1. COMPLEX NUMBERS §5.6

2. OBJECTIVES By the end of today, you should be able to… Identify and graph complex numbers. Add, subtract, and multiply complex numbers.

3. COLLEGE BASKETBALL John Henson shoots a basketball towards a goal with a height of 10 ft. The equation of the ball’s height h at time t is modeled by the quadratic equation h = t 2 + 16. When will the basketball make it in the goal?

4. Remember when you first learned to count? Now, your number system has expanded. You use rational numbers, like ½, and irrational numbers, like. Today, your number system is going to expand to include numbers such as.

5. INTRODUCING… Hey. This is i. i is defined as the number whose square is -1. and An imaginary number is a number in the form a + bi, where b≠0. You’re saying I’m not real?! i

6. PROPERTY: SQUARE ROOT OF A NEGATIVE REAL NUMBER For any positive real number a,.

7. EXAMPLE 1: SIMPLIFYING NUMBERS USING

8. In your graphing calculator, type and choose enter. What does your calculator say? Choose MODE, and then go down to REAL. Move your cursor to the right once, to a + bi, and press ENTER. This mode allows your calculator to work with Type and choose enter. imaginary numbers! You found me!

9. http://www.youtube.com/watch?v=_jxy9bx6SQ8&NR=1&feature=end screen

10. COMPLEX NUMBERS A complex number can be written in the form Where a and b are real numbers, including 0. Real part Imaginary part

11. EXAMPLE 2: WRITE THE COMPLEX NUMBER IN THE FORM

12. COMPLEX NUMBER PLANE The absolute value of a complex number is its distance from the origin on the complex number plane. You can find the absolute value of a complex number by using the Pythagorean Theorem.

13. EXAMPLE 3: FINDING ABSOLUTE VALUE

14. OPERATIONS WITH COMPLEX NUMBERS You can apply the operations of real numbers to complex numbers. If the sum of two complex numbers is 0, then each number is the opposite, or additive inverse, of the other. Find the opposite:

15. EXAMPLE 4: ADDITIVE INVERSE OF A COMPLEX NUMBER

16. ADDING COMPLEX NUMBERS To add or subtract complex numbers, combine the real parts and the imaginary parts separately. Combine “like” terms:

17. EXAMPLE 5: ADDING COMPLEX NUMBER

18. MULTIPLYING COMPLEX NUMBERS For two imaginary numbers, bi and ci, You can multiply two complex numbers of the form a + bi by using the procedure for multiplying binomials. Multiply:

19. EXAMPLE 6: MULTIPLYING COMPLEX NUMBERS

20. FINDING COMPLEX SOLUTIONS Some quadratic equations have solutions that are complex numbers.

21. EXAMPLE 7: FINDING COMPLEX SOLUTIONS