Talk:Complex number

Revert by user:D.Lazard on 13.03.23

Dear D.Lazard, your $$(a+bi)\,i=-b+ai$$ does not fit the current graphic. Nomen4Omen (talk) 15:50, 13 March 2023 (UTC)

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)

Special nature of X^2+1 as ideal

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)

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)

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)

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)

Image of symbol

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)

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)

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)

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)

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 ⁠$\mathbb {C}$⁠ symbol could go in the section § Notation. –jacobolus (t) 17:18, 26 March 2024 (UTC)

 Done, and also for real number, rational number, and integer—Anita5192 (talk) 17:36, 26 March 2024 (UTC)

More justification and intuition, please!

This article is very good from a mathematical perspective. Definitions, algebraic rules and so on are well explained. But I miss more emphasis on WHY. It would be much more inspiring for a reader to go through all those algebraic rules if the reader had a clue about what complex numbers can be used for. There isn't even a mention about a pendulum here!?

Complex numbers are often useful when a phenomenon can change between manifesting itself in two different ways.

One example is a pendulum where energy can be kinetic energy when the pendulum is moving fast at the bottom of its trajectory and positional energy at the highest points of a its trajectory. As the pendulum swings back and forth, the same energy changes in revealing itself in two different dimensions. One of the dimensions can be called "real", and the other then becomes "imaginary".

Another example is in an oscillating electrical circuit, where the same energy can change between revealing itself as a current flowing through a conductor or as a charge in a capacitor.

Another example is sound waves, where a sound measured at one point can change between revealing itself as a air pressure deviation and as a motion in the air.

Thus, two different manifestations or aspects of something can be modelled using just one complex number.

(If something like this had been told me when I started learning about complex number, this would have made my learning easier and more fun) Joreberg (talk) 21:05, 14 April 2024 (UTC)

Some of this is mentioned in the Applications section. It would be great if you could add some of the others you list (with Wikipedia:Reliable sources of course).

In some of your examples, I can't tell whether the object under discussion is the space C of complex numbers or the space R² of pairs of real numbers. This article should restrict its attention to the former. Mgnbar (talk) 11:53, 15 April 2024 (UTC)

I venture that an original motivation for complex numbers is analysis of harmonic oscillators and Fourier analysis in general, so I'd want those featured prominently — which they already are to some extent. —Quantling (talk | contribs) 15:03, 15 April 2024 (UTC)

You seem to be especially interested in uniform circular motion and simple harmonic motion (which can be modeled as a projection of uniform circular motion). These are worth discussing somewhere, but I don't think we should belabor the point at the start of the article. –jacobolus (t) 19:56, 15 April 2024 (UTC)

Question

"Gauss called a^2+b^2 the norm." Shouldn't it be the square of the norm? Pokyrek (talk) 18:06, 30 August 2024 (UTC)

Yes, these days we don't use "norm" the way Gauss did. —Quantling (talk | contribs) —Quantling (talk | contribs) 18:15, 30 August 2024 (UTC)

Nevertheless, a^2+b^2 remains the (field) norm of the complexes over the reals. It is not uncommon in mathematics that a word has different meanings depending on the context (here, it is algebra and number theory versus mathematical analysis. D.Lazard (talk) 08:25, 31 August 2024 (UTC)

"Complex math" and "Complex mathematics" listed at Redirects for discussion

The redirect Complex math has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2024 December 21 § Complex math until a consensus is reached. User:Someone-123-321 (I contribute, Talk page so SineBot will shut up) 09:18, 21 December 2024 (UTC)

The redirect Complex mathematics has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2024 December 21 § Complex mathematics until a consensus is reached. J947 ‡ ^edits 22:49, 21 December 2024 (UTC)

Awkward "abstract algebra" definition

Can someone figure out a clearer and more accessible phrasing than this which pedants will still accept as technically correct? This seems almost deliberately obtuse:

One formal definition of the set of all complex numbers is obtained by taking an extension field $E$ of $\mathbb {R}$ such that the equation $x^{2}+1=0$ has a solution in $E$, calling an arbitrarily chosen solution in $E$ of $x^{2}+1=0$ by the letter $i$, and defining the set of all complex numbers as the subfield $\mathbb {C} =\{z\in E\vert {\text{there exists }}x,y\in \mathbb {R} {\text{ such that }}z=x+iy\}$.

