Abstract
The familiar complex numbers, defined by the relation i² = -1, provide a powerful bridge between algebra and Euclidean geometry. This article explores a profound generalization: number systems defined by p² = -θ₀ + p θ₁ for real parameters θ₀, θ₁. These generalized complex numbers form a unified framework that naturally describes not only elliptic (Euclidean) geometry but also hyperbolic and parabolic geometries. We explore their algebraic properties and geometric interpretations, showing how “circles” in these systems become conic sections in the Cartesian plane and their application in defining generalized exponential and trigonometric functions. This synthesis enriches our understanding of the deep connections between algebra and geometry.