* Add and subtract complex numbers. * Multiply and divide complex numbers. * Simplify powers of i

1. 2.4 Complex Numbers
Chapter 2 Equations and Inequalities

2. Concepts and Objectives
⚫ Objectives for this section:
⚫ Add and subtract complex numbers.
⚫ Multiply and divide complex numbers.
⚫ Simplify powers of i

3. Imaginary Numbers
⚫ As you should recall from Algebra 2, there is no real
number solution of the equation x2 = ‒1, since ‒1 has no
real square root.
⚫ The number i is defined to have the following property:
i2 = ‒1
Thus, and it is called the imaginary unit.
⚫ For a positive real number a, .
1
i = −
a i a
− =

4. a. b. c.
Imaginary Numbers (cont.)
Examples: Simplify.
16
− 70
− 48
−

5. a. b. c.
Imaginary Numbers (cont.)
Examples: Simplify.
16
− 70
− 48
−
16 4
i i
=

6. a. b. c.
Imaginary Numbers (cont.)
Examples: Simplify.
16
− 70
− 48
−
16 4
i i
= 70
i
=

7. a. b. c.
Imaginary Numbers (cont.)
Examples: Simplify.
16
− 70
− 48
−
16 4
i i
= 70
i
= 48
i

8. a. b. c.
Imaginary Numbers (cont.)
Examples: Simplify.
16
− 70
− 48
−
16 4
i i
= 70
i
= 48
i
= 16 3
i
4 3
i
=

9. Complex Numbers
⚫ In the complex number a + bi, a is the real part and bi is
the imaginary part.
⚫ a + bi = c + di if and only if a = c and b = d.
⚫ All properties of real numbers can be extended to
complex numbers.
Complex numbers
a + bi,
a and b are real
Nonreal complex
numbers a + bi, b  0
Real numbers
a + bi, b = 0
Irrational numbers
Rational numbers
Integers
Non-integers

10. Complex Numbers (cont.)
Example: Re-write the expression in standard form a + bi
8 128
4
− + −

11. Complex Numbers (cont.)
Example: Re-write the expression in standard form a + bi
8 128 8 128
4 4
8 64 2
4
8 8 2
2 2 2
4
i
i
i
i
− + − − +
=
− +
=
− +
= = − +

12. The Complex Plane
⚫ We cannot plot complex numbers on a number line as
we can with real numbers; however, we can still
represent them graphically.
⚫ To represent a complex number, we need to address the
two components of the number.
⚫ We use the complex plane, which is a coordinate
system in which the horizontal axis represents the real
component and the vertical axis represents the
imaginary component.
⚫ Complex numbers are the points on the plane, expressed
as ordered pairs (a, b), representing a + bi.

13. The Complex Plane (cont.)
⚫ Example: Plot the complex number 3‒4i on the complex
plane.

14. The Complex Plane (cont.)
⚫ Example: Plot the complex number 3‒4i on the complex
plane.
3
( )
Plot 3, 4
−

15. The Complex Plane (cont.)
⚫ Example: Plot the complex number 3‒4i on the complex
plane.
3
( )
Plot 3, 4
−
‒4
● 3 4i
−

16. Complex Number Operations
For complex numbers a + bi and c + di,
and
⚫ That is, to add or subtract complex numbers, add or
subtract the real parts and add or subtract the complex
parts.
⚫ The product of two complex numbers is found by
multiplying as if the numbers were binomials.
( ) ( ) ( ) ( )
a bi c di a c b d i
+  + = +  +
( )( ) ( ) ( )
a bi c di ac bd ad bc i
+ + = − + +

17. Complex Numbers (cont.)
Examples: Perform the given operation.
⚫
⚫
( ) ( )
4 3 6 7
i i
− + − −
( )( )
2 3 3 4
i i
− +

18. Complex Numbers (cont.)
Examples: Perform the given operation.
⚫
⚫
( ) ( )
4 3 6 7
i i
− + − − ( ) ( )
4 6 3 7
10 10
i i
i
 
= − − + − −
 
= − +
( )( )
2 4
3 3
i i
− + ( ) ( ) ( ) ( )
( )
2
6 8 9 12
6 1
4 4
12
1
3
8
2 3 3
2 3 i i
i i
i
i
i
i
i
= +
= + − −
= −
−
=
−
− −
−

19. Complex Numbers (cont.)
Examples: Perform the given operation.
⚫
⚫
( )
2
4 3i
+
( )( )
6 5 6 5
i i
+ −

20. Complex Numbers (cont.)
Examples: Perform the given operation.
⚫
⚫
( )
2
4 3i
+ 2
16 24 9
16 24 9 7 24
i i
i i
= + +
= + − = +
( )( )
6 5 6 5
i i
+ −
( )
2
36 25
36 25
61, or 61 0
i
i
= −
= − −
= +
These numbers are
examples of complex
conjugates. Their
product is always a
real number.

21. Complex Numbers (still cont.)
⚫ Complex conjugates:
⚫ We can also use this property to divide complex
numbers by multiplying top and bottom by the complex
conjugate of the denominator.
For real numbers a and b,
( )( ) 2 2
a bi a bi a b
+ − = +

22. Complex Numbers (still cont.)
Example: Write each quotient in standard form a + bi
⚫
⚫
3 2
5
i
i
+
−
5 5
3
i
i
−
+

23. Complex Numbers (still cont.)
Example: Write each quotient in standard form a + bi
⚫
⚫
3 2
5
i
i
+
−
( )( )
( )( )
3 2 5
5 5
15 13 2 13 13 1 1
25 1 26 2 2
i i
i i
i i
i
+ +
=
− +
+ − +
= = = +
+
5 5
3
i
i
−
+

24. Complex Numbers (still cont.)
Example: Write each quotient in standard form a + bi
⚫
⚫
3 2
5
i
i
+
−
( )( )
( )( )
3 2 5
5 5
15 13 2 13 13 1 1
25 1 26 2 2
i i
i i
i i
i
+ +
=
− +
+ − +
= = = +
+
5 5
3
i
i
−
+
( )( )
( )( )
5 5 3
3 3
15 20 5 10 20
1 2
9 1 10
i i
i i
i i
i
− −
=
+ −
− − −
= = = −
+

25. Powers of i
⚫ An interesting pattern develops as we look beyond i2:
1
i i
=
2
1
i = −
3 2
i
i i i
= = −

4 2 2
1
i i i
=  =
5 4
i
i i i
=  =
6 4 2
1
i i i
=  = −
7 4 3
i
i i i
=  = −
8 4 4
1
i i i
=  =

26. Powers of i (cont.)
⚫ Since this pattern repeats every fourth power, divide the
exponent by 4 and find where the remainder is on this
list.
Example: Simplify i19
19 3
19 4 4 remainder 3
i i i
 =
= = −
(In case you are using a calculator, and you have forgotten
how 4s work: .25 = R1, .5 = R2, .75 = R3)

27. Classwork
⚫ College Algebra 2e
⚫ 2.4: 6-14 (even); 2.3: 18-26 (even); 2.2: 36-46 (even)
⚫ 2.4 Classwork Check
⚫ Quiz 2.3