# On Hyperbolic Complex Numbers
## 2 The Basic Hyperbolic Complex Algebraic Structure  
### Vector Algebra  
Let  
\[\mathfrak{X}=\{(x,y)^{T}:x\in\mathbb{R},y\in\mathbb{R},x>0,|y|<x\}\]  
denote an event space, that is the set of possible outcomes of an experiment or the light cone in the physical interpretation or just a set of possible observations of interest on which multiplication with positive real numbers is defined to be the function \(\bullet:\mathbb{R}^{+}\times\mathfrak{X}\to\mathfrak{X}\) where \(\lambda\bullet(x,y)^{T}=(\lambda x,\lambda y)^{T},\lambda>0,(x,y)\in\mathfrak{X}\). **Definition 1**. : _The vector addition \(z_{1}\oplus z_{2}:\mathfrak{X}\times\mathfrak{X}\to\mathfrak{X}\) is defined componentwise as_  
\[\left( \begin{array}{c}x_{1}\\ y_{1}\end{array} \right)\oplus\left( \begin{array}{c}x_{2}\\ y_{2}\end{array} \right)=\left( \begin{array}{c}x_{1}+x_{2}\\ y_{1}+y_{2}\end{array} \right).\]  
_If there exists \(z_{3}\in\mathfrak{X}\) such that \(z_{1}\oplus z_{3}=z_{2}\), then we say that \(z_{1}\) can be subtracted from \(z_{2}\) or \(z_{2}\) is additively reachable from \(z_{1}\) or event \(z_{1}\) influences event \(z_{2}\). In this case, we write_  
\[z_{2}\ominus z_{1}=\left( \begin{array}{c}x_{2}-x_{1}\\ y_{2}-y_{1}\end{array} \right)=\left( \begin{array}{c}x_{3}\\ y_{3}\end{array} \right)=z_{3}\]  
_and call the operation \(\ominus:\mathfrak{X}\times\mathfrak{X}\to\mathfrak{X}\) the operation of constrained vector subtraction._  
Let  
\[\mathfrak{Y}=\{(x,y)^{T}:x\in\mathbb{R},y\in\mathbb{R},y>0,|x|<y\}\]  
and assume that vector addition and constrained vector subtraction are defined analogously on \(\mathfrak{Y}\times\mathfrak{Y}\). **Definition 2**. : _The analytical hyperbolic vector product of \(z_{1}=(x_{1},y_{1})^{T}\) and \(z_{2}=(x_{2},y_{2})^{T}\) is defined as_  
\[z_{1}\odot_{h}z_{2}=\left( \begin{array}{c}z_{1}\odot_{h\mathfrak{X}}z_{2} \text{ if }z_{1}\text{ and }z_{2}\text{ are from }\mathfrak{X}\\ \text{and}\\ z_{1}\odot_{h\mathfrak{Y}}z_{2}\text{ if }z_{1}\text{ and }z_{2}\text{ are from }\mathfrak{Y}\end{array} \right) \tag{5}\]  
_where_  
\[z_{1}\odot_{h\mathfrak{X}}z_{2}=\left( \begin{array}{c}x_{1}x_{2}+y_{1}y_{2} \\ x_{1}y_{2}+y_{1}x_{2}\end{array} \right),\text{ and }z_{1}\odot_{h\mathfrak{Y}}z_{2}= \left( \begin{array}{c}x_{1}y_{2}+y_{1}x_{2}\\ x_{1}x_{2}+y_{1}y_{2}\end{array} \right).\]  
_The k'th hyperbolic vector power of \(z=(x,y)^{T}\in\mathfrak{X}\cup\mathfrak{Y}\) and the hyperbolic vector exponential of \(z\) are defined by_  
\[z^{k}=z^{k-1}\odot_{h\mathfrak{X}}z,k\geq 2,z^{1}=z,z^{0}=(1,0)^{T}\]  
_and_  
\[e_{[h]}(z)=\sum_{k=0}^{\infty}\frac{z^{k}}{k!},\]  
_respectively._  
We emphasize that we exclusively use the product \(\odot_{h\mathfrak{X}}\) for defining the hyperbolic exponential of \(z\). For our motivation for doing so, we refer to Remark 2 below. Vector product and exponential can also be understood as arrow product and exponential in the sense of [1]. **Definition 3**. : _The analytical hyperbolic vector product \(z_{1}\odot_{h}z_{2}\) of \(z_{1}=(x_{1},y_{1})^{T}\in\mathfrak{X}\) and \(z_{2}=(x_{2},y_{2})^{T}\in\mathfrak{Y}\) is defined to be a random variable on an abstract probability space that takes the values \(z_{1}\odot_{h\mathfrak{X}}z_{2}\) and \(z_{1}\odot_{h\mathfrak{Y}}z_{2}\) accordingly with probabilities \(p,p\in(0,1)\) and \(1-p\)._  
Notice that  
\[z_{1}\odot_{h\mathfrak{X}}z_{2}\in\mathfrak{X}\text{ if }z_{1} \text{ and }z_{2}\text{ are from }\ X\] \[\text{and}\] \[z_{1}\odot_{h\mathfrak{Y}}z_{2}\in\mathfrak{Y}\text{ if }z_{1} \text{ and }z_{2}\text{ are from }\mathfrak{Y}.\]  
If you look at Definitions 2 and 3 together, the hyperbolic vector product is a function  
\[\odot_{h}:(\mathfrak{X}\times\mathfrak{X})\cup(\mathfrak{Y})\times\mathfrak{Y })\cup(\mathfrak{X}\times\mathfrak{Y})\rightarrow\mathfrak{X}\cup\mathfrak{Y}.\]  
Because the focus of our consideration is \(\mathfrak{X}\), we do not consider \(\odot_{h}\) on combinations of \(-\mathfrak{X}\) and \(-\mathfrak{Y}\). **Example 1**.