Talk:Complex number

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 ${\displaystyle ab}$ and by Luke Walsh distributes ${\displaystyle b}$ to ${\displaystyle a=x+yi}$, i.e. ${\displaystyle ab=xb+yib}$, while Grant Sanderson and Ben Sparks in their visualizations distribute ${\displaystyle a}$ to ${\displaystyle b=u+vi}$, i.e. ${\displaystyle ab=ua+vib}$, but due to commutativity this does really matter. (In other words their presentations of ${\displaystyle ab}$ match mine and Luke Walsh's of ${\displaystyle ba}$.)

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 b to ${\displaystyle a=x+yi}$:

$${\displaystyle a\,b=(x+yi)\,b=xb+yib.}$$

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 a and b is the point that has the same coordinates as the point a in the complex plane but in a new scaled and rotated Cartesian coordinate system that has the x-axis and y-axis going through b and ib and a unit of measure ${\displaystyle |b|}$. ^[1]

Happy for any suggestion. Qerez (talk) 11:31, 20 February 2022 (UTC)

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)

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)

Aside from a lazy reversion of several improvements in my edits, we're back to an opening sentence that's unclear:

"In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted i, called the imaginary unit, and satisfying the equation i2 = −1."

It needs to be clearly expressed, even if experts can wade through it to extract the correct meaning.

• (1) is it "an element" or "a number system" that contains the real numbers and a specific element denoted i, called the imaginary unit? It needs to be unambiguous.
• (2) what is "satisfying" the equation i2 = −1? Is it "the imaginary unit" or "a complex number", or both "the real numbers and a specific element denoted i, called the imaginary unit"?

Tony (talk) 08:08, 4 July 2022 (UTC)

Reverting the introduction of grammar errors is not a lazy revert.

Point (2): it is i that satisfies ${\displaystyle i^{2}=-1.}$ Nothing confusing here.

Point (1): the usual rule is that "that" refers to the last preceding noun, that is, "number system". Also, an element is not a set, and thus does not contain anything.

Nevertheless, I have replaced "that contains the real numbers and a specific element denoted i" with "that extends the real numbers with a specific element denoted i", hoping that this will be clearer. D.Lazard (talk) 09:09, 4 July 2022 (UTC)

In accusing, you appear to have misread my opening above: "Aside from a lazy reversion of several improvements in my edits, we're back to an opening sentence that's unclear:" (my italics).

"an element is not a set" – you're making it up. Depends on scale. In many areas, elements can have internal components. You're not talking just to experts who will be able to decipher your text because they know the subject already. We can't be expected to know what you're assuming.

"the usual rule is that "that" refers to the last preceding noun" – your source for that?

Back to the problematic sentence in the article: "that extends the real numbers with a specific element denoted i, called the imaginary unit, and satisfying the equation i2 = −1". But you say "Nothing confusing here".

If it's i that satisfies the equation, you need at the very least to get rid of the confusing comma after "unit"; or perhaps better, either (a) or (b):

(a) "with a specific element denoted i (called the imaginary unit) that satisfies the equation i2 = −1"

(b) "with a specific element denoted i (called the imaginary unit), which satisfies the equation i2 = −1."

I don't know which of (a) or (b) is correct: you need to decide that, based on whether i inherently satisfies the equation (which), or not inherently (that).

Precision is central, especially at the opening. Tony (talk) 10:08, 4 July 2022 (UTC)

Parentheses are not convenient for an essential clause. The comma is an old-standing typo that I have fixed. D.Lazard (talk) 10:46, 4 July 2022 (UTC)

For what it’s worth, I think this lead section is pretty mediocre, suffering from a common problem of math textbooks of elevating formal definitions ahead of meaning. The #1 key thing to know about the complex numbers is that while multiplication by a "real number" represents scaling, multiplication by a complex number represents both scaling and planar rotation (Needham calls this an "amplitwist"). Here, the words “rotate”, “rotation”, etc. don’t occur until 1200 words into the article, and even then the explanation is that multiplication is "adding the angles" (To lay readers, this does not obviously and immediately come across as meaning rotation.) A proper explanation doesn’t come until about 4000 words into the article. The lead section says that multiplication is "a similarity" with a hyperlink: this is not remotely lay-accessible; most readers are going to skip right past that without understanding what is meant.

The #2 key thing to know about the imaginary unit i (after #1 knowing that it squares to –1) is that it represents some kind of movement or position at a right angle from the "straight" real-number direction. But words like "perpendicular", "orthogonal", "right angle" don’t show up anywhere here, and again, multiplication of a complex number by i corresponds to rotating the position vector counterclockwise by a quarter turn (90°) about the origin doesn’t show up until 1200 words in.

The lead image here is also not sufficient for the article. Just showing that a complex number can be pictured as a 2-dimensional Euclidean point or an arrow from the origin (representing a Euclidean planar vector) in a diagram with axes labeled Re and Im instead of x and y doesn’t explain anything. The key geometrical features of complex numbers are that they add tip-to-tail like a geometric vector and that they multiply by “amplitwisting”. Both of these key features can be easily depicted in a diagram or two.

