Math 71B

1. Math 71B 7.7 – Complex Numbers

2. Does have any solutions that are real numbers? Nope! Let’s define a solution (called the imaginary unit): ______ (so that )

3. Does have any solutions that are real numbers? Nope! Let’s define a solution (called the imaginary unit): ______ (so that )

4. Does have any solutions that are real numbers? Nope! Let’s define a solution (called the imaginary unit): ______ (so that )

5. Does have any solutions that are real numbers? Nope! Let’s define a solution (called the imaginary unit): ______ (so that )

6. Does have any solutions that are real numbers? Nope! Let’s define a solution (called the imaginary unit): ______ (so that )

7. Ex 1.Write as a multiple of :

8. Ex 1.Write as a multiple of :

9. Ex 1.Write as a multiple of :

10. Ex 1.Write as a multiple of :

11. Ex 1.Write as a multiple of :

12. Ex 1.Write as a multiple of :

13. Ex 1.Write as a multiple of :

14. Ex 1.Write as a multiple of :

15. Real and imaginary numbers together make the ________________________, which can all be written in the form: __________ ( and are real #’s; is called the ____________; is called the ________________________) ex:

16. Real and imaginary numbers together make the ________________________, which can all be written in the form: __________ ( and are real #’s; is called the ____________; is called the ________________________) ex: complex numbers

17. Real and imaginary numbers together make the ________________________, which can all be written in the form: __________ ( and are real #’s; is called the ____________; is called the ________________________) ex: complex numbers

18. Real and imaginary numbers together make the ________________________, which can all be written in the form: __________ ( and are real #’s; is called the ____________; is called the ________________________) ex: complex numbers real part

19. Real and imaginary numbers together make the ________________________, which can all be written in the form: __________ ( and are real #’s; is called the ____________; is called the ________________________) ex: complex numbers real part imaginary part

20. Real and imaginary numbers together make the ________________________, which can all be written in the form: __________ ( and are real #’s; is called the ____________; is called the ________________________) ex: complex numbers real part imaginary part

21. Real and imaginary numbers together make the ________________________, which can all be written in the form: __________ ( and are real #’s; is called the ____________; is called the ________________________) ex: complex numbers real part imaginary part

22. Ex 2. Ex 3.

23. Note: When multiplying square roots with negatives inside, pull out the ’s first! Ex 4.

24. Note: When multiplying square roots with negatives inside, pull out the ’s first! Ex 4.

25. Note: When multiplying square roots with negatives inside, pull out the ’s first! Ex 4.

26. Note: When multiplying square roots with negatives inside, pull out the ’s first! Ex 4.

27. Note: When multiplying square roots with negatives inside, pull out the ’s first! Ex 4.

28. Conjugates and Division The conjugate of is __________. To divide complex #’s, we can use the conjugate to help.

29. Conjugates and Division The conjugate of is __________. To divide complex #’s, we can use the conjugate to help.

30. Conjugates and Division Ex 5.Divide and simplify to the form . Ex 6.Divide and simplify to the form .

31. Powers of Ex 7.Simplify: