Hybrid number

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A hybrid number is a generalization of complex numbers [math]\displaystyle{ \left(a+\mathbf{i}b, \mathbf{i}^{2}=-1\right) }[/math], split-complex numbers (or "hyperbolic number") [math]\displaystyle{ \left(a+\mathbf{h}b, \mathbf{h}^2=1\right) }[/math] and dual numbers [math]\displaystyle{ \left(a+\mathbf{\varepsilon} b, \mathbf{\varepsilon}^2 = 0\right) }[/math]. Hybrid numbers form a noncommutative ring. Complex, hyperbolic and dual numbers are well known two-dimensional number systems. It is well known that, the set of complex numbers, hyperbolic numbers and dual numbers are

[math]\displaystyle{ \mathbb{C}=\{\mathbf{z}=x+\mathbf{i}y:\mathbf{i}^2=-1,\,\, x,y\in\mathbb{R}\}, }[/math]
[math]\displaystyle{ \mathbb{P}=\{\mathbf{z}=x+\mathbf{h}y:\mathbf{h}^{2}=1,\,\,x,y\in\mathbb{R}\}, }[/math]
[math]\displaystyle{ \mathbb{D}=\{\mathbf{z}=x+\mathbf{\varepsilon} y : \mathbf{\varepsilon}^2=0,\,\, x,y\in\mathbb{R}\}, }[/math]

respectively. The algebra of hybrid numbers is a noncommutative algebra which unifies all three number systems calls them hybrid numbers.[1], [2], [3].

A hybrid number

[math]\displaystyle{ a+\mathbf{i}b+c\mathbf{\varepsilon} + d\mathbf{h} }[/math]

is a number created with any combination of the complex, hyperbolic and dual numbers satisfying the relation

[math]\displaystyle{ \mathbf{ih=-hi=i}+\mathbf{\varepsilon}. }[/math]

Because these numbers are a composition of dual, complex and hyperbolic numbers, Ozdemir calls them hybrid numbers [1]. A commutative two-dimensional unital algebra generated by a 2 by 2 matrix is isomorphic to either complex, dual or hyperbolic numbers [4]. Due to the set of hybrid numbers is a two-dimensional commutative algebra spanned by 1 and [math]\displaystyle{ \mathbf{i}b+c\mathbf{\varepsilon }+d\mathbf{h} }[/math], it is isomorphic to one of the complex, dual or hyperbolic numbers.

Planar rotations with complex, hyperbolic, and dual numbers

Especially in the last century, a lot of researchers deal with the geometric and physical applications of these numbers. Just as the geometry of the Euclidean plane can be described with complex numbers, the geometry of the Minkowski plane and Galilean plane can be described with hyperbolic numbers. The group of Euclidean rotations SO(2) is isomorphic to the group U(1) of unit complex numbers. The geometrical meaning of multiplying by [math]\displaystyle{ e^{\mathbf{i}\theta}=\cos \theta +\mathbf{i}\sin \theta }[/math] means a rotation of the plane. [5], [6].

The group of Lorentzian rotations [math]\displaystyle{ SO(1,1) }[/math] is isomorphic to the group of unit spacelike hyperbolic numbers. This rotation can be viewed as hyperbolic rotation. Thus, multiplying by [math]\displaystyle{ e^{\mathbf{h}\theta }=\cosh \theta +\mathbf{h} \sinh \theta }[/math] means a map of hyperbolic numbers into itself which preserves the Lorentzian metric. [7], [8], [9], [10] The Galilean rotations can be interpreted with dual numbers. The concept of a rotation in the dual number plane is equivalent to a vertical shear mapping since [math]\displaystyle{ \left( 1+x\mathbf{\varepsilon }\right) \left( 1+y\mathbf{\varepsilon }\right) =1+\left( x+y\right) \mathbf{\varepsilon } }[/math]. The Euler formula for dual numbers is [math]\displaystyle{ e^{\mathbf{\varepsilon }\theta }=1+\mathbf{\varepsilon }\theta }[/math]. Multiplying by [math]\displaystyle{ e^{\mathbf{\varepsilon \theta }} }[/math] is a map of dual numbers into itself which preserves the Galilean metric. This rotation can be named as parabolic rotation [11], [12][13][14], [15], [16], [17], [18]. File:Planar rotations.tif In abstract algebra, the complex, the hyperbolic and the dual numbers can be described as the quotient of the polynomial ring [math]\displaystyle{ \mathbb{R}[x] }[/math] by the ideal generated by the polynomials [math]\displaystyle{ x^2+1, }[/math], [math]\displaystyle{ x^2-1 }[/math] and [math]\displaystyle{ x^{2} }[/math] respectively. That is,

