Talk:Complex number/Archive 4

If there's any such 'i', then '-i' satisfies i^2 = -1 too. You are either stating that i = -i (the complex number plane folds to a half-plane), or (if i != -i) you must choose only one of the two roots, and label it 'i'. But since both roots have the same properties, there is no way to do it without a coin toss. And I don't have a coin. — Preceding unsigned comment added by 213.175.41.130 (talk) 12:01, 17 May 2019 (UTC)

You're right. However, it's a fundamental fact of the mathematics. See Galois theory. Mgnbar (talk) 12:25, 17 May 2019 (UTC)

More elementary: The sentence "i is an indeterminate satisfying i^2 = −1" means: as the equation ${\displaystyle x^{2}=-1}$ does not has any real solution, and as many things would be simpler if it would have a solution, one creates a symbol i such that (formally) ${\displaystyle i^{2}=-1.}$ A simple computation shows that this definition implies that ${\displaystyle -i}$ is another solution of the equation. Therefore everything that can be done with i can also be done by replacing everywhere i by ${\displaystyle -i.}$ This is the complex conjugation, which makes i and ${\displaystyle -i}$ undistinguishable.

In other words, the quoted sentence is a definition that is not ambiguous. But this sentence must not be confused with "i is a square root of –1". The latter is a property, that cannot be taken as a definition, because, as you said, this would be ambiguous. I am not sure whether all textbooks make the distinction clearly. D.Lazard (talk) 12:51, 17 May 2019 (UTC)

@D.Lazard: It seems that you are using the term "indeterminate" in its colloquial sense of "one does not know anything". You even use it in the meaning of "undefined", so e.g. in "The polar angle for the complex number 0 is indeterminate". This is confirmed by your colloquially saying in your response above

The sentence "i is an indeterminate satisfying i^2 = −1" means:
as the equation ${\displaystyle x^{2}=-1}$ does not has any real solution ... one creates a symbol i such that ${\displaystyle i^{2}=-1.}$

This says: an indeterminate is a symbol created for satisfying a certain relation, doesn't it? (Indeed, i is created for satisfying a certain relation. But then it is not an indeterminate!)

And as you can see in the article indeterminate, the term has quite a precise meaning. If X is an indeterminate then R[X] is a polynomial ring^[1] and there is the isomorphism

${\displaystyle {\begin{array}{llll}\varphi \quad \colon \quad &\mathbb {R} [X]&\to \quad &\mathbb {R} ^{(\mathbb {N} _{0})}:={\bigl \{}\left(a_{k}\right)_{k\in \mathbb {N} _{0}}\mid a_{k}\in \mathbb {R} ,a_{k}=0\ {\text{ for almost all }}k{\bigr \}}\\&1&\mapsto &(1,0,0,0,0,\dotsc )\\&X&\mapsto &(0,1,0,0,0,\dotsc )\end{array}}}$

The section Complex number#Construction as a quotient field (and also the section Complex number#Construction as ordered pairs without any X) explain the matter without using the term "indeterminate" at all.

But in general, the article is not in a very good shape. Besides the misuse of the term "indeterminate" there are many almost-repetitions etc etc etc. So I'm so happy to leave its amelioration completely up to you. --Nomen4Omen (talk) 11:34, 10 December 2019 (UTC)

I don't see a substantive conflict here. One forms the polynomial ring R[x], in which x is an indeterminate. And this word "indeterminate" is used (in every source I can remember, although I'm traveling right now and can't cite them) to emphasize that x is not (and does not represent) any element of R. So at a certain point of the construction it is correct, verifiable, and useful to say that x is an indeterminate. Then one mods out by the ideal (x² + 1) to obtain C. And of course we use i instead of x for historical/cultural reasons.

This is all carried out in a later section of the article. The earlier Definition section is arguably a bit loose and intuitive. I would support tightening that section up. Mgnbar (talk) 14:03, 10 December 2019 (UTC)

@Mgnbar: As already said: I agree with your last "This is all carried out in ...".

What I say is: The out-modded x is no longer an indeterminate. (An indeterminate is always algebraically independent. And i is of course algebraically dependent.)

