# Maths - Double Numbers- Datasheet

## Algebra Laws

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

unit element 0 1
commutative yes yes
associative yes yes
distributive over addition - 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 D(2), it has the following properties:

### Cayley Table

The Cayley table is symmetric about its leading diagonal:

 1 D -1 -i 1 1 D -1 -D i D 1 -D -1 -1 -1 -D 1 D -i -D -1 D 1

### Cyclic Notation

The group has two generators each cycling between two elements:

(1,D)(-1,-D)
(1,-1)(D,-D)

### Group Presentation

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

<a,b | a²=1,b²=1,ab=ba>

### Group Representation

A representation using 4 ×4 matricies containing 0 and 1 is:

[
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
,
 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
]

An alternative 2×2 matrix representation containing 0, 1 and -1 is:

[
 -1 0 0 -1
,
 0 1 1 0
]

## Related datasheets

Book Shop - Further reading.

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.