1. 5.9 Complex Numbers Objectives: 1.Add and Subtract complex numbers 2.Multiply and divide complex numbers

2. Imaginary numbers Because we cannot take the square root of a negative number, imaginary number were created. The imaginary number i is defined as the principal square root of -1, or i²=-1 or Pure imaginary numbers are square roots of negative real numbers such as 3i, -4i, and 8i. Factor out a negative one, then reduce the radical. The negative one will become i on the outside of the square root.

3. Examples Simplify.

4. The power of i Simplifying a power of i. Remember: i²=-1 so the following is true: i=i i²=-1 i³=i²· i=-i In order to find out what any power of “i” is, divide the exponent by 4. The remainder is the power to raise i.

5. Complex Numbers 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 b is called the imaginary part, such as 4+3i. (A combination of real and imaginary numbers.) To add or subtract complex numbers, add/subtract the real part and then the imaginary part. (5+2i)+(3-8i)=8-6i(7-3i)-(4+2i)=3-5i

6. Multiplying Complex Numbers Use FOIL method if they are binomials. Example: (3-9i)(2+4i) = 6+12i-18i-36i² =6-6i-36(-1)=42-6i Try this one: (7i-3)(2i+6) 14i²+42i-6i-18 14(-1)+36i-18 -32+36i

7. Dividing Complex Numbers Complex Conjugates: For 3-5i the conjugate is 3+5i. Their product is always a real number. Use this fact to rationalize complex numbers. Example:

8. Solving Equations Using Square Roots 1.Solve for the squared variable. 2.Take the square root of each side. (Remember to use the positive and negative values for the square root) Example:

9. Homework page 274 18-54 even