So how does this algebra compare with other algebras? Since it has 4 'dimensions' lets compare it with quaternions. Both these algebras implement addition by adding corresponding terms, as usual with this type of algebra, the differences (if any) are in the multiplication rules:

The 2D multivector multiplication table is:

a*b |
b.e | b.e1 | b.e2 | b.e12 |

a.e | 1 | e_{1} |
e_{2} |
e_{12} |

a.e1 | e_{1} |
1 | e_{12} |
e_{2} |

a.e2 | e_{2} |
-e_{12} |
1 | -e_{1} |

a.e12 | e_{12} |
-e_{2} |
e_{1} |
-1 |

The multiplication table for quaternions is:

a*b |
b.1 | b.i | b.j | b.k |

a.1 | 1 | i_{} |
j_{} |
k_{} |

a.i | i_{} |
-1 | k_{} |
-j_{} |

a.j | j_{} |
-k_{} |
-1 | i_{} |

a.k | k_{} |
j_{} |
-i_{} |
-1 |

These tables are very similar, there is a difference in the 'square' terms, i.e. the terms on the leading diagonal. In the case of 2D multivector there are 3 positives and 1 negative, in the case of the quaternion there is 1 positive and 3 negatives. Otherwise the tables seem very similar, in both cases the real terms commute and the other terms (not on the leading diagonal) anticommute.

Does anyone know if:

- Is there a way to relate these two types?
- Quaternions always have an inverse, do 2D multivectors always have an inverse?

## Comparison of 2D multivector with Complex Numbers

We can relate 2D multivectors with complex numbers.

If we let c be a complex number:

c = a + i b

where a and b are the real and imaginary parts.

Assume that we can also represent a complex number by a linear sum of two basis vectors:

x = a e_{1} + b e_{2}

When we chek this out, using the multipication rules above, this does not quite work. for instance squaring e_{2} gives +1 not -1.

However if we multiply both sides by e_{1} we get:

e_{1} x = a + b e_{1}e_{2}

This has the correct properties, for instance, e_{1}e_{2} is equivilant to:

e_{1}e_{2} = √-1