Hybrid number

Hybrid multiplication

×	$\mathbf{𝟏}$	$\mathbf{𝐢}$	$\mathit{\epsilon }$	$\mathbf{𝐡}$
$\mathbf{𝟏}$	$\mathbf{𝟏}$	$\mathbf{𝐢}$	$\mathit{\epsilon }$	$\mathbf{𝐡}$
$\mathbf{𝐢}$	$\mathbf{𝐢}$	$\mathbf{-}\mathbf{𝟏}$	$\mathbf{𝟏}\mathbf{-}\mathbf{𝐡}$	$\mathit{\epsilon }\mathit{+}\mathit{i}$
$\mathit{\epsilon }$	$\mathit{\epsilon }$	$\mathbf{𝟏}\mathbf{+}\mathbf{𝐡}$	$\mathbf{𝟎}$	$\mathit{-}\mathit{\epsilon }$
$\mathbf{𝐡}$	$\mathbf{𝐡}$	$\mathit{-}\mathit{\epsilon }\mathit{-}\mathit{i}$	$\mathit{\epsilon }$	$\mathbf{𝟏}$

A hybrid number is a generalization of complex numbers $(a+\mathbf{𝐢}b,{\mathbf{𝐢}}^2=-1)$, split-complex numbers (or "hyperbolic number") $(a+\mathbf{𝐡}b,{\mathbf{𝐡}}^2=1)$ and dual numbers $(a+\mathit{\epsilon }b,{\mathit{\epsilon }}^2=0)$. 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

$\mathbb{ℂ}=\{\mathbf{𝐳}=x+\mathbf{𝐢}y:{\mathbf{𝐢}}^2=-1,x,y\in \mathbb{ℝ}\},$

$\mathbb{\mathbb{P}}=\{\mathbf{𝐳}=x+\mathbf{𝐡}y:{\mathbf{𝐡}}^2=1,x,y\in \mathbb{ℝ}\},$

$\mathbb{𝔻}=\{\mathbf{𝐳}=x+\mathit{\epsilon }y:{\mathit{\epsilon }}^2=0,x,y\in \mathbb{ℝ}\},$

respectively. The algebra of hybrid numbers is a noncommutative algebra which unifies all three number systems, and calls them hybrid numbers., , .

A hybrid number

$a+\mathbf{𝐢}b+c\mathit{\epsilon }+d\mathbf{𝐡}$

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

$\mathbf{𝐢}\mathbf{𝐡}\mathbf{=}\mathbf{-}\mathbf{𝐡}\mathbf{𝐢}\mathbf{=}\mathbf{𝐢}+\mathit{\epsilon }.$

A commutative two-dimensional unital algebra generated by a 2 by 2 matrix is isomorphic to either complex, dual or hyperbolic numbers . Due to the set of hybrid numbers being a two-dimensional commutative algebra spanned by 1 and $\mathbf{𝐢}b+c\mathit{\epsilon }+d\mathbf{𝐡}$, it is isomorphic to one of the complex, dual or hyperbolic numbers.

Planar rotations with complex, hyperbolic, and dual 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 Lorentzian rotations $SO(1,1)$ is isomorphic to the group of unit spacelike hyperbolic numbers. This rotation can be viewed as hyperbolic rotation. , The Galilean rotations can be interpreted with dual numbers. This rotation can be named as parabolic rotation , , .

In abstract algebra, the complex, the hyperbolic and the dual numbers can be described as the quotient of the polynomial ring $\mathbb{ℝ}[x]$ by the ideal generated by the polynomials $x^2+1,$, $x^2-1$ and $x^2$ respectively.

