On the Extension of Complex Numbers

This paper proposes an extension of the complex numbers, adding further imaginary units and preserving the idea of the product as a geometric construction. These `supercomplex numbers\u27, denoted S, are studied, and it is found that the algebra contains both new and old phenomena. It is established that equal-dimensional subspaces of S containing R are isomorphic under algebraic operations, whereby a symmetry within the space of imaginary units is illuminated. Certain equations are studied, and also a connection to special relativity is set up and explored. Finally, abstraction leads to the notion of a `generalised supercomplex algebra\u27; both the supercomplex numbers and the quaternions are found to be such algebras