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 a+b+c+d. 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 i = j = k = 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.

x + 1 = 0    has got the solution  ,    b,c,d,    b + c + d ≠ 0


If a = 0, then z is a purely imaginary quaternion  u = bi  + cj + dk.    u = -(b + c + d)

If  b + c + d = 1 , then  u = -1. With this a quaternion z is:

z = A + Bu ,  A,B        or

        (DeMoivre's formula)

root calculation:

z = p(A + Bu)   with  A + B = 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