1. Chapter 2 – Polynomial and Rational Functions
2.4 – Complex Numbers

2. Some quadratic equations have no real solutions but still can have imaginary solutions.
and
Evaluate to fill in the tables. Look for a pattern: i0 i1 i2 i3 i4 i5 i6 1 i7 i8 i9
i10
i11
i12
i13 -i i 1 -1 i -1 -i -i 1 i 1 -1 i
Imaginary Numbers

3. Complex #s = the set of all real and all imaginary numbers.
If a and b are real numbers, then a complex number can be written in the standard form of a + bi.
a is the real part
bi is the imaginary part
Ex: Simplify (3 – 2i) – (2 – 3i).
CLT (combine like terms)!
3 – 2i – 2 + 3i
1 + i
Complex Numbers

4. Multiplying Complex Numbers
Ex: Simplify .
Write in i form and multiply!
(2i)(4i) = 8i2
= 8(-1) = -8
Ex: Simplify (3 + 4i)(2 – i).
FOIL to get 6 – 3i + 8i – 4i2
= 6 + 5i – 4(-1)
= i
Ex: Simplify (5 + 2i)2
= i + 4i2
= i + 4(-1)
= i
Multiplying Complex Numbers

5. Simplify: (3 – 4i)(2 + 7i)
6 – 28i
6 – 15i i i i

6. Simplify: (9 – 2i)(9+2i) 77 83
81 – 4i i 85

7. Complex Conjugates The complex conjugate of a + bi is a – bi.
Multiplying two complex conjugates results in a real number:
(a + bi)(a – bi) = a2 + b2
Use this fact to get complex numbers out of denominators!
Ex: Write in standard form:
Multiply the entire fraction by the complex conjugate of the denominator!
Complex Conjugates

8. Simplify: (3 – 3i)/(4+2i)

9. The Complex Plane When plotting complex coordinates, remember:
The x-axis is the real axis
The y-axis is the imaginary axis
Plot the following points:
2 + 4i
1 – 2i 3 -i
The Complex Plane