March 3, 2026Open Access

Beyond Imaginary: The J-Number

Key Points

This framework clarifies the relationship between hyperbolic and harmonic analyticity, enhancing mathematical physics applications.
Using split-complex algebra, key features include pairing complex numbers with hyperbolic units to generate new insights.
Analyses utilize Minkowski geometry to streamline Lorentz boosts and facilitate connections with projective space.
This compact approach may unify various mathematical constructs, offering a fresh perspective on rotation and boost.

Abstract

MATHEMATICAL PHYSICS ESSAY: An Exploration of Simple Numbers and Functions This essay develops a rigorous calculus “beyond the imaginary” by pairing the Euclidean unit i (i² = -1) with the hyperbolic unit j (j² = +1). The resulting split-complex (hyperbolic) algebra ties directly to Minkowski geometry via the invariant N (x +jy) = x² + y². . . . On this backbone we package electromagnetism bicomplexly, streamline Lorentz boosts, and connect lines in projective 3-space to Plücker coordinates and twistor incidence. Throughout, a J-discipline preserves j = i² symbolically until the final audit step, clarifying reactive and causal bookkeeping. The result is a compact framework unifying rotation vs boost, harmonic vs hyperbolic analyticity, and field transport along characteristics, with appendices supplying proof-level details.

Beyond Imaginary: The J-Number

Key Points

Abstract

Cite This Study