Even when getting to formalities, this article somewhat picks sides. Defining complex numbers as a quotient ring is not wrong, but it is just one of many possible definitions. The definition from linear algebra (a certain class of matrices), the definition from geometric algebra (the even sub-algebra of the geometric algebra of the Euclidean plane), and others are equally valid. As Clifford (and Grassmann!) figured out, a complex number is best thought of conceptually as the ratio of two planar vectors. Grassmann (1844): “From this all imaginary expressions now acquire a purely ge- ometric meaning, and can be described by geometric constructions.... it is likewise now evident how, according to the meaning of the imaginaries thus discovered, one can derive the laws of analysis in the plane; however it is not possible to derive the laws for space as well by means of imaginaries. In addition there are general difficulties in considering the angle in space, for the solution of which I have not yet had sufficient leisure.”

Wikipedia math articles about basic topics should try to lead with a lay-accessible definition/explanation, ideally with a figure that gets the main point(s) across, and then follow up (possibly soon) after with the formalities. –jacobolus (t) 12:01, 4 July 2022 (UTC)

I agree with Tony that the opening sentence is pretty clumsy as it stands.

I agree with Jacobolus that many Wikipedia math articles are harmed by a textbookish insistence of having a rigorous definition in the opening sentence. I wish that we could get past that, as a community. And I agree that, when several definitions are equally valid, more accessible ones should be presented first.

I disagree with Jacobolus's #1 and #2 key-thing opinions. That geometry does not require the invention of the complex numbers. Complex numbers exist because of algebraic closure.

Anyway, if we were willing to set aside rigor in the opening sentence, we would have much more freedom in writing succinct sentences that satisfy the typical reader's needs. Mgnbar (talk) 13:15, 4 July 2022 (UTC)

(P.S. Okay, in addition to algebraic closure, complex numbers are popular because they provide a handy calculational framework for quantum theory and other physics. Mgnbar (talk) 13:19, 4 July 2022 (UTC))

Why complex numbers were first invented/discovered (solving cubic polynomials) and why mathematicians enjoy them (they are commutative, unlike higher-dimensional analogs) is not the same as why they are important (having an algebra for planar similarity transformations (including inside an algebra for rotations on the circle that doesn’t involve so many transcendental functions as classical trigonometry) is essential all over science, engineering, and mathematics). –jacobolus (t) 13:25, 4 July 2022 (UTC)

Well, I disagree, but maybe we should not debate it here, until it impinges on how we edit the opening sentence. Mgnbar (talk) 13:41, 4 July 2022 (UTC)

(edit conflict) The application of complex numbers to plane geometry is important, but not more than the non-geometric applications, such as, among many, the fundamental theorem of algebra, Euler's formula (and its applications in acoustics and electrical engineering through Fourier analysis), the concept of holomorphic functions, etc. D.Lazard (talk) 13:57, 4 July 2022 (UTC)

“Euler’s formula” (the complex exponential), Fourier analysis, and holomorphic functions are all essentially geometrical, and are useful in math and in applications for geometrical reasons. The complex logarithm is a way of taking the multiplicative structure of complex numbers (scaling and rotation) and transplanting it by a conformal map from the plane to the infinite cylinder, where it turns into complex addition (translation). When limited to the unit circle, this just recovers the traditional way of thinking about goniometry (angle measures) from the more natural multiplicative structure in the complex plane: that multiplicative structure was already implicit in the angle-sum formulas (${\textstyle \cos(a+b)=\cos a\,\cos b-\sin a\,\sin b}$ and ${\textstyle \sin(a+b)=\cos a\,\sin b+\sin a\,\cos b}$) from antiquity, long before anyone defined complex numbers per se. Notice that historically the more general complex logarithm showed up first geometrically as the relation between the stereographic projection and Mercator projection, long before the relevant mathematical formalisms were developed. Fourier analysis is all about representing periodic signals by considering the independent variable to be a point on a conceptual circle (or a conceptual angle measure) and then approximating them as trigonometric polynomials (each term of which represents spinning around the circle at some integer frequency). It turns out periodic (or nearly periodic) motions and signals show up all over the place so this is a very generally useful tool. Historically the first example of this general idea is approximating planetary motions by epicycles. Holomorphic functions are a representation of those maps from the plane to the plane (or more generally between two locally planar surfaces) which locally preserve complex multiplication (similarity transformations). Historically these also first showed up in cartography. –jacobolus (t) 14:24, 4 July 2022 (UTC)

Almost all mathematics can be interpreted geometrically, but the converse is also true, as shown by René Descartes (Cartesian coordinates) and Felix Klein (Erlangen program): all geometry can be interpreted algebraically. To assert, as you do that one direction is more important than the other, you must provide reliable sources. Otherwise this is only your opinion, and Wikipedia is not the place to discuss it. (By the way, your characterization of holomorphic functions is wrong, and I cannot imagine how to interpret geometrically the fundamental theorem of holomorphic functions, which asserts that a complex function is derivable if and only if it equals the sum of its Taylor series.) D.Lazard (talk) 14:55, 4 July 2022 (UTC)

