Math / Turn and stretch
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One move in the plane
Turn and stretch
A pair of numbers can mean one motion: stretch by a length, turn by an angle. Multiplying by that pair does both at once to every point in the plane. That is the job of as a move — not as a mystery root of a negative (komplexa tal). Purpose first: turn and stretch. The letter i comes later as “a quarter turn.”
Origins
Bombelli
1500s algebra stumbled on square roots of negatives while solving cubics. Bombelli learned to compute with them anyway. The plane picture came later.
Gauss
Around 1800: draw the pair as a point / arrow in the plane. Multiplication = rotate and scale. Suddenly the “imaginary” was a move you could see.
quarter turn
Multiply by i twice and you reverse direction — a half turn. So i itself is a quarter turn. That is the everyday picture we keep.
Honesty: we lead with the move. We do not open with √(−1) as a scare.
Drag stretch and turn — watch the arrow
Start with a point (arrow from origin). Choose a stretch factor r and a turn θ. The product arrow is the original, stretched by r and turned by θ. That pair (r, θ) — or its grid form a+bi — is one move (multiplikation = rotation + skalning).
r = 1.5 · θ = 40° · product ≈ (0.57, 1.61)
Grey arrow = original (length 1). Bold = after stretch and turn. Same job as composing a scale with a turn on the circle.
Komplexa tal: z = r(cos θ + i sin θ). Multiplicera = multiplicera r och addera vinklar.
One object, one move
You already stretched and turned similar shapes. This is that move, written as one object.
Gate — four, all required
Hands-on (drag stretch or turn) is 20. Each right gate item is 20.
1. What is this tool doing?
2. Multiplying by i once does what to an arrow?
3. Multiply stretches, add turns — why?
4. Which is a real question?