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An element of a finite-dimensional algebra with a unit element over the field of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048390/h0483901.png" /> (formerly known as a hypercomplex system). Historically, hypercomplex numbers arose as a generalization of complex numbers (cf. [[Complex number|Complex number]]). Operations on complex numbers correspond to geometrical transformations of the plane (translation, rotation, dilation, and combinations of such operations). In trying to construct numbers whose role with respect to three-dimensional space corresponds to the role played by complex numbers with respect to the plane, it became clear that a full analogy is not possible; this gave rise to the development of the theory of systems of hypercomplex numbers.
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A hypercomplex system of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048390/h0483903.png" /> is obtained by introducing a multiplication in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048390/h0483904.png" />-dimensional real space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048390/h0483905.png" /> which satisfies the axioms of an algebra over a field. Let 1 be the unit of a hypercomplex system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048390/h0483906.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048390/h0483907.png" /> be some basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048390/h0483908.png" />. The hypercomplex number
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048390/h0483909.png" /></td> </tr></table>
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An element of a finite-dimensional algebra with a unit element over the field of real numbers  $  \mathbf R $(
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formerly known as a hypercomplex system). Historically, hypercomplex numbers arose as a generalization of complex numbers (cf. [[Complex number|Complex number]]). Operations on complex numbers correspond to geometrical transformations of the plane (translation, rotation, dilation, and combinations of such operations). In trying to construct numbers whose role with respect to three-dimensional space corresponds to the role played by complex numbers with respect to the plane, it became clear that a full analogy is not possible; this gave rise to the development of the theory of systems of hypercomplex numbers.
  
of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048390/h04839010.png" /> is said to be the conjugate hypercomplex number of
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A hypercomplex system of rank  $  n $
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is obtained by introducing a multiplication in the  $  n $-
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dimensional real space  $  \mathbf R  ^ {n} $
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which satisfies the axioms of an algebra over a field. Let 1 be the unit of a hypercomplex system  $  U $
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and let  $  1, i _ {1} \dots i _ {n-} 1 $
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be some basis of  $  \mathbf R  ^ {n} $.
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The hypercomplex number
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048390/h04839011.png" /></td> </tr></table>
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$$
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\overline \alpha \; = a _ {0} - a _ {1} i _ {1} - \dots - a _ {n} i _ {n}  $$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048390/h04839012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048390/h04839013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048390/h04839014.png" /> is some new symbol. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048390/h04839015.png" /> may be converted into a hypercomplex system by defining addition by
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of  $  U $
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is said to be the conjugate hypercomplex number of
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048390/h04839016.png" /></td> </tr></table>
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$$
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\alpha  = a _ {0} + a _ {1} i _ {1} + \dots + a _ {n} i _ {n} .
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$$
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Let  $  U  ^ {(} 2) = \{ u _ {1} + u _ {2} e \} $,
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where  $  u _ {1} , u _ {2} \in U $
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and  $  e $
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is some new symbol. The set  $  U  ^ {(} 2) $
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may be converted into a hypercomplex system by defining addition by
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$$
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( u _ {1} + u _ {2} e) + ( v _ {1} + v _ {2} e)  = \
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( u _ {1} + v _ {1} ) + ( u _ {2} + v _ {2} ) e
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$$
  
 
and multiplication by
 
and multiplication by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048390/h04839017.png" /></td> </tr></table>
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$$
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( u _ {1} + u _ {2} e) ( v _ {1} + v _ {2} e)  = \
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( u _ {1} v _ {1} - \overline{ {v _ {2} }}\; u _ {2} ) +
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( v _ {2} u _ {1} + u _ {2} \overline{ {v _ {1} }}\; ) e.
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$$
  
The hypercomplex system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048390/h04839018.png" /> is called the doubling of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048390/h04839019.png" />.
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The hypercomplex system $  U  ^ {(} 2) $
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is called the doubling of $  U $.
  
 
Examples of hypercomplex systems are: the real numbers, the complex numbers, the quaternions, and the Cayley numbers (in this list each successive system is obtained by doubling the preceding one, cf. [[Quaternion|Quaternion]]; [[Cayley numbers|Cayley numbers]]). Other examples include [[Double and dual numbers|double and dual numbers]], and hypercomplex systems of the form
 
Examples of hypercomplex systems are: the real numbers, the complex numbers, the quaternions, and the Cayley numbers (in this list each successive system is obtained by doubling the preceding one, cf. [[Quaternion|Quaternion]]; [[Cayley numbers|Cayley numbers]]). Other examples include [[Double and dual numbers|double and dual numbers]], and hypercomplex systems of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048390/h04839020.png" /></td> </tr></table>
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$$
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= a _ {0} \cdot 1 +
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\sum _ {\gamma = 1 } ^ { {2 ^ {n} - 1 } }
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a _  \gamma  i _  \gamma  ,
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$$
  
which, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048390/h04839021.png" />, are known as Clifford–Lipschitz numbers (these hypercomplex numbers are elements of the [[Clifford algebra|Clifford algebra]] of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048390/h04839022.png" />). An important example of hypercomplex systems are complete matrix algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048390/h04839023.png" />.
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which, if $  n = 4 $,  
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are known as Clifford–Lipschitz numbers (these hypercomplex numbers are elements of the [[Clifford algebra|Clifford algebra]] of rank $  2  ^ {n} $).  
 +
An important example of hypercomplex systems are complete matrix algebras over $  \mathbf R $.
  
 
The definition of a system of hypercomplex numbers may include the requirement of associativeness of multiplication; one also identifies the concepts of an algebra and a hypercomplex system.
 
The definition of a system of hypercomplex numbers may include the requirement of associativeness of multiplication; one also identifies the concepts of an algebra and a hypercomplex system.

Revision as of 22:11, 5 June 2020


An element of a finite-dimensional algebra with a unit element over the field of real numbers $ \mathbf R $( formerly known as a hypercomplex system). Historically, hypercomplex numbers arose as a generalization of complex numbers (cf. Complex number). Operations on complex numbers correspond to geometrical transformations of the plane (translation, rotation, dilation, and combinations of such operations). In trying to construct numbers whose role with respect to three-dimensional space corresponds to the role played by complex numbers with respect to the plane, it became clear that a full analogy is not possible; this gave rise to the development of the theory of systems of hypercomplex numbers.

A hypercomplex system of rank $ n $ is obtained by introducing a multiplication in the $ n $- dimensional real space $ \mathbf R ^ {n} $ which satisfies the axioms of an algebra over a field. Let 1 be the unit of a hypercomplex system $ U $ and let $ 1, i _ {1} \dots i _ {n-} 1 $ be some basis of $ \mathbf R ^ {n} $. The hypercomplex number

$$ \overline \alpha \; = a _ {0} - a _ {1} i _ {1} - \dots - a _ {n} i _ {n} $$

of $ U $ is said to be the conjugate hypercomplex number of

$$ \alpha = a _ {0} + a _ {1} i _ {1} + \dots + a _ {n} i _ {n} . $$

Let $ U ^ {(} 2) = \{ u _ {1} + u _ {2} e \} $, where $ u _ {1} , u _ {2} \in U $ and $ e $ is some new symbol. The set $ U ^ {(} 2) $ may be converted into a hypercomplex system by defining addition by

$$ ( u _ {1} + u _ {2} e) + ( v _ {1} + v _ {2} e) = \ ( u _ {1} + v _ {1} ) + ( u _ {2} + v _ {2} ) e $$

and multiplication by

$$ ( u _ {1} + u _ {2} e) ( v _ {1} + v _ {2} e) = \ ( u _ {1} v _ {1} - \overline{ {v _ {2} }}\; u _ {2} ) + ( v _ {2} u _ {1} + u _ {2} \overline{ {v _ {1} }}\; ) e. $$

The hypercomplex system $ U ^ {(} 2) $ is called the doubling of $ U $.

Examples of hypercomplex systems are: the real numbers, the complex numbers, the quaternions, and the Cayley numbers (in this list each successive system is obtained by doubling the preceding one, cf. Quaternion; Cayley numbers). Other examples include double and dual numbers, and hypercomplex systems of the form

$$ A = a _ {0} \cdot 1 + \sum _ {\gamma = 1 } ^ { {2 ^ {n} - 1 } } a _ \gamma i _ \gamma , $$

which, if $ n = 4 $, are known as Clifford–Lipschitz numbers (these hypercomplex numbers are elements of the Clifford algebra of rank $ 2 ^ {n} $). An important example of hypercomplex systems are complete matrix algebras over $ \mathbf R $.

The definition of a system of hypercomplex numbers may include the requirement of associativeness of multiplication; one also identifies the concepts of an algebra and a hypercomplex system.

References

[1] I.L. Kantor, A.S. Solodovnikov, "Hyperkomplexe Zahlen" , Teubner (1978) (Translated from Russian)
[2] L.A. Kaluzhnin, "Introduction to general algebra" , Moscow (1973) (In Russian)
How to Cite This Entry:
Hypercomplex number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hypercomplex_number&oldid=18770
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article