1. Definition

The set of hybrid numbers [math]\displaystyle{ \mathbb{K} }[/math], defined as

[math]\displaystyle{ \mathbb{K}=\left\{ a+\mathbf{i}b+c\mathbf{\varepsilon }+d\mathbf{h}:a,b,c,d\in\mathbb{R},\text{ } \mathbf{i}^2=-1,\,\, \mathbf{\varepsilon }^2=0,\,\,\mathbf{h}^2=1,\,\, \mathbf{ih}=-\mathbf{hi} =\mathbf{\varepsilon} + \mathbf{i}\right\} . }[/math]

For the hybrid number [math]\displaystyle{ \mathbf{Z}=a+b\mathbf{i}+c\mathbf{\varepsilon }+d\mathbf{h} }[/math], the number [math]\displaystyle{ a }[/math] is called the scalar part and is denoted by [math]\displaystyle{ S(\mathbf{Z}) }[/math]; [math]\displaystyle{ b\mathbf{i}+c\mathbf{\varepsilon }+d\mathbf{h} }[/math] is called the vector part and is denoted by [math]\displaystyle{ V(\mathbf{Z}) }[/math] ^[1]

The conjugate of a hybrid number [math]\displaystyle{ \mathbf{Z}=a+b\mathbf{i}+c\mathbf{\varepsilon }+d\mathbf{h} }[/math], denoted by [math]\displaystyle{ \overline{\mathbf{Z}} }[/math], is defined as [math]\displaystyle{ \overline{\mathbf{Z}}=S\left( \mathbf{Z}\right) -V\left( \mathbf{Z}\right) =a-b\mathbf{i}-c\mathbf{\varepsilon }-d\mathbf{h} }[/math] as in quaternions. Multiplication operation in the hybrid numbers is associative and not commutative.

2. Character and Type of a Hybrid Number

Let [math]\displaystyle{ \mathbf{Z}=a+b\mathbf{i}+c\mathbf{\varepsilon }+d\mathbf{h} }[/math] be a hybrid number. The real number

[math]\displaystyle{ \mathcal{C}\left( \mathbf{Z}\right) =\mathbf{Z}\overline{\mathbf{Z}}=\overline{\mathbf{Z}}\mathbf{Z}=a^{2}+\left( b-c\right) ^{2}-c^{2}-d^{2} }[/math]

is called the characteristic number of $\mathbf{Z.}</math> We say that a hybrid number;

[math]\displaystyle{ \left\{ \begin{array}{ll} \mathbf{Z}\text{ is spacelike } & \text{if }\mathcal{C}\left( \mathbf{Z}\right) \lt 0; \\ \mathbf{Z}\text{ is timelike} & \text{if }\mathcal{C}\left( \mathbf{Z}\right) \gt 0; \\ \mathbf{Z}\text{ is lightlike} & \text{if }\mathcal{C}\left( \mathbf{Z}\right) =0. \end{array} \right. }[/math]

These are called ""'the characters of the hybrid numbers"'".

Let [math]\displaystyle{ \mathbf{Z}=a+b\mathbf{i}+c\mathbf{\varepsilon }+d\mathbf{h} }[/math] be a hybrid number. The real number

[math]\displaystyle{ \mathcal{\bigtriangleup }\left( \mathbf{Z}\right) =-\left( b-c\right) ^{2}+c^{2}+d^{2} }[/math]

is called the type number of [math]\displaystyle{ \mathbf{Z.} }[/math] We say that a hybrid number;

[math]\displaystyle{ \left\{ \begin{array}{ll} \mathbf{Z}\text{ is elliptic } & \text{if }\mathcal{\bigtriangleup }\left( \mathbf{Z}\right) \lt 0; \\ \mathbf{Z}\text{ is hyperbolic} & \text{if }\mathcal{\bigtriangleup }\left( \mathbf{Z}\right) \gt 0; \\ \mathbf{Z}\text{ is parabolic} & \text{if }\mathcal{\bigtriangleup }\left( \mathbf{Z}\right) =0. \end{array} \right. }[/math]

These are called the \textbf{types of the hybrid numbers}. The vector [math]\displaystyle{ \mathcal{E}_{\mathbf{Z}}=\left( b-c,c,d\right) }[/math] is called hybridian vector of [math]\displaystyle{ \mathbf{Z.} }[/math]

