I.  History

1.  Quadratic and cubic equations

Solution of quadratics, solution of cubics

2.  The Fundamental Theorem of Algebra

3.  The number i

The Fundamental Theorem of Algebra proved!

II.  Mathematics

4.  The complex plane, addition and subtraction

Notation, arithmetic operations on C, parallelogram rule, addition as translation, negation and subtraction

5.  Absolute value

The unit circle, the triangle inequality

6.  Multiplication

Multiplication done algebraically, multiplying a complex number by a real number, multiplication and absolute value, powers of i, roots of unity, multiplying a complex number by i, a geometric interpretation of multiplication

7.  Angles and polar coordinates 

8.  Reciprocals, conjugation, and division

Reciprocals done geometrically, complex conjugates, division

9.  Powers and roots

Powers, roots, more roots of unity