Ideally it should ditch the set-builder notation and as much jargon as possible. –jacobolus (t) 18:31, 24 April 2026 (UTC)

The reference cited is Ahlfors. Maybe he worded this idea better? I don't have that book. Mgnbar (talk) 18:48, 24 April 2026 (UTC)

Ahlfors is pitched at an audience of "first-year graduate and advanced undergraduate students [in pure math]". We should try to aim our version at a broader audience if we can. –jacobolus (t) 19:02, 24 April 2026 (UTC)

Also, Ahlfors' version is clearer and probably even more accessible than ours, and better written, albeit longer:

The equation ⁠$x^{2}+1=0$⁠ has no solution in ⁠$\mathbf {R}$⁠, for ⁠$\alpha ^{2}+1$⁠ is always positive. Suppose now that a field ⁠$\mathbf {F}$⁠ can be found which contains ⁠$\mathbf {R}$⁠ as a subfield, and in which the equation ⁠$x^{2}+1=0$⁠ can be solved. Denote a solution by ⁠$i$⁠. Then ⁠$x^{2}+1=(x+i)(x-i)$⁠, and the equation ⁠$x^{2}+1=0$⁠ has exactly two roots in ⁠$\mathbf {F}$⁠, ⁠$i$⁠ and ⁠$-i$⁠. Let ⁠$\mathbf {C}$⁠ be the subset of ⁠$\mathbf {F}$⁠ consisting of all elements which can be expressed in the form ⁠$\alpha +i\beta$⁠ with real ⁠$\alpha$⁠ and ⁠$\beta$⁠. This representation is unique, for ⁠$\alpha +i\beta =\alpha '+i\beta '$⁠ implies ⁠$\alpha -\alpha '=-i(\beta -\beta ')$⁠; hence ⁠$(\alpha -\alpha ')^{2}=-(\beta -\beta ')^{2}$⁠, and this is possible only if ⁠$\alpha =\alpha '$⁠, ⁠$\beta =\beta '$⁠.

[...]

There are many ways in which such a field can be constructed. The simplest and most direct method is the following: Consider all expressions of the form ⁠$\alpha +i\beta$⁠, where ⁠$\alpha$⁠, ⁠$\beta$⁠ are real numbers while the signs ⁠$+$⁠ and ⁠$i$⁠ are pure symbols (⁠$+$⁠ does not indicate addition, and ⁠$i$⁠ is not an element of a field). These expressions are elements of a field ⁠$\mathbf {F}$⁠ in which addition and multiplication are defined by (1) and (2) (observe the two different meanings of the sign ⁠$+$⁠). The elements of the particular form ⁠$\alpha +i0$⁠ are seen to constitute a subfield isomorphic to ⁠$\mathbf {R}$⁠, and the element ⁠$0+i1$⁠ satisfies the equation ⁠$x^{2}+1=0$⁠; we obtain in fact ⁠$(0+i1)^{2}=-(1+i0)$⁠. The field ⁠$\mathbf {F}$⁠ has thus the required properties; moreover, it is identical with teh corresponding subfield ⁠$\mathbf {C}$⁠, [...]

