10
votes
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?
The short answer is "because you are considering $\mathbb{C}$ and $\mathbb{R}[\epsilon]$ them with different structure", which is an artificial choice.
Maybe to illustrate the point : If I ...
- 42.4k
5
votes
Accepted
Real matrix rings and associative hypercomplex numbers
It's not clear what you mean by "real matrix ring," which could either mean a ring of the form $M_n(\mathbb{R})$ or a real subalgebra of $M_n(\mathbb{R})$; if the latter, this is the same as ...
- 118k
4
votes
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?
Experimenting with Maple, it seems Maple's definition of $0^A$, where $A$ is
a square matrix, will be:
$\bullet\;$If $A$ is diagonalizable, Say $A = Q^{-1} D Q$ with
$D = \operatorname{diag}(a_1,\...
- 41.1k
3
votes
Accepted
The name of special 16-dimensional hypercomplex number
Ther are not sedenions because sedenions are not associative: https://en.wikipedia.org/wiki/Sedenion
I doubt they have a name but this algebra is well understood. The hyperbolic quaternions is just $...
- 12.3k
2
votes
Representing split-complex numbers as intervals and related compactification
There is a compactification of the split-complex numbers. I will denote the split-complex numbers as $\mathbb R^2$ as you suggested. The consequences of such a compactification are described here:
...
- 4,943
2
votes
Accepted
Are there algebras over reals besides complex numbers, where identities, analoguous to $(-1)^i=e^{-\pi}$ and $i^i=e^{-\pi/2}$ hold?
I will address the question in the case $f(z) = (-1)^z = \exp(z \log(-1))$ and $A$ is a finite-dimensional commutative (associative unital) $\mathbb{R}$-algebra, where $\log(-1)$ is a suitable choice ...
- 7,127
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