3. Norms of Hybrid Numbers

Let [math]\displaystyle{ \mathbf{Z}=a+b\mathbf{i}+c\mathbf{\varepsilon }+d\mathbf{h} }[/math] be a hybrid number. The real number

[math]\displaystyle{ \left\Vert \mathbf{Z}\right\Vert =\sqrt{\left\vert \mathcal{C}\left( \mathbf{Z}\right) \right\vert }=\sqrt{\left\vert a^{2}+\left( b-c\right) ^{2}-c^{2}-d^{2}\right\vert } }[/math]

is called the norm of [math]\displaystyle{ \mathbf{Z}. }[/math] Besides, the real number

[math]\displaystyle{ \mathcal{N}\left( \mathbf{Z}\right) =\sqrt{\left\vert \mathcal{\bigtriangleup }\right\vert} = \sqrt{\left\vert - (b-c)^2+c^2+d^2\right\vert } }[/math]

will be called the norm of the hybrid vector of [math]\displaystyle{ \mathbf{Z} }[/math]. This norm definition is a generalized norm definition that overlaps with the definitions of norms in complex, hyperbolic and dual numbers.

• If [math]\displaystyle{ \mathbf{Z} }[/math] is a complex number [math]\displaystyle{ (c=d=0) }[/math], then [math]\displaystyle{ \left\Vert \mathbf{Z}\right\Vert =\sqrt{\left\vert \mathbf{Z}\overline{\mathbf{Z}}\right\vert }=\sqrt{a^{2}+b^{2}} }[/math]

• If [math]\displaystyle{ \mathbf{Z} }[/math] is a hyperbolic number [math]\displaystyle{ (b=c=0) }[/math], then [math]\displaystyle{ \left\Vert \mathbf{Z}\right\Vert =\sqrt{\left\vert \mathbf{Z}\overline{\mathbf{Z}}\right\vert} = \sqrt{\left\vert a^2-d^2\right\vert }, }[/math]

• If [math]\displaystyle{ \mathbf{Z} }[/math] is a dual number [math]\displaystyle{ (b=d=0) }[/math], then [math]\displaystyle{ \left\Vert \mathbf{Z} \right\Vert=\sqrt{a^2}=\left\vert a\right\vert }[/math].

4. Inverse of a Hybrid Number

Using the hybridian product of hybrid numbers, one can show that the equality [math]\displaystyle{ \mathcal{C}\left( \mathbf{Z}_{1}\mathbf{Z}_{2}\right) =\mathcal{C} \left( \mathbf{Z}_{1}\right) \mathcal{C}\left( \mathbf{Z}_{2}\right) }[/math] holds So, timelike hybrid numbers form a group according to the multiplication operation. The inverse of a hybrid number [math]\displaystyle{ \mathbf{Z}=a+b \mathbf{i}+c\mathbf{\varepsilon }+d\mathbf{h,} }[/math][math]\displaystyle{ \left\Vert \mathbf{Z} \right\Vert \neq 0 }[/math] is defined as

[math]\displaystyle{ \mathbf{Z}^{-1}=\dfrac{\overline{\mathbf{Z}}}{\mathcal{C}\left( \mathbf{Z} \right) }. }[/math]

Accordingly, lightlike hybrid numbers have no inverse.

5. Argument of a Hybrid Number

Let [math]\displaystyle{ \mathbf{Z}=a+b\mathbf{i}+c\mathbf{\varepsilon }+d\mathbf{h} }[/math] be a hybrid number. The argument [math]\displaystyle{ \arg \mathbf{Z}=\theta }[/math] of [math]\displaystyle{ \mathbf{Z} }[/math] is defined as follows with respect to its type.

[math]\displaystyle{ \left\{ \begin{array}{ll} \pi -\arctan \dfrac{\mathcal{N}\left( \mathbf{Z}\right) }{a} & \text{if } \mathbf{Z}\text{ is elliptic and } a\lt 0; \\ \arctan \dfrac{\mathcal{N}\left( \mathbf{Z}\right) }{a} & \text{if } \mathbf{Z}\text{ is elliptic and } a\gt 0,\text{ } \\ \ln \left\vert \dfrac{a+\mathcal{N}\left( \mathbf{Z}\right) }{\rho} \right\vert & \text{if } \mathbf{Z}\text{ is nonlightlike hyperbolic;} \\ \dfrac{c}{\left\Vert \mathbf{Z}\right\Vert } & \text{if }\mathbf{Z}\text{ is parabolic.} \end{array} \right. }[/math]

