In two dimensions the geometric algebra, indicated as or also as it is based on Clifford’s theory of algebras, is generated by the bases and which combine by contraction and extension, or by scalar product and external product.

(scalar)

(bivector)

since geometric algebra is NOT commutative, it is also worth calculating:

(opposite bivector)

and the most interesting product of all:

We say interesting because it shows us two things:

- the square of the bivector is -1 and therefore can be assimilated to the imaginary unit , even if for caution we will indicate it with a different symbol: uppercase
- the geometric algebra is closed, i.e. the attempt to extend the elements of degree 2 by means of an external product leads to the collapse in lower degrees

Summarizing, in two dimensions the geometric algebra is formed by the following bases:

which combine to form the multivector

#### The even sub-algebra: generalized complex numbers

It is interesting to note that in we have an even subalgebra, formed by the elements of even degree (scalars and bivectors). It is subalgebra because the elements of even degree are a closed set with respect to addition and multiplication.

The well-known vectors and the even subalgebra, which we have called generalized complex numbers, are two subsets of 2D geometric algebra. Let’s find out what interactions exist between these two worlds.