Negative numbers worked before they were accepted. Chinese rod arithmetic used them for balanced accounts. Indian mathematicians used them for debts. Medieval Islamic algebra used them operationally while denying their reality. European mathematicians through the seventeenth century treated them as useful fictions — producing correct results through illegitimate means. The gap between computational utility and conceptual legitimacy persisted for centuries.
Complex numbers followed a different trajectory (arXiv:2603.21445). They appeared not as tools for solving problems but as unwanted byproducts of algebraic formulas — square roots of negative numbers that had to be manipulated to reach real solutions. Bombelli's rules for operating on them came first; geometric representation (the complex plane) came later; rigorous analysis (Cauchy, Weierstrass) came later still; physical naturalization through quantum mechanics came last.
The paper tracks four distinct processes in the stabilization of both number types: operational use (they work in calculations), formal legitimation (axioms and proofs establish their status), pedagogical normalization (textbooks teach them as natural), and physical naturalization (experiments show they correspond to measurable quantities). These four processes operate on different timescales and in different communities, and a number type can be fully established in one domain while remaining suspect in another.
Franklin's electrical sign convention — positive and negative charge — was pivotal for negative numbers. It made sign physically tangible: not an abstract operation but a property of matter. Electrical engineering performed the same service for complex numbers: impedance, phasors, and complex amplitudes turned √-1 into a routine tool for describing phase relationships in circuits.
The structural insight: mathematical concepts don't become real through a single validation. They become real through convergent legitimation across independent domains. A concept that is operationally useful but formally suspect remains vulnerable. One that is formally rigorous but physically uninterpretable remains abstract. Full naturalization requires all four: it works, it's proven, it's taught, and it's measured. Each legitimation addresses a different kind of doubt. The concept becomes unchallengeable only when no domain of doubt remains unaddressed.