Talk:Complex number

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Why imaginary "unit"?

Latest comment: 7 years ago19 comments8 people in discussion

Since this article garnered a lot of attention because of some traditional, and imho naive, but not without reason, discussion about math articles being only for the elite, I use the chance to ask for some motivation calling i a 'unit'. I did not find an answer within this article and also not within the article "Imaginary unit". Perhaps some illustrating words could be added here.

Sure, i is a unit in the sense of 'being invertible', but this notion of 'unit' is not quite easy to find within WP. I found it hidden in an article on rings. Certainly, i is no 'unit' in the sense of '(multiplicative) identity' or 'unity', but it has a magnitude of ' 1 ', a property shared with many other complex numbers, similar to being invertible. Associating 'imaginary unit' with something like 'meter' being the 'unit' belonging to a dimension of 'length' is absolutely alien to me. I think 'unit' is too laden with associations to leave it simply "unrefined". Purgy (talk) 08:36, 26 October 2017 (UTC)Reply

IMO, this is simply a traditional terminology. Nevertheless, this is a unit (ring theory) of every subring of the complexes that contains it. D.Lazard (talk) 10:02, 26 October 2017 (UTC)Reply

I suppose imaginary root of unity might be more technically correct. Sławomir Biały (talk) 10:37, 26 October 2017 (UTC)Reply

I do think it's just traditional terminology. Rather than meaning "unit" in the sense of "invertible", it seems to mean "unit" as in "basic unit of measurement", as purely imaginary numbers are "measured" in multiples of i. This make some sense because |i| is 1. But I think that in many cases we can avoid the term "unit" here and use more direct wording. — Carl (CBM · talk) 14:55, 26 October 2017 (UTC)Reply

I don't think I've ever said 'imaginary unit' in my life so it shouldn't be too hard to avoid! Dmcq (talk) 14:58, 26 October 2017 (UTC)Reply

I agree with D.Lazard's comment; I could imagine that the name is related to units as in Dirichlet's unit theorem. However, unless there is a clear source backing up this idea or any other idea, I would not write anything in the article itself. Jakob.scholbach (talk) 15:10, 26 October 2017 (UTC)Reply

I was surprised to find imaginary unit the other day. — Carl (CBM · talk) 15:29, 26 October 2017 (UTC)Reply

Is there a more standard way to refer to i? Sławomir Biały (talk)

A little history may help here. The term comes from being invertible, a unit in the ring sense. Gauss introduced it when he talked about the four "units" in the Gaussian integers, 1, −1, i and −i. As with several other terms, once Gauss used it, it became standard to copy him. --Bill Cherowitzo (talk) 23:32, 26 October 2017 (UTC)Reply

Thanks for all the thoughts. Please check my attempt on implementation in the lede. Purgy (talk) 09:28, 27 October 2017 (UTC)Reply

I think this will simply make things more confusing for likely readers. If there is a place to discuss the meaning of the word "unit" in this context, it is surely the article imaginary unit, rather than the first paragraph of the lead. The meaning of the word "unit" is almost irrelevant to the subject of this article. I think we just need to establish that it is a standard way to refer to "i" in prose, but not to belabor the point. Sławomir Biały (talk) 10:43, 27 October 2017 (UTC)Reply

I certainly would have used a "standard way to refer to "i" in prose", if such would have been mentioned above. But, as articulated in my thread starter, I perceived misguidance by the word "unit" in "imaginary unit", which is used throughout the whole article and also in the linked one with this very title, without being sourced at all, so I tried to establish this term at the beginning of this article in a most simple and plausible manner. Omitting the list of 4 values would have taken away the plausibility. Getting rid of the term appears to me not sooo easy as Dmcq seems to think.

