Why Does Multiplying Complex Numbers Rotate Them?

Multiply by i and a number turns a quarter circle. Why? We start with one fact: i times i is negative one — and follow it all the way to the reason multiplication spins the plane. Polar form, a little FOIL, the angle addition formulas falling out on their own, and suddenly it's obvious: multiplying complex numbers multiplies their lengths and adds their angles. Scale and turn. That's all it ever was. A calm, visual walk through one of the prettiest ideas in math. Chapters: 0:00 Multiply by i 0:47 It's a rotation 1:02 Length and angle: polar form 1:51 Multiplying two numbers on the circle 2:13 Expanding the product 2:41 Sorting the real and imaginary parts 3:01 The angle addition formulas appear 3:39 The secret: the angles add 3:52 Scaling and rotating 4:30 Back to i 4:55 Where it all came from 5:09 Two arrows 5:21 Squares, cubes, and roots of unity #maths #complexnumbers #rotation