Talk:Quaternion

The article is currently too technical for non-experts to understand; I am adding a tag to suggest the article be improved to be understandable to non-experts. Betanote4 (talk) 18:16, 5 August 2020 (UTC)

:A certain amount of it is accessible to non-experts, and a certain amount of it isn't. But it's a technical subject, and that's about what we should expect. Unless you can be more specific, the tag isn't really helpful, so I've removed it. –Deacon Vorbis (carbon • videos) 18:30, 5 August 2020 (UTC)

:This article is "Quaternions for Pure Mathematicians". For practical aspects, see Quaternions and spatial rotations, with Quaternions and spatial rotations#Quaternions providing a quick introduction to quaternions. BMJ-pdx (talk) 22:55, 5 June 2023 (UTC)

“ Quaternions are generally represented in the form

a + bi + cj + dk

where a, b, c, and d are real numbers; and i, j, and k are the basic quaternions.”

Why refer to i, j, and k as the “basic quaternions” and not the “standard basis vectors”? I have not seen the term basic quaternion before and did not find any relevant information when looking it up. 76.151.136.63 (talk) 13:40, 26 January 2023 (UTC)

:Firstly, the standard basis of the vector space of the quaternions contains also the real number 1. Secondly, one may understand what are quaternions without knowing vector spaces, bases of vector spaces, and standard bases. So, the change you suggest would make the article unnecessarily more technical (see WP:TECHNICAL). Also, the concept of a vector space has been introduced years after the quaternions, and I guess that bases of vector spaces have been so named after the basic quaternions. D.Lazard (talk) 16:35, 26 January 2023 (UTC)

::I changed basic quaternions to basis vectors or basis elements partly to be consistent with the rest of the article, partly because I found a reference for it, and partly because basic quaternions seems to be nonstandard.—Anita5192 (talk) 17:04, 26 January 2023 (UTC)

:::It's a matter of changing definitions of vector. The use of the word vector in mathematics was originated by Hamilton to refer to the "imaginary" part of a quaternion. But later Gibbs/Heaviside adopted it in their formulation of electrodynamics based on dot and cross products (popularized in the book Vector Analysis). Later, while physicists continue to use the Gibbs/Heaviside concept, mathematicians adopted the same name for the broader concept of a vector space. The mathematician's concept of a "vector" is different enough that applying the word to the imaginary part of a quaternion causes some confusion today. –jacobolus (t) 17:22, 26 January 2023 (UTC)

::::I don't see this in the History section, so perhaps it should be included.—Anita5192 (talk) 17:42, 26 January 2023 (UTC)

:::::There is an additional point of confusion, which is that as the even sub-algebra of the geometric algebra (real Clifford algebra) of Euclidean 3-space, the quaternions are "actually" made up of a scalar ("real") part and a bivector ("imaginary") part. Both Hamilton and Gibbs/Heaviside somewhat conflated the concepts of vectors (line-oriented magnitudes) and bivectors (plane-oriented magnitudes), sometimes calling the latter "pseudovectors" or "axial vectors" because they transform differently than ordinary vectors, the "polar vectors". This is possible in 3 dimensional Euclidean space (but no other dimension) because every plane has a unique perpendicular axis. When you take the cross product of two vectors to get a "pseudovector", it would be conceptually clearer to instead take the wedge product of two vectors to get a bivector, treated as a conceptually different type of object. –jacobolus (t) 20:11, 26 January 2023 (UTC)

The formula for the square root of a quaternion essentially uses the trigonometric identity for the sine of a half angle $\sin(\theta/2) = \sqrt{(1-\cos(\theta))/2}$. The formula looses precision for small angles and should never be used for numerical calculation. This is similar to finding the angle between two vectors using arccos formula, which is generally unacceptable. Arcshinus (talk) 02:50, 15 March 2023 (UTC)

To me, it seems that some things in mathematics are discoveries, and some are inventions. I consider $pi$ and $e$ to be discoveries, since they are fundamental to so much. Matrices I consider to be an invention, since, despite their flexibility and utility value, I've always regarded them as being rather arbitrary (full disclosure: I never did like matrices :). Quaternions also seem to fall into the invention category (more full disclosure: I love quaternions). Complex numbers are harder to so categorize; while the term "imaginary part" may argue for "invention", they are so closely tied to fundamentals (e.g., two-dimensional Euclidean space) that "discovery" also seems accurate. BMJ-pdx (talk) 22:39, 5 June 2023 (UTC)

:Maybe make a blog or social media post out of this instead of chitchatting about it here. Cf. WP:NOTFORUM. –jacobolus (t) 00:46, 6 June 2023 (UTC)

:See Philosophy of Mathematics. --50.47.155.64 (talk) 15:51, 15 August 2023 (UTC)

::Many quantum and particle physicists would say that hypercomplex numbers fully exist in the natural world. LagrangianFox (talk) 18:58, 11 October 2024 (UTC)

I think the sentence: 'Quaternions are generally represented in the form where the coefficients a, b, c, d are real numbers, and 1, i, j, k are the basis vectors or basis elements.'

