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Transcript
The Set of Real
Numbers
KEY CONCEPT
The Real Numbers
The Real Numbers consist of all rational and irrational
numbers.
The Venn Diagram below shows the relationships among the sets of numbers.
Real Numbers
Rational Numbers
Integers
Whole Numbers
Natural Numbers
Irrational Numbers
Classify the following real
numbers.
To decide to which set of numbers a real number belongs, you may need
to rename the number in a different form.
0.54
This is a repeating decimal.
Rational number
44
4
11
Whole number
Integer
Rational number
This is a natural number.
KEY CONCEPT
The Completeness Property for Points
on the Number Line
Every real number corresponds to exactly one point on a
number line and every point on the number line corresponds
to exactly one real number.
The origin of a number line is zero.
Positive and negative numbers are
often called signed numbers.
Points to the left of zero correspond
to negative numbers.
Points to the right of zero correspond
to positive numbers.
Zero is neither positive nor negative.
A number line can help compare and order real numbers.
Graph the following real numbers.
4
2 ,  3 ,  , 2, and  1.7
5
The farther to the right a number is on the number line, the greater it is.
Order the real numbers from least to greatest.
4
 3 ,  1.7, 2 , 2, 
5
The absolute value of a real number, n, is the distance from 0 to n on a
number line.
n
Since distance is always positive, the absolute value of a nonzero
number is always positive.
3.5  3.5
 3.5  3.5
Two real numbers are opposites (or additive inverses) if they are on opposite
sides of 0 and they have the same absolute value.
KEY CONCEPT
For any real number a,
a + (–a) = 0
 (5.6)  5.6
opposite of 5.6.”
and
–a + a = 0
 (5.6)  5.6
opposite of -5.6.”
Find the value of each expression.
 (30)  (12)
What is the
opposite of
-30?
What is the
opposite of
-12?
First find the opposite of each
factor.
Then evaluate the expression.
30 12  360
Find the value of each expression.
 6.9  2.9
What is the
absolute
value of 6.9?
What is the
absolute
value of
2.9?
First find the absolute value of
each number.
Then evaluate the expression.
6.9  2.9  4
Find the value of each expression.
 9  75   675  (675) 675
Find the opposite of the absolute
value of the product of 9 and 75.
First multiply.
Find the opposite.
Find the absolute value.
Find the opposite of the absolute
value of the sum of 3.9 and 5.2.
 3.9  5.2   9.1  (9.1)  9.1
When you apply an operation (for example addition) to any numbers in a
set and the result is also a number in that set, the set is said to be closed
under the operation. This is called the Closure Property.
Determine if each set of numbers is closed under
the indicated operation.
The set of real numbers under addition.
2 1 3
 
8 8 8
 4  3  7
This means to check
whether the sum of any two
real numbers will result in
another real number.
The set of real numbers is closed under addition.
The set of integers under division.
82  4
2  8  0.25
A counterexample shows that
a set of numbers is not closed
under an operation.
This means to check
whether the quotient of any
two integers will result in
another integer.
The set of integers is not closed under division.