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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).

A hypercomplex number is an element of a finite-dimensional algebra over the real numbers that is unital and distributive (but not necessarily associative). Elements are generated with real number coefficients $(a_0, \dots, a_n)$ for a basis $\{ 1, i_1, \dots, i_n \}$. Where possible, it is conventional to choose the basis so that $i_k^2 \in \{ -1, 0, +1 \}$. A technical approach to hypercomplex numbers directs attention first to those of dimension two. Higher dimensions are configured as Cliffordian or algebraic sums of other algebras and are usually obtained with the Cayley-Dickson's construction.

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Why are properties lost in the Cayley–Dickson construction?

Motivating question: What lies beyond the Sedenions? I'm aware that one can construct a hierarchy of number systems via the Cayley–Dickson process: $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S} \subset…
Hooked
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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*} This algebra is four dimensional: using the third relation, one…
Joshua Tilley
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What is the difference between Quaternions and Bicomplex Numbers?

So, I know Quaternions are basically 4 dimensional Complex numbers, and the dimensions can double forever to Octonions, Sedinions, etc. I recently heard about bicomplex numbers, which are also sort of 4 dimensional complex numbers. I looked into it…
RothX
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How are Quaternions derived from Complex numbers or Real numbers?

I understand how complex numbers are derived from real numbers. Namely when you have a sqrt of a negative number you must have an answer of some kind, but this answer cannot be in the real number system, therefore you need another number system…
St.Clair Bij
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Why intuitively do the quaternions satisfy the mixture of geometric and algebraic properties that they do?

[I completely rewrote the question to see if I could make it clearer. The comments below won't make any sense. In fact, my original question has been answered by Eric Wolfsey, so I may restore it.] When you read about the quaternions on Wikipedia…
wlad
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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, where the number of basis elements coincide with…
Francis Ocoma
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Do any significant changes happen in hypercomplex numbers beyond the eight dimensions of the octonions?

They continue in the fashion of powers of 2: reals (1), complex (2), quaternion (4), octonions (8), and then there is sedonions(16), right? And, this keeps going, right? Do any significant changes happen in hypercomplex numbers beyond the eight…
user3146
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Adding a root of $z\bar z=-1$ to $\mathbb C$

This half-serious question is inspired by the answer to my previous one, Want something like Cayley formula for unitary matrices The equation $z^2=-1$ does not have solutions in $\mathbb R$; adding a solution produces $\mathbb C$. The equation…
მამუკა ჯიბლაძე
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What is the combination of Complex, Split-Complex and Dual Numbers

If $a+bi:i^2=-1$ is a complex number, $a+cj:j^2=+1$ is a split-complex number, and $a+d\epsilon:\epsilon^2=0$ is a dual number; what is the term for the combination $a+bi+cj+d\epsilon:i^2=-1,j^2=+1,\epsilon^2=0$?
S-Erase
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Is split-complex $j=i+2\epsilon$?

In matrix representation imaginary unit $$i=\begin{pmatrix}0 & -1 \\ 1 & 0 \end{pmatrix}$$ dual numbers unit $$\epsilon=\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}$$ split-complex unit $$j=\begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix}$$ Given this…
Anixx
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Are the Complex Numbers a subset of the Split Quaternions?

I saw that if $a+bi\in \mathbb C$, it's also an element of the split quaternions ($\mathbb P$), since $a+bi=a+bi+0j+0k$. Does this mean $\mathbb C\subset\mathbb P$? If so, does it follow that all Cayley-Dickenson Constructions are a subset of the…
William C.
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$\epsilon \otimes 1 + 1 \otimes \epsilon$ is a nilcube in $\mathbb R[\epsilon] \otimes \mathbb R[\epsilon]$. What does that mean intuitively?

[EDIT: I know what the notation means, and I can easily show that $x=\epsilon \otimes 1 + 1\otimes \epsilon$ satisfies $x^3=0$ but $x^2 \neq 0$. That's not what this question is about. It might be better to restrict the question to Synthetic…
wlad
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Inquiries around a new number system

We can suppose that we will create a new number system with essentially two imaginaries that do not interact. (Besides this, all quantities are taken to be integers) For example, we have an $i_1$ and an $i_2$. Then we could say $$(a+b i_1)(c+d…
Matt Groff
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Are there "3+ dimensional" complex numbers?

As an engineer, I learned a lot about how to use complex numbers. One way I have heard $i$, the unit complex number, defined is: It is orthogonal to the real number line. Because $\frac{\mathrm{d}}{\mathrm{d}x} e^{x} = e^{x}$, we can clearly see…
William Entriken
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Are dual numbers a special case of grassmann numbers?

Dual numbers are defined in analogy to complex numbers like $$ z = a + \varepsilon b. $$ But instead of $i^2=-1$ it is defined that $\varepsilon^2=0$. The multiplication rule for Grassmann numbers $\theta_i$ is $$ \theta_i\theta_j = - \theta_j…
asmaier
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