# Maths - Quaternions- Datasheet

## Algebra Laws

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

unit element 0 1
commutative yes no
associative yes yes
inverse exists yes yes

## Multiplicative Group

If we ignore addition and treat complex numbers as a group then we have:

### Cayley Table

1 -1 i -i j -j k -k
1 1 -1 i -i j -j k -k
-1 -1 1 -i i -j j -k k
i i -i -1 1 k -k -j j
-i -i i 1 -1 -k k j -j
j j -j -k k -1 1 i -i
-j -j j k -k 1 -1 -i i
k k -k j -j -i i -1 1
-k -k k -j j i -i 1 -1

### Cyclic Notation

As can be seen on the Cayley graph above each generator has two 4-element cycles :

<(1 4 2 3)(5 8 6 7),(1 6 2 5)(3 7 4 8)

### Group Presentation

There are two generators.

<i,j | i² = j², j -1i j = i -1>

where: k=ij

### Group Representation

These are the root of -1 the identity matix. See this page for information about taking roots of a matrix. There are two such generators:

[i] =
 0 -1 1 0
[j] =
 i 0 0 -i

k can be generated from i*j:

[k] =[i][j]=
 0 i i 0