Comparing complex, hyperbolic and dual numbers

Properties	Complex numbers	Hyperbolic numbers	Dual numbers
Algebraic structure	Field	Commutative ring	Commutative ring
Property	$\mathbf{𝐳}=a+\mathbf{𝐢}b,{\mathbf{𝐢}}^2=-1$	$\mathbf{𝐳}=a+\mathbf{𝐡}b,{\mathbf{𝐡}}^2=1$	$\mathbf{𝐳}=a+\mathit{\epsilon }b,{\mathit{\epsilon }}^2=0$
Conjugate	$\overline{\mathbf{𝐳}}=a-\mathbf{𝐢}b$	$\overline{\mathbf{𝐳}}=a-\mathbf{𝐡}b$	$\overline{\mathbf{𝐳}}=a-\mathit{\epsilon }b$
Norm	$\left|\mathbf{𝐳}\right|=\sqrt{a^2+b^2}$	$\left|\mathbf{𝐳}\right|=\sqrt{a^2-b^2}$	$\left|\mathbf{𝐳}\right|=\left|a\right|$
Geometry	Euclidean geometry	Lorentzian geometry	Galilean geometry
Circle	$x^2+y^2=r^2$	$x^2-y^2=\pm r^2$	$\left|x\right|=r$
Rotation type	Elliptic rotation	Hyperbolic rotation	Parabolic rotation
Euler's Formula	$e^{\mathbf{𝐢}\theta }=\cos \theta +i\sin \theta$	$e^{\mathbf{𝐡}\theta }=\cosh \theta +\mathbf{𝐡}\sinh \theta$	$e^{\mathit{\epsilon }\theta }=1+\mathit{\epsilon }\theta$
Argument	$\arg \mathbf{𝐳}=\arctan \frac{b}{a}$	$\arg \mathbf{𝐳}=\ln \frac{|a+b|}{\sqrt{|a^2-b^2|}}$	$\arg \mathbf{𝐳}=\frac{b}{a}$

Definition

The set of hybrid numbers $\mathbb{𝕂}$, defined as

$\mathbb{𝕂}=\{a+\mathbf{𝐢}b+c\mathit{\epsilon }+d\mathbf{𝐡}:a,b,c,d\in \mathbb{ℝ},{\mathbf{𝐢}}^2=-1,{\mathit{\epsilon }}^2=0,{\mathbf{𝐡}}^2=1,\mathbf{𝐢}\mathbf{𝐡}=-\mathbf{𝐡}\mathbf{𝐢}=\mathit{\epsilon }+\mathbf{𝐢}\}.$

For the hybrid number $\mathbf{𝐪}=a+b\mathbf{𝐢}+c\mathit{\epsilon }+d\mathbf{𝐡}$, the number $a$ is called the scalar part and is denoted by $S(\mathbf{𝐪})$; $b\mathbf{𝐢}+c\mathit{\epsilon }+d\mathbf{𝐡}$ is called the vector part and is denoted by $V(\mathbf{𝐪})$

The conjugate of a hybrid number $\mathbf{𝐪}=a+b\mathbf{𝐢}+c\mathit{\epsilon }+d\mathbf{𝐡}$, denoted by $\overline{\mathbf{𝐪}}$, is defined as $\overline{\mathbf{𝐪}}=S\left(\mathbf{𝐪}\right)-V\left(\mathbf{𝐪}\right)=a-b\mathbf{𝐢}-c\mathit{\epsilon }-d\mathbf{𝐡}$ as in quaternions. Multiplication operation in the hybrid numbers is associative and not commutative.

Hybrid Multiplication

×	$\mathbf{𝟏}$	$\mathbf{𝐢}$	$\mathit{\epsilon }$	$\mathbf{𝐡}$
$\mathbf{𝟏}$	$\mathbf{𝟏}$	$\mathbf{𝐢}$	$\mathit{\epsilon }$	$\mathbf{𝐡}$
$\mathbf{𝐢}$	$\mathbf{𝐢}$	$\mathbf{-}\mathbf{𝟏}$	$\mathbf{𝟏}\mathbf{-}\mathbf{𝐡}$	$\mathit{\epsilon }\mathit{+}\mathit{i}$
$\mathit{\epsilon }$	$\mathit{\epsilon }$	$\mathbf{𝟏}\mathbf{+}\mathbf{𝐡}$	$\mathbf{𝟎}$	$\mathit{-}\mathit{\epsilon }$
$\mathbf{𝐡}$	$\mathbf{𝐡}$	$\mathit{-}\mathit{\epsilon }\mathit{-}\mathit{i}$	$\mathit{\epsilon }$	$\mathbf{𝟏}$

