The first great application of complex numbers was in developing AC (alternating current) electricity. Because Gauss and others had developed the arithmetic of this, physicists such as Yugoslavian-American physicist, Michael Pupin (1845-1924), could readily apply complex number arithmetic to provide information and control of AC devices. One great benefit was long distance transmission of electricity from dynamos to plants and homes. By transmitting this at high voltage, the leakage along the line became tolerably cost-effective.

One application of complex numbers is immediately apparent and part of physics teaching. Using the Argand-Wessel-Gauss diagram, a 2-D space can be erected with a "real" abscissa and "imaginary" ordinate, explicating a point by a coordinate (x, iy), with i = (-1)^1/2. This provides for rotation through 360 °, analogous to the "shifting of alternating current".

Physics students further learn that the measure of resistance, R in DC (direct current) electricity must be replaced by impedance, Z which separates into a "real" part for resistance, R, and an "imaginary" part for reactance, involving inductance, L, or capacitance, C. This is bifurcation is never adequately explained in standard physics and mathematics. ("Do it -- without questioning!") Using multivector theory, a few reasons readily emerge.

A complex number is a spinor which is needed to explain rotational dynamics ^[1], another subject not adequately explained in standard texts. A spinor consists of a scalar part (providing a model for the resistance in impedance) and a bivector part (providing a model for the reactance (inductance, capacitance) in impedance). The i = (-1)^1/2 in a complex number is a unit bivector in multivector theory with "built-in counterclockwise rotation". The "compass-pointing" of magnetic inductance in impedance is modeled by this bivector, as is the polarization of the capacitance in impedance. (Polarization is electric dipole moment per unit volume. Magnetization is magnetic dipole moment per unit volume.)

Not explained is that capacitance enters in reciprocal form (1/i2pfC) whereas inductance (i2pfL) does not. Much explaining by mathematicians and physicists yet remain.

Problem: The vector modeling AC electricity rises from the "real" abscissa (modeling resistance) to an increasing fraction of the i (modeling impedance) until it projects onto the plane of this bivector; then this vector moves away from this plane until leaving it entirely at the negative real abscissa. Can this merging from one plane into another, then out, be charted geometrically and arithmetically? Another problem: If this plane-to-plane merging and departing can be charted, does this explain why the spinor -- of which this bivector is a part -- only travels "half as much" as this vector? Is it that "planar skating", so that the spinor moves as "far" as the vector, or even more than the vector? This has not been considered either in conventional physics, which does not treat the complex number as a spinor. And has not been considered by those who recognize the complex number as a spinor.

AC electricity made possible radio transmission (the beginnings of electronics) -- although DC radio transmission is not technologically possible.