–jacobolus (t) 19:09, 24 April 2026 (UTC)

Someone was very insistent on getting this definition in a few weeks ago. They initially put it in as the first definition! I at first removed it, and then (after an edit war) as a compromise put something like it further down in reduced form in the algebraic definitions section. They replaced what I had written with their favored definition (from Ahlfors, they assured me, and therefore beyond reproach). But it seems to me that Ahlfors' task is rather different than what our goal should be here, and just stating that it is the splitting field (unique up to isomorphism) and then constructing it using standard abstract algebra seems like a better approach. Sławomir Biały (talk) 19:29, 24 April 2026 (UTC)

A new editor seems keen on adding this content again, I suspect the same as the old editor. Unclear why someone would be so insistent on this one thing. They also appear to want to add a redux of the main definition to the abstract algebra section. This would be a good place for that editor to explain themselves and get consensus for the edit first. Sławomir Biały (talk) 07:44, 16 June 2026 (UTC)

Still waiting for discussion here from the anonymous account... In case it is not obvious, I oppose adding the same definition we give already a second (and in the case of the most recent edit, a third) time. The reasons for this opposition are obvious: we already include this definition in the article. Please discuss reasons for inclusion here, and build consensus. Sławomir Biały (talk) 15:20, 16 June 2026 (UTC)

So, the same editor still has not weighed in here. I will note that WP:ONUS indicates that the anonymous editor is responsible for building consensus to add the disputed content. In any case, the current consensus is that 100% of the editors who have actually weighed in on the discussion page have objected to it. If there are no objections, I will revert the addition. Sławomir Biały (talk) 15:35, 16 June 2026 (UTC)

@Sławomir Biały, I encourage you let the process at the edit warring noticeboard play out, seek a third opinion or other dispute resolution options, and be mindful of the three revert rule. I'll also ping @Jacobolus and @Mgnbar here as prior participants in the conversation. tony 17:21, 16 June 2026 (UTC)

It seems likely that the TA does not understand how to use article talk pages, I have left them a direct link at their user talk. MrOllie (talk) 18:07, 16 June 2026 (UTC)

I completely and undeniably agree with Jakobulus that Ahlfors' definition is better. It makes the Wikipedia website so much more credible in just using something that is word for word stated by an author who created a textbook which is employed by many universities across the world when teaching graduate complex analysis courses. Any equivalences you can find with definition are great, but Wikipedia is not the place to post uncited proofs or unpublished theorems or even unpublished definitions. The Wikipedia team is adamant about using cited sources ~2026-29788-75 (talk) 18:42, 16 June 2026 (UTC)

Yes, I did not know how to use it. I am new to the wikiepedia cite. If we add in Ahflors' definition of the complex numbers I would be so much more content with the material on the complex numbers wikipedia webpage. And making sure that readers fully understand that the Cayley-Dickson construction used to create a definition of the complex numbers as R^2 with complex addition and complex multiplication would make this webpage so much more unambiguous. You read this highly abstract algebraic definition of the complex numbers which is (more likely than not) equivalent to Ahflors' definition of the complex numbers in the highly celebrated textbook Complex Analysis and then you go down and read the Cayley Dickson construction and realize that Ahlfors never used the definition of an extension field as a field containing a field isomorphic copy of the real numbers with operations which restrict to the operations of the field isomorphic copy of the real numbers; Ahlfors was very deliberate in saying that the complex numbers contain the real numbers rather than "they contain a field isomorphic copy of the real numbers". I understand some abstract algebraists are a little more general with their definition of an extension field, but again, we need to use something that's employed by universities and backed by professional organizations such as the American Mathematical Association and the American Mathematical Society when conducting complex analysis research. ~2026-29788-75 (talk) 18:46, 16 June 2026 (UTC)