6. Polar Form of a Hybrid Number

Let [math]\displaystyle{ \mathbf{Z}=a+b\mathbf{i}+c\mathbf{\varepsilon }+d\mathbf{h} }[/math] be a hybrid number, and [math]\displaystyle{ \theta =\arg \mathbf{Z.} }[/math]

i. If [math]\displaystyle{ \mathbf{Z} }[/math] is elliptic, then [math]\displaystyle{ \mathbf{Z=}\rho \left( \cos \theta +\mathbf{U}\sin \theta \right) }[/math] such that [math]\displaystyle{ \mathbf{U}^2=-1; }[/math]

ii. If [math]\displaystyle{ \mathbf{Z} }[/math] a lightlike hyperbolic, then [math]\displaystyle{ \mathbf{Z}=a\left( 1+\mathbf{U}\right) }[/math] such that [math]\displaystyle{ \mathbf{U}^2=1; }[/math]

iii. If [math]\displaystyle{ \mathbf{Z} }[/math] is spacelike or timelike hyperbolic, then, [math]\displaystyle{ \mathbf{Z}=k\rho \left( \cosh \theta +\mathbf{U}\sinh \theta \right) }[/math] such that [math]\displaystyle{ \mathbf{U}^{2}=1, }[/math] where [math]\displaystyle{ \rho =\left\Vert \mathbf{Z}\right\Vert , }[/math] [math]\displaystyle{ \mathbf{U=}\tfrac{b\mathbf{i}+c\mathbf{\varepsilon }+d\mathbf{h}}{\mathcal{N}\left(\mathbf{Z}\right) } }[/math] and

[math]\displaystyle{ k=\left\{ \begin{array}{ll} 1 & \mathbf{Z}\text{ is timelike and }a\gt 0, \\ -1 & \mathbf{Z}\text{ is timelike and }a\lt 0, \\ \mathbf{U} & \mathbf{Z}\text{ is spacelike and }a\gt 0, \\ -\mathbf{U} & \mathbf{Z}\text{ is spacelike and }a\lt 0, \end{array} \right. }[/math]

for [math]\displaystyle{ k\in \left\{ -1,1,\mathbf{U},-\mathbf{U}\right\} }[/math]

iv. If [math]\displaystyle{ \mathbf{Z} }[/math] is a parabolic hybrid number, then [math]\displaystyle{ \mathbf{Z=}\left\Vert \mathbf{Z}\right\Vert \left( \varepsilon +\mathbf{U}\right) }[/math] where [math]\displaystyle{ \mathbf{U}=\tfrac{V\left( \mathbf{Z}\right) }{\rho }, }[/math] [math]\displaystyle{ \mathbf{U}^{2}=0, }[/math] [math]\displaystyle{ \varepsilon =\sgn\left( S\left( \mathbf{Z}\right) \right) }[/math].

7. De Moivre's Formulas for Hybrid Numbers

De Moivre's formula for hybrid numbers as follows..^[1]. Let [math]\displaystyle{ \mathbf{Z}=a+\mathbf{U}b, }[/math] [math]\displaystyle{ \mathbf{U}^2 \in \left\{ \pm 1,0\right\} }[/math] be a spacelike or timelike hybrid number. If [math]\displaystyle{ \theta =\arg \mathbf{Z} }[/math] and [math]\displaystyle{ \rho =\left\Vert \mathbf{Z}\right\Vert . }[/math]

i. If [math]\displaystyle{ \mathbf{Z} }[/math] is elliptic, then [math]\displaystyle{ \mathbf{Z}^{n}=\rho^n \left( \cos n \theta + \mathbf{U} \sin n\theta \right) , }[/math] [math]\displaystyle{ \mathbf{U}^2=-1; }[/math]

ii. If [math]\displaystyle{ \mathbf{Z} }[/math] is hyperbolic, then [math]\displaystyle{ \mathbf{Z}^{n}=k^n\rho^n\left( \cosh n\theta +\mathbf{U}\sinh n\theta \right) , }[/math] [math]\displaystyle{ \mathbf{U}^2=1; }[/math]

iii. If [math]\displaystyle{ \mathbf{Z} }[/math] is parabolic, then [math]\displaystyle{ \mathbf{Z}^n=\rho^n\left( \varepsilon^n + n\varepsilon^{n-1}\mathbf{U}\right), }[/math] [math]\displaystyle{ \mathbf{U}^{2}=0. }[/math]

If [math]\displaystyle{ \mathbf{Z}=a\left( 1+\mathbf{U}\right) }[/math] is a lightlike hybrid number, then [math]\displaystyle{ \mathbf{Z}^{n}=a^{n}2^{n-1}\left( 1+\mathbf{U} \right) }[/math] where [math]\displaystyle{ \mathbf{U}=\tfrac{V\left( \mathbf{Z}\right) }{\mathcal{N} \left( \mathbf{Z}\right) } }[/math] and [math]\displaystyle{ \mathbf{U}^{2}=1. }[/math]

8. Roots of a Hybrid Number

Let [math]\displaystyle{ \mathbf{W} }[/math] be a hybrid number and [math]\displaystyle{ n\in\mathbb{Z}^{+}. }[/math] The hybrid numbers [math]\displaystyle{ \mathbf{Z} }[/math] satisfying the equation [math]\displaystyle{ \mathbf{Z}^n=\mathbf{W} }[/math] can be found as follows ^[1], ^[2]

i. If [math]\displaystyle{ \mathbf{W}=\rho \left( \cos \theta +\mathbf{U}\sin \theta \right) }[/math] is an elliptic hybrid number, then the roots of [math]\displaystyle{ \mathbf{W} }[/math] are in the form

[math]\displaystyle{ \mathbf{Z}_{m}=\sqrt[n]{\rho }\left( \cos \dfrac{\theta +2m\pi }{n}+\mathbf{U}\sin \dfrac{\theta +2m\pi }{n}\right) }[/math]

for [math]\displaystyle{ m=0,1,2,\ldots,n-1; }[/math]

ii. If [math]\displaystyle{ \mathbf{W=}\rho k\left( \cosh \theta +\mathbf{U}\sinh \theta \right) }[/math] is a spacelike or timelike hyperbolic hybrid number, then the roots of [math]\displaystyle{ \mathbf{W} }[/math] are in the form

[math]\displaystyle{ \left\{ \begin{array}{ll} \sqrt[n]{\rho }\left( \cosh \dfrac{\theta }{n}+\mathbf{U}\sinh \dfrac{\theta}{n}\right) & \text{if }n\text{ is odd,} \\ k\sqrt[n]{\rho }\left( \cosh \dfrac{\theta }{n}+\mathbf{U}\sinh \dfrac{\theta }{n}\right) & \text{if } n \text{ is even, }\mathbf{W}\text{ is timelike and }a\gt 0, \\ \text{no roots} & \text{other cases} \end{array} \right. }[/math]

where [math]\displaystyle{ k\in \left\{ 1,-1,\mathbf{U},-\mathbf{U}\right\} }[/math];

iii. If [math]\displaystyle{ \mathbf{W}=\rho \left( \varepsilon +\mathbf{U}\right) }[/math], [math]\displaystyle{ \varepsilon =\sgn\left( S\left( \mathbf{Z}\right) \right) }[/math] is a parabolic hybrid number, the only root is

[math]\displaystyle{ \mathbf{Z}=\rho \left( 1+\dfrac{\mathbf{U}}{n}\right) }[/math]

where [math]\displaystyle{ \rho =\left\Vert \mathbf{Z}\right\Vert . }[/math]

If [math]\displaystyle{ \mathbf{W}=a\left( 1+\mathbf{U}\right) }[/math] is a lightlike hybrid number, then

[math]\displaystyle{ \mathbf{Z}=\left\{ \begin{array}{cc} \dfrac{\pm \sqrt[n]{2a}}{2}\left( 1+\mathbf{U}\right) & \text{if }n \text{ is even} \\ \dfrac{\sqrt[n]{2a}}{2}\left( 1+\mathbf{U}\right) & \text{if }n\text{ is odd} \end{array} \right. }[/math]

for [math]\displaystyle{ n\in\mathbb{Z}^{+} }[/math] where [math]\displaystyle{ \mathbf{U}=\tfrac{V\left( \mathbf{Z}\right) }{\mathcal{N} \left( \mathbf{Z}\right) } }[/math] and [math]\displaystyle{ \mathbf{U}^{2}=1. }[/math]

