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Questions tagged [hypercomplex-numbers]

A hypercomplex number is an element of a finite-dimensional algebra over the real numbers that is unital and distributive (but not necessarily associative).

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Why intuitively do the quaternions satisfy the mixture of geometric and algebraic properties that they do?

[I completely rewrote the question to see if I could make it clearer. The comments below won't make any sense. In fact, my original question has been answered by Eric Wolfsey, so I may restore it.] ...
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$\epsilon \otimes 1 + 1 \otimes \epsilon$ is a nilcube in $\mathbb R[\epsilon] \otimes \mathbb R[\epsilon]$. What does that mean intuitively?

[EDIT: I know what the notation means, and I can easily show that $x=\epsilon \otimes 1 + 1\otimes \epsilon$ satisfies $x^3=0$ but $x^2 \neq 0$. That's not what this question is about. It might be ...
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Using dual complex numbers for combined rotation and translation

Dual quaternions may be used to perform combined rotations and translations in a single dual quaternion product operation. Translation is performed by placing the displacement, $d$ in vector of the ...
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1answer
63 views

A simple Variation on the Imaginary Unit i

I think a more appropriate tag would have been 'quasicomplex numbers' rather than 'hypercomplex numbers'. I'm normally perfectly comfortable with the correspondence between hyperbolic functions & ...
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Rotation around a whole sphere by multiplying a single hypercomplex number forever?

In quaternion number system, any unit quaternion $\mathbf{q}\in\mathbb{H}$ can be written as $$ \mathbf{q} = \cos \theta + (v_x \mathbf{i} + v_y \mathbf{j} + v_z \mathbf{k})\sin \theta $$ for some $\...
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1answer
117 views

Is it possible to plug hypercomplex numbers into the Riemann Zeta function?

I'm aware of the more detailed question: How to raise a number to a quaternion power However, from a more high-level perspective (read: probably less mathematical, hence my choice for raising ...
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27 views

Dual Number for derivation of tangens to power of arbitrary number

I have recently stumbled across dual numbers and thought they may help me to solve a problem. What I ultimately want to do is root finding of a polynomial of an arbitrary degree. The polynomial is of ...
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33 views

Prerequisites for Hypercomplex analysis.

I'm thinking about buying a book about the subject but I wasn't able to find a list of prerequisites anywhere so decided to ask here. I have taught myself Real and Complex Analysis, Linear and ...
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1answer
72 views

Is there a multivector that's a non-trivial cube root of $1$?

This answer (or this site, in case the answer gets deleted) defines a certain 3-dimensional Real algebra by declaring that $j^3={^-}1$, and that $1,j,j^2$ are linearly independent. (By a simple ...
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1answer
69 views

Hyper complex number $e_{16}$ had a zero divisor.

I was looking into hyper complex numbers https://en.wikipedia.org/wiki/Hypercomplex_number where it stated that under Cayley–Dickson construction for "$\{1,i_1,...i_{2^n-1}\}$... in $16$ or more ...
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4answers
63 views

Adding a complex and its conjugate, both raised to a degree of 4?

$(-1+\sqrt{-3})^4+(-1-\sqrt{-3})^4$ = ? I deduced the complex form: $z=(-1+i\sqrt{3})$ I can see that the question is basically $z^4+\overline z^4$ Now, if we add a complex number and its ...
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152 views

Any references on supercomplex/surcomplex numbers?

I know many text in surreal numbers have went over them but I can't seem to find anything that goes into great detail beyond a small section on them. For instance, in Foundation of Analysis over ...
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1answer
160 views

Residue theorem for split complex numbers

One can show that any function on the split-complex numbers which can be represented by a Laurent series is infinitely differentiable, except at the union of several (shifted) hyperbolas of modulus 0 (...
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Triangle inequality for split-complex numbers

After much research and head-scratching to no avail, I was hoping someone here could shed some light on certain properties of the split-complex numbers, namely: 1) Is there any meaningful analog of ...
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1answer
55 views

A Particular Quotient of Polynomials over Hyperbolic Numbers: is it Anything?

The hyperbolic numbers are a two real dimensional algebra generated by $j$ such that $j^2=1$. In particular, denote $\mathcal{H} = \{ a+bj \ | \ a,b \in \mathbb{R} \}$. Multiplication is done in the ...
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Fourier-like transformation with split-complex numbers?

Usually the Fourier transformation is defined using the imaginary unit $$ F(p) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f(x) e^{ipx} dx = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f(x) \cos ...
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Geometric interpretation of the hyperbolic pythagorean theorem

Multiplying the identity $$ \cosh^2 x - \sinh^2x = 1 $$ with $c^2$ and defining $a^2= c^2\cosh^2$ and $b^2= c^2\sinh^2$ we get a hyperbolic version of the pythagorean theorem: $$ a^2-b^2 = c^2 $$ In ...
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1answer
31 views

Series with complex terms

I am trying to solve the following: for what values of $z\in \mathbb{C} $ the series $\sum_{k=1}^{\infty} \frac{2k+i}{k+2i} z^k $ converges. I transformed the series to the form $\sum_{k=1}^{\infty}...
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2answers
235 views

Dual number $(a+b\varepsilon)$ raised to a dual power, e.g. $(a+b\varepsilon)^{(c+d\varepsilon)}$

