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Questions tagged [split-complex-numbers]

Questions involving split complex numbers, that is numbers of the form $a+bj$ where $j^2=1,j\ne\pm 1$ and $a,b\in\mathbb{R}$.

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Does multiplication by the split-complex number $j$ have a geometric interpretation? [closed]

For complex numbers it's fairly intuitive. i² corresponds to a nice counter-clockwise rotation along the unit circle. rotation1 rotation2 A "rotation" in the split-complex plane however ...
transpant's user avatar
1 vote
1 answer
139 views

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 ...
Rehno Lindeque's user avatar
1 vote
0 answers
72 views

Can split-complex numbers be compared?

Doing one problem on my own I found that split-complex numbers might be useful to solve it but couldn't find any information about comparing two split-complex numbers. Can you always find a minimal ...
epsilonQM's user avatar
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1 answer
114 views

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 ...
Kevin M. Lamoreau's user avatar
2 votes
1 answer
377 views

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 ...
Kevin M. Lamoreau's user avatar
1 vote
2 answers
307 views

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 ...
William Ryman's user avatar
1 vote
1 answer
122 views

How do you explain intuitively that in split-complex numbers, $0^{\frac12+\frac{j}2}=\frac12-\frac{j}2$ and $0^{\frac12-\frac{j}2}=\frac12+\frac{j}2$?

How do you explain intuitively that in split-complex numbers, $0^{\frac12+\frac{j}2}=\frac12-\frac{j}2$ and $0^{\frac12-\frac{j}2}=\frac12+\frac{j}2$? Usually zero to any power is zero, except when ...
Anixx's user avatar
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3 votes
1 answer
151 views

Is the Cauchy-Goursat theorem or even the homotopic invariance theorem valid for fields other than $\Bbb C$ which are similar to $\Bbb R^2$?

Clearly, the Green's Theorem proof does not hold as that relies on the specific conditions of complex differentiability. However, Goursat's proof involving the triangle case and building up from there ...
FShrike's user avatar
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157 views

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 $\arg(a+bj)=\frac12\log \left(\frac{a+b}{a-b}\right)$.
Anixx's user avatar
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Split-complixificatinon of Lie algebras and Lie groups

The complexification of the Lie algebras and groups over $\mathbb{C}$ is a well-studied topic. However, I can not find any reference to the similar process for "split-complexification" over $...
Eddward's user avatar
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2 answers
40 views

Field extension of $\mathbb{Q}$ is the splitting field of two different polynomials?

On the topic of splitting fields at Wolfram Mathworld, one of the examples makes sense but the other doesn't. From https://mathworld.wolfram.com/SplittingField.html: For example, the extension field ...
Hank Igoe's user avatar
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2 votes
1 answer
260 views

Are there Olympiad problems that can be solved using split-complex numbers?

There are various uses for complex numbers in Math Olympiads. In addition, quadratic number fields are sometimes useful, for instance for Pell equations. Are there any Olympiad/contest problems that ...
Federico Poloni's user avatar
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1 answer
262 views

Alternative to complex numbers - Study numbers $j^2=1$, $j \neq 1$ [duplicate]

On p. 24 "Clifford Algebra and Spinors" (2.2 Double Ring of $\:{}^2\mathbb{R}$ of $\mathbb{R}$) by Petri Lounesto the author mentions Study numbers as an equivalent alternative to complex ...
Eddward's user avatar
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51 views

Does the observation that we can raise finite dimensional vector spaces to the power of certain hyperbolic numbers teach us anything?

($U^*$ denotes the dual space of $U$ whenever $U$ is a vector space. I'll work over $\mathbb{R}$ for simplicity) Given a finite-dimensional vector space $V$ and a hyperbolic number $a + bj$ with $a,b \...
goblin GONE's user avatar
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1 answer
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How do you plot $f(x)=x^2$ in the split-complex plane?

What does the graph of $f(x)=x^2$ look like in the split-complex plane? How do you plot it? for split-complex number $x.$ So, I think you have to do $f(a+bj)=(a+bj)^2=a^2+2abj+b^2j^2$ $j^2=+1$ in ...
zeta space's user avatar
0 votes
2 answers
1k views

Complex numbers, set of values for which z will be purely real or imaginary.

A complex number z is given by $ z = \frac{a+i}{a-i}, a∈R$. Determine the set of values of a such that (a). z is purely real; (b). z is purely imaginary. (c). Show that |z| is a constant for all ...
Thomas J.'s user avatar
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0 answers
154 views

In split-complex numbers, why modulus of $z=a+bj$ is $a^2-b^2$ and not $\sqrt{a^2-b^2}$?

A common convention in many algebras is that modulus is $||z||=\exp( R (\ln z))$ where R(z) is the real or scalar part. If we use this convention for split-complex numbers, we have: $$\ln (a+bj)=\...
Anixx's user avatar
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3 votes
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123 views

What shape is the split-complex projective line?

