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
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
33. Complex Numbers
x + iy
Imaginary Numbers Real Numbers
y0 y=0
34. Complex Numbers
x + iy
Imaginary Numbers Real Numbers
y0 y=0
Rational Numbers
35. Complex Numbers
x + iy
Imaginary Numbers Real Numbers
y0 y=0
Rational Numbers
Irrational Numbers
36. Complex Numbers
x + iy
Imaginary Numbers Real Numbers
y0 y=0
Rational Numbers
Fractions
Irrational Numbers
37. Complex Numbers
x + iy
Imaginary Numbers Real Numbers
y0 y=0
Rational Numbers
Fractions Integers
Irrational Numbers
38. Complex Numbers
x + iy
Imaginary Numbers Real Numbers
y0 y=0
Rational Numbers
Fractions Integers
Naturals
Irrational Numbers
39. Complex Numbers
x + iy
Imaginary Numbers Real Numbers
y0 y=0
Rational Numbers
Fractions Integers
Naturals
Zero
Irrational Numbers
40. Complex Numbers
x + iy
Imaginary Numbers Real Numbers
y0 y=0
Rational Numbers
Fractions Integers
Naturals
Zero
Negatives
Irrational Numbers
41. Complex Numbers
x + iy
Imaginary Numbers Real Numbers
y0 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
y0 y=0
Rational Numbers
Pure Fractions Integers
Imaginary Numbers Naturals
x 0, y 0
Zero
Negatives
NOTE: Imaginary numbers
cannot be ordered
Irrational Numbers
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