# On Hyperbolic Complex Numbers
## 2 The Basic Hyperbolic Complex Algebraic Structure
### Complex Analysis  
A classical way to introduce complex numbers is to say that they allow the representation \(x+\mathrm{i}y\) where \(x\) and \(y\) are real numbers and \(\mathrm{i}\) is a formal quantity coming from somewhere other than \(\mathbb{R}\) and satisfying \(\mathrm{i}^{2}=-1\). The reader of such a definition is then tacitly asked to believe that the quantity \(\mathrm{i}\) exists in the sense that it can be assigned a strict mathematical meaning and that the operation of squaring this \"imaginary unit\" agrees or harmonizes with the operation usual in the realm of real numbers. The reader is also left alone with thinking about the possible uniqueness of such a quantity. The answer to this challenge can vary. One can try to take \(\mathrm{i}\) for a vector or a matrix and in each case define a corresponding multiplication rule. However, the usual squaring of vectors and matrixes does not result in the number \(-1\). Therefore people in usual complex number theory sometimes say that they \"identify\" the vector \((0,1)^{2}=(-1,0)\) with the real number \(-1\), without saying, however, what an \"identification\" is other than an equation. However, obviously \((-1,0)\) does not equal \(-1\). In [26], it is alternatively discussed that a complex number has a dual real value nature, i.e., is capable of two values without being able, however, to fully explain it. We recall that Definition 3 of the present work gives a well motivated explicit definition of a two-valued random hyperbolic vector product. In [22; 23; 24; 25], algebraic structures are introduced, within which the \"imaginary unit\" is always explicitly given as a well-defined element from the vector space underlying this structure. Here, we follow this approach as closely as possible and therefore necessarily extend the event space. Just as the imaginary unit \(\mathrm{i}\) does not belong to the real line \(\mathbb{R}\) if one describes complex solutions of quadratic equations over \(\mathbb{R}\), see Section 2.4 and [24], the vector  
\[\mathrm{j}=\left( \begin{array}{c}0\\ 1\end{array} \right)\]  
obviously does not belong to the originally considered event space \(\mathfrak{X}\) but belongs to the set \(\mathfrak{Y}\). Note that  
\[\left( \begin{array}{c}x\\ y\end{array} \right)\odot_{h\mathfrak{X}}\mathrm{j}=\left( \begin{array}{c}y \\ x\end{array} \right)\]  
and  
\[\mathrm{j}\odot_{h\mathfrak{X}}\mathrm{j}=\mathfrak{e}. \tag{9}\]  
**Remark 2**. : _If one follows usual hyperbolic number notation and writes_  
\[\mathrm{j}^{2}=1\]  
_instead of (9), then one is tacitly using the multiplication rule, which is well motivated for elements of \(\mathfrak{X}\) for the element \(\mathfrak{j}\) of \(\mathfrak{Y}\). This motivates our definition of the hyperbolic exponential in Definition 2._  
The hyperbolic vector exponential of \(x\bullet\mathrm{j}\) allows a representation that is reminiscent of [PERSON]'s well-known trigonometric representation of usual complex numbers, where, however, the so called imaginary unit \(\mathrm{i}=\sqrt{-1}\) is replaced with vector \(\mathrm{j}\) and the trigonometric functions sine and cosine are replaced with hyperbolic functions. **Theorem 3**. : _The following hyperbolic Euler-type formula holds true:_  
\[\mathfrak{e}_{[\mathrm{i}]}(x\bullet\mathrm{j})=\left( \begin{array}{c} \cosh x\\ \sinh x\end{array} \right)=\left(\cosh x\right)\bullet\mathrm{c}+\left(\sinh x \right)\bullet\mathrm{j},\,x\in\mathbb{R}. \tag{10}\]Proof. : The rearrangement of the infinite vector sum  
\[\mathfrak{e}+x\bullet\mathfrak{j}+\frac{x^{2}}{2!}\bullet\mathfrak{e}+\frac{x^{3} }{3!}\bullet\mathfrak{j}+\ldots=(1+\frac{x^{2}}{2!}+\frac{x^{4}}{4!}+\ldots) \bullet\mathfrak{e}+(x+\frac{x^{3}}{3!}+\frac{x^{4}}{4!}\ldots)\bullet \mathfrak{j}\]  
is allowed due to its convergence properties. Finally, the definitions of the hyperbolic functions apply.