Complex Numbers & the Argand Plane
A complex number a + bi stops being mysterious the moment you plot it as a point — real part across, imaginary part up. Then the two operations split cleanly: adding is a shift, but multiplying is a turn. First-year, and the secret engine behind waves, rotations and signal processing.
A number you can point at
Put the real part on the horizontal axis and the imaginary part on the vertical one, and every complex number becomes an arrow from the origin. It has a length (its modulus |z|) and a direction (its argument, the angle from the positive real axis). Those two views — a + bi and length & angle — are the whole story.
Add, then multiply 🟠🔵
Drag z₁ (orange) and z₂ (blue). Flip between Add and Multiply and watch the green result. In Add mode you'll see the parallelogram; in Multiply mode, watch the angles stack up.
Two operations, two pictures
Adding is sliding
z₁ + z₂ adds the parts separately, exactly like adding vectors: slide z₁'s arrow along z₂ and the tip lands on the sum. The four points make a parallelogram. Nothing turns.
Multiplying is turning
Lengths multiply, angles add. |z₁z₂| = |z₁|·|z₂| and arg(z₁z₂) = arg z₁ + arg z₂. So multiplying by a number of length 1 is a pure rotation — and that's why i, which sits at 90°, turns everything a quarter-turn.
Watch it directly: multiply anything by i and it swings 90° counter-clockwise. Do it again — another 90° — and you're at 180°, which is × (−1). That's the honest meaning of i² = −1: two quarter-turns make a half-turn. Suddenly the "impossible" square root of −1 is just the instruction "rotate by 90°".