1. IMAGINARY NUMBERS AND DEMOIVRE’S THEOREM Dual 8.3

2. Remember Complex Numbers: a + b i a = real number b = imaginary number Plot the following points. 3 + 4i 2 – i 4i -3

3. Converting Rectangular to Polar Form Polar form of a Complex number: Rem: if x neg add 180 If y only neg. add 360 a (or x) b (or y) a+bi x = r cos (θ) y = r sin (θ) a + b i

4. Example:

5. Converting Polar to Rectangular Form x = r cos (θ) y = r sin (θ) Example:

6. Operations with Complex Numbers: All these can be worked much easier in POLAR FORM.

7. MULTIPLICATION Multiplying in Polar form

8. DIVISION

9. POWERS AND ROOTS Formula:

10. INDIVIDUAL ROOT

11. ALL ROOTS 3 solutions  1 real and 2 imaginary Change to complex Add 360 Conjugate pairs