1. Imaginary & Complex Numbers

2. Once upon a time…

3. -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. -These numbers were devised using an imaginary unit named i.

4. -The imaginary numbers consist of all numbers bi, where b is a real number and i is the imaginary unit, with the property that i² = -1. -The first four powers of i establish an important pattern and should be memorized. Powers of i

5. 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.

6. Powers of i 1.) Find i 23 2.) Find i 2006 3.) Find i 37 4.) Find i 828

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, -3-7i, etc.

8. Simplify. 3.) 4.)5.) -Express these numbers in terms of i.

9. You try… 6. 7. 8.

10. Now that you know how to take the square root of a negative number, try solving this quadratic with completing the square.

11. x 2 – 4x +6 = 0

12. Check with your teacher to see if you got the right answer. Stop Here

13. To multiply imaginary numbers or an imaginary number by a real number, it is important first to express the imaginary numbers in terms of i.

14. Multiplying 9. 10. 11.

15. 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.

16. Add or Subtract 12. 13. 14.

18. Multiplying & Dividing Complex Numbers Part of 7.9 in your book

19. REMEMBER: i² = -1 Multiply 1) 2)

20. You try… 3) 4)

21. 5) Multiply

22. You try… 6)

23. You try… 7)

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

25. Find the conjugate of each number… 8) 9) 10) 11)

26. Divide… 12)

27. 13) You try…