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 does the dual component of a dual number represent the derivative?

Example taken from here. Suppose we want to calculate derivative of $f(x) = 3x + 2$ at the point $x = 4$. We can use dual numbers; $x = 4 + 1\varepsilon$ because the derivative of $x$ is 1; then ...
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Can the traditional space of Euclidean vectors (with dot and cross products) be extended to be algebra over reals and what properties would it have?

There is traditional vector calculus of 3D vectors: https://en.wikipedia.org/wiki/Euclidean_vector#Basic_properties https://en.wikipedia.org/wiki/Vector_algebra_relations https://en.wikipedia.org/...
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What is the argument of the split-complex number? Is it the hyperbolic angle of the number's vector?

In split-complex numbers, what is the analog of argument of the complex numbers? I think, it is $\arg(a+bj)=\frac12\log \left(\frac{a+b}{a-b}\right)$.
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Can different Hypercomplex Dimensions be operated?

Can we make a product or a sum of quaternions with complexes ? Or maybe is it legal to convert complex to higher dimension. Like (5 + 2i) * (-4 + 1i + 4j + 3k)
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Nonassociative algebra's closed under $\sqrt{}$?

Consider a non-associative commutative unital algebra of finite dimension where the product is defined by a Cayley table such that elements are generated with real number coefficients $(a_0, \dots, ...
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Four dimensional field over complex numbers

A guy in Facebook claims he's come up with an algebraic field extension to the complex plane. He's defined the unit multiplications as $i^2=-1$, $j^2=i$ and $k^2=-i$. This implies that $ij=ji=k$, $ik=...
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Are true parabolic numbers possible?

Often the term "parabolic numbers" is applied to dual numbers. But I do not see how they are parabolic. They are better to be called "linear". So, I decided to build true parabolic ...
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Is $(-1)^{\varepsilon}=1+i\pi \varepsilon$?

Let $\varepsilon$ be defined as in the dual numbers. Using the standard matrix representation. I tried $(-1)^{\left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \\ \end{array} \right)}$ and got the ...
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Is there any situation where $x^2 \not= \sqrt{x^4}$

This is new for me so sorry if i am missing something, thanks for any helpful pointers. So I was thinking about this classic equation $E=mc^2$ Then I was, why is there $c^2$. I found it is just ...
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Why does not identity theorem works for Quaternions, Octonions, Sedenions etc?

Big chunk of complex analysis is built around the Identity theorem which in layman terms "allows analytical continuation to happen". Is there more or less intuitive way to understand why ...
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Vectors of complex or hypercomplex numbers?

Is there a field which studies vectors of complex numbers or vectors of hypercomplex numbers in general? Is it useful to think of a mono audio signal as a vector of complex numbers where imaginary ...
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General solution of an ODE from the general solution of a PDE/SDE/SPDE and most general “differential” equation

Is it possible to obtain the general solution of an ODE by solving a PDE, SDE or SPDE? I haven't dived into PDEs, nor SDEs, therefore the question. Another question that arises from my love to ...
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Soft question: References for Dedekind's work on quaternions and hypercomplex numbers?

I've been told that quaternions and hypercomplex numbers in general were very important to Dedekind's early as well as late work. Besides Ferreiros' Labyrinth of Thought, ch. 7 and the references ...
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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 ...
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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 ...
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who pioneered the study of the sedenions?

The nature of this question is pure historical curiosity. I found lots of background information about the discovery of both Imaginary and Complex Numbers, and enough information about the first two ...
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split-quaternion rotation

So, I learn about rotation basic quaternion, and now I am trying to understand split-quaternion and their rotation. As far as I understand, there are different formula, like $q=N(cosh(a)+psinh(a))$. ...
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Has there been any work done on extensions of “norms” to complex/hypercomplex spaces?

Let $x \in \Bbb R^d$ or some other vector space. Define $\|.\|_c$ as $(\sum_{j=1}^dx_j^c)^{\frac{1}{c}}$ where $c \in \Bbb C$ (Complex number) or $\Bbb H$ (other hypercomplex number systems). I was ...
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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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Rotation around a whole sphere by multiplying a single hypercomplex number forever?

In quaternion number system, any unit quaternion $\mathbf{q}\in\mathbb{H}$ can be written as $$ \mathbf{q} = \cos \theta + (v_x \mathbf{i} + v_y \mathbf{j} + v_z \mathbf{k})\sin \theta $$ for some $\...
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Is it possible to plug hypercomplex numbers into the Riemann Zeta function?

I'm aware of the more detailed question: How to raise a number to a quaternion power However, from a more high-level perspective (read: probably less mathematical, hence my choice for raising ...
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Hyper complex number $e_{16}$ had a zero divisor.

