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)
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
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
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
... complex i