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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From algebraic point of view, what are the similarities and differences between these two hypercomplex systems?

I would like to know how compare from the algebraic point of view these two 3-dimensional hypercomplex number systems. 3-dimensional split-complex numbers Take $\mathbb{R}^3$ with Hadamard product. In ...
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Is this 3-dimensional split-complex numbers system viable? What are the properties?

The Wikipedia article n hypercomplex numbers completely lacks any mention of 3-dimensional and 6-dimensional systems, making impression they are impossible. Intuition. Basically, if you add two ...
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Mapping from dual numbers to real numbers

Background I was naively playing around with some interpolation ideas once again and came across the dual numbers as a way to perform differentiation implicitly. Naturally, I thought, okay, perhaps ...
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Quantifying which properties are lost in Cayley–Dickson constructions?

The Cayley–Dickson construction is a method of generalizing the algebraic structures of the Reals to the Complex to the Quaternions, etc. For those interested, discussion on the algebraic properties ...
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Which books/papers are good source for the concept of dual numbers over a ring?

Recently I came across the concept of dual numbers over a ring, $R[\epsilon] = \{a+b\epsilon ~|~ a,b \in R ~\text{and}~ \epsilon^2 = 0 \}$. I want to learn more about these rings. Hence looking for a ...
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Square root of dual number

We know the ring of dual numbers over the real field $\mathbb{R}$, is defined as $\mathbb{R}[\epsilon] = \{a+b\epsilon~|~a,b\in\mathbb{R} ~\text{and}~ \epsilon^2 = 0\}.$ Is the square root of $\...
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Unconventional hypercomplex numbers

I was learning a lot about hypercomplex numbers lately. I've seen articles about complex numbers, double numbers, dual numbers, binarions, quaternions, octonions etc. But one thing in common about all ...
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Construction of dual numbers

Out of curiosity, I was seeing about hypercomplex numbers. In that article, the definition says that, "Where possible, it is conventional to choose the basis so that $i_k^2 \in \{ -1, 0, +1 \}$. ...
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Can $\mathbb{R}^3$ with Hadamard product be represented as matrices?

It is known that $\mathbb{R}^2$ with Hadamard product, represented as pairs or numbers $(a,b)$ with element-wise operations is isomorphic to real matrices of the form $\left( \begin{array}{cc} \frac{...
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'Higher order' complex numbers

I recently learned that complex numbers can also be represented as matrices of the form: $$\begin{pmatrix} x & -y \\ y & x\end{pmatrix}$$ where complex multiplication corresponds to matrix ...
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Are dual-complex number equivalent to complex numbers having dual numbers as coefficient?

Wikipedia article for dual-quaternion explicitly states that, 'they may be constructed in the same way as the quaternions, except using dual numbers instead of real numbers as coefficients'. But, ...
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How to obtain heypercomplex numbers by an operation over complex numbers?

Using subtract operation with natural numbers would yield integers. Similarly, using division operation with ...
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A controversy regarding the generalization of the Sign function to dual numbers

Here is a link to a long discussion regarding generalization of $\operatorname{sign}z$ function to dual numbers. There are basically two proposed versions: $\operatorname{sign}(a+\varepsilon b) = \...
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How do you explain intuitively that in split-complex numbers, $0^{\frac12+\frac{j}2}=\frac12-\frac{j}2$ and $0^{\frac12-\frac{j}2}=\frac12+\frac{j}2$?

How do you explain intuitively that in split-complex numbers, $0^{\frac12+\frac{j}2}=\frac12-\frac{j}2$ and $0^{\frac12-\frac{j}2}=\frac12+\frac{j}2$? Usually zero to any power is zero, except when ...
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Explain Polynomes With HyperComplexs

I learn some of HyperComplex, Polynomes, Tailor series ... for informatics. I know there is no direct one-way to solve (find roots) $n$-th degree polynomial equations. We have methods. But I know that ...
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Why does the dual component of a dual number represent the derivative?

