i and a+bi

V1 — Invent i: a square that is −1.

Brand-new (light bridge: you already know squares of reals never go negative).

Tool gained: i² = −1 (i as a quarter-turn unit, not a scare label).

1. Principle. On the number line, no real square is negative. Extending the plane, one new unit i with i² = −1 unlocks arrows that turn.
2. Model. i² = −1; then i³ = −i, i⁴ = 1 — a four-step cycle like quarter turns.
3. Prediction / Answer. i⁴ = 1, so i²⁰ = (i⁴)⁵ = 1. Transfer: “√(−1)” is this unit, not a mystery ghost.

V2 — a+bi as a plane point / arrow.

Builds on: i² = −1 (V1) — real part + imaginary part.

Tool gained: z = a + bi ↔ point (a,b) / arrow from origin (Re = a, Im = b).

1. Principle. Every complex number is one address in the plane: walk a on the real axis, b on the i-axis.
2. Model. z = a + bi with a,b real. Equal when both parts match.
3. Prediction / Answer. 3−2i sits at (3,−2). Transfer: sensor “east + north” packed as one z.

V3 — Add / conjugate / modulus.

Builds on: a+bi plane tool (V2) — vector add, flip Im, length.

Tool gained: (a+bi)+(c+di)=(a+c)+(b+d)i; conjugate z̄=a−bi; |z|=√(a²+b²).

1. Principle. Add like vectors tip-to-tail. Conjugate flips the imaginary part (mirror in the real axis). Modulus is ordinary plane length.
2. Model. z+w componentwise; z·z̄ = |z|²; |z| = √(a²+b²).
3. Prediction / Answer. |3−4i|=5; (3−4i)+(1+2i)=4−2i. Next: multiply / divide with this kit.