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
i 1
or
i 2 1
i is an imaginary 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
x 2 1
x 1
x i
7. All complex numbers (z) can be written as; z = x + iy
Definitions;
(1) All complex numbers contain a real and an imaginary part
8. 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
9. 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
10. 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
11. 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
12. 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
13. 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
14. 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
z 2 7i
e.g . z 2 7i
18. Complex Numbers
x + iy
Imaginary Numbers
y0
Real Numbers
y=0
Rational Numbers
Irrational Numbers
19. Complex Numbers
x + iy
Imaginary Numbers
y0
Real Numbers
y=0
Rational Numbers
Fractions
Integers
Irrational Numbers
20. Complex Numbers
x + iy
Imaginary Numbers
y0
Real Numbers
y=0
Rational Numbers
Fractions
Integers
Naturals
Zero
Irrational Numbers
21. Complex Numbers
x + iy
Imaginary Numbers
y0
Real Numbers
y=0
Rational Numbers
Fractions
Integers
Naturals
Zero
Negatives
Irrational Numbers
22. Complex Numbers
x + iy
Imaginary Numbers
y0
Real Numbers
y=0
Rational Numbers
Pure
Imaginary Numbers
x 0, y 0
Fractions
Integers
Naturals
Zero
Negatives
Irrational Numbers
23. Complex Numbers
x + iy
Imaginary Numbers
y0
Real Numbers
y=0
Rational Numbers
Pure
Imaginary Numbers
x 0, y 0
Fractions
Integers
Naturals
Zero
Negatives
NOTE: Imaginary numbers
cannot be ordered
Irrational Numbers
24. Basic Operations
As i a surd, the operations with complex numbers are the same as surds
25. Basic Operations
As i a surd, the operations with complex numbers are the same as surds
Addition
4 3i 8 2i
4 i
26. 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
27. 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
28. 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
29. 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
30. 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
Division (Realising The Denominator)
4 3i
8 2i
39. Complex Equations
If two complex numbers are equal, then their real parts are equal and
their imaginary parts are equal
i.e. If a1 b1i a2 b2i
then
a1 a2
b1 b2
40. Complex Equations
If two complex numbers are equal, then their real parts are equal and
their imaginary parts are equal
i.e. If a1 b1i a2 b2i
then
a1 a2
b1 b2
41. Complex Equations
If two complex numbers are equal, then their real parts are equal and
their imaginary parts are equal
i.e. If a1 b1i a2 b2i
then
a1 a2
b1 b2
e.g . (i)
2 3i x iy 4 2i
42. Complex Equations
If two complex numbers are equal, then their real parts are equal and
their imaginary parts are equal
i.e. If a1 b1i a2 b2i
then
a1 a2
b1 b2
e.g . (i)
2 3i x iy 4 2i
2 x 2iy 3ix 3 y 4 2i
43. Complex Equations
If two complex numbers are equal, then their real parts are equal and
their imaginary parts are equal
i.e. If a1 b1i a2 b2i
then
a1 a2
b1 b2
e.g . (i)
2 3i x iy 4 2i
2 x 2iy 3ix 3 y 4 2i
2 x 3 y 4 and 3 x 2 y 2
44. Complex Equations
If two complex numbers are equal, then their real parts are equal and
their imaginary parts are equal
i.e. If a1 b1i a2 b2i
then
a1 a2
b1 b2
e.g . (i)
2 3i x iy 4 2i
2 x 2iy 3ix 3 y 4 2i
2 x 3 y 4 and 3 x 2 y 2
4x 6 y 8
9 x 6 y 6
45. Complex Equations
If two complex numbers are equal, then their real parts are equal and
their imaginary parts are equal
i.e. If a1 b1i a2 b2i
then
a1 a2
b1 b2
e.g . (i)
2 3i x iy 4 2i
2 x 2iy 3ix 3 y 4 2i
2 x 3 y 4 and 3 x 2 y 2
4x 6 y 8
9 x 6 y 6
5x
14
14
x
5
46. Complex Equations
If two complex numbers are equal, then their real parts are equal and
their imaginary parts are equal
i.e. If a1 b1i a2 b2i
then
a1 a2
b1 b2
e.g . (i)
2 3i x iy 4 2i
2 x 2iy 3ix 3 y 4 2i
2 x 3 y 4 and 3 x 2 y 2
14
16
, y
x
5
5
4x 6 y 8
9 x 6 y 6
5x
14
14
x
5