[math]\displaystyle{ \mathbb{C}=\mathbb{R}[x] /\left\langle x^2+1\right\rangle ,\text{ } }[/math]
[math]\displaystyle{ \mathbb{P}=\mathbb{R}[x] /\left\langle x^2-1\right\rangle ,\text{ } }[/math]
[math]\displaystyle{ \mathbb{D}=\mathbb{R}[x] /\left\langle x^2\right\rangle .\text{ } }[/math]

Matrix represantations of the units [math]\displaystyle{ \mathbf{i} }[/math], [math]\displaystyle{ \mathbf{\varepsilon } }[/math], [math]\displaystyle{ \mathbf{h} }[/math] are

[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]


Comparing complex, hyperbolic and dual numbers
Properties Complex numbers Hyperbolic numbers Dual numbers
Algebraic structure Field Commutative ring Commutative ring
Property [math]\displaystyle{ \mathbf{z}=a+\mathbf{i}b, \mathbf{i}^2=-1 }[/math] [math]\displaystyle{ \mathbf{z}=a+\mathbf{h}b, \mathbf{h}^{2}=1 }[/math] [math]\displaystyle{ \mathbf{z}=a+\mathbf{\varepsilon }b, \mathbf{\varepsilon }^{2}= 0 }[/math]
Conjugate [math]\displaystyle{ \overline{\mathbf{z}}=a-\mathbf{i}b }[/math] [math]\displaystyle{ \overline{\mathbf{z}}=a-\mathbf{h}b }[/math] [math]\displaystyle{ \overline{\mathbf{z}}=a-\mathbf{\varepsilon }b }[/math]
Norm [math]\displaystyle{ \left\vert \mathbf{z}\right\vert =\sqrt{a^{2}+b^{2}} }[/math] [math]\displaystyle{ \left\vert \mathbf{z}\right\vert =\sqrt{a^{2}-b^{2}} }[/math] [math]\displaystyle{ \left\vert\mathbf{z}\right\vert = \left\vert a\right\vert }[/math]
Geometry Euclidean geometry Lorentzian geometry Galilean geometry
Circle [math]\displaystyle{ x^2+y^2=r^2 }[/math] [math]\displaystyle{ x^2-y^2=\pm r^2 }[/math] [math]\displaystyle{ \left\vert x\right\vert=r }[/math]
Rotation type Elliptic rotation Hyperbolic rotation Parabolic rotation
Euler's Formula [math]\displaystyle{ e^{\mathbf{i}\theta }=\cos\theta+i\sin\theta }[/math] [math]\displaystyle{ e^{\mathbf{h}\theta }=\cosh\theta +\mathbf{h}\sinh\theta }[/math] [math]\displaystyle{ e^{\mathbf{\varepsilon}\theta }=1+\mathbf{\varepsilon }\theta }[/math]
Argument [math]\displaystyle{ \arg\mathbf{z}=\arctan\tfrac{b}{a} }[/math] [math]\displaystyle{ \arg\mathbf{z}=\ln\tfrac{\left\vert a+b\right\vert }{\sqrt{\left\vert a^2-b^2\right\vert }} }[/math] [math]\displaystyle{ \arg\mathbf{z}=\tfrac{b}{a} }[/math]


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.

