An Ordered Tuple Construction of Geometric Algebras

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 $${\mathbb {G}}^{(p,q)}$$ .