Complex Numbers on the Argand Plane
A complex number z = x + iy is a point (or an arrow) on a plane. Adding slides arrows end to end; multiplying rotates and stretches. Drag the points and see.
Numbers you can draw
Plot z = x + iy at the point (x, y): that is the Argand plane. Its distance from the origin is the modulus |z| = √(x² + y²), and its angle from the positive real axis is the argument arg z. So z = r(cos θ + i sin θ), the polar form. Adding complex numbers adds the arrows like vectors (the parallelogram rule).
Multiplying = rotating
When you multiply, the moduli multiply and the arguments add: |z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg z₁ + arg z₂. Multiplying by i (modulus 1, argument 90°) is a quarter turn, which is why i² = −1. The n-th roots of a number sit evenly around a circle: the roots of unity form a regular polygon.
Yellow = z₁, pink = z₂, green = the result.