Btw, one needs a proper name, and i is one, similar to e (Euler's constant). x is not considered a proper name, but it can be used as name of an indeterminate. --Nomen4Omen (talk) 18:50, 10 December 2019 (UTC)

The term, indeterminate, is traditionally used for a solution not necessarily in the original field of coefficients. Since we don’t know what or where it is, we call it an indeterminate—not a variable.—Anita5192 (talk) 19:10, 10 December 2019 (UTC)

OK, if "indeterminate" can be used in the meaning of "undefined", so e.g. in

"The polar angle for the complex number 0 is indeterminate",

then almost everything is possible. --Nomen4Omen (talk) 08:47, 11 December 2019 (UTC)

No,"indeterminate" (as an adjective) and "undefined" have slightly different meanings. "Undefined" means "has not been defined", or, sometimes, specifically in mathematics, "cannot be correctly defined". Moreover, "undefined" is never used as a noun. On the other hand, "indeterminate", as an adjective, means "whose numerical value cannot be deduced from the general definition". This is the case of the polar angle of zero: if one apply to zero the definition of a polar angle, one has to compute ${\displaystyle \arctan {\frac {0}{0}},}$ which involves an indeterminate form. As a noun, "indeterminate" has a different meaning, and refers to a symbol that is considered and manipulated independently of any numerical value. As such, it differs from a variable, which is a symbol that can represent any numerical value. So, i is an indeterminate (noun), but this does not mean that it is indeterminate (adjective). D.Lazard (talk) 09:51, 11 December 2019 (UTC)

@D.Lazard: Why then is the text "The polar angle for the complex number 0 is indeterminate, but arbitrary choice of the angle 0 is common." still in the article ? --Nomen4Omen (talk) 09:53, 11 December 2019 (UTC)

As said above, "indeterminate" (adjective) means that the value cannot be deduced for the general definition (here, of a polar angle). But this does not mean that a definition for the polar angle of zero cannot be given, for convenience. The sentence, means that when one chooses to attribute a conventional value to the polar angle of zero, a common choice is zero. D.Lazard (talk) 10:42, 11 December 2019 (UTC)

Does any reliable source refer to i as "indeterminate" or "an indeterminate" (after the modding-out)? I'd rather not guess, but I'd guess not. Mgnbar (talk) 15:01, 11 December 2019 (UTC)

I think this pursuit is pretty silly. The term is defined clearly in the article Indeterminate (variable), which is linked in the lead.—Anita5192 (talk) 17:15, 11 December 2019 (UTC)

You may be right in essence, Anita5192. But it is even sillier to mix up a mathematical ring with a wedding ring, only because both are called ring. What I mean: we should be careful in our wording for our readers' sake. --Nomen4Omen (talk) 16:51, 14 December 2019 (UTC)

The Exponentiation section begins as follows:

Euler's formula

Euler's formula states that, for any real number x,

${\displaystyle e^{ix}=\cos x+i\sin x\ }$"

This is already confusing two notations: the one for exponentiation of a complex number e to the complex power ix, and the notation for the application of the exponential function exp(z) to the complex number ix. This is a very bad way to begin the section on exponentiation.50.205.142.35 (talk) 15:10, 17 January 2020 (UTC)

What specifically is your objection? Is it that both notations, ${\displaystyle e^{ix}}$ and ${\displaystyle \exp(z)}$, are used, or that ${\displaystyle e^{ix}}$ is defined before ${\displaystyle \exp(z)}$, or that there are two sections for exponentiation, or something else?—Anita5192 (talk) 16:28, 17 January 2020 (UTC)

Good point for the IP: The whole section was a mess, with circular definitions and even no clear definition of complex exponentiation. I have thus completely restructured the section. In particular, I have split it into a section "Exponential function" and a section "Exponentiation". I have aslo removed the proofs that belong to another article. I hope that the result is clearer. D.Lazard (talk) 18:54, 17 January 2020 (UTC)

I think it is much better now. Thank you for cleaning it up. —Anita5192 (talk) 19:10, 17 January 2020 (UTC)

As a reader interested in the history of the Complex Numbers, I found quite confusing the History section, and I believe that rewriting it in a more chronological order would help to understand how the idea has evolved through centuries. Of course we don't need to be strict in that sense, but at the moment the text starts with Cardano -> Bombelli -> Hamilton (1545 -> 1843), then jumps back to 1st century, then again Cardano, then Euler and his book (1770), back to De Moivre (1730) and then again Euler but in 1748. — Preceding unsigned comment added by Brunko R. (talk • contribs) 05:42, 15 September 2020 (UTC)

I agree, this can definitely be improved. If you are up to it, please go ahead! Jakob.scholbach (talk) 08:05, 15 September 2020 (UTC)

This article is repetitive. How many versions of the pic "C is the complex plane" does one need.? How many times do you have to define RE.?

There should be only a few topic-dedicated pics: "C is the complex plane", "the complex conjugate", "the polar representation".

