Biquaternion

the Biquaternionen are a hypercomplex number system,of William the Kingdon Clifford in that second half 19. Century one described. Before Arthur Cayley had alreadythe quaternions with complex coefficients (thus the quantity <math> \ mathbb {C} \ times \ mathbb {H}< /math>) Biquaternionen calls - of it is not here the speech.

definition

the Biquaternionen are a 8dimensionales hypercomplex number system also the units 1, i, j, k, <math> \ omega< /math>, <math> \ omega< /math> i, <math> \ omega< /math> j, <math> \ omega< /math> k.

Here i, j, k are the units of the quaternions, <math> \ omega^2=1< /math>, and<math> \ omega< /math> kommutiert with i, j, k.

characteristics

the Biquaternionen form a ring with zero-divisors

of each Biquaternion p can as sum of two quaternions q and r be represented as follows: <math> p = q + \ omega r< /math>.

The Biquaternionen is Clifford algebra <math> Cl (3.0, \ mathbb R)< /math>. The connection develops through <math> \ mathbf {i} _1 =i< /math>, <math> \ mathbf {i} _2 =j< /math>, <math> \ mathbf {i} _3 =k< /math> and <math> \ mathbf {i} _1 \ mathbf {i} _2 \ mathbf {i} _3 = \ omega< /math>

The Biquaternionen is the direct sum of the quaternions with itself, <math> \ mathbb H \ oplus \ mathbb H< /math>.

used topics

 

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