From False Roots to Phasors: Negative and Complex Numbers in Mathematics, Physics, and Electrical Engineering

Abstract

Negative and complex numbers are so familiar in modern mathematics, physics, and engineering that it is easy to forget how uncertain their status once was. They did not become established through a single route. This article follows four linked processes in their stabilization: operational use, formal legitimation, pedagogical normalization, and physical naturalization. Negative quantities appear early in Chinese rod arithmetic and Indian debt--fortune rules, were reshaped in medieval Islamic algebra, and remained conceptually unstable in early modern Europe even when they worked in practice. Complex quantities followed a different path: they first appeared as troubling by-products of algebraic formulas, then gained stability through Bombelli's rules, geometric representation, nineteenth-century analysis, and later applications in circuits, wave theory, optics, and quantum mechanics. Franklin's electrical plus and minus helped make sign physically intelligible, while electrical engineering turned impedance and complex amplitudes into routine tools. The broader lesson is that these quantities became natural through repeated interaction among calculation, representation, teaching, and experiment.

Figures (5)

• Figure 1: Analytical map of selected episodes in the stabilization of negative and complex quantities. Horizontal position indicates broad historical period; vertical position indicates the dominant process emphasized in this study: operational use ($U$), formal legitimation ($L$), pedagogical and representational normalization ($N$), and physical naturalization ($P$). Filled circles denote negative quantities; open squares denote complex quantities.
• Figure 2: Geometric comparison of two extensions of ordinary positive magnitude. (a) Negative quantities are represented on an oriented line relative to zero. (b) The equation $x^2+1=0$ has no solution when values are restricted to $\mathbb{R}$. (c) In the complex plane, the same equation has roots $\pm i$, and multiplication by $i$ corresponds to a quarter-turn. This geometric reformulation was central to nineteenth-century legitimation and pedagogy of complex quantities.
• Figure 3: Schematic reconstruction of Franklin's sign logic for electrical states. Positive and negative denote excess and defect relative to a reference amount, and transfer tends toward restoration of balance. The figure restates the relational semantics of Franklin's one-fluid framework in modern form.
• Figure 4: Three technical settings in which complex quantities became standard analytical objects. (a) In steady-state alternating-current analysis, a rotating-vector representation encodes magnitude and phase, with the sinusoid recovered by projection onto the real axis. (b) In the complex $s$-plane, poles encode decay and oscillation; a conjugate pair corresponds to a damped sinusoid. (c) In an absorbing medium, a complex refractive index yields both phase advance and exponential attenuation. These representations were central to the normalization and physical naturalization of complex quantities in physics and engineering.
• Figure 5: Complex-frequency excitation as waveform matching in the complex-frequency plane. Under the $e^{-i\omega t}$ convention, passive poles lie in the lower half-plane and the corresponding time-reversed scattering zeros lie in the upper half-plane. A finite-duration waveform matched to such a zero suppresses the outgoing response during the excitation window and increases stored energy, which underlies virtual critical coupling and related effects.