Why not having the most simple definition based upon that repeated pic "C is the complex plane" as foundation of the article.?

Full of amateurish subsections (equality: not wrong, but so obviously obvious that no-one needs that in writing).

Mathematically out-dated: the article clings to a "i = a formal variable with i^2=-1" approach to C while "C=IRxIR plus a new multiplication" which is taught at university introductory algebra and analysis courses is suppressed.

Mathematically incomplete: discussion of C as a field.

Mathematically amateurish: discussion of arg: should be limited to saying that the complex plane needs to be cut by a ray from 0, e.g., -[0,oo]. For the sake of efficient presentation. Plus once "mod 2Pi" remark. Arctan/atan2/etc can be discussed elsewhere, e.g., on the arctan/atan2 pages.

Treatment of EXP is a nightmare.

Multiplication in polar form should use EXP.

Holomorphic functions is a nightmare.

No-one needs these colorful shiny self-satisfying "graph pics".

History should be a separate article. History's timeline is a mess.

Applications should be a separate article.

LMSchmitt 23:04, 19 September 2020 (UTC)

You make some good points, but it also covers way too many disparate issues to be useful and comes across at least to me as just, "this bad." This article does need a restructuring though, so before we start to look at the quality of the content *within* the sections, might you first suggest a re-sectioning of the article? _{Integral Python} ^{click here to argue with me} 17:56, 20 September 2020 (UTC)

The article has been reset by an editor such that:

a) the most simple definition C=IRxIR plus a new multiplication is suppressed (only mentioned later in the article)

b) The definition as linear polynomials is obviously not understood, no one needs to handle degree 3,4... polynomials in "i".

c) The so-called formal definition is not thee only one. That's possibly the classical one, but also the least simple one.

I had taken the time to list 3 equivalent definitions including the most simple one which is taught at university in beginner's courses. I had let stay other people's contributions while they do not reflect modern university teaching in analysis courses. The indeterminate definition is overly complicated. The "formal" definition is "algebra", but is not needed in an "analysis" course. However an editor (which in my professional opinion (Prof.Dr.rer.nat.habil. (math)) lacks understanding of the matter, not a personal attack ;-) ) has reset my edit (structuring these definitions) to previous, bad and slightly false efforts.

I will revert these reverts by the editor, if this post does not trigger strong arguments FOR the editor's reversal within 24h.

See the "Definition" section in "my" old version: https://en.wikipedia.org/w/index.php?title=Complex_number&oldid=979141833

LMSchmitt 23:04, 19 September 2020 (UTC)

PS: I found an error in my writing "degree 1" should be either "degree 0,1" or "degree at most 1".

LMSchmitt 01:39, 20 September 2020 (UTC)

What you added was poorly written, and made the first main section of the article very difficult to wade through. Right off the bat, you've thrown around terms like "isomorphism" and "vector space", which are completely unnecessary for a basic definition. See also MOS:NOTED – while "Note that ..." is pretty typical in mathematical writing, it shouldn't be used here. What you added also essentially duplicated material that's already given later in the article.

Frankly, I do think what's there isn't ideal, and it the article could do with some restructuring. Saying that complex numbers are polynomials isn't wrong, but it's more meaningful to say they're equivalence classes of polynomials. Of course, this is just what's being done with the quotient of ${\displaystyle \mathbb {R} [x]}$, but starting out this way is too technical. What's probably best is to just present them as usual as numbers of the form a + bi, right along with a description of the various arithmetic operations on them. This can segue into a discussion of the main arithmetic properties – that multiplication is associative and commutative, etc.

This article is definitely suffering from bloat over years of fiddling and it could really use a makeover. However, your edits made the article worse, not better, and so reverting them was appropriate. Trying to saddle readers with multiple definitions right off the bat, especially ones that are only superficially different (like presenting as ordered pairs, which just isn't an enlightening or useful way to work with complex numbers) isn't a good approach.

On a side note, just saying "this isn't a personal attack" doesn't make it so. Calling into question my understanding of the material before you've even seen any discussion from me kinda is one, disclaimer or no. Moreover, this sort of "explain yourself within 24 hours and if I'm not satisfied I'm going to put my edits back" approach just isn't how things work around here. It's the weekend, and there's no deadline. This is a high-visibility article, and a lot of other folks are going to see this. It's worth giving this some time and seeing if other people want to weigh in.  –Deacon Vorbis (carbon • videos) 00:29, 20 September 2020 (UTC)

Sorry. Apologies, if my statement was understood as a personal attack.

