What is the symbol for split-complex numbers?

Question

$\mathbb{R}$ 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?

Answer 1

The letter $D$, either plain or in blackboard bold ($\mathbb D$), denotes these numbers. The D stands for "double number", an alternative name (the split-complex numbers have been given many names).

An example of this usage is Antonuccio (1994):

An informal yet instructive way of introducing the complex number system $\mathbb C$ to a newcomer is to postulate the existence of 'numbers' having the form $$z = x + iy\tag{1.1}$$ where $x$ and $y$ are real numbers and $i$ is a commuting variable satisfying the relation $$i^2=-1.\tag{1.2}$$ If we write $i^2=+1$ instead of $i^2=-1$, what kind of number system do we end up with? To explore this possibility we will consider 'numbers' of the form $$w = t + jx \tag{1.3}$$ where $t$ and $x$ are real and $j$ is a commuting variable satisfying the relation $$j^2 = +1\tag{1.4}$$ The algebra defined by $(1.3)$ and $(1.4)$ will be denoted by the symbol $D$…

Answer 2

Another notation is $\Bbb C'$, but I'm not sure how standard it is. Although in general one can take $\alpha,\beta\in \Bbb R$ and set $$\Bbb C_{\alpha,\beta} =\{a+b\mathfrak{u}\mid a,b\in\Bbb R\mbox{ and } \mathfrak{u}^2=\alpha+\beta\mathfrak{u}\},$$so that $\Bbb C_{-1,0} =\Bbb C$, $\Bbb C_{1,0} = \Bbb C'$, etc.. The safe way to proceed when writing is to choose a notation, explain it in the beginning of your text, and use it consistently.

Answer 3

It is $\mathbb{D}$ as in here:

Image source: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.527.356&rep=rep1&type=pdf

But for many it may be confusing, for instance some may thing it means dual numbers.