# Maths - Complex Numbers- Datasheet

## Algebra Laws

Complex Numbers over the real numbers are a 'field' they have the following properties:

unit element 0 1
commutative yes yes
associative yes yes
inverse exists yes yes

R[x]/<x²+1>

## 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:

### Cayley Table

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:

 1 i -1 -i 1 1 i -1 -i i i -1 -i 1 -1 -1 -i 1 i -i -i 1 i -1

### Cyclic Notation

The group is shown as a single cycle:

(1,i,-1,-i)

### Group Presentation

There is only one generator which when applied n times cycles back to the identity.

<i | i 4 =1>

### Group Representation

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:

[i] =
 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0

An alternative 2×2 matrix using positive and negative reals is:

[i] =
 0 1 -1 0