# 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 [itex] \ 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, [itex] \ omega< /math>, [itex] \ omega< /math> i, [itex] \ omega< /math> j, [itex] \ omega< /math> k.

Here i, j, k are the units of the quaternions, [itex] \ omega^2=1< /math>, and[itex] \ 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: [itex] p = q + \ omega r< /math>.

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

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