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. Acknowledgements: Jan Philip Solovej, Ryszard Nest and my sponsor Niels Gronbaek, Dept. of Mathematics, University of Copenhagen, have been most helpful in commenting on the work presented in this paper. Also I sincerely thank the referee on the paper for the useful suggestions for improvement. Finally, I am very grateful to my former teacher of mathematics, Christian Thune Jacobsen, Gl. Hellerup Gymnasium (and DTU), who has supported me throughout the project. Page 126 RHIT Undergrad. Math. J., Vol. 13, No. 2
Publication Year: 2012
Publication Date: 2012-01-01
Language: en
Type: article
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