1. Complex Numbers
MAΘ

2. What is a Complex Number?
i is the square root of -1
Form: a + bi
Conjugate of a + bi is a - bi
In the complex plane, the x-axis has real numbers and the y-axis has purely complex numbers

3. Using Complex Numbers n solutions to any degree n polynomial What is ?
What about ?
And ?
Solve these two equations
What solutions do we know for
Challenge (don’t try): what is

4. Common Techniques
Each complex number has an imaginary and a real part -- you can usually get two equations out of this.
Example: (a + bi)(c + di) is real. What does that mean about a, b, c, and d?

5. Imaginary Roots Graph of Only crosses x-axis twice -- only two
real roots
Others are imaginary
Notice that the imaginary roots come in conjugate pairs.

6. Polar Form Can express as What is i in this notation?
How do we go from rectangular, (a+bi), to polar, (r,Θ)?

7. Multiplying Them Multiply Magnitudes, Add Angles! So what is
How do we use this to find
given
Find with two ways

8. Exponential Form Given z = a + bi, factor out r, from (r,Θ).
z = r(cos Θ + i sin Θ)
Euler’s Formula states that
Therefore z = reiΘ

9. Roots of Unity What are the solutions of ?
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.