Tagged Questions
1
vote
0answers
23 views
“adding” numbers $\in\mathbb{U\left(2\right)}\times \mathbb{R}^+_0$
Let a unitary number be one that corresponds to a matrix of the form:
$$\left(
\begin{array}{cc} w+i x & y+i z \\ -e^{i p} (y-i z) & e^{i p} (w-i x)
\end{array}
\right)$$
This is analogous to ...
3
votes
1answer
57 views
Some questions about quaternions.
It is possible make something like complexification of a real vector space using quaternions?
If yes, it's similar to complex case or there are considerable differences?
Has been studied a quaternion ...
2
votes
2answers
109 views
Proof that Quaternion Algebras are simple
I have a proof that every quaternion algebra over a field $A=\left(\frac{a,b}{F}\right)$ is simple, i.e. has no nontrivial two-sided ideals, which appeals to the algebraic closure of $F$ and the ...
4
votes
2answers
97 views
$x^2+1=0$ uncountable many solutions [duplicate]
Possible Duplicate:
Why are the solutions of polynomial equations so unconstrained over the quaternions?
Coudl someone explain me the following: Why should $x^2+1=0$ have uncountable ...
2
votes
1answer
104 views
Unit elements in Hurwitz quaternions
Hurwitz quaternions are defined as:
$$H_u = \left\{a+bi+cj+dk\mid (a,b,c,d) \in\mathbb{Z}\mbox{ or }(a,b,c,d) \in\mathbb{Z}+\frac{1}{2}\right\}$$
(that is, all integer or half integer quaternions).
...
37
votes
5answers
784 views
Why are the solutions of polynomial equations so unconstrained over the quaternions?
An $n$th-degree polynomial has at most $n$ distinct zeroes in the complex numbers. But it may have an uncountable set of zeroes in the quaternions. For example, $x^2+1$ has two zeroes in $\mathbb ...
5
votes
1answer
116 views
Why are properties lost in the the Cayley-Dickson construction?
Motivating question: What lies beyond the Sedenions?
I'm aware that one can construct a hierarchy of number systems via the Cayley-Dickson process:
$$\mathbb{R} \subset \mathbb{C} \subset ...
13
votes
1answer
357 views
What lies beyond the Sedenions
In the construction of types of numbers, we have the following sequence:
$$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S}$$
or:
$$2^0 \mathrm{-ions} \subset ...
2
votes
2answers
146 views
what is the tensor product $\mathbb{H\otimes_{R}H}$
I'm looking for a simpler way of thinking about the tensor product: $\mathbb{H\otimes_{R}H}$, i.e a more known algbera which is isomorphic to it.
I have built the algebra and played with it for a ...
3
votes
1answer
286 views
Why is quaternion algebra 4d and not 3d?
Why is quaternion algebra 4D and not 3D? Complex algebra is 2D and what is known as quaternion algebra jumps to 4D.
$ i^2 = j^2 = k^2 = ijk = -1 $
Using $1, i, j,$ and $k$ as the base (where ...
3
votes
1answer
958 views
Rotate Quaternion A by 180 degrees
Suppose you have an arbitrary quaternion - call it A - how do you rotate it by 180 degrees?
Is there a way to do this without convert to angle-axis representation, i.e., keep it within the ...
20
votes
2answers
781 views
Why are the only division algebras over the real numbers the real numbers, the complex numbers, and the quaternions?
Why are the only (associative) division algebras over the real numbers the real numbers, the complex numbers, and the quaternions?
Here a division algebra is an associative algebra where every ...