Should read: 'Quaternions are generally represented in the form where the coefficients a, b, c, d are real numbers, and i, j, k are the basis vectors or basis elements.'

That is, the '1, ' before the 'i' should be deleted. Is that correct? MathewMunro (talk) 09:09, 15 February 2024 (UTC)

:The introduction is correct, quaternions form a vector space of dimension 4 over the reals. However, the modern concept of a vector space was not elaborated when Hamilton introduced quaternions, and this may make terminology slightly confusing. Indeed, 1 is a vector (element) in the vector space of all quaternions, but is not a "vector quaternion", the vector quaternions being those quaternions for which $a=0;$ they form a vector space of dimension 3. D.Lazard (talk) 09:43, 15 February 2024 (UTC)

The section on "P.R. Girard's 1984 essay..." is full of references to the author. I'm too busy, but someone needs to clean that up or delete the entire ugly self-promotion. Verdana♥Bold 10:52, 25 February 2024 (UTC)

:Paragraph removed. D.Lazard (talk) 12:07, 25 February 2024 (UTC)

::Wait, why delete this section. From what i see and find, the references are indeed correct. How can this be self-promotion? Mwcb (talk) 12:01, 2 March 2024 (UTC)

:::Details on a specific author does not belong to this article. For not being promotional, an independent source is needed. Such a source must discuss the importance, if any, of the results. Without that, the paragraph is there only for promoting an author. D.Lazard (talk) 12:38, 2 March 2024 (UTC)

What does the term "extends" mean in the first sentence of this article? Comfr (talk) 18:52, 17 April 2024 (UTC)

:It means that the complex numbers can be considered as a subset of the quaternions, with the same behavior as quaternions that they have as complex numbers, but quaternions also include additional elements which combine compatibly with the existing ones. –jacobolus (t) 20:33, 17 April 2024 (UTC)

::Because this is a technical article, technical terms should be well defined. Unfortunately, the technical term "extends" appears in the article without a definition, which motivated me to hyperlink "extends" to Field extension.

::User:Quantling correctly observed that Quaternion are not a field. Quaternion are not commutative, which is a required property of a field.

::Vector products and also not commutative, however "vector fields" exist.

::The article Field extension does not say that a field extension might not be a field. Should that be fixed? Comfr (talk) 22:14, 20 April 2024 (UTC)

:::See ring extension for an appropriate link.— Rgdboer (talk) 00:01, 21 April 2024 (UTC)

:::Two more on topic: Complexification#Dickson doubling and Cayley%E2%80%93Dickson construction. — Rgdboer (talk) 00:09, 21 April 2024 (UTC)

:::To clarify: there are (at least) two meanings of field in mathematics. In algebra it means a set with a commutative addition and commutative multiplication operation and various additional properties; this is the present discussion. In analysis, it means a function defined on a manifold (including manifolds like ordinary Euclidean spaces); and if the result (image) of the function is a vector (at each point of the manifold) then it is called a vector field. —Quantling (talk | contribs) 13:52, 22 April 2024 (UTC)

:::In this context, "extends" does not need to be interpreted as a precise technical term; the ordinary English meaning of the word is plenty clear. I would not bother wiki-linking it to anything. –jacobolus (t) 18:24, 22 April 2024 (UTC)

An underlying extension is the group extension from {1, i, –1, –i } ≅ ℤ₄ to the quaternion group ℚ₈. The extension is not uniquely determined and can lead to the dihedral group of order 8 which lies under the coquaternion ring. Rgdboer (talk) 00:09, 28 April 2024 (UTC)

Overheard in a 19th century classroom:

:Student: We are familiar with complex numbers, but now you want to introduce some things even more complicated than these complex ones. What do you propose to call such things?

:Teacher: Well, hypercomplex numbers, of course. — Rgdboer (talk) 01:02, 5 May 2024 (UTC)

The image showing the cycles of multiplication appears to be incorrect. In particular, the arrows in the three outer cycles should be inverted.

For example, starting at positive j, cycling along the blue path counter clockwise (xk):

• j * k = -i (graph shows positive i)
• i * k = j (graph shows negative j)
• -j * k = i (graph shows negative i)
• -i * k = -j (graph shows positive j)

The same is true for the outer red and green cycles. However, inverting the direction fixes the error. 2620:1F7:93F:425:0:0:32:14F (talk) 12:51, 6 June 2024 (UTC)

:The diagram is correct. As you can see in several places in the article (for example, Multiplication of basis elements), jk = i, ik = –j, –jk = –i, and –ik = j.—Anita5192 (talk) 13:25, 6 June 2024 (UTC)

[https://www.researchgate.net/publication/268244724_Quaternionic_linear_and_quadratic_equations This link] studies a certain family of quadratic equations. In general, a homogeneous quaternionic polynomial of degree of n is of the form

$\backslash sum\_i\; a\_\{i,1\}xa\_\{i,2\}x\backslash cdots\; a\_\{i,n\}xa\_\{i,n+1\},$

and there can be arbitrarily many terms in the summation.

