1. Complex Numbers
Solving Quadratics

2. Complex Numbers
Solving Quadratics
x2 1  0

3. Complex Numbers
Solving Quadratics
x2 1  0
x 2  1
no real solutions

4. Complex Numbers
Solving Quadratics
x2 1  0
x 2  1
no real solutions
In order to solve this equation we define a new number

5. Complex Numbers
Solving Quadratics
x2 1  0
x 2  1
no real solutions
In order to solve this equation we define a new number
i   1 or i 2  1
i is an imaginary number

6. Complex Numbers
Solving Quadratics
x2 1  0
x 2  1
no real solutions
In order to solve this equation we define a new number
i   1 or i 2  1
i is an imaginary number
x 2  1
x   1
x  i

7. e.g . x 2  3 x  7  0

8. e.g . x 2  3 x  7  0
 3  9  28
x
2
 3   19

2

9. e.g . x 2  3 x  7  0
 3  9  28
x
2
 3   19

2
 3  19i

2

10. e.g . x 2  3 x  7  0
 3  9  28
x
2
 3   19

2
 3  19i

2
 3  19i  3  19i
x or x 
2 2

11. e.g . x 2  3 x  7  0 x 2  3x  7  0
 3  9  28
x OR
2
 3   19

2
 3  19i

2
 3  19i  3  19i
x or x 
2 2

12. e.g . x 2  3 x  7  0 x 2  3x  7  0
2
x 3  19
 3  9  28    0
x OR  2 4
2
 3   19

2
 3  19i

2
 3  19i  3  19i
x or x 
2 2

13. e.g . x 2  3 x  7  0 x 2  3x  7  0
2
x 3  19
 3  9  28    0
x OR  2 4
2 2
 3   19  x  3  19 i 2
    0
2  2 4
 3  19i

2
 3  19i  3  19i
x or x 
2 2

14. e.g . x 2  3 x  7  0 x 2  3x  7  0
2
x 3  19
 3  9  28    0
x OR  2 4
2 2
 3   19  x  3  19 i 2
    0
2  2 4
 3  19i  3 19  3 19 
 x  i  x   i  0
2  2 2  2 2 
 3  19i  3  19i
x or x 
2 2

15. e.g . x 2  3 x  7  0 x 2  3x  7  0
2
x 3  19
 3  9  28    0
x OR  2 4
2 2
 3   19  x  3  19 i 2
    0
2  2 4
 3  19i  3 19  3 19 
 x  i  x   i  0
2  2 2  2 2 
 3  19i  3  19i
x or x  3 19 3 19
2 2 x  i or x    i
2 2 2 2

16. All complex numbers (z) can be written as; z = x + iy

17. All complex numbers (z) can be written as; z = x + iy
Definitions;

18. All complex numbers (z) can be written as; z = x + iy
Definitions;
(1) All complex numbers contain a real and an imaginary part

19. All complex numbers (z) can be written as; z = x + iy
Definitions;
(1) All complex numbers contain a real and an imaginary part
Re z   x
Im z   y

20. All complex numbers (z) can be written as; z = x + iy
Definitions;
(1) All complex numbers contain a real and an imaginary part
Re z   x
Im z   y
e.g . z  3  5i

21. All complex numbers (z) can be written as; z = x + iy
Definitions;
(1) All complex numbers contain a real and an imaginary part
Re z   x
Im z   y
e.g . z  3  5i
Re z   3

22. All complex numbers (z) can be written as; z = x + iy
Definitions;
(1) All complex numbers contain a real and an imaginary part
Re z   x
Im z   y
e.g . z  3  5i
Re z   3 Im z   5

23. All complex numbers (z) can be written as; z = x + iy
Definitions;
(1) All complex numbers contain a real and an imaginary part
Re z   x
Im z   y
e.g . z  3  5i
Re z   3 Im z   5
(2) If Re(z) = 0, then z is a pure imaginary number

24. All complex numbers (z) can be written as; z = x + iy
Definitions;
(1) All complex numbers contain a real and an imaginary part
Re z   x
Im z   y
e.g . z  3  5i
Re z   3 Im z   5
(2) If Re(z) = 0, then z is a pure imaginary number
e.g . 3i,6i

