Lesson 12 of 48
Complex numbers
Rotation, not imagination.
“Imaginary” is a historical insult that stuck. Multiplying by i is a quarter turn of the plane, and once you see it geometrically the algebra stops being arbitrary. Signals, waves and eigenvalues all cash this in.
Do this
Show on the plane that multiplying by i turns a point a quarter turn. Then compute (1 + i)^8 by hand using only rotation and scaling — no binomial expansion.
The question that unlocks the next lesson
Geometrically, what does multiplying a point by i do?
- AReflects it across the x-axis
- BRotates it 90° counter-clockwise about the origin
- CDoubles its distance from the origin
- DMoves it one unit up