Personally, I do not think Ahlfors' construction is the right one to use in the article, and one author is not dispositive here. The main problem is that it presupposes that "a field ⁠$\mathbf {F}$⁠ can be found which contains ⁠$\mathbf {R}$⁠ as a subfield, and in which the equation ⁠$x^{2}+1=0$⁠ can be solved." While certainly correct, the problem is that it introduces an auxilary field F which is not part of the basic construction. Once one strips this away, one arrives at the more common characterization of $\mathbb {C}$ as the splitting field of $x^{2}+1$ over $\mathbb {R}$. Or, what amounts to the same thing, as the field $\mathbb {R} (i)$ obtained by adjoining a primitive fourth root of unity to the reals. Also, why did you delete the reference to JS Milne? You did so with the edit summary "The reference which was provided does not provide an explicit definition of the complex numbers, like Lars V. Ahlfors does. Please provide a citation with an explicit definition." But that source does give a definition, on the indicated page: "We define ℂ to be the splitting field of 𝑋² + 1 over ℝ." Sławomir Biały (talk) 19:11, 16 June 2026 (UTC)

The AMS and AMA do not "back" mathematical definitions. However, they sometimes publish books whose authors use one or another definition. Universities also do not "employ" definitions. What they do is hire researchers whose published books and articles pick one or another definition. For something like complex numbers, which specific definition is convenient depends on the context.

contain the real numbers rather than "they contain a field isomorphic copy of the real numbers" – the former is essentially a sloppy shorthand for the latter; in practice, it's convenient to call the complex number ⁠$a+0i$⁠ a "real number" and not bother writing the ⁠$0i$⁠ part or worry about the details of how the phrase "real number" is defined. –jacobolus (t) 19:24, 16 June 2026 (UTC)

Either way, you understand that I am emphasizing that the definition given is not backed up by any scholarly textbooks or publishers of academic textbooks. Lars V. Ahlfors' definition, however, is published in a book and is still being used to this day to teach complex analysis to students in an effort to help them publish research in complex analysis and complex dynamical systesms and quantum dynamics; it is sucessful. This is in contrast to the current definition on wikipedia which has no scholarly textbook listed as an official source that uses that definition to conduct complex analysis. It is uncredible. Even if it is equivalent to Ahlfors' definition, until you find an academic publishing that establishes that equivalence of definitions it is still uncredible. ~2026-29788-75 (talk) 19:38, 16 June 2026 (UTC)

Here's Loomis & Sternberg (1990) with a more careful phrasing:

The mapping ⁠$x\mapsto \langle x,0\rangle$⁠ is an isomorphic injection of the field ⁠$\mathbb {R}$⁠ into the field ⁠$\mathbb {C}$⁠. It clearly preserves sums, and the reader can check in his mind that it also preserves products. It is conventional to identify ⁠$x$⁠ with its image ⁠$\langle x,0\rangle$⁠, and so to view ⁠$\mathbb {R}$⁠ as a subfield of ⁠$\mathbb {C}$⁠.

–jacobolus (t) 21:38, 16 June 2026 (UTC)

@Jacobolus, you need to cite a textbook that says so and quote it. Nobody on the internet should just take the word of someone from the internet who could be posing as "Steven G. Krants" or "Terence Tao" and expect to make it big within the research community and start publishing successful and impactful research papers. ~2026-29788-75 (talk) 19:40, 16 June 2026 (UTC)

Regardless, Ahlfors said something very literal, and, when using Ahlfors' definition, the researchers 500 years from now who use Ahlfors' textbook will most likely take it in the literal fashion until Ahlfors rises up from the grave and says otherwise.  ~2026-29788-75 (talk) 19:42, 16 June 2026 (UTC)

Ahlfors is not some sacred text that must inform the coverage of all topics on Wikipedia related to complex numbers. Generations of wise mathematicians are not sagely stroking their beards over how wisely Ahlfors defined the complex numbers. Get a grip. Sławomir Biały (talk) 19:55, 16 June 2026 (UTC)

Slawomir, the point is, your "sworn by definition" is NOT backed up by a scholarly textbook or paper. Realize that mathematics topics on Wikipedia are trying to promote information that is backed by textbooks and papers. They do not prefer unbacked sources, and this not now or ever the place for unbacked information. If you cannot cite something to back what you're saying then chances are it doesn't belong on this website. Lars V. Ahlfors' textbook has an explicit and working definition that we can cite as a community which tons of mathematicians enjoy and use to conduct research and teach classes, so it does belong on this website. ~2026-29788-75 (talk) 21:20, 16 June 2026 (UTC)

