OpenAha
00
All courses
Math · Advanced

Complex Numbers

Mathematicians once called them "imaginary", but complex numbers turned out to describe rotation, waves and electrical circuits beautifully. Learn to picture them as points, discover that multiplication rotates, arrive at Euler's famous eiπ+1=0e^{i\pi} + 1 = 0, see why every polynomial has a root, and use complex maps to design an airfoil.

7 lessons~35 min0/7 complete
1

A New Kind of Number

i, and the complex plane.

  1. The Square Root of −1Meet i
  2. The Complex PlaneNumbers as points
2

Multiplication Is Rotation

Polar form and Euler’s formula.

  1. Multiplying RotatesLengths multiply, angles add
  2. Euler’s Formulae^(iπ) + 1 = 0
  3. Roots of UnityPolygons from equations
3

Complex Numbers at Work

Why polynomials have roots, and functions as maps.

  1. Why Every Polynomial Has a RootCounting roots by winding
  2. Complex Functions as MapsSquaring, inverting and designing wings

Courses that build on this

Ready for more? These courses use what you learn in Complex Numbers.

Math · Intermediate

Fractals & Chaos

Infinite detail, fractional dimensions, the butterfly effect, and the road to chaos.

8 lessons3 chapters