I'm working on some code which utilizes Newton's method, and I would like to take advantage of dual numbers to simplify taking the derivative. I've worked out a class definition ...
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1answer
121 views

Adding a root of $z\bar z=-1$ to $\mathbb C$

This half-serious question is inspired by the answer to my previous one, Want something like Cayley formula for unitary matrices The equation $z^2=-1$ does not have solutions in $\mathbb R$; adding a ...
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1answer
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Systems with Principle Roots of Unity

Over the complexes, it's possible to have a principle root of unity - in other words, a value $\omega$ with $\omega^n = 1$, and satisfying: $$\sum_{i=0}^{n-1}{ \omega^{ij} } = 0, j \in \{1, 2, \dots, ...
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1answer
85 views

Is $f(a+b\epsilon)=f(a)+b\epsilon f'(a)$ true for non-analytic smooth functions of dual argument

Does $f(a+b\epsilon)=f(a)+b\epsilon f'(a)$ remain true for non-analytic smooth functions of dual number argument? Where $ \epsilon^2=0$, $\epsilon \neq 0$ and $a,b \in \mathbb R$. I found proofs ...
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Are integers ever used in calculations involving hypercomplex numbers?

Integers are of course used in calculations involving complex numbers. But are integers ever used in calculations involving quaternions, octonions, etc.? In essence, are integers ever used in ...
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1answer
696 views

Are there “3+ dimensional” complex numbers? [duplicate]

As an engineer, I learned a lot about how to use complex numbers. One way I have heard $i$, the unit complex number, defined is: It is orthogonal to the real number line. Because $\frac{\mathrm{d}}{...
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1answer
846 views

How to calculate sin/cos/tan of a Quaternion?

I would like to learn about Quaternions. I've read this article: https://en.wikipedia.org/wiki/Quaternion Most of the article was not hard to understand, except the (Exponential, logarithm, and power)...
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1answer
1k views

How are Quaternions derived from Complex numbers or Real numbers?

I understand how complex numbers are derived from real numbers. Namely when you have a sqrt of a negative number you must have an answer of some kind, but this answer cannot be in the real number ...
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1answer
400 views

The “argument” of a quaternion

My question is pretty simple. I've been trying to read a pretty introductory text on Clifford algebras, and I encountered how they define the "argument" of a quaternion as an ordered quadruple ...
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1answer
61 views

Is there a generaliztion of phasors to n-argument sine waves?

Phasors, from what I can understand, are a way of representing the addition of and scalar multiplication of sine waves with constant frequency (so that only phase and amplitude differ), and this is ...
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1answer
63 views

Proving that $\mathbb{C}$ has a natural definition

I would like to prove a theorem in complex analysis which states: Let $K$ be a commutative field. We suppose that $L$ is a sub-field of $K$ and that $K$ is thus a vector space of finite dimension $...
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1answer
1k views

What is the difference between Quaternions and Bicomplex Numbers?

So, I know Quaternions are basically 4 dimensional Complex numbers, and the dimensions can double forever to Octonions, Sedinions, etc. I recently heard about bicomplex numbers, which are also sort of ...
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Cayley-Dickson construction: a general rule for multiplying imaginary units?

The Cayley–Dickson construction (see refs below) is a way of generating 'algebras' (in the loose sense) of increasing size over the reals, obtaining a sequence of algebras $\mathbb R = R_0 \...
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2answers
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Are dual numbers a special case of grassmann numbers?

Dual numbers are defined in analogy to complex numbers like $$ z = a + \varepsilon b. $$ But instead of $i^2=-1$ it is defined that $\varepsilon^2=0$. The multiplication rule for Grassmann numbers $\...
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1answer
435 views

Relationship between Levi-Civita symbol and Grassmann numbers?

The multiplication rule for Grassmann numbers $\theta_i$ is $$ \theta_i\theta_j = - \theta_j \theta_i $$ so that $\theta_i\theta_i = 0$. Multiplying three Grassmann numbers yields $$ \theta_i\theta_j\...
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1answer
135 views

Why formal power series are not considered a system of hypercomplex numbers?

One can introduce a constant $\chi$ following all numerical properties and two additional operations $\delta$ and $\circ$, like the following: $\delta a=0$ if a is a standard number $\delta \chi=1$ ...
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1answer
389 views

What is the combination of Complex, Split-Complex and Dual Numbers

If $a+bi:i^2=-1$ is a complex number, $a+cj:j^2=+1$ is a split-complex number, and $a+d\epsilon:\epsilon^2=0$ is a dual number; what is the term for the combination $a+bi+cj+d\epsilon:i^2=-1,j^2=+1,\...
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Is split-complex $j=i+2\epsilon$?

In matrix representation imaginary unit $$i=\begin{pmatrix}0 & -1 \\ 1 & 0 \end{pmatrix}$$ dual numbers unit $$\epsilon=\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}$$ split-...
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How quickly can we multiply hypercomplexes?

If we start with a hypercomplex number with $2^n$ entries, how quickly can we multiply it by another hypercomplex number, modulo a prime? EXAMPLE For example, with $n=1$, we get the complex numbers. ...
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207 views

Construction of Hyper-Complex Numbers

How does one construct a hyper-complex number multiplication table? For example: Quarternions: ...
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2answers
3k views

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}...