2. Complex Numbers Any number in form a+bi, where a and b are real numbers and i is imaginary. What is an imaginary number?

3. Once upon a time…

4. -In the set of real numbers, negative numbers do not have square roots. -Imaginary numbers were invented so that negative numbers would have square roots and certain equations would have solutions.

5. Definition of imaginary numbers: It's any number you can imagine

6. a + bi Complex Numbers real imaginary 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.

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

8. Simplify complex numbers Remember 28

10. Answer: -i

11. Powers of i

12. Divide the exponent by 4 No remainder: answer is 1. remainder of 1: answer is i. remainder of 2: answer is –1. remainder of 3:answer is –i. Divide the exponent by 4 No remainder: answer is 1. remainder of 1: answer is i. remainder of 2: answer is –1. remainder of 3:answer is –i.

13. When adding or subtracting complex numbers, combine like terms.

14. Multiplying complex numbers. To multiply complex numbers, you use the same procedure as multiplying polynomials.

15. Examples-6: FOIL

16. Answer: 21-i

17. Conjugates In order to simplify a fractional complex number, use a conjugate. What is a conjugate?

18. Conjugate -The conjugate of a + bi is a – bi -The conjugate of a – bi is a + bi

19. Find the conjugate of each number… a) b) c) d)

20. Example-7:

21. Example-8: Realize the denominator of:

23. Discriminant of a Quadratic Equation

24. Fundamental Theorem of Algebra

25. Complex Conjugate Zeros

26. Quick survey I feel I understand “complex Number” a) Very well b) With some review, I’ll be good c) Not really d) Not at all

27. Pair-work