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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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 ...
2 votes
1 answer
83 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 ...
1 vote
2 answers
95 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 ...
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Representing split-complex numbers as intervals and related compactification

Since there is an isomorphism between split-complex numbers and $\mathbb{R}^2$ with element-wise operations, of the following form $a + bj \leftrightarrow (a - b, a + b)$, one can think about a split-...
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Modulus of the "pair" $(0,\infty)$ seems to be a finite value?

It is known that the split-complex numbers are isomorphic to the pairs of real numbers in the following way: $a + bj \leftrightarrow (a - b, a + b)$ with operations defined on the pairs element-wise. ...
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1 vote
1 answer
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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 ...
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1 vote
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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 ...
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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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1 answer
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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 $...
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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 ...
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1 answer
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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 ...
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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 ...
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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 \...
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1 vote
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 ...
0 votes
2 answers
568 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 ...
0 votes
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86 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)=\...
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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 ...
4 votes
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Maximal extension of domain of $f(x)=\sum_{n=1}^\infty n^{\frac{1}{\log_n(x)}}$ using analytic continuation

Given $$ f(x)=\sum_{n=1}^\infty n^{\frac{1}{\log_n(x)}}=\sum_{n=1}^\infty e^{\frac{\ln^2(n)}{\ln(x)}} .$$ By inspection this is a sum of nonlinear hyperbolas, over $n$. It's because $\ln(x)\ln(y)=\ln^...
4 votes
1 answer
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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 \\ ...
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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....
2 votes
3 answers
162 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?
6 votes
2 answers
251 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 ...
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53 votes
4 answers
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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 $...
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5 votes
2 answers
310 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 \...
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3 votes
1 answer
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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 ...
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3 votes
1 answer
378 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 (...
4 votes
1 answer
215 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 ...
3 votes
1 answer
92 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 ...
2 votes
1 answer
63 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 ...
3 votes
0 answers
264 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 ...
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5 votes
1 answer
491 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 ...
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0 votes
0 answers
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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 ...
2 votes
1 answer
328 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 ...
1 vote
0 answers
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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?
13 votes
2 answers
4k 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 ...
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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 ...