That being said, if you can find an actual scholarly textbook or paper that explicitly uses that exact same definition, then I have no reason to argue. Until then, it shouldn't even be argued that actual definitions being used in scholarly research papers and being used in textbooks belong on this website in their unaltered form. If you alter it, then you need to cite something or you are publishing unbacked and uncredible and unreliable information on Wikipedia; simply giving high school teachers another reason to say "I don't want anybody citing this website". ~2026-29788-75 (talk) 21:23, 16 June 2026 (UTC)

I have no idea what you are talking about. Please try to state things simply, about specific sentences in the article, without these ridiculous histrionics. Sławomir Biały (talk) 06:26, 17 June 2026 (UTC)

I think they're looking for further sources which explicitly define complex numbers using a quotient ring or matrices. These shouldn't be too hard to find.

Most analysis sources which give a formal definition use pairs of real numbers with component-wise addition and an explicitly defined multiplication operation. Some others say start with real numbers and then throw in an extra number ⁠$i={\sqrt {-1}}$⁠ and apply the usual rules of arithmetic; this is effectively the algebraic approach but without introducing as many abstract concepts.

I do think we can do some amount of rephrasing here, both in introductory sections and in a formal definitions section. –jacobolus (t) 06:40, 17 June 2026 (UTC)

We already cite sources about the quotient ring view, and I checked the Bourbaki source myself. The basic definition seems adequately supported by the Spiegel source. Ironically, it is also supported by the very Ahlfors souce that the IP is pushing. I think that possibly the IP seems to be confused in the belief that we have to repeat sources verbatim. We are rather explicitly not supposed to do that. Sławomir Biały (talk) 06:47, 17 June 2026 (UTC)

Nobody here is impersonating Terence Tao. I don't understand what you are trying to say. –jacobolus (t) 20:03, 16 June 2026 (UTC)

Abstract and algebraic definitions

I partially agree with the concerns, in the preceding thread, of a TA: the beginning of § Abstract and algebraic definitions was unnecessarily technical and explains nothing useful. Moreover, the existence of many definitions that produce complex numbers of very different nature may be confusing for some readers.

I rewrote this paragraph for clarifying this. The new version is impired by the theory of universal properties without any explicit reference to category theory. So I believe that the new version may be useful for many readers.

I removed some citations that seem not to be relevant to the new version. Although I am convinced that the definition of the complex numbers as the solution of a universal property shoud have been published somewhere, I am unable to provide an explicit reference. Please add one if you know one. D.Lazard (talk) 09:28, 17 June 2026 (UTC)

Overall, I think directionally the edit is good, but I would leave the universal characterization out probably. I think more often in field theory, a splitting field is understood up to isomorphism. I think that language should probably be restored somewhere, and I believe it can easily be sourced. Sławomir Biały (talk) 10:02, 17 June 2026 (UTC)

A bit of trim and reorder to the lead

The lead section was getting pretty unwieldy and rambling. I gave it a bit of a trim, tried to re-order a few of the points in the first few paragraphs for narrative flow, and e.g. briefly glossed "real number" for accessibility. (I didn't think e.g. a detailed discussion of the geometric meaning of each arithmetical operation belongs in the lead section.) Other folks may want to weigh in, in case anything I chopped seems essential to keep in the lead, or in case anything I write seems confusing or inaccurate. I think the lead could still be usefully extended by further summary material, e.g. about complex analysis or about applications of complex numbers in math and science. –jacobolus (t) 19:19, 18 September 2026 (UTC)

Semi-protected edit request on 20 September 2026

To clarify the description of the multiplication operations, change the current description FROM: $$(a+bi)\cdot (c+di)=ac-bd+(ad+bc)i.$$

