Talk:Complex number

message: 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)
replies: text: 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)
thread_title:  History of applications 

message: An editor has identified a potential problem with the redirect :Complex getal and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 February 15#Complex getal until a consensus is reached, and readers of this page are welcome to contribute to the discussion.  ~~~~User:1234qwer1234qwer4 (talk) 20:28, 15 February 2022 (UTC)
thread_title:  "Complex getal" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] 

message: An editor has identified a potential problem with the redirect :Nombre complexe and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 February 15#Nombre complexe until a consensus is reached, and readers of this page are welcome to contribute to the discussion.  ~~~~User:1234qwer1234qwer4 (talk) 20:29, 15 February 2022 (UTC)
thread_title:  "Nombre complexe" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] 

message: I'm looking for a better source for the geometric interpretation of multiplication, preferably a textbook. This interpretation seems to be well known e.g. it featured in this desmos course by  Luke Walsh on complex numbers: #REDIRECT [], Grant Sanders heavily uses in his video on complex numbers: #REDIRECT []. Grant refers to #REDIRECT [] by Ben Sparks. But I did not find a text book that features it. The only source beside these I found was deemed not reliable enough. 
Note: The illustration below for ab and by Luke Walsh distributes b to a = x + yi, i.e. ab = xb + yib, while Grant Sanderson and Ben Sparks in their visualizations distribute a to b = u + vi, i.e. ab = ua + vib, but due to commutativity this does really matter. (In other words their presentations of ab match mine and Luke Walsh's of ba.) 
This is a bit (together with two pictures) I'd like to add in the section Complex_number#Multiplication_and_square.
:The distributive property of multiplication over addition can be used to visualize multiplication geometrically. In particular, distributing  to a = x + yi:
::a\,b  = (x+yi)\, b= xb+ yib.
: and using the visualization of complex numbers in the complex plane, multiplication can be given the following geometric interpretation: the product of two complex numbers  and  is the point that has the same coordinates as the point  in the complex plane but in a new scaled and rotated Cartesian coordinate system that has the x-axis and y-axis going through  and  and a unit of measure |b|. 
Happy for any suggestion.
Qerez (talk) 11:31, 20 February 2022 (UTC)
replies: text: The proposed addition would certainly be confusing and too technical for most readers (I, a professional mathematician, have difficulties for understanding the logic behind the explanation), and the reference (a communication in a minor conference) is certainly not a reliable source for a subject covered by hundreds of textbooks. D.Lazard (talk) 12:02, 20 February 2022 (UTC)
replies: text:  I had more or less the same objections and suggested user Qerez to come here. My comments are available at User talk:Qerez. - DVdm (talk) 12:38, 20 February 2022 (UTC)
thread_title:  Geometric interpretation of the multiplication of complex numbers 

message: Dear D.Lazard, your (a + bi)\, i = -b + ai does not fit the current graphic. Nomen4Omen (talk) 15:50, 13 March 2023 (UTC)
replies: text: I do not know what you call the “current graphic”, as there is no figure illustrating the rotation that is the subject of this section. The central dot is not used in this article, and the real and imaginary parts are called a and b in all previous sections. So changing a and b to x and y could be confusing. D.Lazard (talk) 16:23, 13 March 2023 (UTC)
thread_title:  Revert by [[user:D.Lazard]] on 13.03.23 

message: From Google Gemini: 
Clarifying Isomorphism and Topological Properties:
Isomorphism in the context of fields refers to structural equivalence. Two fields are considered isomorphic if there exists a one-to-one correspondence between their elements that preserves all the field operations (addition, subtraction, multiplication, and division) under the corresponding operations in the other field.
Topological properties, such as connectedness and local compactness, are distinct characteristics from the purely algebraic structure of a field. These properties are related to the way elements in a field are "close" to each other and how sets of elements within the field can be organized.
In the case of constructing the complex numbers, using different irreducible polynomials (like X^2 + a and X^2 + b) leads to isomorphic fields in the sense that their algebraic structure (addition, subtraction, multiplication, and division) is identical. However, these isomorphic fields might not inherit the same topological properties from the real numbers when constructing them as quotient fields.
Example:
* As we discussed previously, the field of complex numbers (C) constructed using the polynomial X^2 + 1 exhibits both connectedness and local compactness as a topological field.
* However, if we construct a field using a different irreducible polynomial, like X^2 + 2, the resulting quotient field would be isomorphic to the complex numbers in terms of its algebraic structure but would not be connected or locally compact.
 
