1. Complex Numbers

2. What is a complex number?
i is the square root of -1
What would the square root of -3 be? -24?
Complex plane:
x-axis = real numbers
y-axis = imaginary numbers
points are written as a+bi

3. Conjugates
Useful for rationalizing
Conjugate of a+bi is a-bi

4. Polynomials
A polynomial of degree n has n solutions, real or imaginary
What are the 4 solutions to x^4 = 1?

5. Exercises
If (a+bi)(c+di) is a real number, what equations do we have in a, b, c, and d?
What is the square root of i? (Hint: Use the fact that complex numbers are written as a+bi)

6. Polar coordinates The point a+bi can also be
represented in polar form.
How?

7. Operations Multiply the r’s, add the angles!
What are the solutions to x^9 = 1 in polar form?

8. Roots of Unity What are the solutions of x^9 = 1?
Let x = (r,Θ). Then x9 = (r9, 9Θ), and
1 = (1, 0 + 2πk).
These two are equal if r9 = 1 or 9Θ = 2πk.
r must be a positive integer by definition, so
r = 1, and Θ = 2πk/9.
Thus x = (1, 2πk/9), where k = 0, ±1, ±2, etc.