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Questions tagged [quaternions]

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

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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 $\mathbb C$,...
97
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7answers
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Why are There No “Triernions” (3-dimensional analogue of complex numbers / quaternions)? [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 ...
50
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2answers
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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 ...
46
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3answers
7k 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 ...
35
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5answers
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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 ...
35
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6answers
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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?
34
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5answers
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How do you rotate a vector by a unit quaternion?

Given a 3-variable right-handed vector v that is a translation measured in local space and a unit quaternion representing an orientation from local to world space, how do you use the quaternion to ...
33
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5answers
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Why should quaternions exist?

Why do quaternions exist? I want to believe they exist, but all I can think of are reasons they should not exist. These are my reasons. The quaternions are defined by the following equation: $$i^2 =...
32
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7answers
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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,...
32
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8answers
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Real world uses of Quaternions?

I've recently started reading about Quaternions, and I keep reading that for example they're used in computer graphics and mechanics calculations to calculate movement and rotation, but without real ...
31
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8answers
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Why is the complex number $z=a+bi$ equivalent to the matrix form $\left(\begin{smallmatrix}a &-b\\b&a\end{smallmatrix}\right)$ [duplicate]

Possible Duplicate: Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1 & 0\end{smallmatrix}\right)$ to $i$ On Wikipedia, it says that: Matrix representation ...
30
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2answers
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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 ...
27
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2answers
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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 $2^n$ ...
25
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4answers
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Why don't quaternions contradict the Fundamental Theorem of Algebra? [duplicate]

I don't pretend to know anything much about the Fundamental Theorem of Algebra (FTA), but I do know what it states: for any polynomial with degree $n$, there are exactly $n$ solutions (roots). Well, ...
25
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3answers
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Is the set of quaternions $\mathbb{H}$ algebraically closed?

A skew field $K$ is said to be algebraically closed if it contains a root for every non-constant polynomial in $K[x]$. I know that this is true for $\mathbb{C}$, which is the algebraic closure of $\...
22
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3answers
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How to rotate one vector about another?

Brief 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: vector ...
21
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2answers
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Why are properties lost in 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 \mathbb{H}...
20
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1answer
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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 ...
20
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3answers
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If there are any 3nion, 5nion, 7nion, 9nion, 10nion, etc.

The quaternion/octonion extend the complex numbers, which extend the real numbers. So we go: 1-tuple: Real numbers. 2-tuple: Complex numbers. 4-tuple: Quaternions. 8-tuple: Octonions. The Wikipedia ...
18
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4answers
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Quaternions: why does ijk = -1 and ij=k and -ji=k

Currently i am studying quaternions. I do understand that i, j and k are imaginairy numbers. so $i^2 = j^2 =k^2 = -1$. But I could not understand this: $$\begin{matrix}ij=k,&ji=-k,\\jk=i,&kj=...
18
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4answers
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How to form a mental image of $\mathbf {ijk}=-1$ in quaternions?

$\mathbf {ijk}=-1$ is part of the famously stone-carved formula by Sir William Rowan Hamilton, allowing the multiplication of triplets. There is already an intuition question on the topic of ...
18
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2answers
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Is there an algebraic closure for the quaternions?

This post is a sequel of: Is the set of quaternions $\mathbb{H}$ algebraically closed? This answer shows that: 1. $\mathbb{H}$ is algebraically closed for the polynomials of the form $\sum a_r x^r$...
18
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1answer
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The Quaternions and $SO(4)$

I am interested in the map $\phi:S^3 \times S^3 \to GL_4(\mathbb{R})$ given as follows: Let $(p,q) \in S^3 \times S^3$. We identify $p$ and $q$ as real quaternions with unit norms and define $\phi(p,...
15
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1answer
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Exponential Function of Quaternion - Derivation

The equation for the exponential function of a quaternion $q = a + b i + c j + dk$ is supposed to be $$e^{q} = e^a (\cos(\sqrt{b^2+c^2+d^2})+\frac{(b i + c j + dk)}{\sqrt{b^2+c^2+d^2}} \sin(\sqrt{b^2+...
15
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1answer
375 views

Why is the dimension of $SL(2,\mathbb{H})$ equal to $15$?

Let me ask a very basic question which is inspired by reading M. Atiyah's "Geometry and physics of knots". Could you explain me (or give a reference to) the definition of the special linear group $SL(...
15
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1answer
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How to raise a number to a quaternion power

Now, I know that it's (relatively) easy to calculate, say, $r^{a+bi}$ (using the fact that, for $z_1, z_2\in \mathbb{C}, {z_1}^{z_2}=e^{z_2\ln(z_1)}$ and $\ln(z_1$) can just be found using: $\ln(a+bi)...
14
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1answer
326 views

Is my paper on a number system that allows arithmetic on 3D vectors useful?

I have constructed a number system similar to the quaternions, but with three dimensions, not four, ie vectors of the form $(x, y, z)$. It has fairly well-behaved multiplication and division and every ...
13
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4answers
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Euler angles and gimbal lock

Can someone show mathematically how gimbal lock happens when doing matrix rotation with Euler angles for yaw, pitch, roll? I'm having a hard time understanding what is going on even after reading ...
13
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2answers
619 views

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 ...
13
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2answers
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Combining rotation quaternions

If I combine 2 rotation quaternions by multiplying them, lets say one represents some rotation around x axis and other represents some rotation around some arbitrary axis. The order of rotation ...
13
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3answers
790 views

Binary tetrahedral group and $\rm{SL}_2(\mathbb F_3)$

The binary tetrahedral group $\mathbb T$ is an interesting 24-element group. For instance it can be expressed as the subgroup $$ \mathbb T = \left\{ \pm 1, \pm i, \pm j, \pm k, \dfrac{\pm 1 \pm i \pm ...
13
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1answer
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Square roots of quaternions

In class we saw that $-1$ has infinitely many square roots in the ring of quaternions. Is it possible to compute the square roots of a given nonreal quaternion? What do they look like?
13
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1answer
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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 ...
12
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2answers
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Rotation in 4D?

Rotation in 2D by an angle $t$ can be performed using $$R=\begin{pmatrix}\cos(t) &-\sin(t) \\ \sin(t) & \cos(t)\end{pmatrix}$$ matrix. But, if I want to rotate a point or vector in 4D, is ...
12
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3answers
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Does anyone know any resources for Quaternions for truly understanding them?

I've been studying Quaternions for a week, on my own. I've learned various facts about them but I still don't understand them. My goal is to understand rotation quaternions specifically. I don't want ...
12
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3answers
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Matrix representation of the Quaternions?

Can anyone explain how why the matrix representation of the quaternions using real matrices is constructed as such?
11
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2answers
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Proving that $\mathbb R^3$ cannot be made into a real division algebra (and that extending complex multiplication would not work)

I am trying to solve the following exercise: Prove that complex multiplication does not extend to a multiplication on $\mathbb R^3$ so as to make $\mathbb R^3$ into a real division algebra. I ...
11
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2answers
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Is there a relationship between the cross product and quaternion multiplication?

I've just been introduced to the Kronecker delta, $\delta_{ij}$, along with the alternating tensor, $\varepsilon_{ijk}$ (in vector calculus). Motivation for the question: I've been introduced to ...
11
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1answer
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Quaternion Rings

Let $R$ be a commutative ring. Define the Hamilton quaternions $H(R)$ over $R$ to be the free $R$-module with basis $\{1, i, j, k\}$, that is, $$H(R)=\{a_0+a_1i+a_2j+a_3k\;\;:\;\;a_l\in R\}.$$ and ...
11
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0answers
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Does the exceptional Lie algebra $\mathfrak{g}_2$ arise from the isometry group of any projective space?

I learned from Baez's notes on octonions that the classical simple Lie algebras can be identified with the Lie algebras of isometry groups of projective spaces over $\mathbb{R}, \mathbb{C}$ and $\...
10
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4answers
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what is the relation between quaternions and imaginary numbers?

I understand the idea behind complex and imaginary numbers. I am trying to understand quaternions. What is the relation between imaginary (or complex) numbers, and quaternions?
10
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2answers
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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 ...
10
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3answers
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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 ...
10
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2answers
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$x^2+1=0$ 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 infinite ...
10
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1answer
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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 $q(t),q(0)$ ...
10
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3answers
420 views

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, ...
10
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1answer
772 views

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 $-1$ ...
10
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0answers
569 views

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 ...
9
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4answers
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Motivation for construction of cross-product (Quaternions?)

I'm trying to present a narrative that brings the (3D) Cross Product into existence. "Given two vectors $\mathbf u$, $\mathbf v$, how to construct a vector perpendicular to both?" ... looks like a ...
9
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3answers
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What is a good geometric interpretation of quaternion multiplication?

I understand that the formula for quaternion multiplication of $q_1=(s_1,\vec{v_1})$ by $q_2=(s_2,\vec{v_2})$ $q_1q_2=(s_1s_2-\vec{v_1}\cdot\vec{v_2}, \vec{v_1} \times\vec{v_2} + \vec{v_1}s_2 + \vec{...