Norm and type of a hybrid number

The real number

${△}\left(\mathbf{𝐪}\right)=-{(b-c)}^2+c^2+d^2$

is called the type number of $\mathbf{𝐪}\mathbf{.}$ We say that a hybrid number:

$\{\begin{array}{ll}\mathbf{𝐪}\text{ is elliptic }& \text{if }{△}\left(\mathbf{𝐪}\right)<0;\\ \mathbf{𝐪}\text{ is hyperbolic}& \text{if }{△}\left(\mathbf{𝐪}\right)>0;\\ \mathbf{𝐪}\text{ is parabolic}& \text{if }{△}\left(\mathbf{𝐪}\right)=0.\end{array}$

The real number

$\Vert \mathbf{𝐪}\Vert =\mathbf{𝐪}\overline{\mathbf{𝐪}}=\overline{\mathbf{𝐪}}\mathbf{𝐪}=\sqrt{|a^2+{(b-c)}^2-c^2-d^2|}$

is called the norm of $\mathbf{𝐪}.$ This norm definition is a generalized norm definition that overlaps with the definitions of norms in complex, hyperbolic and dual numbers.

• If $\mathbf{𝐪}$ is a complex number $(c=d=0)$, then $\Vert \mathbf{𝐪}\Vert =\sqrt{\left|\mathbf{𝐪}\overline{\mathbf{𝐪}}\right|}=\sqrt{a^2+b^2}$

• If $\mathbf{𝐪}$ is a hyperbolic number $(b=c=0)$, then $\Vert \mathbf{𝐪}\Vert =\sqrt{\left|\mathbf{𝐪}\overline{\mathbf{𝐪}}\right|}=\sqrt{|a^2-d^2|},$

• If $\mathbf{𝐪}$ is a dual number $(b=d=0)$, then $\Vert \mathbf{𝐪}\Vert =\sqrt{a^2}=\left|a\right|$.

The matrix representation of hybrid numbers

Just as split quaternions can be represented as 2x2 real 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 $\mathbb{𝕂}$ is isomorphic to $2\times 2$ matrix rings ${\mathbb{𝕄}}_{2\times 2}$. So, each hybrid number can be represented by a 2 by 2 real matrix. Thus, one can do operations and calculations in the hybrid numbers using the corresponding matrices. The map $\phi :{\mathbb{𝕂}\mathbb{\to }\mathbb{𝕄}}_{2\times 2}$ is a ring isomorphism where

$\phi (a+b\mathbf{𝐢}+c\mathit{\epsilon }+d\mathbf{𝐡})=\left[\begin{array}{cc}a+c& b-c+d\\ c-b+d& a-c\end{array}\right]$

for $\mathbf{𝐪}=a+b\mathbf{𝐢}+c\mathit{\epsilon }+d\mathbf{𝐡}\in \mathbb{𝕂}$. Also, the real matrix

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

corresponds to the hybrid number

$\mathbf{𝐪}\mathbf{=}\left(\frac{a+d}{2}\right)+\left(\frac{a+b-c-d}{2}\right)\mathbf{𝐢}+\left(\frac{a-d}{2}\right)\mathit{\epsilon }+\left(\frac{b+c}{2}\right)\mathbf{𝐡}$

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

Let $A$ be a 2 by 2 real matrix corresponding to the hybrid number $\mathbf{𝐪},$ then there are the following equalities.

• $\Vert \mathbf{𝐪}\Vert =\sqrt{|\det A|},$

• ${△}\left(\mathbf{𝐪}\right)={\left(\frac{\mathrm{tr}A}{2}\right)}^2-\det A,$

• $\underset{A}{△}=(\mathrm{tr}A{)}^2-4\det A=4{△}\left(\mathbf{𝐪}\right)$ is discriminant of the characteristic polynomial of $A$

• ${\mathbf{𝐪}}^{-1}$ exists if and only if $\det (A)\ne 0$.

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

This article "Hybrid number" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Hybrid number. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

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