For example, $(2-i)(3+4i)=2\cdot 3-((-1)\cdot 4)+(2\cdot 4+(-1)\cdot 3)i=10+5i.$ In particular, this includes as a special case the fundamental formula $$i^{2}=i\cdot i=-1.$$

TO:

$$(a+bi)\cdot (c+di)=ac+bi*di+(ad+bc)i.$$ By definition, $$i^{2}=i\cdot i=-1.$$ so we can rewrite the above equation as: $$(a+bi)\cdot (c+di)=ac-bd+(ad+bc)i.$$

For example, $(2-i)(3+4i)=2\cdot 3-((-1)\cdot 4)+(2\cdot 4+(-1)\cdot 3)i=10+5i.$ ~2026-50833-54 (talk) 22:32, 20 September 2026 (UTC)

I did something similar to that. What do you think? —Quantling (talk | contribs) 23:05, 20 September 2026 (UTC)

"is" vs. "is called"

@Sapphorain et al., IMHO, there is overuse of "is called" in this article. Sure, every sentence that uses a mathematical term could say that with "is called":

In mathematics, a number is called a complex number if it is of the form ⁠$a+bi$⁠, where a and b are what are called real numbers, and i is what is called the imaginary unit, a number that satisfies the equation ⁠$\textstyle i^{2}=-1$⁠; the number a is called the real part of ⁠$a+bi$⁠, and b is called the imaginary part.

The complex numbers form what is called a number system which extends the real numbers and supports the same basic arithmetic operations known as addition, subtraction, multiplication, and division, satisfying laws known as the associative, commutative, and distributive laws. This makes the complex numbers what is called a field. The set of complex numbers is denoted by either of the symbols $\mathbb {C}$ or C.

But I think turning what is a mathematics focus into a linguistics focus is not doing the reader any favors. I'd rather see:

In mathematics, a complex number is a number of the form ⁠$a+bi$⁠, where a and b are real numbers, and i is the imaginary unit, a number that satisfies the equation ⁠$\textstyle i^{2}=-1$⁠; the number a is the real part of ⁠$a+bi$⁠, and b is the imaginary part.

The complex numbers form a number system which extends the real numbers and supports the same basic arithmetic operations of addition, subtraction, multiplication, and division, satisfying the same associative, commutative, and distributive laws. This makes the complex numbers a field. The set of complex numbers is denoted by either of the symbols $\mathbb {C}$ or C.

In most cases, I find is called and similar (e.g., is known as) as word clutter that does not add to a reader's comprehension. Thoughts? —Quantling (talk | contribs) 14:58, 22 September 2026 (UTC)

Your comparison is an absurdist straw man and it's distracting from whatever point you were trying to make. –jacobolus (t) 17:13, 22 September 2026 (UTC)

I would like the lede to say the number a is the real part of ⁠$a+bi$⁠, and b is the imaginary part. I find is called makes me start thinking about the linguistic history of mathematical terminology rather than the mathematics itself. That's the negative I'm trying to avoid. What is the positive that you associate with the is called approach? —Quantling (talk | contribs) 17:57, 22 September 2026 (UTC)

If you say straight out that "a is the real part and b is the imaginary part", then readers who don't know what that means are going to be misled/confused, because both numbers are equally "real" or "imaginary" in the sense of physical reality. (And even more confusing, both are "real numbers"; neither is "imaginary" in the mathematical sense.)

These labels are made up words, and their connection to the ordinary lay definition of "real" and "imaginary" is an incidental detail of linguistic history, not a factual/mathematical feature. The whole point, in this specific case, is to get someone thinking about the linguistic history of mathematical terminology, because the terminology is weird and frankly bad.

