't matter how this is written and authors are free to do as they like, and they do. My main objection is that by sourcing this statement it makes it appear as if there is some controversy concerning the notation when there isn't. A better way to deal with the weasel word issue would be to rewrite the sentence without it – for example – By the commutivity of multiplication, complex numbers are written as either a + bi or a + ib. Bill Cherowitzo (talk) 17:30, 6 October 2014 (UTC)

I have no objection to removing the reference altogether. As I already indicated, it is trivially verifiable by consulting any of a number of standard reference textbooks. Sławomir Biały (talk) 19:05, 6 October 2014 (UTC)