This document discusses complex numbers. It defines the imaginary unit i as the number whose square is -1. Complex numbers are expressed as a + bi, where a is the real part and bi is the imaginary part. All real number properties extend to complex numbers. Operations like addition, subtraction, multiplication, division and powers of i are demonstrated. Complex conjugates are used to simplify expressions and divide complex numbers. Examples are provided to simplify expressions and perform operations on complex numbers.

1. 1.3 Complex Numbers
Chapter 1 Equations and Inequalities

2. Concepts and Objectives
⚫ Understand basic concepts of complex numbers
⚫ Perform operations on complex numbers

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, .
1i = −
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 4i i= 70i=

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

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

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

9. a. b. c.
Imaginary Numbers (cont.)
Examples: Simplify.
16− 70− 48−
16 4i i= 70i= 48i
16 3
4 4
34i=

10. Imaginary Numbers (cont.)
⚫ Products or quotients with negative radicands are
simplified by first rewriting . Then the
properties of real numbers are applied, together with
the fact that i2 = ‒1.
Use this definition before using other rules for
radicals. In particular, the rule is valid
only when c and d are not both negative. Example:
c d cd=
( )( )4 9 36 6− − = = ( )( ) 2
4 9 2 3 6 6i i i− − = = = −
a i a− =

11. Imaginary Numbers (cont.)
Examples: Simplify each answer
a) b)
c) d)
7 7− − 6 10− −
20
2
−
−
48
24
−

12. Imaginary Numbers (cont.)
Examples: Simplify each answer
a) b)
c) d)
( )
2
2
7 7 7 7
7
1 7 7
i i
i
− − =
=
= − = −
2
6 10 6 10
60
1 4 15
2 15
i i
i
− − =
=
= −
= −
20 20
2 2
20
10
2
i
i
−
=
−
= =
48 48
24 24
48
2
24
i
i i
−
=
= =

13. 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

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

15. 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
− + − − +
=
− +
=
− +
= = − +

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

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

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

19. 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 5i 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.

20. Complex Numbers (still cont.)
⚫ Complex conjugates:
⚫ No more kittens need to die!
⚫ 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+ − = +

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

22. 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
−
+

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
−
+
( )( )
( )( )
5 5 3
3 3
15 20 5 10 20
1 2
9 1 10
i i
i i
i i
i
− −
=
+ −
− − −
= = = −
+

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

25. 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
 =
= = −

26. Classwork
⚫ College Algebra
⚫ Page 109: 18-36, pg. 88: 26-42, pg. 41: 58-68 (all
even)