LMSchmitt 01:32, 20 September 2020 (UTC)

Setting aside the personal attack, there is some value in adding the R^2-with-multiplication definition to the Definition section. Ideally it would be supported by Wikipedia:Reliable sources rather than some lecture notes on a mathematician's personal web page. Mgnbar (talk) 01:43, 20 September 2020 (UTC)

Dear Deacon Vorbis, I will copy your writing and then COMMENT on it.

DV: What you added was poorly written, and made the first main section of the article very difficult to wade through.  

COMMENT: No: it was written from a modern 21rst century understanding of teaching C and handling C. This is not poorly written, though it may be improvable. (see my lapse PS above).

What I added (mainly Definition 1) made the understanding of C simpler.

It boils down to a familiar set from High School, the plane IR^2, and a new multiplication of points in IR^2 with easily understandable operations (+,*).

DV: Right off the bat, you've thrown around terms like "isomorphism" and "vector space", which are completely unnecessary for a basic definition.  

COMMENT: -- isomorphism: yes one could just write that these 3 definitions yield the same (as I have) or identical objects. I prefer telling people the truth that there is an isomorphism (the proper wording). In case of defs 1<->2, it's (a, b)-> a+ib. -- vector space: I find your comment a contradiction in itself. You allow and like terms like "polynomial" and "indeterminate variable". "polynomial" and "vector space" are taught in high school, and it is a natural assumption that a reader which is supposed to understand "polynomial" also understands "vector space". From my 40y teaching experience, I know that an "indeterminate variable", just a symbol that "miraculously" is declared a number to be a much more difficult concept to grasp. Using the vector space structure of IR^2 is also efficient in that one saves the explicit formulation of addition in C. -- In essence, in my opinion, you apply different standards here. The sentence which describes the formal definition at the end of the disputed section is certainly much more abstract than my use of "isomorphism" and "vector space". But that's ok.

DV: See also MOS:NOTED – while "Note that ..." is pretty typical in mathematical writing, it shouldn't be used here.

COMMENT: In regard to "remember that", I agree that it is instructional. In regard to "note that" I disagree. It means "dear reader, may I explicitly point you attention to this important fact". It is polite and a good form of emphasis. But we can disagree here. Why not using typical math jargon in a math article where everything else is typical math style is beyond me. In German, we say etepetete.

DV: What you added also essentially duplicated material that's already given later in the article.

COMMENT: Yes, found it necessary and still find it necessary to rewrite the article with a modern understanding of C. My contribution was a first step. And it needed to be up-front. However, I didn't want to erase other people's contributions. Again, you are applying double standards here in my opinion: this article is excessively duplicating various items which you accept, but my duplicate item is deleted. Note that my definition 1 of C is complete, while the other section explaining C this way is not.

DV: Frankly, I do think what's there isn't ideal, and it the article could do with some restructuring.

COMMENT: The article needs to be rewritten from scratch.

DV: Saying that complex numbers are polynomials isn't wrong, but it's more meaningful to say they're equivalence classes of polynomials.

COMMENT: From my understanding, "Saying that complex numbers are polynomials" is plain wrong. C has dimension 2 over IR. The vector space of polynomial has dimension oo over IR. It gets correct, if you say that they are linear polynomial with the reduction rule i^2=-1. -- What the author of Definition 2 does not understand is that he actually gives a full definition of C, and a quotient construction is not needed. C is the set of linear polynomials with coefficients in IR in a variable i and the reduction rule i^2=-1 which is only needed for multiplication. -- Yes: it's more meaningful to say they're equivalence classes of polynomials.

DV: Of course, this is just what's being done with the quotient of ${\displaystyle \mathbb {R} [x]}$, but starting out this way is too technical.  

COMMENT: agreed.

DV: What's probably best is to just present them as usual as numbers of the form a + bi, right along with a description of the various arithmetic operations on them. This can segue into a discussion of the main arithmetic properties – that multiplication is associative and commutative, etc.

COMMENT: I asked some colleagues about this. Here is one answer:

<< I really despise the "definition" of complex numbers as "something of the form a+ib where blablabla". This is 18th century style mathematics. >>

I completely concur. You propose an outdated approach. You propose an approach where in "imaginary number" i (a symbol) is miraculously introduced that behaves like and suddenly is a number while at the same time the "variable" or "indeterminate" in (linear) polynomials. That's very bad from both educational and scientific standard. i is simply the point (0,1) in the plane, and that's much better to understand for a beginner.

