An Ordered Tuple Construction of Geometric Algebras

Abstract

In this paper we will present a new construction of any real geometric (Clifford) algebra ${\mathbb {G}}^{(p,q)}$ with signature (p, q) where $p+q=n$ by defining a product on the vector space ${\mathbb {R}}^{(2^n)}$ in a manner similar to Gauss’ ordered pair construction of the complex numbers (${\mathbb {C}}$) and Hamilton’s ordered quadruple construction of the quaternions (${\mathbb {H}}$). We will motivate the definition of a geometric product on ${\mathbb {G}}^{(p,q)}$ by generalizing the ordered tuple definition of multiplication on each of ${\mathbb {C}}$ and ${\mathbb {H}}$. Similar to the way in which Gauss obtains the basis $\{1, i\}$ from the ordered pair definition of multiplication on ${\mathbb {C}}$, we will likewise derive a basis of monomials for ${\mathbb {G}}^{(p,q)}$ by multiplying those ordered $2^n$ tuples that generate ${\mathbb {G}}^{(p,q)}$.