Unit 10
Complex numbers.
The number i and the complex plane: multiplying as turning, Euler’s formula, and a square wave built from sines.
Before this unit
The unit, lesson by lesson
Lessons
3 lessons · 17 practice questions · about an hour
Multiplying by i
The number i, whose square is minus 1, and the complex plane: multiplying by i as a quarter turn, the powers of i, a complex number by length and angle, and a number that three times over makes minus 1.
Turns, stretches and Euler’s formula
Multiplying complex numbers multiplies lengths and adds angles: powers as spirals, doubling time on the complex plane, a spiral that closes in on zero, and Euler’s formula, in which e to the iπ equals minus 1.
Adding sines
Every repeating wave is a sum of sines: circles spinning on circles, the square wave’s odd sines and their sizes, the triangle wave that converges fast, and the bump beside a jump that never goes away.