1. Complex Numbers 22 11 Definitions Graphing 33 Absolute Values

2. Imaginary Number (i)  Defined as:  Powers of i 2

3. Complex Numbers  A complex number has a real part & an imaginary part.  Standard form is: Real part Imaginary part Example: 5+4i

4. Definitions  Pure imaginary number  Monomial containing i  Complex Number  An imaginary number combined with a real number  Always separate real and imaginary parts 4

5. The Complex plane Imaginary Axis Real Axis

6. Graphing in the complex plane

7. Absolute Value of a Complex Number  The distance the complex number is from the origin on the complex plane.  If you have a complex number the absolute value can be found using:

8. Examples 1. 2.

9. Simplifying Monomials  Simplify a Power of i  Steps  Separate i into a power of 2 or 4 taken to another power  Use power of i rules to simplify i into -1 or 1  Take -1 or 1 to the power indicated  Recombine any leftover parts 9

10. Operations  Simplify a Power of i Simplify 10

11. Simplifying Monomials Example  Square Roots of Negative Numbers Simplify 11

12. Addition & Subtraction  Add and Subtract Complex Numbers  Treat i like a variable Simplify 12

13. Ex: Addition & Subtraction Examples

14. Multiplying Complex Numbers  Multiply Pure Imaginary Numbers  Steps  Multiply real parts  Multiply imaginary parts  Use rules of i to simplify imaginary parts 14

15. Monomial Multiplication Example  Multiply Pure Imaginary Numbers Simplify 15

16. Multiplication Example  Multiply Complex Numbers Simplify 16

17. Solving ax 2 +b=0  Equation With Imaginary Solutions Solve Note: ± is placed in the answer because both 4 and -4 squared equal 16 17

18. Multiply the numerator and denominator by the complex conjugate of the complex number in the denominator. 7 + 2i 3 – 5i The complex conjugate of 3 – 5i is 3 + 5i. Multiplying Complex Numbers

19. Dividing Complex Numbers  Divide Complex Numbers  No imaginary numbers in the denominator!  i is a radical  Remember to use conjugates if the denominator is a binomial Simplify 19

21. Division Example Simplify 21

22. 7 + 2i 3 – 5i 21 + 35i + 6i + 10i 2 9 + 15i – 15i – 25i 2 21 + 41i – 10 9 + 25 (3 + 5i) 11 + 41i 34

23. Try These. 1.(3 + 5i) – (11 – 9i) 2.(5 – 6i)(2 + 7i) 3.2 – 3i 5 + 8i 4. (19 – i) + (4 + 15i)

24. Try These. 1.(3 + 5i) – (11 – 9i) -8 + 14i 2.(5 – 6i)(2 + 7i) 52 + 23i 3.2 – 3i –14 – 31i 5 + 8i 89 4. (19 – i) + (4 + 15i) 23 + 14i