PlaySpectrum

Helper · book companion path

Prior: Multiply, divide & distance · Next: De Moivre & powers · Shelf: Kurs 4

Math / Helper companion / Polar form

Uncut · book companion · ~218–227

Modulus becomes r — multiply by adding angles

Polar form

Score 0 / 100

Climb from |z|: name arg θ, write z = r(cos θ + i sin θ), then multiply by multiplying r and adding θ. That polar multiply Tool is what powers will repeat. (polär form · argument · belopp)

Vortex

V1 — Polar: r and arg from |z|.

Builds on: |z| and (a,b) plane Tools (intro) — plus angle sense from trig.

Tool gained: z = r(cos θ + i sin θ) with r = |z|, θ = arg(z) (principal branch care).

  1. Principle. Same arrow as length and heading: stretch r, turn θ from the positive real axis.
  2. Model. a = r cos θ, b = r sin θ; r = √(a²+b²).
  3. Prediction / Answer. 1+i → r=√2, θ=π/4. Transfer: same point, navigation form.
Vortex

V2 — Multiply in polar: r’s multiply, θ’s add.

Builds on: polar form (V1) + geometric meaning of a+bi multiply (ops).

Tool gained: r₁∠θ₁ · r₂∠θ₂ = (r₁ r₂)∠(θ₁+θ₂).

  1. Principle. One move stretches then turns; composing two moves multiplies stretches and adds turns — trig addition is why.
  2. Model. Product modulus = product of moduli; product arg = sum of args (mod 2π).
  3. Prediction / Answer. √2∠π/4 · √2∠π/4 = 2∠π/2 = 2i. Transfer: cascade two phase shifts.
Vortex

V3 — Divide / convert back.

Builds on: polar multiply Tool (V2) — inverse is 1/r and −θ.

Tool gained: z/w in polar = (r_z/r_w)∠(θ_z−θ_w); convert polar ↔ a+bi when needed.

  1. Principle. Undoing a move shrinks by 1/r and turns the other way — same Tool as multiply, run backward.
  2. Model. z/w = (r_z/r_w)(cos(θ_z−θ_w)+i sin(θ_z−θ_w)).
  3. Prediction / Answer. 2i / (√2∠π/4) = √2∠π/4 = 1+i. Next: powers = repeat multiply.

1. Poor dump → rewrite

Poor explanation

Book dump: “cis θ” stamped with no link to |z|. Our rewrite: r is the modulus Tool; multiply is compose stretch+turn; divide is the inverse move.

Tool · weapon

Weapons: z=r(cosθ+i sinθ) → r₁r₂∠(θ₁+θ₂) → divide by 1/r and −θ. Next: De Moivre / zⁿ.

2. Hands-on — r, θ → a+bi and product

Set r and θ; read a+bi and a second factor’s product in polar.

…

3. Fake → real

Fake → real

Fake

“Write 1+i in polar form.” No job.

Real

A robot arm command is stretch √2 and turn 45°. Packing that as √2∠π/4 (=1+i) is the same polar Tool — now a motion packet.

Gate — four, all required

1. Reconstruct: polar form of z uses…

2. Why does polar multiply build on |z| / arg (climb)?

3. Predict: (2∠π/3)·(3∠π/6) = ?

4. Transfer: undo a gain 4∠π/5. The inverse move is…