25. All complex numbers (z) can be written as; z = x + iy
Definitions;
(1) All complex numbers contain a real and an imaginary part
Re z   x
Im z   y
e.g . z  3  5i
Re z   3 Im z   5
(2) If Re(z) = 0, then z is a pure imaginary number
e.g . 3i,6i
(3) If Im(z) = 0, then z is a real number

26. All complex numbers (z) can be written as; z = x + iy
Definitions;
(1) All complex numbers contain a real and an imaginary part
Re z   x
Im z   y
e.g . z  3  5i
Re z   3 Im z   5
(2) If Re(z) = 0, then z is a pure imaginary number
e.g . 3i,6i
(3) If Im(z) = 0, then z is a real number
3
e.g . ,  , e,4
4

27. All complex numbers (z) can be written as; z = x + iy
Definitions;
(1) All complex numbers contain a real and an imaginary part
Re z   x
Im z   y
e.g . z  3  5i
Re z   3 Im z   5
(2) If Re(z) = 0, then z is a pure imaginary number
e.g . 3i,6i
(3) If Im(z) = 0, then z is a real number
3
e.g . ,  , e,4
4
(4) Every complex number z = x + iy, has a complex conjugate

28. All complex numbers (z) can be written as; z = x + iy
Definitions;
(1) All complex numbers contain a real and an imaginary part
Re z   x
Im z   y
e.g . z  3  5i
Re z   3 Im z   5
(2) If Re(z) = 0, then z is a pure imaginary number
e.g . 3i,6i
(3) If Im(z) = 0, then z is a real number
3
e.g . ,  , e,4
4
(4) Every complex number z = x + iy, has a complex conjugate
z  x  iy

29. All complex numbers (z) can be written as; z = x + iy
Definitions;
(1) All complex numbers contain a real and an imaginary part
Re z   x
Im z   y
e.g . z  3  5i
Re z   3 Im z   5
(2) If Re(z) = 0, then z is a pure imaginary number
e.g . 3i,6i
(3) If Im(z) = 0, then z is a real number
3
e.g . ,  , e,4
4
(4) Every complex number z = x + iy, has a complex conjugate
z  x  iy
e.g . z  2  7i

30. All complex numbers (z) can be written as; z = x + iy
Definitions;
(1) All complex numbers contain a real and an imaginary part
Re z   x
Im z   y
e.g . z  3  5i
Re z   3 Im z   5
(2) If Re(z) = 0, then z is a pure imaginary number
e.g . 3i,6i
(3) If Im(z) = 0, then z is a real number
3
e.g . ,  , e,4
4
(4) Every complex number z = x + iy, has a complex conjugate
z  x  iy
e.g . z  2  7i z  2  7i

31. Complex Numbers
x + iy

32. Complex Numbers
x + iy
Real Numbers
y=0

33. Complex Numbers
x + iy
Imaginary Numbers Real Numbers
y0 y=0

34. Complex Numbers
x + iy
Imaginary Numbers Real Numbers
y0 y=0
Rational Numbers

35. Complex Numbers
x + iy
Imaginary Numbers Real Numbers
y0 y=0
Rational Numbers
Irrational Numbers

36. Complex Numbers
x + iy
Imaginary Numbers Real Numbers
y0 y=0
Rational Numbers
Fractions
Irrational Numbers

37. Complex Numbers
x + iy
Imaginary Numbers Real Numbers
y0 y=0
Rational Numbers
Fractions Integers
Irrational Numbers

38. Complex Numbers
x + iy
Imaginary Numbers Real Numbers
y0 y=0
Rational Numbers
Fractions Integers
Naturals
Irrational Numbers

39. Complex Numbers
x + iy
Imaginary Numbers Real Numbers
y0 y=0
Rational Numbers
Fractions Integers
Naturals
Zero
Irrational Numbers

40. Complex Numbers
x + iy
Imaginary Numbers Real Numbers
y0 y=0
Rational Numbers
Fractions Integers
Naturals
Zero
Negatives
Irrational Numbers

41. Complex Numbers
x + iy
Imaginary Numbers Real Numbers
y0 y=0
Rational Numbers
Pure Fractions Integers
Imaginary Numbers Naturals
x  0, y  0
Zero
Negatives
Irrational Numbers