Nobody says that the symbolic/algebraic structure of complex numbers is unimportant, but only that (a) this article is currently woefully deficient in describing the basic geometry (especially for an encyclopedia article aimed at a wide audience for many of whom visual/spatial reasoning is clearer than symbol twiddling), and (b) complex numbers are fundamentally the algebra of rotation-and-scaling transformations of the plane, which can (as one possibility) serve as their basic definition and which provides much of their power. Many other situations in mathematics turn out to have the same structure as rotation-and-scaling of the plane, and we can learn much by interpreting other questions as a kind of conceptual plane and operating on them there using the tools of complex numbers. As for holomorphic functions: a holomorphic function f:U → V is a function with a complex derivative, which means that if you zoom in on a very tiny piece of U, the function f applied to that tiny piece looks locally complex-linear, i.e. like a similarity transformation. f(z + dz) - f(z) = dz f′(z), which is to say, displacements in f(z) preserve the complex-multiplicative structure present among very small displacements of z.

As for sources, Tristan Needham thought this interpretation of the meaning of holomorphic functions was so important that he literally wrote a whole book about it (Needham (1997), Visual Complex Analysis)! At least some other people agree – the book has been cited >1000 times in the academic literature. –jacobolus (t) 18:17, 4 July 2022 (UTC)

Before discussing lead sentences, I would like to hear your opinion on whether to merge imaginary numbers into this page. If we didn't merge, I think we can omit some explanations about imaginary numbers. Also, I think the section of Relations and operations will be explained before section of visualization.--SilverMatsu (talk) 15:36, 4 July 2022 (UTC)

There's not much at Imaginary number, and I'm not sure what should be added. The imaginary numbers do have their own existence as the Lie algebra of the unitary group U(1), but that's not much. Is there any other reason why imaginary numbers should be treated on their own, separately from the complex numbers? Mgnbar (talk) 17:25, 4 July 2022 (UTC)

Thank you your reply. I haven't found any reason other than the example you gave. It seems that wikipedia already has an imaginary unit as a separate article from the imaginary number. If merge, there isn't much to add to this article, I think just redirecting the Imaginary number to this article will complete the merge. --SilverMatsu (talk) 05:40, 6 July 2022 (UTC)

I don’t think they should necessarily be merged. Among other potential topics, Imaginary number seems like a better place to talk at greater length about the confusion about whether "real" and "imaginary" numbers are really "real" or "imaginary" under the conventional lay connotations of those terms. There can also be some discussion about which quantities should best be thought of as "pure imaginary" and why (for example, angle measures are naturally "imaginary” (bivector-valued) quantities, which is why we exponentiate them as ${\displaystyle \exp \theta i}$). Imaginary number should probably link to (and discuss) bivector, and maybe include some material about the “imaginary” part of a quaternion and its (somewhat confusing) use in representing both vectors and bivectors. –jacobolus (t) 08:33, 6 July 2022 (UTC)

If more place is needed for explaining the lay connotations of "real" and "imaginary", then "complex" and "hypercomplex" must also be included. As far as I know (without historical references), the word "complex" is used because the complex numbers have two parts, as opposite to (simple) numbers that have a single part. "Hypercomplex" means thus more than two parts.

IMO this information has its place in this article. D.Lazard (talk) 09:22, 6 July 2022 (UTC)

Yes this is correct, “complex” is used in the sense of “composite” (etymologically, complect = “braid together”). “Hypercomplex” numbers “split complex” numbers, “dual” numbers, (and possibly Clifford algebras, etc.) should probably also just be called “complex” by standards of ordinary language, but I suspect that by the time the term “hypercomplex” was introduced, the idea of “complex numbers” had already become ingrained as a single chunk referring only to one specific number system with most users of the term not ever thinking too hard about what “complex” means as a plain-language word per se.

Talking about the history of the name “imaginary” is relevant enough here at complex number to briefly mention, but belaboring the point here starts to become a distraction, whereas I think you could easily put an extended history/language lesson at imaginary number. –jacobolus (t) 19:38, 6 July 2022 (UTC)

• I arrived at the article last week to remind myself of what a complex number is, after encountering it in a YouTube physics video. I didn't find the opening helpful. (But of course it must become too technical for me, further down.) Tony (talk) 12:11, 8 July 2022 (UTC)

Dear D.Lazard, your $${\displaystyle (a+bi)\,i=-b+ai}$$ 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)

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)

This article and the articles for real number, rational number, integer, and possibly others, have a large image of the symbol (in this article, ${\displaystyle \mathbb {C} }$) 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 ⁠${\displaystyle \mathbb {C} }$⁠ 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)

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)

"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)

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)