DV: This article is definitely suffering from bloat over years of fiddling and it could really use a makeover. However, your edits made the article worse, not better, and so reverting them was appropriate. Trying to saddle readers with multiple definitions right off the bat, especially ones that are only superficially different (like presenting as ordered pairs, which just isn't an enlightening or useful way to work with complex numbers) isn't a good approach.

COMMENT: NO. My edit was the meaningful beginning to present C from a modern understanding, and not a make-belief 18th century understanding. My edits made the article better. Reverting them was not appropriate. In fact, reverting shows lack of understanding what a modern approach to C in education currently is. "Trying to saddle readers with multiple definitions right off the bat" is giving them meaningful proper information. They can chose what they like better.

You are seemingly presenting falsehoods [A],[B] here:

[A] Before my editing, the definition with linear polynomials is not recognized as a proper definition in the article and is bloated by claiming to need higher order polynomials in i and higher order powers of i for reduction. Furthermore, it points to the quotient construction as the only formal definition. Taking that the quotient construction is the "formal" definition, there is a substantial difference between handling {C=IR^2 plus multiply} and handling the quotient construction. (a) The former needs understanding of IR^2 from high school and understanding +,* from elementary school. (b) The latter needs understanding of the commutative ring of polynomials over a formal variable i, the definition of ideal, and the definition of a quotient of a ring by an ideal, equivalence relations and equivalence classes. IT IS ABSOLUTELY UNTRUE that these approaches are only "superficially different". Method (a) can be taught in 90mins with all details. Method (b) needs several lectures.

[B] Many of the comments in the article point to diagrams and the complex plane which seems so useful. This is exactly working with (a, b) type coordinates in the plane. Right.? — The C=IR^2 approach is the way C is introduced to beginning math students and is (as proved above) much simpler than the quotient ring construction which is promoted in this article and seemingly by yourself. Right.? You should try to understand that the former approach involves much simpler basic objects, and is thus better to understand for beginners. You should try to understand that in the former approach i=(0,1) emerges naturally as base vector, and one can discover/check that i^2=-1. The difference is that i and i^2=-1 are not abstractly postulated. The computation

(a, b) = (a, 0)+(0,b) = a (1,0)+b(0,1)= a*1+b*i=a+bi

is then done to make life easier. Again in the new approach not with a make-belief i. Here, the shortcut „if "a" is not inside a 2D vector, then a=(a, 0)“ is used as a convenient convention. -- Even the two definitions „C=IR^2“ and „linear polynomials“ are conceptionally very different from each other, even if they have a simple isomorphism (a, b)->a+bi. The linear polynomials need to postulate/understand the indeterminant variable/symbol i. The „C=IR^2“ approach needs only very elemental, well-defined objects.

Your position is essentially against the modern approach of teaching math at university level. — Preceding unsigned comment added by LMSchmitt (talk • contribs) 06:23, 20 September 2020 (UTC)

"Teaching" is not part of an encyclopedia's (hence Wikipedia's) purpose (and if it were it would not be at the "university level"). Paul August ☎ 10:16, 20 September 2020 (UTC)

Dear Paul August, You didn't get the point:

[a] Every encyclopedia should be accurate in presenting the most recent established facts. In math, this implies not presenting math in the style of the 1700s. However, the point of the article seems to be to present the complex numbers from a very old-fashioned perspective as I have outlined above, and  a colleague has confirmed. The modern approach (at university level, not taught at school usually) is simpler, more precise, more well-defined, easier to understand than what is presented in the current version of the article.

[b] Complex numbers are usually not taught in high school. They are taught at the beginning of university study. Thus, understanding complex numbers is naturally "understanding at university level." Today's university does it better and simpler as the present WP article. ---- And please, don't try to convince me that the quotient space construction of the complex numbers outlined in one section of this article is not university level. The article uses many instances of concepts at university level, so I find your comment self-contradictory.

[c] Please note, that I said something about an attitude of a person, namely ignoring modern scientific development. Accepting the modern view which I advertise, means accepting another view how to present the matter. But my latter sentence doesn't comment on how to present the matter. ---- The probem is more like "presenting physics like Newton and deliberately ignoring Einstein." That wouldn't be good encyclopedia.

LMSchmitt 12:17, 20 September 2020 (UTC)

You know, it's extremely difficult to respond to an 8.5K post. My original response was a little over 2K, and I was feeling like it was already way too long. So, I'm going to focus on what seems to be your main complaint here – that we should immediately define complex numbers as ordered pairs along with the operations thereon.

