# 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 ...
• 9,036
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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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### 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 $f: \mathbb{C} \to \mathbb{C}$ ...
• 3,968
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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 ...
• 9,036
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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-...
• 9,036
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• 9,036
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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-...
• 9,036
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 ...