Warm Up November 18, 2013

1. Warm Up November 18, 2013 Complete the square 1. x2 + 2x – 80 = 0

2. Imaginary Numbers

3. Squaring Numbers What is ? 16 What is -? 16 *When you multiply 2 negatives, you still get a positive number.

4. Square Rooting Numbers What is +6 or -6 What is ? hmmmmm… *Here arrives the need for a new number, any ideas?

5. Introducing Imaginary Numbers Remember radical is an expression that uses a root. The sign is An imaginary number, i, is defined to be and can be used to simplify negative radicals.

7. Examples 7i 10i 5i

8. Steps to Simplify Radicals STEP #1: Remove the √(−1)= i. Step #2: Look for “squares”. Take the square root out of the radical. Step #3: Leave anything else under the radical.

9. 36 64 Remember “Squares” 121 144 25 9 49 4 16 81 100

10. Examples i i 3i i i 5i i i 4i

11. Property of i = = -1 = =1

12. Examples -15

13. Feeling like you are in an imaginary world??? Be patient Keep trying The more you see it, the easier it gets

15. Introducing Complex Numbers A complex number is a number with 2 parts: real part and the imaginary part written as: a + bi

16. Examples -4 + 13i

17. Examples 10 + i 2 5 + 3i 8 -8 4 + i 4 + 2i

18. Introducing Conjugates The conjugateof a complex number has the same real part, but an opposite imaginary part.

19. Examples

20. Vocabulary Review Real Number Radical Imaginary Number Complex Number Conjugate

21. Look for a “square” • Real Part? • Imaginary Part? - • Conjugate Remove the “i”

22. Look for a “square” • Real Part? • Imaginary Part? • Conjugate Remove the “i”

23. Look for a “square” • Real Part? • Imaginary Part? • Conjugate Remove the “i”

24. Real Part? • Imaginary Part? • Conjugate Combine like terms

25. (5 + 2i) (5 + 2i) • Simplify • Real Part? • Imaginary Part? • Conjugate Distribute

26. 2 2 2 • Look for a “square” • Simplify • Real Part? • Imaginary Part? Remove the “i” • Conjugate

27. WOW!!! Nice Job working hard today