Review of Real Numbers

1
Review of Real Numbers
Chapter 1
2
1.1

• Tips for Success in Mathematics

3
Getting Ready for This Course
4
General Tips for Success
Continued
5
General Tips for Success
6
Using This Text
Continued
7
Using This Text
8
Getting Help
9
Preparing for and Taking an Exam
Continued
10
Preparing for and Taking an Exam
11
Managing Your Time
12
1.2

• Symbols and Sets of Numbers

13
Set of Numbers

• Natural numbers 1, 2, 3, 4, 5, 6 . . .
• Whole numbers 0, 1, 2, 3, 4 . . .
• Integers . . . 3, -2, -1, 0, 1, 2, 3 . . .
• Rational numbers the set of all numbers that
  can be expressed as a quotient of integers, with
  denominator ? 0
• Irrational numbers the set of all numbers that
  can NOT be expressed as a quotient of integers
• Real numbers the set of all rational and
  irrational numbers combined

14
Equality and Inequality Symbols
15
The Number Line
A number line is a line on which each point is
associated with a number.
16
Order Property for Real Numbers

• For any two real numbers a and b, a is less than
  b if a is to the left of b on the number line.
• a line.
• a b means a is to the right of b on a number
  line.

Insert between the following pair of
numbers to make a true statement.
17
Absolute Value
The absolute value of a real number a, denoted by
a, is the distance between a and 0 on the
number line.
4 4
5 5
18
1.3

• Fractions

19
Numerators and Denominators
A quotient of two numbers is called a fraction.
numerator
denominator
20
Simplifying Fractions
To simplify fractions we can simplify the
numerator and the denominator.
factors
product
A fraction is said to be simplified or in lowest
terms when the numerator and denominator have no
factors in common other than 1.
21
Prime and Composite Numbers
A prime number is a natural number, other than 1,
whose only factors are 1 and itself.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
The first 10 prime numbers
A natural number, other than 1, that is not a
prime number is called a composite number. Every
composite number can be written as a product of
prime numbers
22
Product of Primes
Example Write the number 24 as a product of
primes.
24 4 ? 6
Write 24 as the product of any two whole numbers.
If the factors are not prime, they must be
factored.
2 ? 2
2 ? 3
When all of the factors are prime, the number has
been completely factored.
24 2 ? 2 ? 2 ? 3
23
The Fundamental Principal of Fractions
The Fundamental Principal of Fractions If is
a fraction and c is a nonzero real number, then
24
Multiplying Fractions
To multiply two fractions, multiply numerator
times numerator to obtain the numerator of the
product.
Multiply denominator times denominator to obtain
the denominator of the product.
25
Multiplying Fractions
Example Multiply.
Multiply numerators.
Multiply denominators.
Simplify the product by dividing the numerator
and the denominator by any common factors.
26
Dividing Fractions
Two fractions are reciprocals of each other if
their product is 1.
27
Dividing Fractions
Example Divide.
28
Fractions with the Same Denominator
To add or subtract fractions with the same
denominator, combine numerators and place the sum
or difference over the common denominator.
29
Equivalent Fractions
Equivalent fractions are fractions that represent
the same quantity.
30
Equivalent Fractions
Example Write as an equivalent fraction with
a denominator of 20.
31
Fractions without the Same Denominator
To add or subtract fractions without the same
denominator, first write the fractions as
equivalent fractions with a common denominator
The least common denominator (LCD) is the
smallest number both denominators will divide
evenly into.
32
Fractions without the Same Denominator
Example
33
1.4

• Introduction to Variable Expressions and Equations

34
Exponents
Exponential notation is used to write repeated
multiplication in a more compact form.
exponent
34
base
The expression 34 is called an exponential
expression.
Example Evaluate 26.
35
Order of Operations
36
Using the Order of Operations
Example
Simplify the expression.
Simplify numerator and denominator separately
Divide.
Add.
Simplify.
37
Evaluating Algebraic Expressions
38
Determining Whether a Number is a Solution
7 is not a solution.
39
Translating Phrases
40
Translating Phrases
Example
Write as an algebraic expression. Let x to
represent the unknown number.
a.) 5 decreased by a number b.) The quotient of
a number and 12
5
decreased by
a number

Translate
5
x
The quotient of
a number
and
12
Translate
x
?
12
41
1.5

• Adding Real Numbers

42
Adding Real Numbers

• Adding two numbers with the same sign
• Add their absolute values.
• Use their common sign as the sign of the sum.

