1. Manipulate real and complex numbers and solve equations
AS 91577

2. Worksheet 1

3. Quadratics
General formula:
General solution:

4. Example 1
Equation cannot be factorised.

5. Using quadratic formula
We use the substitution
A complex number

6. The equation has 2 complex solutions
Imaginary
Real

7. Equation has 2 complex solutions.

8. Example 2

9. Example 2

10. Example 2

11. Adding complex numbers
Subtracting complex numbers

12. Example

13. Example

14. (x + yi)(u + vi) = (xu – yv) + (xv + yu)i.
Multiplying Complex Numbers
(x + yi)(u + vi) = (xu – yv) + (xv + yu)i.

15. Example

16. Example

17. Example 2

18. Conjugate If
The conjugate of z is If
The conjugate of z is

19. Dividing Complex Numbers

20. Example

21. Example

22. Example

23. Solving by matching terms
Match real and imaginary
Real
Imaginary

24. Solving polynomials Quadratics: 2 solutions 2 real roots
2 complex roots

25. If coefficients are all real, imaginary roots are in conjugate pairs

26. If coefficients are all real, imaginary roots are in conjugate pairs

27. Cubic
Cubics: 3 solutions
3 real roots
1 real and 2 complex roots

28. Quartic Quartic: 4 solutions 2 real and 2 imaginary roots 4 real roots

29. Solve the quadratic for the other solutions
Solving a cubic
This cubic must have at least 1 real solutions
Form the quadratic.
Solve the quadratic for the other solutions
x = 1, -1 - i, 1 + i

30. Finding other solutions when you are given one solution.
Because coefficients are real, roots come in conjugate pairs so
Form the quadratic i.e.
Form the cubic:

31. Argand Diagram

32. Just mark the spot
with a cross

33. Plot z = 3 + i z

36. z = i
z = -1
z =1
z = -i

39. Multiplying a complex number by a real number. (x + yi) u = xu + yu i.

41. Multiplying a complex number by i. z i = (x + yi) i = –y + xi.

42. Reciprocal of z
Conjugate

43. Rectangular to polar form
Using Pythagoras
Modulus is the length
Argument is the angle
Check the quadrant of the complex number

44. Modulus is the length

45. Example 1
Rectangular form
Polar form

46. Example 2

47. Example 3

48. Converting from polar to rectangular

49. Multiplying numbers in polar form
Example 1

50. Multiplying numbers in polar form
Example 2
Take out
multiples of

51. Remove all multiples of

52. De Moivre’s Theorem
Example 1

53. De Moivre’s Theorem
Example 2
Take out
multiples of

54. Solving equations using De Moivre’s Theorem
1. Put into polar form
2. Add in multiples of
4th root 81
3. Fourth root
4. Generate solutions
Letting n = 0, 1, 2, 3
Divide angle by 4

55. Take note:

56. Useful websites Good general level
Advanced level
Good general level- Also gives proofs
Problems at 3 levels