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## Functions of a hypercomplex variable

### Hypercomplex components

#### Absolute value

For an arbitrary element in an arbitrary hypercomplex number system , , the modulus of , where is an involution called conjugation. The properties of conjugation dictate the properties of the modulus.

is an absolute value if it is a real-valued function that satisfies

• Non-negativity ()
• Positive-definiteness ()
• Multiplicativity ()
• and Subadditivity ()

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#### Distance function

• Complex Euclidean metric ([ complex map] / [ 3D plot])
• Complex Euclidean metric squared ([ complex map] / [ 3D plot])
• Dual spacetime interval ([ dual map] / [ 3D plot])
• Split-complex spacetime interval ([ split-complex map] / [ 3D plot])

#### Commutator

• Quaternionic commutator (vectors / [ sliders] / [ 3D rotations])
• Octonionic commutator ([ vectors] / [ sliders] / [ 3D rotations])

#### Anticommutator

• Quaternionic anticommutator (vectors / [ sliders] / [ 3D rotations])
• Octonionic anticommutator ([ vectors] / [ sliders] / [ 3D rotations])

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