# 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 ...
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
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### 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
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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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1 vote
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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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### 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 ...
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 ...
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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....
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### 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?
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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### 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 ... 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 (...
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 ...
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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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### 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 ...
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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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### 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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### 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 ...
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### 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
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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?
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 ...