1. Warm-Up: December 13, 2011  Solve for x:

2. Complex Numbers Section 2.1

3. Imaginary Unit, i

4. Imaginary Numbers  Multiples of i are imaginary numbers.  Square roots of negative numbers are imaginary.

5. Complex Numbers  Any number that can be written in the form “ a+b i ” is a complex number. This is called the standard form.  In “a+bi” both a and b are real numbers  a is called the real part  b is called the imaginary part  A complex number is simplified iff it is written in standard form, “ a+b i ”

6. Operations with Complex Numbers  Addition: Combine like terms  Add the real parts, add the imaginary parts  Subtraction: Combine like terms  Subtract the real parts, subtract imaginary parts  Don’t forget to distribute the negative  Multiplication: Use distributive property or FOIL  Replace i 2 with -1  Always write final answer in standard form, “ a+b i ”

7. Example 1 (like HW #1-8)

8. You-Try #1 (like HW #1-8)

9. Example 2 (like HW #9-20)

10. You-Try #2 (like HW #9-20)

11. Complex Conjugates  The complex conjugate of “ a+b i ” is “ a-b i ”  Switch the sign of the imaginary part  The product of a complex number and its complex conjugate is:

12. Warm-Up: December 6, 2010

13. Warm-Up: December 7, 2010

14. Dividing Complex Numbers 1. Multiply the numerator and denominator by the complex conjugate of the denominator 2. FOIL the numerator and simplify 3. Denominator becomes a 2 +b 2 4. Write the result in standard form

15. Example 3 (like HW #21-28)

16. Warm-Up: December 14, 2011

17. Homework Questions?

18. You-Try #3 (like HW #21-28)

19. Principal Square Root of Negative Numbers  Product and quotient properties of square roots do not apply to square roots of negative numbers.  Take out the i before multiplying or dividing square roots

20. Example 4 (like HW #29-44)  Perform the indicated operations and write the result in standard form

21. You-Try #4 (like HW #29-44)  Perform the indicated operations and write the result in standard form

22. Quadratic Formula  It works to find complex solutions of quadratic equations, not just real ones.

23. Example 5 (like HW #45-50)  Solve for x. Express solutions in standard form.

24. You-Try #5 (like HW #45-50)  Solve for x. Express solutions in standard form.

25. Assignment  Page 252 #1-49 Every Other Odd