9. The Matrix Representation of Hybrid Numbers

Just as complex numbers and quaternions can be represented as matrices, so can hybrid numbers. There are at least two ways of representing hybrid numbers as real matrices in such a way that hybrid addition and multiplication correspond to matrix addition and matrix multiplication. The hybrid number ring [math]\displaystyle{ \mathbb{K} }[/math] is isomorphic to [math]\displaystyle{ 2\times 2 }[/math] matrix rings [math]\displaystyle{ \mathbb{M}_{2\times 2} }[/math]. So, each hybrid number can be represented by a 2 by 2 real matrix. Thus, it can be done operations and calculations in the hybrid numbers using the corresponding matrices.^[1][2][3] The map [math]\displaystyle{ \varphi :\mathbb{K\rightarrow M}_{2\times 2} }[/math] is a ring isomorphism where

[math]\displaystyle{ \varphi \left( a+b\mathbf{i}+c\mathbf{\varepsilon }+d\mathbf{h}\right) = \left[ \begin{array}{cc} a+c & b-c+d \\ c-b+d & a-c \end{array} \right] }[/math]

for [math]\displaystyle{ \mathbf{Z}=a+b\mathbf{i}+c\mathbf{\varepsilon }+d\mathbf{h}\in \mathbb{K} }[/math]. Also, the real matrix

[math]\displaystyle{ A=\left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] , }[/math]

corresponds to the hybrid number

[math]\displaystyle{ \mathbf{Z=}\left( \dfrac{a+d}{2}\right) +\left( \dfrac{a+b-c-d}{2}\right) \mathbf{i}+\left( \dfrac{a-d}{2}\right) \mathbf{\varepsilon }+\left( \dfrac{b+c}{2}\right) \mathbf{h} }[/math]

According to this ring isomorphism, matrix represantations of the units 1, [math]\displaystyle{ \mathbf{i} }[/math], [math]\displaystyle{ \mathbf{\varepsilon } }[/math], [math]\displaystyle{ \mathbf{h} }[/math] are as follows :
[math]\displaystyle{ \mathbf{1\leftrightarrow }\left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right] , }[/math] [math]\displaystyle{ \mathbf{i\leftrightarrow }\left[ \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right] , }[/math] [math]\displaystyle{ \mathbf{\varepsilon \leftrightarrow }\left[ \begin{array}{cc} 1 & -1 \\ 1 & -1 \end{array} \right] , }[/math] [math]\displaystyle{ \mathbf{h\leftrightarrow }\left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right] }[/math]

Let [math]\displaystyle{ A }[/math] be a 2 by 2 real matrix corresponding to the hybrid number [math]\displaystyle{ \mathbf{Z,} }[/math] then there are the following equalities.

• [math]\displaystyle{ \left\Vert \mathbf{Z}\right\Vert =\sqrt{\left\vert \det A\right\vert }, }[/math]

• [math]\displaystyle{ \mathcal{\bigtriangleup }\left( \mathbf{Z}\right) =\left( \tfrac{\operatorname{tr}A}{2}\right)^2-\det A, }[/math]

• [math]\displaystyle{ \bigtriangleup_A=(\operatorname{tr}A)^2-4\det A=4\mathcal{\bigtriangleup }\left( \mathbf{Z}\right) }[/math] is discriminant of the characteristic polynomial of [math]\displaystyle{ A }[/math]

• [math]\displaystyle{ \mathbf{Z}^{-1} }[/math] exists if and only if [math]\displaystyle{ \det(A) \neq 0 }[/math].

10. The Logarithm of a Hybrid Number

Logarithm function for elliptic and hyperbolic hybrid numbers can be defined as

[math]\displaystyle{ \ln \mathbf{Z}=\ln \left\vert \mathbf{Z}\right\vert +\mathbf{V}\theta. }[/math]

And, the logarithm of parabolic hybrid numbers is not defined. The identity [math]\displaystyle{ \log \left( \mathbf{Z}_{1}\mathbf{Z}_{2}\right) =\log \mathbf{Z}_{1}+\log\mathbf{Z}_{2} }[/math] which is well known for the real numbers, is not correct for the hybrid numbers, since [math]\displaystyle{ \mathbf{Z}_{1}\mathbf{Z}_{2}\neq \mathbf{Z}_{2}\mathbf{Z}_{1}. }[/math]

11. Euler's Formulas for the Hybrid Numbers

Using the serial expansions of exponential, hyperbolic and trigonometric functions, we can express the Euler formulas of unit hybrid numbers as follows.