# 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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### 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.] ...
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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 ...
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### Using dual complex numbers for combined rotation and translation

Dual quaternions may be used to perform combined rotations and translations in a single dual quaternion product operation. Translation is performed by placing the displacement, $d$ in vector of the ...
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### A simple Variation on the Imaginary Unit i

I think a more appropriate tag would have been 'quasicomplex numbers' rather than 'hypercomplex numbers'. I'm normally perfectly comfortable with the correspondence between hyperbolic functions & ...
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### How to calculate sin/cos/tan of a Quaternion?

I would like to learn about Quaternions. I've read this article: https://en.wikipedia.org/wiki/Quaternion Most of the article was not hard to understand, except the (Exponential, logarithm, and power)...
1answer
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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 ...
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### The “argument” of a quaternion

My question is pretty simple. I've been trying to read a pretty introductory text on Clifford algebras, and I encountered how they define the "argument" of a quaternion as an ordered quadruple ...
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### Is there a generaliztion of phasors to n-argument sine waves?

Phasors, from what I can understand, are a way of representing the addition of and scalar multiplication of sine waves with constant frequency (so that only phase and amplitude differ), and this is ...
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