complex numbers

Complex numbers are an extension to the real numbers. They can be seen as two dimensional vectors where also multiplication is defined.  With them problems can be solved, where real numbers fail. E. g. roots of negative numbers. In physics they are useful in describing oscillation in short form (in circuits).

We define:         ;  where iČ = -1 ,   a ... real part ,    b ... imaginary part


With the extension into the 2nd dimension the property of relation is lost, means "bigger than" and "smaller than" make no sense:

a)    i < 0    | * i                                    b)    i > 0    | * i

       iČ = -1 > 0     wrong                            iČ = -1 > 0     wrong


Representation as matrices: