Grade and antigrade

The grade of a basis element in a geometric algebra is equal to the number of basis vectors present in its factorization. An arbitrary element whose components all have the same grade is also said to have that grade.

The antigrade of a basis element is equal to the number of basis vectors absent from its factorization.

The grade of an element $$\mathbf x$$ is denoted by $$\operatorname{gr}(\mathbf x)$$, and the antigrade is denoted by $$\operatorname{ag}(\mathbf x)$$. In an n-dimensional geometric algebra, it is always the case that


 * $$\operatorname{gr}(\mathbf x) + \operatorname{ag}(\mathbf x) = n$$.

Scalars have grade 0, and antiscalars have grade n. Vectors have grade 1, and antivectors have antigrade 1. Bivectors have grade 2, and trivectors have grade 3.