Problem Set 12: Complex Numbers

1. Explain how to add complex numbers.

2. What is the basic principle in multiplication of complex numbers?

3. Give an example to show the product of two imaginary numbers is not always imaginary.

4. What is a characteristic of the plot of a real number in the complex plane?

For the following exercises, evaluate the algebraic expressions.

5. If f(x)=x2+x4, evaluate f(2i).

6. If f(x)=x32, evaluate f(i).

7. If f(x)=x2+3x+5, evaluate f(2+i).

8. If f(x)=2x2+x3, evaluate f(23i).

9. If f(x)=x+12x, evaluate f(5i).

10. If f(x)=1+2xx+3, evaluate f(4i).

For the following exercises, determine the number of real and nonreal solutions for each quadratic function shown.

11.
Graph of a parabola intersecting the real axis.

12.
Graph of a parabola not intersecting the real axis.
For the following exercises, plot the complex numbers on the complex plane.

13. 12i

14. 2+3i

15. i

16. 34i

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

17. (3+2i)+(53i)

18. (24i)+(1+6i)

19. (5+3i)(6i)

20. (23i)(3+2i)

21. (4+4i)(6+9i)

22. (2+3i)(4i)

23. (52i)(3i)

24. (62i)(5)

25. (2+4i)(8)

26. (2+3i)(4i)

27. (1+2i)(2+3i)

28. (42i)(4+2i)

29. (3+4i)(34i)

30. 3+4i2

31. 62i3

32. 5+3i2i

33. 6+4ii

34. 23i4+3i

35. 3+4i2i

36. 2+3i23i

37. 9+316

38. 4425

39. 2+122

40. 4+202

41. i8

42. i15

43. i22

For the following exercises, use a calculator to help answer the questions.

44. Evaluate (1+i)k for k=4, 8, and 12. Predict the value if k=16.

45. Evaluate (1i)k for k=2, 6, and 10. Predict the value if k=14.

46. Evaluate (1+i)k(1i)k for k=4, 8, and 12. Predict the value for k=16.

47. Show that a solution of x6+1=0 is 32+12i.

48. Show that a solution of x81=0 is 22+22i.

For the following exercises, evaluate the expressions, writing the result as a simplified complex number.

49. 1i+4i3

50. 1i111i21

51. i7(1+i2)

52. i3+5i7

53. (2+i)(42i)(1+i)

54. (1+3i)(24i)(1+2i)

55. (3+i)2(1+2i)2

56. 3+2i2+i+(4+3i)

57. 4+ii+34i1i

58. 3+2i1+2i23i3+i