# 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

## Related datasheets

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

 Symmetry and the Monster - This is a popular science type book which traces the history leading up to the discovery of the largest symmetry groups.

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