Why is the grade of the wedge product of two arbitrary blades the sum of the two blades' grades independently?

Question

I'm reading Geometric Algebra For Computer Science, An Object Oriented Approach to Geometry and it says that this is true of any two arbitrary blades.

$\ grade( \textbf{ A} \wedge \textbf{B})=grade( \textbf{ A} )+grade( \textbf{B})$

However, it seems like this is wrong, since

$\ 0=grade( (e_1 \wedge e_2) \wedge (e_2 \wedge e_3)) \\=grade(e_1 \wedge e_2) +grade (e_2 \wedge e_3)\\ =grade(e_1)+ grade( e_2) +grade (e_2) +grade( e_3)=4$

Is this formula incorrect or am I using it incorrectly, and how?

Answer 1

I don't think there's any issue here. $(e_1 \wedge e_2) \wedge (e_2 \wedge e_3)$ is the zero 4-vector. It should not be confused with the zero scalar, although in geometric algebra, we can and often do use the same symbol (0) to denote any zero $k$-vector, or whole linear combinations of these zero $k$-vectors.

I would say the grade of the zero 4-vector is still 4.

Answer 2

You ask: "Why is the grade of the wedge product of two arbitrary blades the sum of the two blades grades independently?"

The answer is that this is the definition of the wedge product. Why is this the definition? Because it is useful.

About the grade of 0. If the grade-k multivectors are to be a vector space (highly desirable), then the grade-k multivectors must contain a zero.