1. Introduction to Complex Numbers Adding, Subtracting, Multiplying Complex Numbers

2. Complex Numbers (a + bi) Natural (Counting) Numbers Whole Numbers Integers Rational Numbers Real Numbers Irrational #’s Imaginary #’s

3. Definition of pure imaginary numbers:

4. Any positive real number b, where i is the imaginary unit and bi is called the pure imaginary number.

5. Definition of pure imaginary numbers: i is not a variable it is a symbol for a specific number

6. Simplify each expression.

7. Remember Simplify each expression. Remember

9. Simplify. To figure out where we are in the cycle divide the exponent by 4 and look at the remainder.

10. Simplify. Divide the exponent by 4 and look at the remainder.

11. Simplify. Divide the exponent by 4 and look at the remainder.

12. Simplify. Divide the exponent by 4 and look at the remainder.

13. Complex Numbers are written in the form a + bi, where a is the real part and b is the imaginary part. a + bi real part imaginary part

14. When adding complex numbers, add the real parts together and add the imaginary parts together. (3 + 7i) + (8 + 11i) real part imaginary part 11 + 18i

15. When subtracting complex numbers, be sure to distribute the subtraction sign; then add like parts. (5 + 10i) – (15 – 2i) –10 + 12i 5 + 10i – 15 + 2i

16. When multiplying complex numbers, use the FOIL method. (3 – 8i)(5 + 7i) 71 – 19i 15 + 21i – 40i – 56i 2 15 – 19i + 56 Remember, i 2 = –1

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

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

19. Homework  Basics of Complex Numbers

20. The quadratic formula

22. From the above example when the number under the square root sign is zero there is only 1 solution.

24. We do not have real roots. We have 2 complex roots

26. This leads to the following observation. Since the discriminant is zero, the roots are real and equal.

27. Homework  Basics of Complex Numbers  Discriminant