# Questions tagged [hypercomplex-numbers]

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### Extending reals with logarithm of zero: properties and reference request

If we take logarithmic function, we can see that its real part at zero approaches negative infinity with the same rate and sign from any direction on the complex plane, while the Cauchy main value of ...

2
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1
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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?
...

6
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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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1
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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 ...

0
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1
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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} &...

1
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1
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346
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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-...

3
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2
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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 ...

4
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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-...