I'm aware that the complex projective line is a sphere, the Riemann sphere, and the dual projective line is a cylinder; but I can't find anything mentioning what shape the split-complex projective ...
Seth Schmidt-O'Hainle's user avatar
4 votes
1 answer
478 views

matrix representations of complex, dual and split-complex numbers

For complex, dual and split-complex numbers there are matrix representations: $$a+b \cdot i \equiv a\begin{pmatrix} 1 & 0\\0 & 1 \\ \end{pmatrix} +b\begin{pmatrix}0 & -1\\1 & 0 \\ ...
Sascha's user avatar
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1 answer
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Split numbers and linear independence of multiplicative inverses

My understanding is that the product of ijk may equal +1 when working with split-quaternions. What are good examples of systems defined such that the product of two (not three) linearly independent (i....
bblohowiak's user avatar
2 votes
3 answers
291 views

What is the symbol for split-complex numbers?

$\mathbb{R}$ is the set of the real numbers, $\mathbb{C}$ is the set of the complex numbers, but is there also a symbol for the split-complex numbers?
Illuminatio2718's user avatar
6 votes
2 answers
318 views

Square roots of $j$ and $ε$

I know how to find the square root of the imaginary unit $i$, but I'm still learning about split-complex and dual numbers. I can't find any info anywhere about the square roots of $j$ and $ε$, if they ...
A.J.'s user avatar
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54 votes
4 answers
5k views

How many "super imaginary" numbers are there?

How many "super imaginary" numbers are there? Numbers like $i$? I always wanted to come up with a number like $i$ but it seemed like it was impossible, until I thought about the relation of $...
EEVV's user avatar
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5 votes
2 answers
483 views

compute $\log_e(j)$ of split complex number $j$

I am trying to calculate the value of $\ln j$ where $j^2=1, j\ne\pm1$($j$ is split complex). This is how I did it: given $e^{j\theta}=\cosh\theta+j\sinh\theta$ I can set $\cosh\theta=0\implies \...
Holo's user avatar
  • 10k
3 votes
1 answer
253 views

Caculus of $j^j$, $i^j$ and $j^i$ (where $i^2=-1$, $j^2=1$, $j \neq -1$ and $j\neq 1$), and others like $Ln(i+j)$

First of all, I am do math only for fun, and I am only amateur, so excuse me because of my lack on knowledge in this field. I was curious about some answers and I want to know if my judgment was ok or ...
user avatar
3 votes
1 answer
504 views

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 (...
WhiteboardFunk's user avatar
4 votes
1 answer
261 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 ...
WhiteboardFunk's user avatar
3 votes
1 answer
107 views

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 ...
James S. Cook's user avatar
2 votes
1 answer
80 views

Is there a meaningful way to define $ij$, where $i$ is the imaginary unit and $j$ is the split-complex unit?

At the moment I'm treating $ij=ji$ as its own quantity and reducing it where possible, as in $(ij)^2 =-1$ and $e^{ij} = \cos(j) + i\sin(j) = \cosh(i) + j \sinh(i)$. From the first identity seems like ...
Will Bolden's user avatar
3 votes
0 answers
388 views

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 ...
asmaier's user avatar
  • 2,674
5 votes
1 answer
863 views

How to do calculus with split-complex (hyperbolic) numbers?

TL;DR: how do I define a "split-holomorphic" function? As far as I've heard, there is a notion of split-complex numbers: $z = x+uy$, with $u\not\in \Bbb R $ and $u^2 = 1$. One defines the ...
Ivo Terek's user avatar
  • 78.4k
0 votes
0 answers
129 views

uniquess of hyperbolic numbers

I'm trying to prove, the uniqueness of hyperbolic numbers, like the complex numbers are unique, but since hyperbolic numbers aren't a field, I can't use the ideas of this. Are there theorems of ...
user194502's user avatar
3 votes
1 answer
387 views

Reference Request: Split-Complex Numbers

Does anyone have a recommendation for a good book on split-complex numbers? If it also covers dual numbers or the relation between split-complex numbers and special relativity or Minkowski 4-space or ...
user157624's user avatar
1 vote
0 answers
182 views

Relation between hyperbolic numbers and hyperbolic functions

Is there any relation between the hyperbolic (split-complex) numbers and hyperbolic trig functions? Or are they just named similarly by accident?
user157624's user avatar
15 votes
2 answers
6k views

What are the uses of split-complex numbers?

The set of Complex numbers can be used in lots of domains like geometry, vectorial calculations, solving equation with no real solution etc. But what are the uses of split-complex number that can't be ...
moray95's user avatar
  • 1,047
1 vote
1 answer
348 views

Split-complex numbers and their possible application

Suppose that there is number $a+jb$ where $j^2=1$ and the whole number is split-complex number. We want to set this number to satisfy the following: A) $(a+jb)(a+jb) = k(c+jd)$ where $k$ is fixed ...
user72658's user avatar