Talk:Quaternion

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Regarding section on Multiplication of basis elements[edit]

We should include further proofs rather than just say. "All the other possible products can be determined by similar methods". Remember although this may seem obvious to those familiar with quaternions, the article should aim at those who aren't. Even just one poof where one of the sides is negative for example to prove ji =-k

ijk = -1 Multiply across by ij

iijjk= -1ij

(substituting -1 for the squares of i and j we get..)

(-1)(-1)k = -ij

k=-ij

Feel free to correct my proof, but I think the other proves are not strictly obvious and therefore needed. — Preceding unsigned comment added by Iantierney (talkcontribs) 22:19, 17 June 2015 (UTC)

The first case given is intended to do that. I don't see how a second example is going to help. The negative sign is the least of the problem – in your case you've just added i and j in the middle; you cannot do that. However, the teaching basic rules of noncommutative algebra (keeping track of multiplying on the left or right) is not the function of an encyclopaedia article. I don;t agree that anything more should be included. —Quondum 23:00, 17 June 2015 (UTC)
In that case, put in a reference to the "basic rules of noncommutative algebra". I agree that something more is needed, because I didn't get it, and I'm smarter than your average bear (seriously). — Preceding unsigned comment added by JohnL4 27709 (talkcontribs) 18:09, 16 July 2015 (UTC)
The section mentioned seems to introduce the ideas at an appropriate level. Perhaps a specific change should be suggested? The argument above makes the obvious mistake of incorrectly assuming that ijij=iijj, which is not allowed due to noncommutativity. Mark MacD (talk) 07:56, 20 July 2015 (UTC)

Is an algebra invented or discovered?[edit]

In the summary it says "In fact, the quaternions were the first noncommutative division algebra to be discovered". Is it not an invention?

No, the quaternions have been invented, but the fact that they form a division algebra is a discovery. D.Lazard (talk) 08:32, 13 January 2016 (UTC)

"isomorphic as a set"[edit]

The article includes the following sentence:

- As a set, the quaternions H are isomorphic to R4, a four-dimensional vector space over the real numbers.

While this is technically true, I don't think this sentence means what it is supposed mean: being "isomorphic as a set" is another way of saying "has the same cardinality as", as isomorphisms in Set are arbitrary bijections. One might as well state that "as a set, the quaternions H are isomorphic to the Cantor set. --Letkhfan (talk) 19:09, 28 April 2016 (UTC)

Good point, that's clearly not the intention. I changed it; do you think the article now conveys the intended meaning? Ozob (talk) 23:22, 28 April 2016 (UTC)

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Reduced form?[edit]

Given a quaternion a + bi + cj + dk, dividing by a yields the quaternion 1 + b/ai + c/aj + d/ak, or equivalently 1 + pi + qj + rk (where p=b/a, q=c/a, and r=d/a). What is this called? A "normalized" or "reduced" quaternion, perhaps? — Loadmaster (talk) 17:29, 4 August 2016 (UTC)

Not come across it and I struggle to think of an application. Normalised is usually reserved for dividing by the norm, i.e. by √(a2 + b2 + c2 + d2), which is is useful for e.g. using quaternions to rotate.--JohnBlackburnewordsdeeds 17:48, 4 August 2016 (UTC)
Why searching for a specific name for quaternions for which the real part is one, when complex numbers for which the real part is one do not have any specific name? D.Lazard (talk) 17:54, 4 August 2016 (UTC)
I was probably thinking of the unit quaternion, i.e., a quaternion with a norm of 1, which is exactly what JohnBlackburne mentions above. So while dividing the quaternion components by a to get a real part of 1 might be useful, it's obviously far more useful if the whole quaternion has a magnitude/norm of 1. — Loadmaster (talk) 16:37, 8 August 2016 (UTC)