Commenting further on the lede, I think the citation of Penrose in its whole length is only justified to refute the, imho, unsubstantiated objection by Bob31416 to "real". I perceive it a bit, say, flowery. I do hope that my changes are not considered rubbish, wholesale, but certainly they are improvable. Purgy (talk) 16:47, 27 October 2017 (UTC)Reply

I actually think a better solution would be to try to remove "imaginary unit" from the lead. It can be introduced in context later in the article. I've taken a swing at this. If we can agree that the original wording of complex numbers being just as real as real numbers is an adequate summary of the scientific literature, I'm certainly open to saying things in that much shorter way. The only reason I included the quotation at length was, of course, that a certain editor seemed to be pushing a WP:FRINGEPOV. Sławomir Biały (talk) 16:59, 27 October 2017 (UTC)Reply

I agree, I don't think that the term adds anything here and hinting at its origins will definitely raise more questions than it answers. I mentioned it above to give a little historical background, but never intended it to be used in the article. It provides the answer to why these and not the other roots of unity are called units, but the context is Gaussian integers and that is not something that should be brought up in the lead of this article. While I'm at it, I also dislike the use of "imagined solution" in the first line, even with the scare quotes. This is repeating (at least according to Gauss) the mistake that Descartes made in terminology. I would prefer all mentions of imaginary to be dropped into the second paragraph where the issue can be dealt with. And, referring to the existence of imaginaries as a "settled question" seems to me to be too blunt a statement, making it sound like the issue was dealt with head-on. A better phrasing would be something like, "Mathematicians' unease with the concept was gradually dispelled by ...". --Bill Cherowitzo (talk) 16:58, 27 October 2017 (UTC)Reply

Any objections to just this? Significantly shorter and easier to read, IMO. Sławomir Biały (talk) 17:03, 27 October 2017 (UTC)Reply

I don't think 'indeterminate' is a good word, it is correct as Indeterminate (variable) but it has too much other baggage and is used more often in indeterminate form, otherwise I think your changes were good. Dmcq (talk) 17:24, 27 October 2017 (UTC)Reply

The question whether mathematical objects are discovered, invented, created, etc. is a deep issue in mathematical philosophy, and there is no reason our article here should weigh in on it. So IMO we should avoid claiming that complex numbers were "invented", "created", etc. There is almost always a more neutral way to discuss the situation without opening that can of worms. — Carl (CBM · talk) 17:29, 27 October 2017 (UTC)Reply

My issues with the lead have been dealt with, so I am okay with it. I totally agree with Carl, but would like to point out that the notation associated with a concept is always "invented" (whether or not we can trace to its origins) and that might be a way to avoid stepping in that can. --Bill Cherowitzo (talk) 17:41, 27 October 2017 (UTC)Reply

The literature has no problem with the term "imaginary unit"—see Google Books—so I don't see any reason why Wikipedia should shun it. - DVdm (talk) 19:49, 29 October 2017 (UTC)Reply

Obviously, it is not WP's but my problem to associate too many possible meanings with the word 'unit', and being unwilling to assume that the context suffices for most readers to lightheartedly brush over different technical meanings of 'unit'. In no way I want WP to shun the term "imaginary unit", but I want to see it, say, introduced, be it as a sourced traditional, with some associated historical meaning, or as cited definition taken from some handbook.

Meanwhile, I am inclined to take the "imaginary unit" as baptized, neither for Gaussian integers, nor for ring units, but for its valuation and for being in an "imaginary dimension", like j and k. Nothing to shun, but worth an encyclopedic info, imho. Purgy (talk) 08:10, 30 October 2017 (UTC)Reply

Misuse of the English word "real"

Latest comment: 7 years ago11 comments4 people in discussion

Recently a sentence was deleted on the spurious grounds that it is a misuse of the English word "real". It seems to me that this is a vital sentence for the lead, intended to convey that the is nothing imaginary about imaginary numbers, to a lay audience. I don't quite understand the objection to the use of the English word "real". Real numbers are a thing, and imaginary numbers are too. But imaginary numbers aren't less real, in the colloquial sense of the word, than real numbers. (One could argue that real numbers are just as imaginary as imaginary nbers, but this fails to convey the point as clearly). If there is a different, better way to say the same thing, please propose it. But I don't see how getting rid of the word "real" (or "inaginary") can convey the same message to a lay reader. Sławomir Biały (talk) 19:28, 26 October 2017 (UTC)Reply

In the first paragraph of the lead is,

"Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of our description of the natural world.^[1]"

1. Roger Penrose (2005). The road to reality: a complete guide to the laws of our universe. Knopf.

To begin with, what is the page number of the source? --Bob K31416 (talk) 19:29, 26 October 2017 (UTC)Reply

