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).
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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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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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Do exists an inverse Cayley–Dickson construction for deducing lower-dimensional number systems?
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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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 ...
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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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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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Geometric explanation of Fueter-Sce-Qian theorem and similar situations
In Clifford analysis there is a fundamental theorem due to Fueter and extended by Sce and Qian that says (in modern terminology) that the given a slice regular function $f:\mathbb{R}^{m+1}\to\mathbb{R}...
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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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colinearity between complex vectors
You can find if two rea-valued vectors $x$ and $y$ in $\Re^n$ are colinear by calculating.
$\cos(\theta) = \frac{\langle x, y \rangle}{\lvert x\rvert\,\lvert y\rvert} $
For complex vectors ($z \in C^n$...
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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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Would some of the solutions to this function be considered hypercomplex numbers?
Consider the following function;
$$f(x,y) = \sqrt{x} + \sqrt{y}$$
If this function were to be plotted onto a 3-dimensional co-ordinate space, then the x and y axes would be orthogonal to each other.
...
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Properties of a 3D Julia set from squaring a 3D number?
Consider a commutative 3D algebra $T$ where the nonreal units $x,y$ satisfy
$$x^2 = A_1 + A_3 y $$
$$xy = 1 + B_2 x + B_3 y$$
$$y^2 = C_1 + C_3 y$$
where all the parameters $A_1,A_3,B_2,..$ are real.
...
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Factorisation of the sum of $n$ squares using hypercomplex numbers
After finding out that you could factor $a^2 + b^2$ as $(a+bi)(a-bi)$ using complex numbers I wondered if there were any useful factoring tricks using the quaternions or octonions and after some ...
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What is known about Pathions, Chingons, Routons and Voudons?
It is well known that the quaternions, octonions, and sedenions are well studied, but I don't find any articles or books in which other hypercomplex numbers are studied. Does anyone know a book or an ...
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How can one generalize the split-complex algebra to a coordinate space with an arbitrary number of asymptotes?
Given that split-complex numbers generates 2D coordinates with 4 asymptotes (when multiplying) that look as follows:
The arrows along the hyperbola indicate a positive direction for boosting points ...
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Numeric system where iterated square roots of 1 have integer parts
Let us define $s(n)$ as any solution to the equation $x^2 = n$, such that $x \neq n$. I am looking for a numeric system such that $s(s(s\cdots s(1)))$ is always an integer or is composed of integer ...
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Is there any practical use for octonions? [closed]
Quaternions are useful for describing orientation/ rotations in 3- dimensions, however is there much practical use for an 8-dimensional base hyper complex number id est Octonions?
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Iterating Hamilton's construction $\Bbb R\to \Bbb C$ yields what type of algebraic structure on $\Bbb C^2$ (e.g. occurs in nested complex datatypes).
The C++ language provides a general template-based implementation of complex numbers. Normally, these are using to manipulate numbers of the form $a + b i$, where $a, b$ are chosen as either 32 or 64 ...
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Can we extend hypercomplex numbers more?
In the hypercomplex numbers we describe units as $e_n$ for example,
$e_0 = 1, e_1 = i, e_2 = j, e_3 = k ...$
if x is an hypercomplex we can present as
$x = x_0e_0+x_1e_1+x_2e_2+x_3e_3+x_4e_4+...$
Can ...
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Is there an analogue of quaternions best described by tensors? [closed]
The transformations of a quaternion are best described by matrix multiplication, as are the transformations of any other hyper-complex number system I’ve heard of.
Is there an analog of the quaternion ...
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What are the properties of square-matrix algebra with this equivalence class?
Consider the set of all square matrices with the following equivalence class:
$\mathbf{A}\sim\mathbf{A}\otimes\mathbf{I}_n$,
for each $n$, where $\otimes$ is Kronecker product and $\mathbf{I}_n$ is ...
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The exact relationship between non-Euclidean Laguerre geometries and the planar algebras
There are three planar hypercomplex number systems, which we shall denote $\mathbb C, \mathbb D, \mathbb R^2$. It is known that the projective lines over these number systems correspond to the ...
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How to take the square root of a dual number ($\sqrt{a + b \cdot \varepsilon}$ with $a, ~b \in \mathbb{R}$ and $\varepsilon^{2} = 0$)? [duplicate]
How to take the square root of a dual number:
$\sqrt{\Xi}$ with $\begin{align*}
a, ~b &\in \mathbb{R}\\
\varepsilon^{2} &= 0\\
\varepsilon &\ne 0\\
\\
\Xi &:= a + b \cdot \varepsilon\\
...
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Spectrum of dual numbers is a projective scheme
Let $k$ be a field, and consider $X$ the spectrum of the dual numbers $k[x]/(x^2)$. This affine scheme is a closed subscheme of the affine line. Moreover, the affine line is open in the projective ...
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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 ...
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How do you find the 3rd roots of hypercomplex imaginary units?
In my instance I want to find the 3rd root of the octonion imaginary unit $e_4$.
I am working on simplifying the octonion
$$
o=5+2e_1 \sqrt[3]{e_4}+3e_2 \sqrt[3]{e_4}+2e_3 \sqrt[3]{e_4}
$$
$$
=5+...
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Highly polytopic algebras
(Update: I ended up posting two answers for this, each addressing different things. I feel the second one is more deserving to be accepted if nobody posts a better answer.)
Aside from the complex ...
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Cubic number system
In $\Bbb{H}$, the numbers $1$, $i$, $j$, $k$, and their respective negatives can all be seen as the vertices of a cross-polytope or orthoplex. This is true for anything that uses an orthonormal basis, ...
