Scalars and antiscalars

A scalar in a geometric algebra is an element having grade 0. Scalars are just ordinary real numbers, and they do not involve any basis vectors.

The basis element representing the unit scalar is denoted by $$\mathbf 1$$, a boldface number one. The unit scalar $$\mathbf 1$$ is the multiplicative identity of the geometric product.

For a general element $$\mathbf a$$, the notation $$a_{\mathbf 1}$$ means the scalar component of $$\mathbf a$$.

An antiscalar in a geometric algebra is an element having antigrade 0. Antiscalars are multiples of the volume element given by the wedge product of all basis vectors.

The basis element representing the unit antiscalar is denoted by $$\large\unicode{x1D7D9}$$, a double-struck number one. The unit antiscalar $$\large\unicode{x1D7D9}$$ is the multiplicative identity of the geometric antiproduct.

For a general element $$\mathbf a$$, the notation $$a_{\large\unicode{x1D7D9}}$$ means the antiscalar component of $$\mathbf a$$.