<p>Hypercomplex numbers universally adopt a linear combination format based on imaginary units, demonstrating extensive applications in modern physics, geospatial mapping, and remote sensing. This paper presents three representations of hypercomplex numbers and restates Hurwitz’s (1898) theorem through fundamental expansion principles. Critically, imaginary units exhibit a bijective correspondence with mutually orthogonal unit vectors, where the multiplication table of these units forms the core of hypercomplex algebra. While Cayley’s (1845) generator exhibits subjectivity, we propose a novel generator that produces symmetric multiplication tables, significantly enhancing expansion efficiency. Key conclusions indicate that imaginary units are intrinsically interpretable as directional vectors, facilitating cross-disciplinary applications; hypercomplex dimensions are exclusively positive integer powers of 2; hypercomplex multiplication equates to the vector-geometric product; and this product satisfies associativity solely in 3D vector spaces.</p>