What you will learn in Complex Numbers

1. 1. Why the Real Numbers Are Not Enough

   A quadratic that refuses to have roots, and the smallest possible thing you can add to fix it.

2. 2. Complex Arithmetic and the Conjugate

   Adding, multiplying, and the one trick that makes division possible.

3. 3. The Argand Plane

   Every complex number is a point, and every operation is a motion.

4. 4. Modulus and Argument

   A distance and a direction — and the quadrant mistake almost everyone makes.

5. 5. Polar and Exponential Form

   The same number written for multiplying instead of for adding — and a piece of notation that owes a proof.

6. 6. De Moivre's Theorem

   One rotation done n times, and the trigonometric identities that fall out of it.

7. 7. Roots of Unity and n-th Roots

   n solutions, always at the corners of a regular polygon, always adding to zero.

8. 8. Complex Roots of Real Polynomials

   Why non-real roots always come in pairs, and what happens when you break the rule.

9. 9. Loci in the Complex Plane

   Conditions on z, drawn — and the same curves module 14 builds from x and y.