Apart from being one of the central themes of the entire book, in particular pages 72 and 73. Undoubtedly other sources can be found to satisfy your objections. But I still don't clearly understand the nature of your objection to the sentence. In your edit summary, you appeared to object to the word choice "real", but declined to offer details. Here you are asking for a page number (which should be done with one of the standard citation templates, not by deletion). Would you like some other sources? Sławomir Biały (talk) 20:14, 26 October 2017 (UTC)Reply

I looked on 72 and 73 and I didn't see the idea that complex numbers are as "real" as the real numbers. Please provide an excerpt from that source that you think expresses that idea. --Bob K31416 (talk) 20:35, 26 October 2017 (UTC)Reply

Quoting that source at length: "Presumably this suspicion arose because people could not ‘see’ the complex numbers as being presented to them in any obvious way by the physical world. In the case of the real numbers, it had seemed that distances, times, and other physical quantities were providing the reality that such numbers required; yet the complex numbers had appeared to be merely invented entities, called forth from the imaginations of mathematicians who desired numbers with a greater scope than the ones that they had known before. But we should recall from §3.3 that the connection the mathematical real numbers have with those physical concepts of length or time is not as clear as we had imagined it to be. We cannot directly see the minute details of a Dedekind cut, nor is it clear that arbitrarily great or arbitrarily tiny times or lengths actually exist in nature. One could say that the so-called ‘real numbers’ are as much a product of mathematicians’ imaginations as are the complex numbers. Yet we shall Wnd that complex numbers, as much as reals, and perhaps even more, Wnd a unity with nature that is truly remarkable. It is as though Nature herself is as impressed by the scope and consistency of the complex-number system as we are ourselves, and has entrusted to these numbers the precise operations of her world at its minutest scales. In Chapters 21–23, we shall be seeing, in detail, how this works." :::: Sławomir Biały (talk) 20:44, 26 October 2017 (UTC)Reply

(edit conflict) This is what Penrose says on page 73: "One could say that the so-called 'real numbers' are as much a product of mathematician's imaginations as are the complex numbers."

So in fact Penrose says that "the real numbers are as "imaginary" as the complex numbers". Apparently this was re-interpreted here as that "the complex numbers are as "real" as the real numbers." So I can more or less agree with Bob's remark. This could be a slight case of wp:synthesis. I propose we turn it around again, in order to stay closer to the source. A good compromise i.m.o. - DVdm (talk) 20:52, 26 October 2017 (UTC)Reply

No, this is a counterpoint. He is saying that someone could say this, but in fact: "complex numbers, as much as reals, and perhaps even more, find a unity with nature that is truly remarkable. It is as though Nature herself is as impressed by the scope and consistency of the complex-number system as we are ourselves, and has entrusted to these numbers the precise operations of her world at its minutest scales." That seems pretty clear cut as it is, but for added context see where he blathers on in the introduction about being a strict Platonist (pinging @Trovatore:, our friendly neighborhood fundamentalist), making the question of the reality of the real numbers a something of a settled thing for him. I think a better compromise would be simply to quote Roger Penrose in the first paragraph of the lead. A bit unconventional, but he expresses it far better than any of us will. Sławomir Biały (talk) 21:06, 26 October 2017 (UTC)Reply

I'm okay with that. - DVdm (talk) 21:09, 26 October 2017 (UTC)Reply

I don't see the excerpt supporting the statement which is apparently OR. Also, regarding another part of the statement, "complex numbers are regarded in the mathematical sciences as" – it appears that you are concluding that it is generally accepted in the mathematical sciences, which also isn't mentioned in the excerpt you gave from the source. Also, you use scare quotes for "real" because you have no clear definition of what you mean by that term. In any case, the sentence is poor writing that isn't informative, and certainly shouldn't be in the lead, or anywhere in the article.

But hey, this is Wikipedia and any kind of crappy writing can get into articles if there are editors determined to do it. So this is my last message about the subject, and have fun! --Bob K31416 (talk) 22:03, 26 October 2017 (UTC)Reply

Which is why we have now compromised, and included an exact quotation from the source. I still feel that the original phrasing was a reasonable summary of that source. I even supplied a quotation that supported what I still feel is reasonable. You haven't proposed any constructive alternative. Instead, the entire premise of this thread seems to be the WP:FRINGE belief that there is some controversy over the existence of complex numbers. To dispel the false notion that there is any such controversy in mathematics, it is clearly necessary to point this out, especially as certain editors seem keen to banish such important details in a thinly disguised attempt to subvert the neutral point of view policy. You're now explicitly questioning whether the existence of the complex numbers is generally accepted in mathematics. Clearly you are edging closer to the brink in this discussion. Sławomir Biały (talk) 22:22, 26 October 2017 (UTC)Reply

While I'm sure I have seen the statement about complexes being as real as reals in print, I can't find a reference. However, an alternative to Penrose might be an 1831 passage by Gauss where he talks about a mysterious obscurity attached to imaginary numbers due to a poor choice of notation (i.e., "imaginary"), which would vanish if better choices had been made. Gauss also originated the term "complex number" so there might be a possibility of leading into this quote from that point. --Bill Cherowitzo (talk) 23:44, 26 October 2017 (UTC)Reply

(a+bi)^n

Latest comment: 7 years ago3 comments3 people in discussion

can we add the formula for (a+bi)^n? Jackzhp (talk) 08:55, 11 January 2018 (UTC)Reply

Look here, please. Purgy (talk) 09:10, 11 January 2018 (UTC)Reply

However, it should be probably better to put the simpler and important case of integer exponents before the general case. D.Lazard (talk) 09:42, 11 January 2018 (UTC)Reply

Definition of the argument

Latest comment: 6 years ago5 comments3 people in discussion

Section § Conjugate, contains the formula

${\displaystyle \quad \operatorname {arg} ({\overline {z}})\equiv -\operatorname {arg} (z){\pmod {2\pi }}.}$ 

HaydenWong changed ${\displaystyle \equiv }$  into =, and this has been reverted by Deacon Vorbis. IMO, both versions are wrong, and the formula should be simply

${\displaystyle \quad \operatorname {arg} ({\overline {z}})=-\operatorname {arg} (z).}$ 

However, this depends on the chosen definitions for an angle and for the argument of a complex number. The argument of a complex number is defined, later, in § Polar form, but this definition is a mess, and, in particular, suggests that 0 and 2π are different angles. Also, in this section, the formula with multiple cases should better be replaced by the equivalent formula:

The argument of z = x + iy is ${\displaystyle \arctan {\frac {y}{x+|z|}}}$  except if z is a negative real number, where arg z = π, and |z| = –x.

Thus, for fixing this particular formula, one has to define the argument of a complex number before the conjugate and to cleanup the section § Polar form. I'll not restructuring this article myself for lack of time. Someone is willing? D.Lazard (talk) 14:01, 11 December 2018 (UTC)Reply

A while ago I attempted to improve on the elementary operations, but shied away from touching the polar form (a small caveat was reverted), because I am not aware what might be the nowadays most appropriate form to touch this subtle matter. Furthermore, I had a very disappointing encounter while trying to bring some consistency to the atan2 article, so I won't touch anything arctanny. As an aside, I stranded in my efforts to find sources for a consistent denotation and an agreed upon selection of principal branches in the inverse trigs, e.g., I found the use of Sin and Arcsin as opposed to sin and arcsin rather arcane in the superficially scanned literature. Sadly, I am not sure about the reasons why the congruence is considered wrong, I certainly lack a rigorous definition of arg.

As a result, I don't feel capable to touch the addressed problem. Purgy (talk) 16:32, 11 December 2018 (UTC)Reply

I'm confused. The range of ${\displaystyle \arctan }$  is ${\displaystyle (-\pi /2,\pi /2)}$ , so how can ${\displaystyle \arctan(y/(x+|z|))}$  give the correct values for ${\displaystyle \arg }$  on the left side of the complex plane?

I agree that the polar form should be introduced before the Conjugate subsection mentions facts about it. Mgnbar (talk) 01:41, 12 December 2018 (UTC)Reply

- As a proxy: Simply, a factor of 2 is missing (the derivation might be based on the half-angle).

