Section 2'4 Complex Numbers

1
Section 2.4 Complex Numbers
2
What you should learn

• How to use the imaginary unit i to write complex
  numbers
• How to add, subtract, and multiply complex
  numbers
• How to use complex conjugates to write the
  quotient of two complex numbers in standard form
• How to find complex solutions to quadratic
  equations

3
Real Number System

• 1, 2, 3, 4,
• How many natural numbers are there?

Natural
4
Real Number System

• 0, 1, 2, 3, 4,
• How many whole numbers are there?

Natural
Whole
5
Real Number System
Natural

• ...-3, -2, -1, 0, 1, 2, 3,
• How many integers are there?

Whole
Integers
6
Real Number System

• Fractions
• How many rational numbers are there?

Natural
Whole
Integers
Rational
7
Real Number System
Natural

• How many irrational numbers are there?

Whole
Integers
Irrational
Rational
8
Real Number System
Natural

• Each set is a subset of the Real Number System.
• The union of all these sets forms the real number
  system.
• The number line is our model for the real number
  system.

Whole
Integers
Irrational
Rational
Real Numbers
9
Definition of Square Root

• If a2 n then a is a square root of n.
• 42 (4)(4) 16
• ? 4 is a square root of 16
• (-4)2 (-4)(-4) 16
• ? -4 is a square root of 16

10
What square root of -16?

• Whatever it is it is not on the real number line.

11
Definition of i
The number i is such that
Imaginary Unit
12
Complex Numbers
Imaginary
REAL
Complex
13
Definition of a Complex Number

• If a and b are real numbers, the number a bi is
  a complex number, and it is said to be written in
  standard form.
• If b 0 then the number a bi a is a real
  number.
• If b ? 0, then the number a bi is called an
  imaginary number.
• A number of the form bi, where b ? 0 is called a
  pure imaginary number.

14
Examples
15
If you square a radical you get the radicand
2
2
Whenever you have i2 the next turn you will have
-1 and no i.
16
Equality of Complex numbers

• If a bi c di, then a c and b d.

17
Is a negative times a negative always positive?
Trick question. This is not a negative times a
negative.
18
Example
19
Example
20
Example
21
Example
Cancel the i factor
22
Add
Collect like terms.
23
Subtract
First distribute the negative sign.
Now collect like terms.
24
Multiplication
F
O
I
L
25
Simplify each expression. Express your answer in
form.
F-O-I-L
Recall i2-1
Combine like terms.
Combine like terms.
26
Write in the form
2
Multiply by the conjugate factor.
27
Powers of i
Anything other than 0 raised to the 0 is 1.
Anything raised to the 1 is itself.
28
Simplify as much as possible.
29
Use the Quadratic Formula
30
Solve x2 9 0
31
Match the type of complex number with its
definition.

• Real Number
• Imaginary Number
• Pure Imaginary Number

• a bi, a ? 0, b ? 0
• a bi, a 0, b ? 0
• a bi, b 0

32
Fill in the blanks

• The imaginary unit i is defined as i ____,
• Where i2 _________________ .

33
Fill in the blanks

• If a is a positive number the
• _________ ________ root of the negative number
  a is defined as

Where is i?
34
Fill in the blanks

• The numbers a bi and a bi are called
  _______ ________ and their product is the real
  number _____________ .

35
4. Find real numbers a and b such that the
equation is true.
36
10. and 14. Write the complex number in standard
form.
37
57. Perform the operation and write the result in
standard form.
38
67. Use the Quadratic Formula to solve the
quadratic equation.
39
Homework Section 2.41-79, 83 odd