I was looking into hyper complex numbers https://en.wikipedia.org/wiki/Hypercomplex_number where it stated that under Cayley–Dickson construction for "$\{1,i_1,...i_{2^n-1}\}$... in $16$ or more ...
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Adding a complex and its conjugate, both raised to a degree of 4?

$(-1+\sqrt{-3})^4+(-1-\sqrt{-3})^4$ = ? I deduced the complex form: $z=(-1+i\sqrt{3})$ I can see that the question is basically $z^4+\overline z^4$ Now, if we add a complex number and its ...
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Any references on supercomplex/surcomplex numbers?

I know many text in surreal numbers have went over them but I can't seem to find anything that goes into great detail beyond a small section on them. For instance, in Foundation of Analysis over ...
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Residue theorem for split complex numbers

One can show that any function on the split-complex numbers which can be represented by a Laurent series is infinitely differentiable, except at the union of several (shifted) hyperbolas of modulus 0 (...
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168 views

Triangle inequality for split-complex numbers

After much research and head-scratching to no avail, I was hoping someone here could shed some light on certain properties of the split-complex numbers, namely: 1) Is there any meaningful analog of ...
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A Particular Quotient of Polynomials over Hyperbolic Numbers: is it Anything?

The hyperbolic numbers are a two real dimensional algebra generated by $j$ such that $j^2=1$. In particular, denote $\mathcal{H} = \{ a+bj \ | \ a,b \in \mathbb{R} \}$. Multiplication is done in the ...
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Fourier-like transformation with split-complex numbers?

Usually the Fourier transformation is defined using the imaginary unit $$ F(p) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f(x) e^{ipx} dx = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f(x) \cos ...
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Geometric interpretation of the hyperbolic pythagorean theorem

Multiplying the identity $$ \cosh^2 x - \sinh^2x = 1 $$ with $c^2$ and defining $a^2= c^2\cosh^2$ and $b^2= c^2\sinh^2$ we get a hyperbolic version of the pythagorean theorem: $$ a^2-b^2 = c^2 $$ In ...
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Series with complex terms

I am trying to solve the following: for what values of $z\in \mathbb{C} $ the series $\sum_{k=1}^{\infty} \frac{2k+i}{k+2i} z^k $ converges. I transformed the series to the form $\sum_{k=1}^{\infty}...
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Dual number $(a+b\varepsilon)$ raised to a dual power, e.g. $(a+b\varepsilon)^{(c+d\varepsilon)}$

I'm working on some code which utilizes Newton's method, and I would like to take advantage of dual numbers to simplify taking the derivative. I've worked out a class definition ...
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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 ...
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Systems with Principle Roots of Unity

Over the complexes, it's possible to have a principle root of unity - in other words, a value $\omega$ with $\omega^n = 1$, and satisfying: $$\sum_{i=0}^{n-1}{ \omega^{ij} } = 0, j \in \{1, 2, \dots, ...
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Is $f(a+b\epsilon)=f(a)+b\epsilon f'(a)$ true for non-analytic smooth functions of dual argument

Does $f(a+b\epsilon)=f(a)+b\epsilon f'(a)$ remain true for non-analytic smooth functions of dual number argument? Where $ \epsilon^2=0$, $\epsilon \neq 0$ and $a,b \in \mathbb R$. I found proofs ...
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Are integers ever used in calculations involving hypercomplex numbers?

Integers are of course used in calculations involving complex numbers. But are integers ever used in calculations involving quaternions, octonions, etc.? In essence, are integers ever used in ...
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Are there “3+ dimensional” complex numbers? [duplicate]

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}}{...
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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)...
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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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Proving that $\mathbb{C}$ has a natural definition

I would like to prove a theorem in complex analysis which states: Let $K$ be a commutative field. We suppose that $L$ is a sub-field of $K$ and that $K$ is thus a vector space of finite dimension $n$ ...
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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 ...
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Cayley-Dickson construction: a general rule for multiplying imaginary units?

The Cayley–Dickson construction (see refs below) is a way of generating 'algebras' (in the loose sense) of increasing size over the reals, obtaining a sequence of algebras $\mathbb R = R_0 \...
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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 $\...
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Relationship between Levi-Civita symbol and Grassmann numbers?

The multiplication rule for Grassmann numbers $\theta_i$ is $$ \theta_i\theta_j = - \theta_j \theta_i $$ so that $\theta_i\theta_i = 0$. Multiplying three Grassmann numbers yields $$ \theta_i\theta_j\...
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Why formal power series are not considered a system of hypercomplex numbers?

One can introduce a constant $\chi$ following all numerical properties and two additional operations $\delta$ and $\circ$, like the following: $\delta a=0$ if a is a standard number $\delta \chi=1$ ...
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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,\...