This is a non-starter. Writing (a, b) is only superficially different from writing a + bi, and not having the i present makes the multiplication formula, in particular, much more opaque. It's the exact same definition, but with different notation. Just as (if not more) important is that it's notation that's just not really used in practice, either in introductory material, or even in a modern complex analysis text. If you're so focused on a modern definition, the obvious choice is as ${\displaystyle \mathbb {R} [x]/(x^{2}+1).}$ This can be briefly mentioned early on, but anything more than that is too technical and should be reserved for later in the article.

This is a pretty fundamental article, and we should be making it as accessible as possible. This means we shouldn't be expecting someone reading it to be a university student. A good place to aim here is probably someone with at most a reasonable grasp of high school algebra (at least for the beginning of the article; it's okay to get more advanced as we go). We also have to be very careful because education terminology is different around the world, as is when concepts are generally introduced. I, for example, first saw complex numbers in high school algebra, well before I got to college. Maybe that's different elsewhere, but as an encyclopedia, we're really trying to make as much of this understandable to as wide an audience as possible. –Deacon Vorbis (carbon • videos) 14:26, 20 September 2020 (UTC)

I would second what Deacon Vorbis says here. Specifically high school is the correct level for us to be aiming for, and the "a + bi definition" is better for our purposes here than the "(RxR, +, *) definition". And by the way I too was introduced to complex numbers in high school. Paul August ☎ 16:05, 20 September 2020 (UTC)

I also second this: High Schoolers do not learn Complex numbers as an extension of the Real numbers into the plane. They learn it first and foremost through the introduction of i as the square root of negative one, then are later shown that such a treatment gives rise to a convenient treatment of the plane. These two ideas are both important and should be given treatment, but extensive formaliation of these two pretty simple concepts right at the beginning of the article only serves to confuse the reader. _{Integral Python} ^{click here to argue with me} 17:49, 20 September 2020 (UTC)

Dear Deacon Vorbis, thanks for your reply.

Unfortunately, what you write shows that you do not understand the points that I am so desperately trying to make.

Obviously, writing (a, b) is only superficially different from writing a + bi as strings. But this is not the reason for my proposal. The "(IR^2, +, *) definition" builds on well known objects for people with high school level knowledge (the plane, and +,* in IR). The statement

The complex numbers is the Euclidean plane together with a special multiplication for points in that plane.

is a very simple, complete, accurate description and certainly easily understandable for every layman. This is concrete and doesn't need any postulates. The imaginary unit is simply the point i=(0,1) in the plane. That is concrete and straightforward. The  a + bi definition first introduces an indeterminate (means: not exactly known, established, or defined ;-) ) variable/symbol i (in order to define polynomials) which then miraculously (by definition, though not exactly known) is a fixed number and solves i^2=-1. That's a lot of concepts in the package. Explaining why one can actually do that takes time. This is also considered 1700s math (see previous post/reply).

The multiplication formula can be motivated in a clear manner in about two sentences and 2 formulas (3 lines):  Suppose we have a field which contains IR and a number i satisfying.... Then we would have the following multiplication rule.... Now, we implement the latter insight as follows: ---- Note that this is a thought experiment for motivation and very different from postulating i. ---- Thus, any opaqueness can be removed easily.

Falsehood: It's the exact same definition, but with different notation. ---- One definition needs "new" postulates about i, the more modern one does not. Only the algebraic handling of real coefficients is the same.

Falsehood: ordered pair approach not used. ---- See that the ordered pair approach is used in practice in intro texts: https://people.math.gatech.edu/~cain/winter99/ch1.pdf [1]. This is the optimal approach to define and explain C. With the computation

(a, b) = (a, 0)+(0,b) = a (1,0)+b(0,1)= a*1+b*i=a+bi __(*)

a, b in IR, i=(0,1) 2nd base vector, one then switches to the a+bi notation knowing that i=(0,1). Note that also the other reference which I posted uses the "(IR^2, +, *) definition". Modern books on analysis assume that the reader knows C and the above computation (*). And they proceed from there.

Falsehood: For a modern definition, the obvious choice is ${\displaystyle \mathbb {R} [x]/(x^{2}+1).}$ ---- ${\displaystyle \mathbb {R} [x]/(x^{2}+1)}$ is a beautiful math construction. However, it is NOT', ABSOLUTELY NOT an obvious choice. It can be a choice for an Algebra class, an exercise after introducing the quotient construction for rings. As outlined above in detail, ${\displaystyle \mathbb {R} [x]/(x^{2}+1)}$ needs a substantial amount of concepts to be explained (3<= hours lecture time with all details), while the "(IR^2, +, *) definition" needs no new concepts beyond high school math (1.5> hours lecture time with all details). Observe that the above modern reference [1] uses the "(IR^2, +, *) definition". Note that also the other reference which I posted uses the "(IR^2, +, *) definition".

