1. Complex Numbers Essential Question: How do you perform operations on complex numbers? Demonstrated in writing on a summary at the end of the notes.

2. EQ: How do you perform operations on complex numbers? Demonstrated in writing on a summary at the end of the notes. i 1 =_______ i 5 =_______________ i 2 =_______ i 6 =_______________ i 3 =_______ i 7 =_______________ i 4 =_______ i 8 =_______________

3. EQ: How do you perform operations on complex numbers? Demonstrated in writing on a summary at the end of the notes. Imaginary Numbers The answer to a problem with an imaginary solution should never have an i with an exponent greater than one. If you have an answer with an i with an exponent larger than 1 divide the exponent by 4. The remainder will be 0, 1, 2, or 3.

4. EQ: How do you perform operations on complex numbers? Demonstrated in writing on a summary at the end of the notes. If the remainder is… replace the piece with the i with… 0 1 2 3

5. Simplify Example 1 i 50 Example 3 i 132 Example 5 i 256 Example 2 i 29 Example 4 i 151 Example 6i 167

6. Simplify Example 7 3i 3 – 5i 2 + 16i 9 Example 9 Example 11 Example 13 Example 8 Example 10 Example 12

7. Complex Numbers Definition of a Complex Number: A complex number is any number that can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit; a is called the real part and i is called the imaginary part. The piece with the i always comes last!!! Operations with Complex Numbers: To add complex numbers: add the real parts and add the imaginary parts To multiply complex numbers: distribute or use FOIL. Once you have done this replace any i 2 with -1 and simplify.

8. EQ: How do you perform operations on complex numbers? Demonstrated in writing on a summary at the end of the notes. Simplify. Example 13(8 – 5i) + (2 + i)Example 14(4 + 2i) + (5 – 7i) Example 15(4 + 7i) – (2 + 3i)Example 16(9 – 6i) – (3 – 5i) Example 17 (3 + i)(5 – i)Example 18 4i (- 6 + i) Example 19 (9 – 2i)(-4 + 7i) Example 20 i(9 – i)

9. EQ: How do you perform operations on complex numbers? Demonstrated in writing on a summary at the end of the notes. Solve each equation. Example 214x 2 + 36 = 0 Example 22x 2 + 81 = 0

10. EQ: How do you perform operations on complex numbers? Demonstrated in writing on a summary at the end of the notes. Solve each equation. Example 23a 2 + 72 = 0 Example 24 x 2 + 20 = 0

11. EQ: How do you perform operations on complex numbers? Demonstrated in writing on a summary at the end of the notes. Complex Conjugates and Quotients It is possible to have the product of two complex numbers be a real number. This happens when you multiply pairs of complex conjugates. The complex conjugate of a + bi is a - bi. When you need to take the quotient of two complex numbers multiply the numerator and the denominator of the fraction by the complex conjugate of the denominator. There should not be any i’s left in the denominator.

12. EQ: How do you perform operations on complex numbers? Demonstrated in writing on a summary at the end of the notes. Multiply each by its complex conjugate. Example 25 3-5i Example 26 3 + i

13. EQ: How do you perform operations on complex numbers? Demonstrated in writing on a summary at the end of the notes. Simplify each expression.

14. EQ: How do you perform operations on complex numbers? Demonstrated in writing on a summary at the end of the notes. Simplify each expression. Example 30Example 31

15. HW #4 Practice 4-6 (1-_______ )