- I am not absolutely sure about the sequence. Personally, I prefer to look at the complex numbers as entities on their own, primarily neither as a sum of real and i×imaginary part, nor as modulus×exp(i×arg). Complex numbers may be added, subtracted, multiplied and divided by non-zeros AND -as a new feature- conjugated. This conjugation, by relying on its properties, allows for defining real- and imaginary parts, modulus and, via the multivalued arctan- the argument, which is not unique. Of course, the conjugation can be mirrored also in the newly introduced real/imaginary and in the modulus/argument scenario. This reverses the long standing, probably more common route, but I think it gained increasing acceptance over time. Just a personal remark. Purgy (talk) 08:20, 12 December 2018 (UTC)Reply

Yes, I forgot the factor 2.

I agree that complex numbers should be viewed as entities on their own. A way of reaching this goal would be to rename as "Basic operations" the section § Elementary operations, and to include at its beginning subsections on the real-valued unary operations (real and imaginary part, modulus and argument). D.Lazard (talk) 10:22, 12 December 2018 (UTC)Reply

Not the same as a vector in 2 dimensions as multiplication defined differently

Latest comment: 6 years ago5 comments3 people in discussion

The article now uses the word vector a few times, implying that a complex number is the same as a vector in two dimensions. I would question this identification, on the ground that the product of two complex numbers is defined differently from both the scalar product and the vector product of two vectors in vector multiplication. In fact the product of two complex numbers is a complex number in the same complex plane as the factors, whereas the scalar product of two 2-D vectors is of course a scalar, and the vector product is a vector perpendicular to the plane of the vectors being multiplied.

Would it be better to remove all mention of vectors from this article? Or perhaps to add a section explaining why a complex number is an object different from a vector in two dimensions? Dirac66 (talk) 22:34, 14 February 2019 (UTC)Reply

But ${\displaystyle \mathbb {C} }$  is a (two-dimensional) vector space over ${\displaystyle \mathbb {R} .}$  Moreover, addition of complex numbers and multiplication by a real number correspond to the normal vector space operations of addition and scalar multiplication. This is a standard notion and shouldn't really be removed. I'm not sure much extra explanation is needed. The dot product is a bi- (or sesqui-)linear form, not a normal multiplication; and the cross product is something a bit special in 3 dimensions, not 2. Maybe just a quick note could be added, but not much. –Deacon Vorbis (carbon • videos) 22:58, 14 February 2019 (UTC)Reply

Perhaps at the end of the section Multiplication, we could add a sentence such as: This product is different from both the scalar product and the vector product of simple vectors in 3-dimensional space. This would help readers who have learned about scalar and vector products in first-year physics but not in math, by clarifying that we are now not talking about the same thing. Dirac66 (talk) 01:36, 15 February 2019 (UTC)Reply

My edit is reasoned by: I think the vector space property is quite elementary, is used in the top pic, and should therefore be addressed already in the lead, and not only way down in the article. I prefer to keep the binary operation "complex multiplication" free of both the form and the quick and dirty cross-product. Purgy (talk) 09:19, 15 February 2019 (UTC)Reply

Yes, I think the paragraph you have added explains the point quite well. Thank you. Dirac66 (talk) 02:26, 16 February 2019 (UTC)Reply

"i is an indeterminate satisfying i^2 = −1" is ambiguous

Latest comment: 5 years ago14 comments5 people in discussion

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)Reply

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

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)Reply

@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)Reply

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)Reply

@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)Reply

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)Reply

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)Reply

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)Reply

@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)Reply

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)Reply

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)Reply

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)Reply

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)Reply

References

1. Addition in R[X] is defined componentwise and multiplication by Convolution#Discrete convolution.

Unfortunate start to Exponentiation section

Latest comment: 5 years ago4 comments3 people in discussion

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)Reply

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)Reply

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)Reply

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

Chronology of History

Latest comment: 4 years ago2 comments2 people in discussion

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)Reply

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)Reply

Definition of C

Latest comment: 4 years ago11 comments5 people in discussion

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)Reply

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)Reply

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)Reply

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

LMSchmitt 01:32, 20 September 2020 (UTC)Reply

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)Reply

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)Reply

"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)Reply

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)Reply

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)Reply

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)Reply

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)Reply

Worst math article ever

Latest comment: 4 years ago2 comments2 people in discussion

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)Reply

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)Reply