THE PHASOR — a dart

THE PHASOR a spinning complex number — multiply to rotate, and frequencies add

A phasor is a complex number on the unit circle, e^(iθ) = cosθ + i·sinθ, and it turns signal processing into arithmetic. To rotate by an angle you multiply by a phasor; because e^(iα)·e^(iβ) = e^(i(α+β)), frequencies add under multiplication — the whole of mixing, modulation, and the Fourier transform rides on that one fact. No trigonometry is needed to spin one: multiply (a,b)·(c,d) = (ac−bd, ad+bc) and the magnitude is preserved exactly. The phasor is the carrier of the phase channel in its purest form.

THE TECHNIQUE (a,b)·(c,d)=(ac−bd, ad+bc); rotate by multiplying; |z| held

The demo rotates a unit phasor (0.6,0.8) twice by complex multiply and shows it equals the phasor squared — frequencies add, magnitude stays 1: live demo

HISTORY & CREDIT Steinmetz 1893 · phasor calculus

“Rotations need sines and cosines.” — a complex multiply rotates exactly, and composing two multiplies adds their angles. cited

the object · e^(iθ) on the unit circle — magnitude 1, angle the phase.
the law · multiply to rotate; e^(iα)e^(iβ)=e^(i(α+β)) — frequencies add.
1893 · Charles Proteus Steinmetz — phasor calculus for AC circuits.

A number whose whole job is to turn — rotate by multiplying, and angles add themselves. phase channel

RECOMMEND FOR I-13 rotate by multiply, on the compiler

On the canonical compiler, rotating (1,0) twice by (0.6,0.8) gives (−0.28, 0.96), exactly the phasor squared; magnitude stays 1:

$ i13 run cr_phasor.i13 # (a,b)*(c,d) rotation RUN OK · 132 step(s) · peak stack 5 · call depth 1 r2x = -0.28 r2y = 0.96 -- (1,0) rotated twice d2x = -0.28 d2y = 0.96 -- phasor squared: identical freqs_add = 1 mag2 = 1 mag_is_one = 1

Recommend as a keeper shot — the phase channel’s pure carrier. The phasor is the object that holds the phase: a magnitude and an angle in one complex number, rotated by multiplication with the angle (frequency) as the carried quantity. Under David’s PHASE TEST it is the cleanest instance of “carry an orthogonal deterministic channel while |z| is fixed.” The honest counter (for the panel): “frequencies add” is e^(iα)e^(iβ)=e^(i(α+β)), a forall-pinned algebraic identity (B39/confluence-shaped), and the phase channel it carries is the same carried-channel principle as generative-dual. New instance, or new axis? Put to the panel.