LMSchmitt 20:41, 20 September 2020 (UTC)

[User:Deacon Vorbis|Deacon Vorbis] I made quality input which every mathematician would confirm. YET you revert my inputs due to personal issues. For every change, I gave a detailed input and comment and reason. YET you reverted them all. You even reverted to WRONG statements. You must be terminated as an editor. LMSchmitt 18:59, 30 September 2020 (UTC)

I will be happy to discuss any content disputes once the personal attacks stop. I realize English isn't your native language, but saying someone must be terminated can be construed as a death threat.  –Deacon Vorbis (carbon • videos) 19:17, 30 September 2020 (UTC)

Sorry, you are bending reality here to a point where it becomes totally rediculous: you and I know that "terminated as an editor" means "terminated on the job as editor" (which I maintain). Concluding that this may be perceived as a death threat is odd (to put it very mildly). I have read and heard many, many times that someone was terminated on the job, i.e., fired. That's common English terminology, and I have NOT been threatening you at all at any point in time, and the issue of a death threat is absurd and void. For the record, that you should be terminated as an editor is a valid opinion based on many complaints about you including mine. It's not a personal attack. I am not attacking you as a person. You could say that I am attacking you on quite unfounded editorial conduct (in my opinion). I also repeat that I feel harassed by you for personal reasons. That's MY feeling. I have documented this elsewhere, and I suppose you know where. It's not the worst harassment that ever happened, but it hurts. LMSchmitt 21:12, 30 September 2020 (UTC)

I appreciate your offer to discuss matters. LMSchmitt 21:12, 30 September 2020 (UTC)

Point taken that someone (Convey) uses the term "complex number system". However, this is rarely used in practice (in discussions, in lectures, in papers), even though it makes a nice title for a chapter. (I googled it, found several). I have in 45 years (almost) never seen it in lectures, papers, books, even though I have seen Convey's book at some point in time (must have ignored "system"). Every use of "complex number system", I would find "out dated". For modern references, see (https://www.youtube.com/watch?v=SP-YJe7Vldo) (https://home.cc.umanitoba.ca/~thomas/Courses/ComplexRSDT.pdf) (https://59clc.files.wordpress.com/2012/08/real-and-functional-analysis-lang.pdf p.4). The article title is "complex numbers". The intro talks about "complex numbers". That terminology should be maintained throughout the article. A math novice should not be confused with oscillating terminology: "complex numbers" and then "complex number system" (undefined at the point of first encounter) and then again "complex numbers"... But, "(also called "complex number system)" could be in the top intro somewhere (since it's used), but then should be abandoned. Note that the first sentence in the article that uses "complex number system" uses "real numbers" and not "real number system" -- bad and inconsistent. LMSchmitt 21:12, 30 September 2020 (UTC)

@LMSchmitt and Deacon Vorbis: While both of you may have some grounds for complaint with each others behavior, you both really need to try to put those complaints aside. Please. This page is really only for discussion of the content of the article (I know that Deacon Vorbis as a long time editor knows this). In my view some of the changes LMSchmitt has proposed are reasonable. However I would suggest that from now on LMSchmitt should suggest individual changes on this talk page, and gain consensus for each change before including them in the article. Paul August ☎ 21:37, 30 September 2020 (UTC)

@Paul August: Point taken. Reasonable. LMSchmitt 10:18, 1 October 2020 (UTC)

@Deacon Vorbis: It is totally amusing to see that Conway [a] uses a Chapter Headline "The Complex Number System" as you pointed out,  [b] after the headlines immediately starts using the term "complex numbers" (on the same page) as everybody does regularly, and as I have proposed as standard here, [c] introduces C the modern way, i.e., as the Euclidean plane with a special multiplication, as I have also proposed as standard here, and finally [d] sets i=(0,1) explicitly as I have outlined in previous suppressed edits. I  again suggest that this page is reverted as I have proposed, which is supported by expert treatise. (https://math.unice.fr/~nivoche/pdf/Conway1.pdf) LMSchmitt 10:18, 1 October 2020 (UTC)

@DVdm: You write: "no need for that. We usually write a+bi without factoring out intergers", but strangely enough you let the article continue: "Since the real and imaginary part of 5 + 5i are equal ...".

