Questions tagged [hypercomplex-numbers]

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Can a non-distributive algebraic system be represented as matrices?

For instance, can the following 4-dimensional "number system" (which I would call "anti-split numbers") be represented as matrices? It is commutative and associative but not ...
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Exponent of the scalar part of the finite part of the logarithm of an object, or hypermodulus

I will call it "hypermodulus". In simple words, hypermodulus is the exponent of the scalar part of the finite part of the logarithm of the object: $H(A)=\exp (\operatorname{scal} \...
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Anti-dual numbers and what are their properties?

I have asked this question before in Math.SE. It got upvotes but no answer so far. In this post user William Ryman asked what would happen if we try to build "complex numbers" with shapes ...
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Why we can analytically define $ε$ in dual numbers so to distinguish $ε$ from $-ε$ but cannot do so in complex and split-complex numbers?

Why we can analytically augment the algebraic definition of $ε$ in dual numbers so to distinguish $ε$ from $-ε$ but cannot do so in complex and split-complex numbers? Is there a theorem on this? ...
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Lemniscate numbers and others - what would be the properties?

Recently I noticed a question about systems of alternative "complex" numbers, defined by shapes other than circle, hyperbola and straight lines. I even made and answer, telling that this is ...
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Is there a residue sum formula in quaternionic analysis?

In complex analysis, there is a formula involving the residues of complex functions that one can employ to find the value of certain infinite series. If the function $f: \mathbb{C} \to \mathbb{C} $ ...
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What are the properties of 3-dimensional split-complex numbers?

I have often encountered claims that 3-dimensional numbers are impossible. But it seems to me that $\mathbb{R}^3$ with Hadamard multiplication should in fact behave quite similar to split-complex ...
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Representing split-complex numbers as intervals and related compactification

Since there is an isomorphism between split-complex numbers and $\mathbb{R}^2$ with element-wise operations, of the following form $a + bj \leftrightarrow (a - b, a + b)$, one can think about a split-...
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Can we say that everywhere where it makes sense $\log_0 x=0^x$? Are they equal, the function is self-inverse? If so, what is deep intuition behind it? [closed]

It makes little reason to speak about $0^x$ and $\log_0 x$ on the set of real numbers, but in matrices, it seems, the expressions coincide, for instance, $0^ \left( \begin{array}{cc} \frac{1}{2} &...
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Are there algebras over reals besides complex numbers, where identities, analoguous to $(-1)^i=e^{-\pi}$ and $i^i=e^{-\pi/2}$ hold?

Are there algebras over real numbers (with exponentiation), such that there is such $z$ that does not include components in $\mathbb{C}$ (or in a subset isomorphic to $\mathbb{C}$), for which $(-1)^z\...
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Can the notion of algebraic closedness be generalized to the rings with zero divisors?

Is there a notion of rings that are algebraically closed except for the roots of polynomials with coefficients that are divisors of zero? For instance, it seems that any polynomial of non-zero-divisor-...
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The name of special 16-dimensional hypercomplex number

Let's consider the following number: $n = n_0 + n_1\textbf{i} + n_2\textbf{j} + n_3\textbf{k}$ Where $\{1,\textbf{i},\textbf{j},\textbf{k}\}\subset\mathbb{H}$ and in turn each component can be written ...
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4 votes
1 answer
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Function theory of a hyperbolic variable

I've found quite a number of articles on the basics of function theory in one hyperbolic (split-complex, dual, duplex, motro,..) variable, perhaps the most notable being http://arxiv.org/PS_cache/math-...
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