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).
111 questions
2 votes
2 answers
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Exponentiation of a Dual Tessarine to another Dual Tessarine
I have been searching for a way to describe a dual tessarine $A$ to the power of another dual tessarine $B$, but I can't find anything about it that's consistent enough to work. To provide a brief ...
1 vote
1 answer
102 views
Could hypercomplex systems analogous to generalized complex numbers be constructed with a higher order relation instead of a quadratic relation?
A hypercomplex number system is an algebra that expands the real numbers by adding a unit that is distinct from one and negative one. The most well known hypercomplex number system is the complex ...
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2 answers
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Do "Infinitesimal Quaternions", a type of 4D hypercomplex numbers with 3 nilpotent units, exist? [closed]
The Question Do 4-dimensional "Infinitesimal Quaternions", a quaternion/hyperbolic quaternion-like type of hypercomplex numbers with 3 dual/nilpontent units, exist, and, if so, how do they ...
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-6 votes
1 answer
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Is there a General Formula for the Natural Logarithm of a Tessarine, a combination of both Split-Complex and Complex Numbers?
The Question I simply need to find the natural logarithm of a Tessarine number, there's nothing else to my problem. (Tessarines are a 4-dimensional combination of complex and split-complex numbers.) ...
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0 votes
1 answer
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What is the dual number equivalent of Cayley-Dickson construction for hypercomplex numbers?
Cayley-Dickson construction defines general forms of complex multiplication and conjugate: $$ (a,b)^* = (a^*, -b) \\ (a,b)(c,d) = (ac-d^*b,da+bc^*) $$ By applying these recursively, progressively ...
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18 votes
5 answers
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What is this finite dimensional algebra?
Fix a field $k$. Consider the (non-commutative, associative) $k$-algebra $A$ with generators $x$, $y$ subject to the relations \begin{align*} x^2&=x\\ y^2&=y\\ x-xy-yx+y&=1 \end{align*} ...
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2 votes
0 answers
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Do Euler's formula $e^{i\theta} = \cos(\theta) + i\sin(\theta)$ generalize to solid angles?
Is there a known generalization of Euler's formula $e^{i\theta} = \cos(\theta) + i\sin(\theta)$ for solid angles or higher-dimensional angles? If not, how might one go about establishing such a ...
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1 vote
1 answer
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Full rotating property of quaternionic polynomial
Does a quaternionic polynomial $Q\in\mathbb{H}[t]$ exist with the property that for any quaternionic imaginary unit $I \in \mathbb{S}=\{ q\in\mathbb{H} \mid q^2=-1 \}$ it holds $$Q(I)\notin\mathbb{C}...
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2 votes
1 answer
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Do exists an inverse Cayley–Dickson construction for deducing lower-dimensional number systems? [closed]
Is there a known inverse or reverse Cayley–Dickson construction that enables deduction of numbers in the reverse order, from higher-dimensional to lower-dimensional sets? For example, starting from ...
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0 votes
1 answer
101 views
Is this hypercomplex system sound? If so what is it called?
sorry for the bad image quality, I had to improvise Definiton of a hypercomplex number [it's in the tag] : A hypercomplex number is an element of a finite-dimensional algebra over the real numbers ...
1 vote
1 answer
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Real usage of “pure” quaternions in stereometry?
There are two major categories of the "quaternions". It is well-known that a (nonzero) versor represents a three-dimensional rotation operator. A versor is a unit quaternion or a normalized ...
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1 vote
1 answer
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Imagining an exponential "hypercomplex" system
I recently learned about hypercomplex systems that are taken over the reals, i.e. the dual numbers for which $j^2=1$, $j≠1$, and the dual numbers for which $ε^2=0$, $ε≠0$. These number systems, along ...
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2 votes
2 answers
187 views
What is this field in $\mathbb{R}^4$ that contains both the real and complex numbers called?
Note: this question is wrong – this is not a field, though it is not obvious why it wouldn't be. So, I (first year undergraduate mathematics student) was looking around the internet and found an ...
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-1 votes
1 answer
104 views
Is there a direct limit in the category of rings for hypercomplex numbers [closed]
I recently learned about the concept limits in categories. From R we can construct C the H etc... by iterating the Cayley-Dickson construction. My question is: Can we construct a (non-associative)ring ...
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0 votes
1 answer
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Would some of the solutions to this function be considered hypercomplex numbers?
Consider the following function; $$f(x,y) = \sqrt{x} + \sqrt{y}$$ If this function were to be plotted onto a 3-dimensional co-ordinate space, then the x and y axes would be orthogonal to each other. ...