My question: Is this really better than factoring out? Or isn't it kind of essentially the same?? –Nomen4Omen (talk) 17:04, 16 December 2020 (UTC)

Note: this is about . I don't think so. The article says "Since the real and imaginary part of 5 + 5i are equal, ..." It does not say "Since the real and imaginary part of 5(1+i) are equal, ..." Your edit introduced an irrelevant step in the current context. - DVdm (talk) 18:20, 16 December 2020 (UTC)

OK, I guess that everybody understands in principle what’s going on. But let's take:

${\displaystyle z:=(57+i)^{44}\cdot (239+i)^{7}\cdot (682-i)^{12}\cdot (12943+i)^{24}}$

And here, instead of continuing:

${\displaystyle =28443813229566464959121350868488998532782432775233378650470299001276631906199753372073371258470859604903135356065242009605577311191019082746792441237018711497341882932232692837715148925781250000000000000000000000000000000000000}$${\displaystyle +28443813229566464959121350868488998532782432775233378650470299001276631906199753372073371258470859604903135356065242009605577311191019082746792441237018711497341882932232692837715148925781250000000000000000000000000000000000000\,i.}$

Since the real and imaginary part of ${\displaystyle z}$ are equal, the argument of that number is 45 degrees.

especially because one has great interest in a big ${\displaystyle n,}$ it would almost be easier to write:

${\displaystyle =n\cdot (1+i)}$ with ${\displaystyle n=2.84438\dotso \cdot 10^{226},}$

so that ${\displaystyle \arg z=\arg n+\arg(1+i)=0+\arg(1+i)=\arctan 1={\tfrac {\pi }{4}}=45}$ degrees. –Nomen4Omen (talk) 19:40, 16 December 2020 (UTC)

By the way, sorry for that silly typo in my edit summary. Feel free to correct it in the section header. - DVdm (talk) 23:38, 16 December 2020 (UTC)

OK, done.

What I really wanted to say I've also changed by this edit: ${\displaystyle n=2.84438\dotso \cdot 10^{226}}$ (s.a.) –Nomen4Omen (talk) 09:08, 17 December 2020 (UTC)

What is that? --  Denelson83  04:21, 9 January 2021 (UTC)

${\displaystyle \mathbb {C} ^{\times }:=\mathbb {C} \setminus \{0\}}$ is the (multiplicative) group of units in ${\displaystyle \mathbb {C} ,}$ i.e. the set of all (multiplicatively) invertible elements in ${\displaystyle \mathbb {C} .}$ [See e.g. § Unit (ring theory)#Group of units​.] Another wide-spread notation for the very same is ${\displaystyle \mathbb {C} ^{*}.}$ –Nomen4Omen (talk) 10:01, 9 January 2021 (UTC)

The article currently uses the calligraphic mathcal font for the Re and Im operators:

${\displaystyle {\mathcal {Re}}(z)}$ and ${\displaystyle {\mathcal {Im}}(z)}$

While not exactly wrong this strikes me as unusal and unconventional, and I think

${\displaystyle \operatorname {Re} (z)}$ and ${\displaystyle \operatorname {Im} (z)}$

is much more common. – Tea2min (talk) 10:10, 16 April 2021 (UTC)

We could also use

${\displaystyle \mathrm {Re} (z)}$ and ${\displaystyle \mathrm {Im} (z)}$

which looks the same on Chrome.— Anita5192 (talk) 16:39, 16 April 2021 (UTC)

The macro \mathrm must be avoided for operators, since it generates wrong spaces. Compare ${\displaystyle a\mathrm {Re} (z)}$ (with \mathrm) with ${\displaystyle a\operatorname {Re} (z)}$ (with \operatorname). Things are even worse if parentheses are omitted for the argument: ${\displaystyle \mathrm {Re} z}$ vs. ${\displaystyle \operatorname {Re} z.}$ Similarly \operatorname must also be used (in many cases) with \mathcal (for example \operatorname{\mathcal{Re}}). Compare ${\displaystyle a{\mathcal {Re}}(z)}$ with ${\displaystyle a\operatorname {\mathcal {Re}} (z).}$ D.Lazard (talk) 09:04, 17 April 2021 (UTC)

This article could use a secton on the history of the applications of complex numbers (which I myself am completely UNqualified to develop). For example, I came to this article, out of curiosity, to learn how and when complex numbers were first used to describe electrical impedance, and other electromagnetic phenomena. (I still do not know...) Acwilson9 (talk) 05:56, 9 January 2022 (UTC)

You probably have a good point, but I don't think the article specifically needs an entire section devoted to "the history of applications." It already has a history section and an applications section. Perhaps the history section could describe the history of the applications mentioned elsewhere in the article.—Anita5192 (talk) 07:07, 9 January 2022 (UTC)