Talk:Quaternion

Overly technical

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 four-dimensional

We're missing h, it's call the Hamilton set for a reason, Hamilton was a human...

$$a\ w\ \mathbf {h} +b\ x\ \mathbf {i} +c\ y\ \mathbf {j} +d\ z\ \mathbf {k} ,$$ NJB (talk) 04:52, 6 January 2023 (UTC)

It’s hard to tell what you are trying to say. But as a general rule, Wikipedia follows whatever the commonly accepted convention is in reliable sources (or when there are multiple common convention, picks one and mentions the alternatives). Do you have a reliable source for your "h" here? If you are just throwing out ideas, you may want to write a blog post or self-published paper, as you are unlikely to find support for their inclusion here. –jacobolus (t) 07:25, 6 January 2023 (UTC)

It's basic primary school mathematics, you shouldn't need a source for $h$, as it is the original human number system otherwise known as $\mathbb {R}$. It's always there, but it's most often omitted due to a combination of short hand mathematical notation and/or ignorance about lateral numbers.

To add two signed numbers, such as $+1+-1$, most people would just rewrite the equation using subtraction and say that the value is $0$, but this because they were taught short hand arithmetic notation starting in primary school. Even when dealing with only real numbers, 0i is always still there as part of the equation, it's also just simply omitted in short hand notation.

$+1+-1=(+1+0i)+(-1+0i))=0+0i$

So naturally in the equation above, the $0$ alone by itself is on the $h$ axis, $\mathbb {R}$, which is omitted in short hand notation, but the more formal general form for this equation using quaternions is as follows:

$(1h+0i+0j+0k)+(-1h+0i+0j+0k)=0h+0i+0j+0k=0$

${\textstyle 1=h^{2}i^{2}j^{2}k^{2}=({\sqrt {-1}})^{2}({\sqrt {-1}})^{2}({\sqrt {-1}})^{2}({\sqrt {-1}})^{2}}$

${\textstyle h^{2}={\frac {1}{i^{2}j^{2}k^{2}}}={\frac {1}{(-1)(-1)(-1)}}=-1}$

${\textstyle i^{2}={\frac {1}{h^{2}j^{2}k^{2}}}={\frac {1}{(-1)(-1)(-1)}}=-1}$

${\textstyle j^{2}={\frac {1}{h^{2}i^{2}k^{2}}}={\frac {1}{(-1)(-1)(-1)}}=-1}$

${\textstyle k^{2}={\frac {1}{h^{2}i^{2}j^{2}}}={\frac {1}{(-1)(-1)(-1)}}=-1}$

NJB (talk) 16:51, 6 January 2023 (UTC)

Regardless of all that stuff, you still need a reliable source, and YouTube is not a reliable source.—Anita5192 (talk) 17:04, 6 January 2023 (UTC)

(a) This Youtube video is well made and worth showing to students but it does not support your claims.

(b) What you have written here is not the way people use quaternions or other number systems in practice. Again, feel free to invent whatever number system you want in your own writings (self-published papers, blog posts, etc.). However, it’s not relevant to this Wikipedia article. Let’s try to keep discussion focused on improving the article; Wikipedia talk pages are not a general-purpose forum. –jacobolus (t) 18:03, 6 January 2023 (UTC)

In his notation, $(1h+0i+0j+0k)\times (1h+0i+0j+0k)=-1h+0i+0j+0k$. That is not how multiplication in quaternions works.--agr (talk) 18:47, 6 January 2023 (UTC)

I was not attempting to multiple them, it was just simple addition on a system of polynomial equations, here is a better example..

$+a(x-y)^{2}h^{2}+0(x-y)^{2}i^{2}+0(x-y)^{2}j^{2}+0(x-y)^{2}k^{2}$

$-a(x-y)^{2}h^{2}+0(x-y)^{2}i^{2}+0(x-y)^{2}j^{2}+0(x-y)^{2}k^{2}$

There is an implied symbolic coefficient, in this instance the coefficient of $i$, $j$, and $k$ is $0$, so when multiplied those all evaluate to $0$.

$+a(x-y)^{2}h^{2}$

$-a(x-y)^{2}h^{2}$

The only difference here is $a+-a$, so you can use addition here. NJB (talk) 21:33, 6 January 2023 (UTC)

Your expressions here are incoherent because you have not defined a, x, y, or h. But don’t bother to define them; it’s a waste of your (and everyone’s) time. If you have a question (Example question: "why when writing complex numbers don't mathematicians give a symbolic name to the real unit h ≡ 1, so a complex number could be written xh + yi with real part x and imaginary part y, so there would be symbolic symmetry between real and imaginary parts?" Example answer: "You are welcome to do it that way if you want. Most mathematicians like to skip redundant symbols where possible to reduce clutter.") perhaps take it to Wikipedia:Reference desk/Mathematics or try some mathematics discussion forum like reddit or mathematics stack exchange. –jacobolus (t) 22:30, 6 January 2023 (UTC)

Why? a, x, and y have no relevance to the topic at hand, we are talking about our four-demential quaternion number system, the entirety of the equations including the Arabic numerals themself are symbolic representations of abstract ideas. We're dealing with algebra here, and anyone who has passed college algebra knows that you can have coefficients and variables in an equation.

What you call clutter I call lack of completeness and attention to detail, you've just literally admitted the point that I was originally attempting to make, that the wikipedia article is using sloppy shorthand notation in its attempt to rigorously define what a quaternion even is. If you don't see that as a problem then I don't know what to tell you. I have no problem with people using shorthand notation, I do it all the time, but in an encyclopedic article that is attempting to rigorously define something, you have to at least discuss the formal long form notation at least one, and ideally also provide wiki links to all of the assumed knowledge that is necessary to understand it. NJB (talk) 20:05, 7 January 2023 (UTC)

Wikipedia is using the standard notation used by literally every source about this topic for the past 150 years. You can run a weird crusade against conventions you dislike somewhere else, but the job of Wikipedia is to describe encyclopedic topics as established in reliable sources, which is what this article currently does.

Pretty much anyone who does mathematics can think of several notation conventions they dislike for one reason or another, but if they want to change the conventions they can make those arguments in blog posts, journal papers, textbooks, etc.; speculative conversations about possible non-standard notation conventions don’t belong in wikipedia.

"I don't know what to tell you." – that’s fine, you don't need to tell us anything. This whole conversation is off topic and should end ASAP. –jacobolus (t) 21:35, 7 January 2023 (UTC)

Biquaternions are eight-dimensional

See biquaternion for the use of h as a square root of minus one which commutes with i, j, and k. The algebra of biquaternions is only four dimensions when considered over the field of complex numbers x + yh. Biquaternions provide a representation of Minkowski space and Lorentz transformations described by Ludwik Silberstein in 1914, but the original algebra comes from Hamilton's Lectures on Quaternions (1853). Rgdboer (talk) 03:32, 8 January 2023 (UTC)

Why refer to i, j, and k as “basic quaternion”?

“ 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)

Square roots of arbitrary quaternions

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)

Discovery or invention?

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)

Error in the introduction?

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)

What the eff?

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)

Extend

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–Dickson 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)

Errors in product graph image

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)