For more information about complex numbers see this page.
Complex Numbers over the real numbers are a 'field' they have the following properties:
|distributive over addition||-||yes|
As an Extension Field to Real Numbers
As a Multiplicative Group
If we ignore addition and treat complex numbers as a group then the group is equivalent to a cylcic group of order 4, it has the following properties:
The Cayley table is symmetric about its leading diagonal. For a cyclic group the table can be drawn with same terms on the bottom-left to top-right diagonals:
For more information about Cayley table see this page.
For more information about Cayley graph see this page.
The group is shown as a single cycle:
For more information about cyclic notation see this page.
There is only one generator which when applied n times cycles back to the identity.
<i | i 4 =1>
For more information about group presentation see this page.
This is the 4th root of the identity matix (such that lesser roots are not identity). See this page for information about taking roots of a matrix. One a matrix which will do this is an 4×4 matrix of this form:
An alternative 2×2 matrix using positive and negative reals is:
For more information about group representation see this page.