1. Drill #81: Solve each equation or inequality

2. Drill #83: Simplify each expression

3. 5-9 Complex Numbers
Objective: To simplify square roots containing negative numbers and to add, subtract, and multiply complex numbers

4. (1.) i Definition: i is called the imaginary unit.
What is the value of I squared?

5. (2.) Pure Imaginary Numbers
Definition: For any positive number b,
where i is the imaginary unit, and bi is called a pure imaginary number.
Example:

6. Evaluating the Square Root of Negative Numbers*
To find the square root of negative numbers:
1. First separate the negative
2. Evaluate each root separately and multiply

7. More Examples
Ex1.
Ex2.

8. Powers of I *

9. Finding Powers of i* Powers of i are cyclical. They repeat after
To find :
1. Divide n by 4 and keep only the remainder r
2. , where r is the remainder of n/4
Note:

10. Example (Powers of i)
Find the following:
Ex Ex2. Ex3.

11. (3.) Complex Numbers Definition: A number in the form of a + bi
where a and b are real numbers and i is the imaginary unit.
a is called the real part.
b is called the imaginary part.

12. Adding Complex Numbers*
To add complex numbers:
1. add the real parts together (this is the real part of sum)
2. add imaginary parts together (this is the imaginary part of the solution).
(a + bi) + (c + di) = (a + c) + (b + d)i
Ex: (5 + 6i) + (2 + 3i) = 7 + 9i

13. Adding Complex Numbers
Examples:
Ex1.
(2 + 9i) + (3 + 4i)
Ex2.
(5 + 6i) - (2 + 3i)
Ex3.

14. Multiplying Complex Numbers*
Definition: To multiply imaginary numbers you need to FOIL.
(a + bi)(c + di) =
= ac (first) + adi (outside) + bci (inside) + bd (last)
= (ac - bd) + (ad + bd)i
Ex: (2 + 3i)(4 + 5i) = 2(4) + 2(5i) + 3i(4) + 3i(5i)
= i + 12i – 15
= i

15. Multiplying Complex Numbers
(2 + 3i)( 1 + 4i)
Ex2.
(6 + 2i)( 3 – 2i)
Ex3.
(3 – 5i)(3 + 5i)

16. (4.) Complex Conjugates Definition: Numbers of the form
a + bi and a – bi are called complex conjugates.
The product of complex conjugates is:
Example: 3 + 2i and 3 – 2i are complex conj.

17. The Product of Complex Conjugates*

18. Dividing by Complex Numbers* Rationalizing Complex Denominators
To rationalize a complex denominator you need to multiply the numerator and denominator by the complex conjugate.
Example: Simplify

19. Solving 2nd Equations* (no 1st degree term)
To solve equations:
1. Isolate the square term.
2. Take the (+/-) square root of both sides.
Example:

20. (5.) Equal Complex Numbers
Definition: a + bi = c + di
if and only if a = c and b = d.
The real parts must be equal and the imaginary parts must be equal.

21. Equal Complex Numbers*
Find values of x and y for which each equation is true: