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Helper · book companion path

Prior: i and a+bi · Next: Polar form · Shelf: Kurs 4

Math / Helper companion / Multiply, divide & distance

Uncut · book companion · ~210–217

Spend intro Tools — multiply, clear denom, |z−w|

Multiply, divide & distance

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Use the intro kit: multiply binomials with i²=−1, divide by multiplying the conjugate, read distance as |z−w|. Polar form waits until modulus is a Tool you already trust. (multiplikation · division · avstånd)

Vortex

V1 — Multiply in a+bi using i² = −1.

Builds on: a+bi and i²=−1 (intro Tools).

Tool gained: (a+bi)(c+di) = (ac−bd) + (ad+bc)i.

  1. Principle. Expand like binomials; every i² becomes −1 and folds into the real part.
  2. Model. Re = ac−bd, Im = ad+bc.
  3. Prediction / Answer. (1+2i)(3+i) = 1+7i. Transfer: compose two plane moves before polar language.
Vortex

V2 — Divide via conjugate.

Builds on: multiply Tool + conjugate / |w|² (intro).

Tool gained: z/w = (z w̄) / |w|² (w ≠ 0).

  1. Principle. Clearing the denominator means multiply top and bottom by conjugate so the bottom becomes a positive real |w|².
  2. Model. z/w = z w̄ / (w w̄) = z w̄ / |w|².
  3. Prediction / Answer. (1+i)/(1−i) = i. Transfer: undo a scale-and-turn once you can multiply.
Vortex

V3 — Distance in the plane.

Builds on: |z| and subtraction as add-of-negative (intro).

Tool gained: distance between z and w is |z − w|.

  1. Principle. The arrow from w to z is z−w; its length is ordinary plane distance.
  2. Model. d(z,w) = |z−w| = √((a−c)²+(b−d)²).
  3. Prediction / Answer. | (3+4i) − (0+0i) | = 5. Next: polar form reuses |z| as r.

1. Poor dump → rewrite

Poor explanation

Book dump: FOIL formulas and “multiply by conjugate” as chants. Our rewrite: i² folds the cross term; conjugate clears the denom to |w|²; distance is modulus of a difference.

Tool · weapon

Weapons: (ac−bd)+(ad+bc)i → z/w = z w̄/|w|² → |z−w|. Next: polar r, θ.

2. Hands-on — live product vs trap

Multiply (a+bi)(c+di) live; compare with a naive “multiply parts separately” trap.

…

3. Fake → real

Fake → real

Fake

“Compute (2+i)(3−i).” Drill with no job.

Real

Two filter stages multiply as (2+i)(3−i)=7−i. The composite gain/phase for the board is that product — same multiply Tool, now a cascade job.

Gate — four, all required

1. Reconstruct: (2+i)(1+3i) = ?

2. Why does divide build on conjugate (climb)?

3. Predict: |(4+i)−(1−2i)| = ?

4. Transfer: two sensor offsets z₁, z₂. Separation length is…