Color Algebra / "A Complex Field Structure for Color: \mathbb{K} \cong \mathbb{C} with Negative Colors as Full Algebraic Members"

Abstract. We propose a complete algebraic eld structure for color, grounded in the physics of light rather than pigment. Colors are identied with elements of the complex numbers K ∼= C, where the modulus L encodes luminance and the argu- ment θ encodes hue. Black (zero luminance) serves as the additive identity, and white (full, neutral light) serves as the multiplicative identity consistent with additive color mixing in optics. Negative colors, dened as elements with L < 0, are admitted as full algebraic members, analogous to the introduction of negative numbers or complex numbers in the extension of the real line. We show that two invisible (negative) colors can multiply to produce a visible color, paralleling the identity i × i = −1. The resulting structure satises all eld axioms: commuta- tivity, associativity, distributivity, and the existence of additive and multiplicative inverses for all non-zero elements. This framework inherits the full machinery of complex analysis derivatives, integrals, Taylor series, trigonometric identities, and Cauchy's theorem at no additional cost. We situate this work relative to existing literature and discuss open questions regarding the physical interpretation of negative colors.