We begin by discussing why we should care about Clifford Algebra. (If you want an overview of how Clifford Algebra actually works, skip to section 2.)
1. It is advantageous to use Clifford algebra, because it gives a unified view of things that otherwise would need to be understood separately:
• The real numbers are a subalgebra of Clifford algebra: just throw away all elements with grade > 0. Alas this doesn’t tell us much beyond what we already knew.
• Ordinary vector algebra is another subalgebra of Clifford algebra. Alas, again, this doesn’t tell us much beyond what we already knew.
• The complex numbers are another subalgebra of Clifford algebra, as discussed in reference 1. This gives useful insight into complex numbers and into rotations in two dimensions.
• Quaternions can be understood in terms of another subalgebra of Clifford algebra, namely the subalgebra containing just scalars and bivectors. This is tremendously useful for describing rotations in three or more dimen