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Sullivan Algebra and
Trigonometry: Section R.1
Real Numbers
Objectives of this Section
• Classify Numbers
• Evaluate Numerical Expressions
• Work with Properties of Real Numbers
Describing Sets of Numbers
The Roster Method
The roster method is used to list the elements in
a set. For example, we can describe the set of
even digits as follows:
E = {0, 2, 4, 6, 8}
Describing Sets of Numbers
Set Builder Notation
Set Builder notation is used to describe a set of
numbers by defining a property that the
numbers share. For example, we can describe
the set of odd digits as follows:
O = {x | x is an odd digit}
Subsets of the Real Numbers
The Rational Numbers
A rational number is a number that can be
expressed as a quotient a/b . The integer a is
called the numerator, and the integer b, which
cannot be 0, is called the denominator.
All rational numbers can be written as a decimal
that either terminates or repeats. For example:
1/3 = 0.3333…
Subsets of the Rational Numbers
Subsets of the Rational Numbers can be
found by letting the denominator equal 1.
• The
Natural Numbers: {1, 2, 3, 4, …}
• The Whole Numbers: {0, 1, 2, 3, 4, …}
• The Integers: {… -3, -2, -1, 0, 1, 2, 3, …}
Subsets of the Real Numbers
The Irrational Numbers
Numbers in which the decimal neither terminates
nor repeats are called irrational numbers.
Examples of irrational numbers include:
  3.14159
2  1.41421
The set of all rational and irrational numbers
form the set of real numbers.
Approximations
In practice, the decimal representation
of an irrational number is given as an
approximation.
Truncation: Drop all of the digits that follow the
specified final digit in the decimal.
Rounding: Identify the specified digit in the decimal. If
the next digit is 5 or more, add one to the final digit.
If the next digit is 4 or less, leave the final digit as
is. Then, truncate following the final digit.
Order of Operations
1. Begin with the innermost parenthesis and
work outward. Remember that in dividing two
expressions the numerator and denominator
are treated as if they were in parenthesis. For
example:
3 2
 3  2   10
10
Order of Operations
2. Perform multiplication and divisions,
working left to right. For example:
24  4  6   66  36
3. Perform additions and subtractions,
working from left to right.
Properties of Real Numbers
Commutative Properties
Addition: a + b = b + a
Multiplication: ab = ba
Associative Properties
Addition: a + (b + c) = (a + b) + c
Multiplication: a(bc) = (ab)c
Properties of Real Numbers
Distributive Property
a(b + c) = ab + ac
(a+b)c = ac+bc
Example
4(x + 2) = 4(x) + 4(2)
= 4x + 8
Properties of Real Numbers
Identity Properties
0+a=a+0=a
a(1) = 1(a) = a
Additive Inverse Property
a + (- a) = - a + a = 0
Multiplicative Inverse Property
 1  1
a    a 1
 a  a
if
a0
Properties of Real Numbers
Multiplication by Zero
a (0) = 0
Division Properties
0
0
a
a
1 if
a
a 0
Properties of Real Numbers
Rules of Signs
a(-b) = -(ab) = (-a)b (-a)(-b) = ab - ( -a) = a
a a a
 
b b b
a a

b b
Properties of Real Numbers
Cancellation Properties
ac bc
ac
a

bc
b
implies a  b
if
if
c 0
b  0, c  0
Zero – Product Property
If ab = 0, then a = 0 or b = 0 or, both.
Properties of Real Numbers
Arithmetic of Quotients
a c
ad  bc


b d
bd
a c ac
 
b d bd
a
b  a  d  ad
c b c bc
d
if b  0 , d  0
if b  0, d  0
if b  0, c  0, d  0
Arithmetic of Quotients: Example 1
3 2 3  2 3  2
      
5 3 5  3 5 3
9  10
3 3 2 5
  
  
15 15
5 3 3 5
9  ( 10)
1


15
15
Arithmetic of Quotients: Example 2
12
5  12  20  12  20
3
5 3
5 3
20
4  3 5 4

 44
5 3
 16