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Hypercomplex Numbers of the Form $a + ib$ Where $i^2 = p + iq$
Recently, I've been interested in hypercomplex number systems, and I have come across the three main 2-dimensional algebras:
\begin{align*}
i^2 &= -1 \\
j^2 &= 1 \\
\varepsilon^2 & = 0
\...
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Plotting tessarines in (pseudo-Euclidean) $R^{2,2}$ or $C^{1,1}$; conventions around which axes correspond to which signs in the metric signature
[This is a slightly tweaked re-posting of a question I asked in April 2020, that subsequently got auto-deleted as an "abandoned question." Hopefully launching a second attempt now (over two ...
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Relationship between tessarines, complex numbers and split-complex numbers (and between tessarines, complex numbers and (complex numbers) themselves)
[This is an expanded re-posting of a question I asked in April 2020, that subsequently got auto-deleted as an "abandoned question." Hopefully launching a second attempt now (over two years ...
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Were these hypercomplex number sets already defined in the literature for some $n\neq 3$? Are they associative?
I've been studying about hypercomplex numbers for the past few days, and then a question arised:
It looks quite natural to me to define the following set (based on the definition of quaternions):
$X = ...
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What are the properties of this new characteristic of mathematical objects?
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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Why we cannot divide one zero divisor by another one? Or can we?
For instance, in split-complex numbers we definitely have $2\cdot(j/2+1/2)=j+1$, which is absolutely valid. Can we then say that $\frac{j+1}{j/2+1/2}=2$? If not, why we cannot define it this way?
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Anti-dual numbers and what are their properties?
In this post user William Ryman asked what would happen if we try to build "complex numbers" with shapes other than circle or hyperbola in the role of a "unit circle".
Here I ...
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What comes in the next several K-D steps after the sedenions, and what is lost?
Wikipedia and elsewhere seem to say that one can keep on extrapolating forever in hypercomplexification, but that you progressively lose operation-equative symetries or whatever you call, e.g. ...
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If Complex Numbers Describe a Circle and Split-complex Numbers Describe a Hyperbola, Can One Make a Hypercomplex Number System to Describe any Shape?
I was thinking about other complex-like systems the other day, and I decided to define a number $o$ such that $o^2 = 1, o \ne \pm 1$. I wondered if there was a formula like Euler's formula for this ...
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What is $\sqrt{-1}$ (or $\sqrt{-j}$) in the Hyperbolic (Split-complex) Numbers?
Given a number system such that $j^2 = 1, j \ne \pm 1$, what would be the solution to $z^2 + 1 = 0$? Are the hyperbolic numbers not closed under taking roots unlike the complex numbers?
My assumption ...
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From algebraic point of view, what are the similarities and differences between these two hypercomplex systems?
I would like to know how compare from the algebraic point of view these two 3-dimensional hypercomplex number systems.
3-dimensional split-complex numbers
Take $\mathbb{R}^3$ with Hadamard product. In ...
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Is this 3-dimensional split-complex numbers system viable? What are the properties?
The Wikipedia article n hypercomplex numbers completely lacks any mention of 3-dimensional and 6-dimensional systems, making impression they are impossible.
Intuition. Basically, if you add two ...
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Mapping from dual numbers to real numbers
Background
I was naively playing around with some interpolation ideas once again and came across the dual numbers as a way to perform differentiation implicitly. Naturally, I thought, okay, perhaps ...
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Inverse of Grassmann variables
Let $\theta$ be a Grassmann variable. I know that $1/\theta$ is not defined but $\frac{1}{1-\theta}=1+\theta$. My question is simple: is $\frac{1}{\bar{\theta}\theta}$ defined and if so how? My guess ...
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Quantifying which properties are lost in Cayley–Dickson constructions?
The Cayley–Dickson construction is a method of generalizing the algebraic structures of the Reals to the Complex to the Quaternions, etc.
For those interested, discussion on the algebraic properties ...
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Which books/papers are good source for the concept of dual numbers over a ring?
Recently I came across the concept of dual numbers over a ring, $R[\epsilon] = \{a+b\epsilon ~|~ a,b \in R ~\text{and}~ \epsilon^2 = 0 \}$. I want to learn more about these rings. Hence looking for a ...
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Square root of dual number
We know the ring of dual numbers over the real field $\mathbb{R}$, is defined as $\mathbb{R}[\epsilon] = \{a+b\epsilon~|~a,b\in\mathbb{R} ~\text{and}~ \epsilon^2 = 0\}.$ Is the square root of $\...
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Unconventional hypercomplex numbers
I was learning a lot about hypercomplex numbers lately. I've seen articles about complex numbers, double numbers, dual numbers, binarions, quaternions, octonions etc.
But one thing in common about all ...
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Construction of dual numbers
Out of curiosity, I was seeing about hypercomplex numbers. In that article, the definition says that, "Where possible, it is conventional to choose the basis so that $i_k^2 \in \{ -1, 0, +1 \}$. ...
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Can $\mathbb{R}^3$ with Hadamard product be represented as matrices?
It is known that $\mathbb{R}^2$ with Hadamard product, represented as pairs or numbers $(a,b)$ with element-wise operations is isomorphic to real matrices of the form $\left(
\begin{array}{cc}
\frac{...
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'Higher order' complex numbers
I recently learned that complex numbers can also be represented as matrices of the form: $$\begin{pmatrix} x & -y \\ y & x\end{pmatrix}$$ where complex multiplication corresponds to matrix ...
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