# 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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### 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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### 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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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 $\... 4 votes 4 answers 703 views ### Unconventional hypercomplex numbers I was learning a lot about hypercomplex numbers lately. I've seen articles about complex numbers, double numbers, dual numbers, binarions, quaternions, octonions etc. But one thing in common about all ... 3 votes 1 answer 318 views ### Construction of dual numbers Out of curiosity, I was seeing about hypercomplex numbers. In that article, the definition says that, "Where possible, it is conventional to choose the basis so that$i_k^2 \in \{ -1, 0, +1 \}$. ... 3 votes 1 answer 153 views ### Can$\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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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 ...