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Abstract

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', 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'; both the supercomplex numbers and the quaternions are found to be such algebras.

Author Bio

Nicholas Houghton is a second year undergraduate student of mathematics at the University of Copenhagen in Denmark. The presented paper was written mainly from Fall 2011 to Spring 2012 and is the result of work spontaneously begun in September 2009, refined through the following years. Nicholas has wide interests in mathematics and also a major interest in physics - he hopes to combine these throughout his study. He can be reached through e-mail in English, German or Danish.

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