Hybrid number
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]
respectively.
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] |
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.
| × | [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] |
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].
| [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
- Associative algebra
- Complex number
- Biquaternion
- Clifford algebra
- Complex number
- Conversion between quaternions and Euler angles
- Division algebra
- Dual number
- Dual quaternion
- Euler angles
- Exterior algebra
- Geometric algebra
- Hyperbolic quaternion
- Hypercomplex number
- Octonion
- Pauli matrices
- Quaternion
- Quaternion variable
- Quaternionic matrix
- Quaternions and spatial rotation
- Rotations in 4-dimensional Euclidean space
- Split-quaternion
References
- ↑ 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.
- ↑ 2.0 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)
- ↑ 3.0 3.1 3.2 Özdemir M., Finding n-th Roots of a 2×2 Real Matrix Using De Moivre's Formula, Adv. in Applied Clifford Algebras, 29:2, (2019)
- ↑ Lavrentiev M.A., Shabat B.V., Problems of hydrodynamics and their mathematical models. Moscow, Nauka, 416 p., (Russian) (1973).
- ↑ Yaglom I.M., Complex Numbers in Geometry, Academic Press, (1968).
- ↑ Yaglom I.M., A simple non-Euclidean geometry and its physical basis. Heidelberg Science Library. Springer-Verlag, New York, (1979).
- ↑ Catoni F., Cannata R., Catoni V., Zampetti P., Two-dimensional Hypercomplex number and related trigonometries, Advances in Applied Clifford Algebras, Vol.14, Issue 1, 47–68, (2004).
- ↑ Catoni F., Boccaletti D.,Cannata R., Catoni V., Nichelatti E., and Zampetti P., The Mathematics of Minkowski Space-Time: With an Introduction to Commutative Hypercomplex Numbers, Birkhäuser, Basel, (2008).
- ↑ Catoni F., Cannata R., Catoni V., Zampetti P.: Hyperbolic trigonometry in two-dimensional space-time geometry. Nuovo Cimento B, 118(5), 475 (2003).
- ↑ Rooney J., On the three types of complex number and planar transformations, Environment and Planning B, Volume 5, pages 89–99, (1978).
- ↑ Kisil Vladimir V., Induced Representations and Hypercomplex Numbers, Advances in Applied Clifford Algebras, Vol.23, Issue 2, pp 417–440, (2013)
- ↑ Kisil Vladimir V., Erlangen program at large-2: Inventing a wheel. The parabolic one. Trans. Inst. Math. of the NAS of Ukraine, pages 89–98, (2010).
- ↑ Kisil Vladimir V., Erlangen program at large-1: Geometry of invariants. SIGMA, Symmetry Integrability Geom. Methods Appl. 6 (076):45, (2010).
- ↑ Rooney J., Generalised Complex Numbers in Mechanics, Advances on Theory and Practice of Robots and Manipulators, 55–62, (2014).
- ↑ Harkin A. A., Harkin J. B., Geometry of Generalized Complex Numbers, Mathematics Magazine 77(2):118–29 (2004)
- ↑ Fischer I., Dual-Number Methods in Kinematics, Statics and Dynamics. CRC Press, (1999).
- ↑ Borota N. A., Flores E., and Osler T.J., Spacetime numbers the easy way, Mathematics and Computer Education 34: 159–168 (2000).
- ↑ Veldkamp G.R., On the use of dual numbers, vectors and matrices in instantaneous, spatial kinematics, Mechanism and Machine Theory, Volume 11, Issue 2, pages 141–156, (1976)
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