Add the following numbers. 3 (5)
8
43
Adding Real Numbers

• Adding two numbers with different signs
• Subtract the smaller absolute value from the
  larger absolute value.
• Use the sign of the number whose absolute value
  is larger as the sign of the sum.

Add the following numbers. 4 12
8
44
Additive Inverses

• Opposites or additive inverses are two numbers
  the same distance from 0 on the number line, but
  on opposite sides of 0.
• The sum of a number and its opposite is 0.
• If a is a number, ( a) a.

Add the following numbers. 15 15
0
45
1.6

• Subtracting Real Numbers

46
Subtracting Real Numbers

• Subtracting real numbers
• If a and b are real numbers, then a b a (
  b).

Subtract the following numbers. ( 5) 6 (
3)
( 5) ( 6) 3
8
47
Complementary Angles
Complementary angles are two angles whose sum is
90o.
Find the measure of the following complementary
angles.
x 150 2x 90 150 x 90 x
60 x 60 and 150 2x 30
48
Supplementary Angles
Supplementary angles are two angles whose sum is
180o.
Find the measure of the following supplementary
angles.
x x 78 180 2x 78 180 2x 102 x
51 and x 78 129
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1.7

• Multiplying and Dividing Real Numbers

50
Multiplying Real Numbers
The product of two numbers with the same sign is
a positive number.
Example Multiply. 75 ( 3)
75 ( 3) 225
The result will always be positive.
51
Multiplying Real Numbers
The product of two numbers with different signs
is a negative number.
Example Multiply. 6(4)
6(4) 24
Example Multiply. 12( 9)
The result will always be negative.
12( 9) 108
52
Zero as a Factor
If b is a real number, then b 0 0. Also 0
b 0.
Example Multiply. 6 0
6 0 0
Example Multiply. 0 125
0 125 0
53
Evaluating Exponents
Example Evaluate (? 3)3.
(? 3)3 (? 3)(? 3)(? 3) ? 27
Odd exponent Negative result
Example Evaluate (? 2)4.
(? 2)4 (? 2)(? 2) (? 2) (? 2) 16
Even exponent Positive result
54
Quotient of Two Real Numbers
If a and b are real numbers and b is not 0, then
The product or quotient of two numbers with the
same sign is a positive number. The product or
quotient of two numbers with different signs is a
negative number.
55
Multiplying or Dividing Real Numbers
56
Zero as a Divisor or Dividend
1. The quotient of any nonzero real number and 0
is undefined. In symbols, if
is undefined.
Example Divide. 6 0
There is no answer because division by 0 is
undefined.
57
Evaluating Expressions
Example Evaluate the expression 4 (42 13)4
3.
4 (42 13)4 3
Evaluate the exponent inside the parentheses.
4 (16 13)4 3
Work inside the parentheses.
4 (3)4 3
4 81 3
Evaluate the exponent.
85 3
Add.
Subtract.
82
58
1.8

• Properties of Real Numbers

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Commutative and Associative Property

• Commutative property
• Addition a b b a
• Multiplication a b b a

• Associative property
• Addition (a b) c a (b c)
• Multiplication (a b) c a (b c)

60
Using the Properties
Example Simplify. (5 x) ? 3
(5 x) ? 3
5 (x ? 3)
5 (? 3) x
2 x
Example Simplify. 4(7x)
4(7x)
( 47)x
28x
61
Distributive Property

• Distributive Property of Multiplication Over
  Addition
• a(b c) ab ac
• Identities
• 0 is the identity element for addition.
  a 0 a and 0 a a.
• 1 is the identity element for multiplication.
  a 1 a and 1 a a.

62
Using the Distributive Property
Example Simplify. (?5)(x ? 3y)
(?5)(x ? 3y)
(?5)(x)
?
(?5)(3y)
?5x ? (?15y)
?5x 15y
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Inverses

• Inverses
• The numbers a and a are additive inverses or
  opposites of each other because their sum is 0.
  a ( a) 0