42. Complex Numbers
x + iy
Imaginary Numbers Real Numbers
y0 y=0
Rational Numbers
Pure Fractions Integers
Imaginary Numbers Naturals
x  0, y  0
Zero
Negatives
NOTE: Imaginary numbers
cannot be ordered
Irrational Numbers

43. Basic Operations

44. Basic Operations
As i a surd, the operations with complex numbers are the same as surds

45. Basic Operations
As i a surd, the operations with complex numbers are the same as surds
Addition

46. Basic Operations
As i a surd, the operations with complex numbers are the same as surds
Addition
4  3i    8  2i 

47. Basic Operations
As i a surd, the operations with complex numbers are the same as surds
Addition
4  3i    8  2i 
 4  i

48. Basic Operations
As i a surd, the operations with complex numbers are the same as surds
Addition Subtraction
4  3i    8  2i 
 4  i

49. Basic Operations
As i a surd, the operations with complex numbers are the same as surds
Addition Subtraction
4  3i    8  2i  4  3i    8  2i 
 4  i

50. Basic Operations
As i a surd, the operations with complex numbers are the same as surds
Addition Subtraction
4  3i    8  2i  4  3i    8  2i 
 4  i  12  5i

51. Basic Operations
As i a surd, the operations with complex numbers are the same as surds
Addition Subtraction
4  3i    8  2i  4  3i    8  2i 
 4  i  12  5i
Multiplication

52. Basic Operations
As i a surd, the operations with complex numbers are the same as surds
Addition Subtraction
4  3i    8  2i  4  3i    8  2i 
 4  i  12  5i
Multiplication
4  3i  8  2i 

53. Basic Operations
As i a surd, the operations with complex numbers are the same as surds
Addition Subtraction
4  3i    8  2i  4  3i    8  2i 
 4  i  12  5i
Multiplication
4  3i  8  2i 
 32  8i  24i  6i 2

54. Basic Operations
As i a surd, the operations with complex numbers are the same as surds
Addition Subtraction
4  3i    8  2i  4  3i    8  2i 
 4  i  12  5i
Multiplication
4  3i  8  2i 
 32  8i  24i  6i 2
 32  32i  6

55. Basic Operations
As i a surd, the operations with complex numbers are the same as surds
Addition Subtraction
4  3i    8  2i  4  3i    8  2i 
 4  i  12  5i
Multiplication
4  3i  8  2i 
 32  8i  24i  6i 2
 32  32i  6
 26  32i

56. Basic Operations
As i a surd, the operations with complex numbers are the same as surds
Addition Subtraction
4  3i    8  2i  4  3i    8  2i 
 4  i  12  5i
Multiplication Division (Realising The Denominator)
4  3i  8  2i 
 32  8i  24i  6i 2
 32  32i  6
 26  32i

57. Basic Operations
As i a surd, the operations with complex numbers are the same as surds
Addition Subtraction
4  3i    8  2i  4  3i    8  2i 
 4  i  12  5i
Multiplication Division (Realising The Denominator)
4  3i  8  2i  4  3i   8  2i 
 32  8i  24i  6i 2 
 8  2i   8  2i 
 32  32i  6
 26  32i

58. Basic Operations
As i a surd, the operations with complex numbers are the same as surds
Addition Subtraction
4  3i    8  2i  4  3i    8  2i 
 4  i  12  5i
Multiplication Division (Realising The Denominator)
4  3i  8  2i  4  3i   8  2i 
 32  8i  24i  6i 2 
 8  2i   8  2i 
 32  32i  6
 32  8i  24i  6
 26  32i 
64  4

59. Basic Operations
As i a surd, the operations with complex numbers are the same as surds
Addition Subtraction
4  3i    8  2i  4  3i    8  2i 
 4  i  12  5i
Multiplication Division (Realising The Denominator)
4  3i  8  2i  4  3i   8  2i 
 32  8i  24i  6i 2 
 8  2i   8  2i 
 32  32i  6
 32  8i  24i  6
 26  32i 
64  4
 38  16i

68
 19 8
  i
34 34

60. Exercise 4A; 1 to 16 evens