I haven't taken a math course for 40 years now, and this surprises me, but I don't find it implausible. I was taught that X^2 + 1 was not algebraically special, and I thought that meant everything, with no consideration that topology might think different. 
Bear in mind that yesterday Gemini insisted: 
 
As a result of this shift, the Euler gamma function doesn't directly match the factorial values at positive integers. For example:
* Γ(1) = Γ(2) = 1, while 1! = 1.
* Γ(2) = Γ(3) = 2, while 2! = 2.
* Γ(3) = Γ(4) = 6, while 3! = 6.
 
This nonsense continued until I confronted it rather sternly: 
Your error runs deeper than that. Somehow you have processed the true statement that Gamma(n) = (n-1)! as implying that Gamma(n) = Gamma(n-1) which immediately leads to nonsense. From Legendre, the correct recurrence for Gamma is that Gamma(z+1) = z * Gamma(z).
At which point Gemini found the light: 
You're absolutely right, and I apologize for the repeated errors in my previous responses. My understanding of the relationship between the Gamma function and factorial function was flawed, leading to inaccurate statements and misleading information.
My Mistakes:
* I incorrectly stated that the Euler Gamma Function (Γ(z)) and the factorial function (n!) directly interpolate each other, leading to the erroneous claim that Γ(2) = 1.
* I falsely implied that the true statement Γ(n) = (n-1)! somehow implied Γ(n) = Γ(n-1), which is not only incorrect but also mathematically illogical.
Correct Information:
* The Euler Gamma Function extends the concept of factorial to non-integer and complex numbers. It does not directly interpolate the factorial function at positive integers due to a specific definition involving a shift in the argument (Γ(n) ≠ n! for most positive integers n).
* The correct recurrence relation for the Gamma function is Γ(z + 1) = z * Γ(z), as you accurately pointed out, referencing Legendre's work.
 
If Gemini's topological claims rise above the hallucination floor, it seems like something this article might profit from pointing out as a footnote-ish addition to the bottom of the relevant section concerning the extra topological specialness of X2 + 1. — MaxEnt 01:46, 28 February 2024 (UTC)
replies: text: Presently, no AI can be a reliable source for mathematical assertions, and this is not a place for discussing their ability to state fallacies. D.Lazard (talk) 02:38, 28 February 2024 (UTC)

text: Questions about AI nonsense should be removed to Wikipedia:Reference desk/Mathematics to the extent they belong on Wikipedia at all. Some other forum would probably be better still. –jacobolus (t) 03:04, 28 February 2024 (UTC)

text: I am thinking that "if we construct a field using a different irreducible polynomial, like X^2 + 2, the resulting quotient field ... would not be connected or locally compact." is just false.  But sure, if you can find a reliable notable source to back this statement then we could add it. —Quantling (talk | contribs) 15:12, 28 February 2024 (UTC)
thread_title:  Special nature of X^2+1 as ideal 

message: This article and the articles for real number, rational number, integer, and possibly others, have a large image of the symbol (in this article, \mathbb C) even though the symbol is defined in the lead. This is redundant. I think these large images should be removed.—Anita5192 (talk) 14:18, 26 March 2024 (UTC)
replies: text: As a general rule, good images enhance the visual appeal of Wikipedia. Without them, Wikipedia is paragraphs of text, which are informative but not attractive. This specific image is not very appealing, and it's certainly not useful. So the question for me is: Is it better than nothing? And I guess my answer is a weak "no". Mgnbar (talk) 14:41, 26 March 2024 (UTC)
replies: text: I agree that good images enhance the visual appeal of Wikipedia.  However, I approve of images of real objects, scenery, mathematical objects, etc. that help readers visualize the subject matter.  I think an image of a symbol is usually redundant, especially if it is already in the text.—Anita5192 (talk) 15:22, 26 March 2024 (UTC)
replies: text: If this article had no images at all, and no prospects for good images, then I might support having this image in the article, just for the sake of having something. But that doesn't apply here, so I agree with you. Mgnbar (talk) 16:41, 26 March 2024 (UTC)

text: Speaking as the person who substantially wrote the current version of our article Blackboard bold, I think you should feel free to take it out. We can make better images describing and explaining complex numbers. If really necessary an image of a  symbol could go in the section . –jacobolus (t) 17:18, 26 March 2024 (UTC)

text: , and also for real number, rational number, and  integer—Anita5192 (talk) 17:36, 26 March 2024 (UTC)
thread_title:  Image of symbol