hypercomplex numbers 4D

Hypercomplex numbers are like quaternion an extension to the complex number space into the fourth dimension. They can be seen as four dimensional vectors (with one scalar and a vector in three space). In physics they are also used in relativity, because of the bicomplexity they have the same properties as complex numbers.

We define:         ;     wobei iČ = jČ = -1        kČ = ijk = 1

                         ;     conjugate hypercomplex

multiplication (column * row)

  i j k
i -1 k -j
j k -1 -i
k -j -i 1

properties:

Attention:   

We use the notation:

                    A and B are classical complex

                              

As seen, each hypercomplex number can be split into two complex numbers. That's very helpful for function mapping, because for every function is:

So each hypercomplex function can be seen as two complex function. (See functions)

Representation as matrices:

                       

Representation as complex matrices:

                       

... complex i