# quaternions

Quaternions are an extension to the complex numbers into the fourth dimension. They can be seen as four dimensional vectors (with one scalar and a vector in three space). With them you can solve problems, where complex numbers fails. E.g. the factorisation of a²+b²+c²+d². In physics it's good for relativity, in computer science you can rotate objects faster than with matrices and it's better for interpolation of movements.

We define:    ;     where i² = j² = k² = ijk = -1

;     conjugate quaternion

multiplication (column * row)

 i j k i -1 k -j j -k -1 i k j -i -1

Because of the extension into the fourth dimension the commutativity is lost as well as the fundamential theorem of algebra.

x² + 1 = 0    has got the solution  ,    b,c,d,    b² + c² + d² ≠ 0

#### properties:

If a = 0, then z is a purely imaginary quaternion  u = bi  + cj + dk.    u² = -(b² + c² + d²)

If  b² + c² + d² = 1 , then  u² = -1. With this a quaternion z is:

z = A + Bu ,  A,B        or

(DeMoivre's formula)

root calculation:

z = p(A + Bu)   with  A² + B² = 1  and  p

If u and v are purely imaginary quaternions then

is the dot product,    the cross product

#### Representation as matrices:

#### Representation as complex matrices:

... complex i