Talk:Quaternion

message: In , we are identifying the quaternion k with an element of the even subalgebra, but we are neglecting that in Cl3,0(R) we are treating r as a vector, not as a bivector.  Thus, the two formulae, while being superficially the same, are not "identical".  A proper discussion would need to include a mention of the mapping of r to its Hodge dual.  —Quondum 12:31, 15 May 2020 (UTC)
replies: text: I think you are misinterpreting what is written there, but it’s confusingly written. The "two" here means the two algebras, not the two formulas. I tried to rewrite for clarity. –jacobolus (t) 07:51, 6 January 2023 (UTC)
thread_title:  "In fact, the two are identical, if we make the identification ..." 

message: 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)
replies: text: 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)

text: 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)
thread_title:  Overly technical 

message: 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)
thread_title:  Biquaternions are eight-dimensional 

message: “ 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)
replies: text: I don't see this in the History section, so perhaps it should be included.—Anita5192 (talk) 17:42, 26 January 2023 (UTC)
replies: text: 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)
thread_title:  Why refer to i, j, and k as “basic quaternion”? 

message: 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)
thread_title:  Square roots of arbitrary quaternions 

message: 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)
replies: text: 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)

text: See Philosophy of Mathematics. --50.47.155.64 (talk) 15:51, 15 August 2023 (UTC)
thread_title:  Discovery or invention? 

message: 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)
replies: text: 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)
thread_title:  Error in the introduction? 

message: 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)
replies: text: Paragraph removed. D.Lazard (talk) 12:07, 25 February 2024 (UTC)
replies: text: 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)
replies: text: 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)
thread_title:  What the eff?