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 product on the vector space {R}}^{(2^n)}$$ in manner similar to Gauss’ ordered pair complex numbers ( {C}}$$ ) and Hamilton’s quadruple quaternions {H}}$$ ). We motivate definition generalizing tuple multiplication each . Similar way which Gauss obtains basis $$\{1, i\}$$ from , likewise derive monomials for multiplying those $$2^n$$ tuples that generate