Example taken from here. Suppose we want to calculate derivative of $f(x) = 3x + 2$ at the point $x = 4$. We can use dual numbers; $x = 4 + 1\varepsilon$ because the derivative of $x$ is 1; then ...
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Can the traditional space of Euclidean vectors (with dot and cross products) be extended to be algebra over reals and what properties would it have?

There is traditional vector calculus of 3D vectors: https://en.wikipedia.org/wiki/Euclidean_vector#Basic_properties https://en.wikipedia.org/wiki/Vector_algebra_relations https://en.wikipedia.org/...
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What is the argument of the split-complex number? Is it the hyperbolic angle of the number's vector?

In split-complex numbers, what is the analog of argument of the complex numbers? I think, it is $\arg(a+bj)=\frac12\log \left(\frac{a+b}{a-b}\right)$.
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Can different Hypercomplex Dimensions be operated?

Can we make a product or a sum of quaternions with complexes ? Or maybe is it legal to convert complex to higher dimension. Like (5 + 2i) * (-4 + 1i + 4j + 3k)
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Nonassociative algebra's closed under $\sqrt{}$?

Consider a non-associative commutative unital algebra of finite dimension where the product is defined by a Cayley table such that elements are generated with real number coefficients $(a_0, \dots, ...
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Four dimensional field over complex numbers

A guy in Facebook claims he's come up with an algebraic field extension to the complex plane. He's defined the unit multiplications as $i^2=-1$, $j^2=i$ and $k^2=-i$. This implies that $ij=ji=k$, $ik=...
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Is $(-1)^{\varepsilon}=1+i\pi \varepsilon$?

Let $\varepsilon$ be defined as in the dual numbers. Using the standard matrix representation. I tried $(-1)^{\left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \\ \end{array} \right)}$ and got the ...
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Is there any situation where $x^2 \not= \sqrt{x^4}$

This is new for me so sorry if i am missing something, thanks for any helpful pointers. So I was thinking about this classic equation $E=mc^2$ Then I was, why is there $c^2$. I found it is just ...
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Why does not identity theorem works for Quaternions, Octonions, Sedenions etc?

Big chunk of complex analysis is built around the Identity theorem which in layman terms "allows analytical continuation to happen". Is there more or less intuitive way to understand why ...
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Vectors of complex or hypercomplex numbers?

Is there a field which studies vectors of complex numbers or vectors of hypercomplex numbers in general? Is it useful to think of a mono audio signal as a vector of complex numbers where imaginary ...
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Soft question: References for Dedekind's work on quaternions and hypercomplex numbers?

I've been told that quaternions and hypercomplex numbers in general were very important to Dedekind's early as well as late work. Besides Ferreiros' Labyrinth of Thought, ch. 7 and the references ...
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Do any significant changes happen in hypercomplex numbers beyond the eight dimensions of the octonions?

They continue in the fashion of powers of 2: reals (1), complex (2), quaternion (4), octonions (8), and then there is sedonions(16), right? And, this keeps going, right? Do any significant changes ...
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Are the Complex Numbers a subset of the Split Quaternions?

I saw that if $a+bi\in \mathbb C$, it's also an element of the split quaternions ($\mathbb P$), since $a+bi=a+bi+0j+0k$. Does this mean $\mathbb C\subset\mathbb P$? If so, does it follow that all ...
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who pioneered the study of the sedenions?

The nature of this question is pure historical curiosity. I found lots of background information about the discovery of both Imaginary and Complex Numbers, and enough information about the first two ...
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split-quaternion rotation

So, I learn about rotation basic quaternion, and now I am trying to understand split-quaternion and their rotation. As far as I understand, there are different formula, like $q=N(cosh(a)+psinh(a))$. ...
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Has there been any work done on extensions of "norms" to complex/hypercomplex spaces?

Let $x \in \Bbb R^d$ or some other vector space. Define $\|.\|_c$ as $(\sum_{j=1}^dx_j^c)^{\frac{1}{c}}$ where $c \in \Bbb C$ (Complex number) or $\Bbb H$ (other hypercomplex number systems). I was ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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