Drag two complex numbers z and w around the plane. Watch addition behave like adding arrows, and multiplication behave like rotating and stretching: the angles add and the lengths multiply. Then look at powers and roots of z.
How the model works
A complex number z = x + iy is a point in the plane. In polar form, z = r(cos θ + i sin θ) = re^{iθ}, where r = |z| is its length and θ = arg z its angle.
Additionz + w adds coordinates: the parallelogram rule for vectors.
Multiplicationzw = r₁r₂ e^{i(θ₁ + θ₂)}: lengths multiply and angles add. Multiplying by i rotates by 90°.
Conjugatez̄ = x − iy reflects in the real axis; z z̄ = |z|².
Powers zⁿ trace a spiral; if |z| = 1 they walk around the unit circle. Roots: the n solutions of wⁿ = z sit evenly around a circle of radius |z|^{1/n}, spaced 360°/n apart (de Moivre).
Angles are shown between −180° and 180° (the principal argument).
Guided experiments
Multiply by i. Put w = i and move z.
Prediction: zw is always z rotated 90° anticlockwise, with the same length.
Powers on the unit circle. z = cos 30° + i sin 30°, show powers.
Prediction: z, z², … step 30° at a time around the circle and z¹² = 1 exactly.
Cube roots of 8. z = 8, show the 3 roots.
Prediction: 2, and 2 at ±120°: −1 ± 1.732i.
Challenge
Without a calculator, what is (1 + i)⁸?
Reveal the explanation
|1 + i| = √2 and arg = 45°. Then (1 + i)⁸ has length (√2)⁸ = 16 and angle 8 × 45° = 360°, so it equals 16. Set z = 1 + i, show 8 powers, and read the last one.
References
OpenStax, Precalculus 2e, §8.5 “Polar Form of Complex Numbers”.