1. Warm-up:
HW: Pg 258 (7, 11, 18, 20, 21, 32, 34, all, 66, 68, 78, 79, 80, 85, 86)

2. Imaginary & Complex Numbers
Objective:
Identify, add, subtract, multiply, and divide imaginary and complex numbers
Finding complex solutions of a Quadratic Equation

3. Imaginary Unit
Imaginary numbers occur when a quadratic equation has no roots in the set of real numbers.
The reason for the name "imaginary" numbers is that when these numbers were first proposed several hundred years ago, people could not "imagine" such a number. 

4. It is said that the term "imaginary" was coined by René Descartes in the seventeenth century and was meant to be a derogatory reference since, obviously, such numbers did not exist.   Today, we find the imaginary unit being used in mathematics and science.  Electrical engineers use the imaginary unit (which they represent as j ) in the study of electricity. .

5. Complex Number System Reals Rationals (fractions, decimals) Integers
Imaginary
i, 2i, -3-7i, etc.
Rationals
(fractions, decimals)
Integers
(…, -1, -2, 0, 1, 2, …)
Irrationals
(no fractions)
pi, e
Whole
(0, 1, 2, …)
Natural
(1, 2, …)

6. Simplify.
-Express these numbers in terms of i.
recall:
3.)

7. *For larger exponents, divide the exponent by 4, then use the remainder as your exponent instead.
Example:

8. Multiplying 4. 5. 6.

9. a + bi imaginary real Complex Numbers
The complex numbers consist of all sums a + bi, where a and b are real numbers and i is the imaginary unit. The real part is a, and the imaginary part is bi.

10. Add or Subtract 9.
10.
11.

11. Multiplying & Dividing Complex Numbers

12. Multiply
12)

13. Multiply.
13)

14. Example:

15. Complex Solutions of a Quadratic Equation:

16. Complex Conjugates:
The product of two complex numbers can be a real number.
This occurs with pairs of complex numbers of the form a + bi and a – bi, called complex conjugates

17. Dividing Complex Numbers

18. Sneedlegrit: Solve: x2 – 2x + 2 = 0
HW: Pg 258 (7, 11, 18, 20, 21, 32, 34, all, 66, 68, 78, 79, 80, 85, 86)