Questions tagged [dual-numbers]
For questions involving dual numbers. Dual numbers are numbers of the form $a + b \cdot \varepsilon \wedge \left\{ a,\, b \right\} \in \mathbb{R} \wedge \varepsilon \ne 0 = \varepsilon^{2}$.
36 questions
3 votes
1 answer
116 views
What is the motivation behind split-complex and dual numbers?
I encountered the concepts of split-complex and dual numbers through these videos, and am struggling to understand the reason/motivation for defining them. The argument presented in these videos is ...
- 9,031
0 votes
1 answer
63 views
What is the dual number equivalent of Cayley-Dickson construction for hypercomplex numbers?
Cayley-Dickson construction defines general forms of complex multiplication and conjugate: $$ (a,b)^* = (a^*, -b) \\ (a,b)(c,d) = (ac-d^*b,da+bc^*) $$ By applying these recursively, progressively ...
- 135
0 votes
0 answers
74 views
Is there a Polar form for Dual Numbers
Lets say you have a number $a+b\epsilon$ where $\epsilon^2=0$ but $\epsilon\neq0$, you could express this number as $a(1+\frac{b}{a}\epsilon)=ae^{\frac{b}{a}\epsilon}$ You could draw out a plane where ...
- 39
0 votes
0 answers
84 views
Does the construction of automatic differentiation by the dual numbers have any cousins?
The dual numbers are used to effectively construct the whole world of differentiation over extremely complicated algorithms, beginning from the basic idea of a number $a + b$ _ and letting _$^2 = 0$. ...
- 2,184
6 votes
1 answer
189 views
Is the derivative of $f$ at $x$ always given by the dual part of $f(x+\epsilon)$?
I recently learnt about dual numbers, of the form $u + v\varepsilon$ where $u, v \in \mathbb R$ and $\varepsilon \notin \mathbb R$ is such that $\epsilon^2 = 0$. These numbers are used for automatic ...
- 237
4 votes
0 answers
67 views
Infinite differentiability for complex, split and dual functions
This is the sequel to my previous question, inspired by Anixx's comment: Holomorphicity for complex, split and dual functions In it, Qiaochu Yuan helped establish what it means for these functions to ...
- 715
4 votes
1 answer
150 views
Holomorphicity for complex, split and dual functions
The set of complex numbers is: $$\mathbb{C}=\left\{a+bi:a,b\in\mathbb{R},i\notin\mathbb{R},i^{2}=-1\right\}$$ The set of split numbers is: $$\mathbb{D}=\left\{a+bj:a,b\in\mathbb{R},j\notin\mathbb{R},j^...
- 715
2 votes
0 answers
292 views
Does wheel theory help give meaning to some division of polynomials?
I apologise if this question is a little too vague (e.g., "meaningful"). I have included the soft-question tag for good measure. Motivation: This is motivated by one of those annoying memes ...
- 48.5k
3 votes
3 answers
336 views
Matrix function derivative. Introduction
The author of this question was close to determining the derivative of the function of dual variable, when we consider matrices isomorphic (algebraically and topologically) to dual numbers: $$(a+\...
- 169
2 votes
1 answer
109 views
What's the definition of dual number at perspect of exterior algebra?
In Dual Number it said that "It may also be defined as the exterior algebra of a one-dimensional vector space with $\varepsilon$ as its basis element." But I can't find the detailed rigorous ...
1 vote
1 answer
109 views
Imagining an exponential "hypercomplex" system
I recently learned about hypercomplex systems that are taken over the reals, i.e. the dual numbers for which $j^2=1$, $j≠1$, and the dual numbers for which $ε^2=0$, $ε≠0$. These number systems, along ...
- 33
3 votes
1 answer
113 views
L'Hopital's rule with dual numbers
Background: For the dual numbers, we extend the reals with an additional unit vector $\epsilon$ subject to the constraint that $\epsilon^2 = 0$. We can write dual numbers as $x_0 + x_1 \epsilon$ for $...
- 330
2 votes
0 answers
67 views
Closest rigid-body pose and velocity along a screw motion to a given pose
Given two rigid-body poses given by dual quaternions ${\bf q}_1$ and ${\bf q}_2$, where ${\bf q}_1 = e^{{\bf v} t}$ is some pose along a screw motion. The screw is given by dual vector (Pluecker ...
- 91
1 vote
1 answer
93 views
How is it possible that time can be represented as abstract element?
I read about differentiation using dual numbers (and about NSA/SDG approaches to differentiation) and I have question. Let we have function that represents position $x$ at time $t$: $x(t)$ If we use ...
- 1,159
1 vote
0 answers
89 views
Raising a dual quaternion to a real number power [duplicate]
I'm having trouble finding the power of a dual quaternion ($Q=q_r+q_dε$) raised to some real number $n$: $Q^n=(q_r+q_dε)^n=?$ It is known that for a quaternion $q=e^{\vec{u}}=\cos(|\vec{u}|)+\frac{\...