Hybrid Multiplication
× [math]\displaystyle{ \mathbf{1} }[/math] [math]\displaystyle{ \mathbf{i} }[/math] [math]\displaystyle{ \boldsymbol{\varepsilon } }[/math] [math]\displaystyle{ \mathbf{h} }[/math]
[math]\displaystyle{ \mathbf{1} }[/math] [math]\displaystyle{ \mathbf{1} }[/math] [math]\displaystyle{ \mathbf{i} }[/math] [math]\displaystyle{ \boldsymbol{\varepsilon } }[/math] [math]\displaystyle{ \mathbf{h} }[/math]
[math]\displaystyle{ \mathbf{i} }[/math] [math]\displaystyle{ \mathbf{i} }[/math] [math]\displaystyle{ \mathbf{-1} }[/math] [math]\displaystyle{ \mathbf{1-h} }[/math] [math]\displaystyle{ \boldsymbol{\varepsilon +i} }[/math]
[math]\displaystyle{ \boldsymbol{\varepsilon } }[/math] [math]\displaystyle{ \boldsymbol{\varepsilon } }[/math] [math]\displaystyle{ \mathbf{1+h} }[/math] [math]\displaystyle{ \mathbf{0} }[/math] [math]\displaystyle{ \boldsymbol{-\varepsilon} }[/math]
[math]\displaystyle{ \mathbf{h} }[/math] [math]\displaystyle{ \mathbf{h} }[/math] [math]\displaystyle{ \boldsymbol{- \varepsilon -i} }[/math] [math]\displaystyle{ \boldsymbol{\varepsilon} }[/math] [math]\displaystyle{ \mathbf{1} }[/math]

File:Hybrid number.tif

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]

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].

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.

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]

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].

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]

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], [3]

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]

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][3][2] 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].
Classification of matrices
[math]\displaystyle{ A }[/math] [math]\displaystyle{ \det A\gt 0 }[/math] [math]\displaystyle{ \det A=0 }[/math] [math]\displaystyle{ \det A\lt 0 }[/math]
[math]\displaystyle{ (\operatorname{tr}A)^2\lt 4\det A }[/math] Timelike elliptic [math]\displaystyle{ \emptyset }[/math] [math]\displaystyle{ \emptyset }[/math]
[math]\displaystyle{ (\operatorname{tr}A)^2=4\det A }[/math] Timelike parabolic Lightlike parabolic [math]\displaystyle{ \emptyset }[/math]
[math]\displaystyle{ (\operatorname{tr}A)^2\gt 4\det A }[/math] Timelike hyperbolic Lightlike hyperbolic Spacelike hyperbolic

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]

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.

Type of hybrid number Euler formula
[math]\displaystyle{ \mathbf{Z} }[/math] is timelike hyperbolic [math]\displaystyle{ \mathbf{Z}=e^{\mathbf{V}\theta }=\cos \theta +\mathbf{V}\sin \theta }[/math]
[math]\displaystyle{ \mathbf{Z} }[/math] is spacelike hyperbolic [math]\displaystyle{ \mathbf{Z}=\mathbf{V}e^{\mathbf{V}\theta }=\sinh \theta +\mathbf{V}\cosh\theta }[/math]
[math]\displaystyle{ \mathbf{Z} }[/math] is parabolic [math]\displaystyle{ \mathbf{Z}=e^{\mathbf{V}\theta }=\varepsilon +\mathbf{V}\theta }[/math], [math]\displaystyle{ \varepsilon=\sgn\left( S\left( \mathbf{Z}\right) \right) }[/math]

See also


  1. 1.0 1.1 1.2 1.3 1.4 1.5 Ozdemir, M. (2018). "Introduction to Hybrid Numbers". Applied Clifford Algebras 28:11, 2018.. doi:10.1007/s00006-018-0833-3. 
  2. 2.0 2.1 G. Dattoli, S. Licciardi, R. M. Pidatella, E. Sabia, Hybrid Complex Numbers: The Matrix Version, Adv. in Applied Clifford Algebras, 28:58, (2018)
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