# Tagged Questions

For questions about the quaternions: a noncommutative four dimensional division algebra over the real numbers.

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### Why are There No “Triernions”? [duplicate]

Since there are complex numbers (2 dimensions) and quaternions (4 dimensions), it follows intuitively that there ought to be something in between for 3 dimensions ("triernions"). Yet no one uses ...
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### 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 $\mathbb C$,...
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### 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 ...
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### Can Euler's identity be extended to quaternions?

Euler's identity is $e^{i \pi} + 1 = 0$, a special case of the Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$, where $\theta = \pi$ (half-turn of the unit circle). It is commonly described ...
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### What do we lose passing from the reals to the complex numbers?

As normed division algebras, when we go from the complex numbers to the quaternions, we lose commutativity. Moving on to the octonions, we lose associativity. Is there some analogous property that we ...
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### What is the motivation for quaternions?

I know imaginary numbers solve $x^2 +1=0$, but what is the motivation for quaternions?
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### How can one intuitively think about quaternions?

Quaternions came up while I was interning not too long ago and it seemed like no one really know how they worked. While eventually certain people were tracked down and were able to help with the issue,...
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### Quaternionic veronese Embedding

I know that the complex projective line $\mathbb{C}P^1$ can be embedded in the complex projective space $\mathbb{C}P^n$ (Veronese embedding). For example, $\mathbb{C}P^1\rightarrow\mathbb{C}P^3$ is ...
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### What specific algebraic properties are broken at each Cayley-Dickson stage beyond octonions?

I'm starting to come around to an understanding of hypercomplex numbers, and I'm particularly fascinated by the fact that certain algebraic properties are broken as we move through each of the $2^n$ ...
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### Quaternion distance

I am using quaternions to represent orientation as a rotational offset from a global coordinate frame. Is it correct in thinking that quaternion distance gives a metric that defines the closeness of ...
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### Why should I consider the components $j^2$ and $k^2$ to be $=-1$ in the search for quaternions?

I'm reading a paper about Hamilton's discovery of quaternions and it explains why he failed in his 'theory of triplets' where he tried to make a vector with $3$ dimensions, as an analogy to the ...
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### How do quaternions represent rotations?

I wonder how $qvq^{-1}$ gives the rotated vector of $v$. Is there any easy-to-understand proof for it? I was on Wikipedia, but I could not understand the proof there because of the conversions. Why ...
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### Are j and k on different imaginary planes than i?

I'm trying to understand Quaternions. So I understand that a Quaternion is written like $xi+yj+zk+w$. I also understand that $i^2 = j^2 = k^2 = ijk = -1$, and how that can be used to derive equations ...
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### How to rotate one vector about another?

Breif Having given 2 non-parallel vectors: a and b, is there any way by which I may rotate a about b such that b acts as the axis about which a is rotating. Question Given: vector a and b To find: ...
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### Quaternions as an Algebra

I'm lacking some vital understanding about quaternions and algebras in general. If we first define $V=\{a+bi+cj+dk|a,b,c,d\in\mathbb{R}\}$. Then we define scalar multiplication, vector multiplication, ...
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### Quaternion integration

If the angular velocity is changing continuously, the following holds true $q(t)=q(0)\exp\left({\int_{0}^{t}\frac{q_\omega(\tau)}{2}\ d\tau}\right) \tag 1$ Specifications and Data $q(t),q(0)$ ...
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### Using Padé approximants for the quaternion exponential

On a lark, I wanted to know if one can use Padé approximants to compute the exponential $\exp(z)$ of a quaternion $z=a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}$. Since Mathematica has a package meant for ...
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### Is a 3D Mandelbrot-esque fractal analogue possible?

I understand that (unlike complex numbers) there's no consistent 3 dimensional number system (even 4D loses some nice properties). Regardless, I'm wondering if there might be a 'trick' to create a 3D ...
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### Clarify: “$S^0$, $S^1$ and $S^3$ are the only spheres which are also groups”

The zero, one, and three dimensional spheres $S^0$, $S^1$ and $S^3$ are in bijection with the sets $\{a\in \mathbb{K}:|a|=1\}$ for $\mathbb{K} = \mathbb{R}, \mathbb{C}, \mathbb{H}$ respectively. The ...
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### Quaternion group associativity

Consider the eight objects $\pm 1, \pm i, \pm j, \pm k$ with multiplication rules: $ij=k,jk=i,ki=j,ji=-k,kj=-i,ik=-j,i^2=j^2=k^2=-1$, where the minus signs behave as expected and $1$ and $-1$ ...
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### Why do Quaternions and octonions exist?

Ok so I have known about imaginary numbers for quite some time now. I also understand why we want them to exist (to have a solution for $x^2=-1$). I also remember reading that the complex numbers are ...
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### Is it possible to use the imaginary components of quaternions to facilitate calculation of vector cross products?

It has come to my attention that the cross products of the vectors $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ are almost identical to the products of the imaginary components of quaternions $i$, $j$,...
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### Can quaternions be useful for integrals?

Lets assume we want to find a closed form for $\int_0^1 f(x) dx$ where $f(x)$ is a real-analytic function. There are many techniques to find that. Some include contour integration on the complex ...
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### 4 dimensional numbers

I've tought using split complex and complex numbers toghether for building a 3 dimensional space (related to my previous question). I then found out using both together, we can have trouble on the ...
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### Square roots of $-1$ in quaternion ring

In this Wikipedia page it is said that the square roots of -1 in the quaternion ring are the elements of the imaginary sphere. I don't understand why this is so. I don't understand the system that's ...
### Why do division algebras always have a number of dimensions which is a power of $2$?
Why do number systems always have a number of dimensions which is a power of $2$? Real numbers: $2^0 = 1$ dimension. Complex numbers: $2^1 = 2$ dimensions. Quaternions: $2^2 = 4$ dimensions. ...