Example of a linear polynomial: $ax+xb=ax1+1xb$;

Example of a quatratic polynomial: $ax^2+xbx+x^2c=ax1x1+1xbx1+1x1xc$. 129.104.244.74 (talk) 11:42, 26 May 2025 (UTC)

It seems to me that any function of real or complex numbers that can be equated to a power series can be seamlessly extended to a function of quaternions, via the power series. This would apply to trigonometric and hyperbolic functions, etc. If this is true and notable, should we add mention of it to the article?

Furthermore, in the case of complex numbers, such a power series can often be extended beyond its radius of convergence via analytic continuation. If there is a similar concept for quaternions, should we mention that too?

Thoughts? —Quantling (talk | contribs) 16:24, 16 June 2025 (UTC)

: I think the power series of a quaternion corresponds fairly trivially to that of a complex number with its imaginary part equal to the norm of the vector part of the quaternion. Any geometric interpretations, of the complex number, should have a corresponding geometric interpretation in the plane generated by the quaternion and its powers. This makes sense as the quaternions can be derived by the Cayley–Dickson construction from complex numbers so contains multiple copies of them.

: I don't see any quaternion specific applications though. If you try to generalise it to quaternions not in the same plane then things stop working like complex numbers. Even simple formula like e^(a+b) = e^a * a^b stop working as multiplication is no longer commutative. --2A04:4A43:900F:FA65:253D:1E14:39A3:1BCC (talk) 14:47, 17 June 2025 (UTC)

::I'd have to put some effort into understanding your first paragraph, but if you would summarize the topic for the article itself, that sounds like an improvement to me. Yes, non-commutativity would mean that the exponential function so defined would not behave as we might first have expected, and similarly for other functions. I'd like to see mention of something along these lines in the article too. So long as this isn't original research but an actual reflection of what mathematicians have done ... I'd like to see it in the article. —Quantling (talk | contribs) 15:15, 17 June 2025 (UTC)

:::It definitely is original research as I did not use any sources writing that. But again I think it's a trivial consequence of how quaternions contain the complex numbers, so the properties of complex numbers apply to such a subset. Much like the complex numbers contain the real numbers so you can do real number math within them. It's not very interesting using complex numbers for that though. --2A04:4A43:900F:FA65:253D:1E14:39A3:1BCC (talk) 16:18, 17 June 2025 (UTC)

::::Find a reliable source for your conjecture. Hawkeye7 (discuss) 19:20, 17 June 2025 (UTC)

:::::Which "conjecture" are we talking about here? You are asking whether the exponential function of a quaternion can be defined as a power series (or defined some other way and proven to be equal to a power series)? Yes, that is standard and easy to find references for.

:::::The functions sinh and cosh are just the even and odd parts of the exponential function, which can be, trivially, the even and odd terms of the power series for exp. Defining quaternion-valued sine or cosine isn't really very insightful in my opinion, but you can probably find someone doing it if you look around. What you can do is take the sinh or cosh of an "imaginary" quaternion (more generally, bivector) and pull out the "imaginary" direction (unit bivector) out front which leaves the sine or cosine of a real quantity.

:::::The lack of commutativity of multiplication means that, as the IP editor mentioned, exponential identities must be treated carefully. There are certainly sources about this, both for quaternions per se and for more general kinds of multivectors. –jacobolus (t) 20:08, 17 June 2025 (UTC)

:::::Here's a source from the 1870s, {{jstor|25138496}}. –jacobolus (t) 20:18, 17 June 2025 (UTC)

::::::So, the question I have is whether this is important enough for the article. I think a brief mention is called for. For example, we could change the lead sentence of the Exponential, logarithm, and power functions section from {{!tq|Given a quaternion,}} to {{tq|A function of a quaternion can be defined from a power series. For example, given a quaternion,}}. That touches on the topic about as lightly as I can imagine, excepting that it is the lead sentence of that section. We could do more. —Quantling (talk | contribs) 20:23, 17 June 2025 (UTC)

:::::::Apologies. I was put on the defensive by "it's a trivial consequence of how quaternions contain the complex numbers, so the properties of complex numbers apply to such a subset." The quaternions are not a subset, so some properties of complex numbers, like commutativity, do not apply to quaternions.

:::::::Returning to your point: Can "any function of real or complex numbers that can be equated to a power series can be seamlessly extended to a function of quaternions, via the power series"? Hawkeye7 (discuss) 00:56, 18 June 2025 (UTC)

::::::::Firstly, a power series defines a function only if the variable commutes with the coefficients. So, the question makes sense only for series with real coefficients. As mentioned in the article, the quaternion form a Banach algebra over the reals. Series on Banach algebas have been widely studied, and all general results on Banach algebras apply to quaternions. I am not sure whether there are results specific to quaternions. D.Lazard (talk) 11:10, 18 June 2025 (UTC)

:::::::::Good to know about the coefficients being real numbers. Would it be okay if I change the lead sentence of the Exponential, logarithm, and power functions section from {{!tq|Given a quaternion,}} to {{tq|A function of a quaternion can be defined from a power series with real coefficients. For example, given a quaternion,}}. —Quantling (talk | contribs) 13:40, 18 June 2025 (UTC)