Abstract
All the commutative hypercomplex number systems can be associated with a geometry. In two dimensions, by analogy with complex numbers, a general system of hypercomplex numbers \(\{ z = x + uy;\;u^2 = \alpha + u \beta;\;x, y, \alpha, \beta \in {\mathbf{R}};\;u \notin {\mathbf{R}}\} \) can be introduced and can be associated with plane Euclidean and pseudo-Euclidean (space-time) geometries.
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