To quote Sapphorain's edit summary:

It may be simpler, but also more likely to be confusing. It is already a very unfortunate terminology that a real number should be qualified as « imaginary », so we should at least, in order to avoid confusion, state that «it is CALLED the imaginary part of» instead of stating that « it IS the imaginary part of »

–jacobolus (t) 18:09, 22 September 2026 (UTC)

Hmm. All of these are names, but I don't see you arguing for In mathematics, a number is called a complex number if .... Surely, a quaternion or a matrix is pretty complex to many, but would you say that, for our audience, "complex number" without "is called" is easier to handle than "imaginary part" without "is called"? In this article we have the advantage that complex number, real part, and imaginary part are all in bold, which (to me) is a pretty strong signal that these numbers/parts are not merely complicated, in reality, or illusory. —Quantling (talk | contribs) 18:23, 22 September 2026 (UTC)

You are right, nobody is arguing for that, which is why the straw man proposal of putting "is called" in front of every jargon word is very distracting and silly. –jacobolus (t) 20:31, 22 September 2026 (UTC)

We agree on "complex" but not on "imaginary", so I'm trying to figure out how you see those two as different (i.e., without "is called" and with, respectively) where I see them as the same (i.e., both not needing "is called"). —Quantling (talk | contribs) 23:38, 22 September 2026 (UTC)

When you say one thing is "real" and another thing is "imaginary" to someone who has never heard of this topic, they start thinking that the first one is definite and tangible and the second one is fictional or part of a dream or hallucination. They are unlikely to immediately guess that the first one means "number somewhere in a linear continuum" and the second one means "a number that might represent a rescaled 90° rotation". (It's especially confusing because we are saying the real part is a real number and the imaginary part is ... also a real number....)

If you say that some number is "complex", and someone doesn't know what that jargon means, they might be reminded of a building complex, a psychological complex, a geological complex, a protein complex, or perhaps, following the lay definition, just think it might have a lot of parts.... but they are less likely to be dramatically misled than with the terms "real"/"imaginary". I don't think the "known as complex" or "called complex" phrasing is particularly helpful (though it might be, in some contexts; feel free to propose more possible variants). But again, I think it would be a good idea to put a brief bit of explanation about where the term complex comes from, since it might help readers. –jacobolus (t) 01:06, 23 September 2026 (UTC)

We could add further explanation of the term "complex" somewhere if you want. That would probably be helpful, since readers could certainly think think that "complex numbers" must be a very difficult or complicated topic.

Complex, from Latin com- and plectō, meaning something like "braid together", is intended to imply that the two parts of a "complex number" are to be taken as a single unified object. It isn't supposed to mean anything about how difficult or intricate the subject is. The words "complex" and "complicated" originally had quite distinct meanings, but they are often used loosely in ordinary speech, and, perhaps also because they sound similar, people confuse the two. –jacobolus (t) 20:36, 22 September 2026 (UTC)

I didn't know about "complex vs. complicated"; thank you, good to know! Perhaps this article would benefit from a etymology section where we discuss the linguistics: why is each mathematical concept called what it is called? But that's almost a separate discussion I think ... unless you are saying that adding such a section would affect how many times we should then use "is called" in the lede. —Quantling (talk | contribs) 23:44, 22 September 2026 (UTC)

In general I agree that "is called" is often superfluous. But in this particular case the phrase might help readers understand that these words are not being used in their colloquial sense but rather in an artificial mathematical jargon sense. So please leave them in. Regards, Mgnbar (talk) 20:19, 22 September 2026 (UTC)

Thank you. This unfortunate and illogical terminology defining a real number as the imaginary part of a complex number cannot be changed anymore. But at least there should be a clear warning that this is just a disastrous convention. So the problem of repetition and an overuse of « is called » is the least of my worries. For any layman, and for many mathematicians like myself, this definition doesn’t make sense: so at least let’s keep the precision that if « it is », it’s because « it has been called thus ». --Sapphorain (talk) 20:41, 22 September 2026 (UTC)

Your feelings are much stronger than mine. I don't think that it's a disastrous convention, and I don't think that we need a clear warning. But at least we agree on the micro-issue at hand. Cheers, Mgnbar (talk) 21:00, 22 September 2026 (UTC)