# 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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### '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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### 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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### 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 ...