Download Sets of Real Numbers (0-2)

Sets of Real Numbers (0-2)
Objective: Classify and use real numbers.
Sets of Real Numbers
• A number line can be used to show the sets of natural
numbers, whole numbers, integers, and rational
numbers.
• Values greater than 0, or positive numbers, are listed to
the right of 0, and values less than 0, or negative
numbers, are listed to the left of 0.
Sets of Real Numbers
Number Set
Description
Examples
Natural Numbers
Counting Numbers
1, 2, 3, . . .
Whole Numbers
Counting Numbers
and Zero
0, 1, 2, 3, . . .
Integers
Whole Numbers
and their Opposites
. . ., -3, -2, -1, 0,
1, 2, 3, . . .
Rational Numbers
Numbers that can be
expressed in the form a/b,
where a and b are integers
and b  0.
1
23
1 , 0.5,
, 0.3
4
41
Irrational Numbers
Numbers that cannot be
expressed as terminating or
repeating decimals, or in the form
a/ , where a and b are integers and
b
b  0.
Real Numbers
Rational and
Irrational Numbers
Together
3, 15,  4 10, 
All Numbers
Graph (If Possible)
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
Venn Diagram
Real Numbers
Rational Numbers
Integers
Whole Numbers
Natural
Numbers
Irrational Numbers
Example 1
• Name the set or sets of numbers to which each real
number belongs.
9
10
121 = 11
18
Rational Number
Natural Number
Irrational Number
Real Number
Whole Number
Real Number
Integer
Rational Number
Real Number
Graphing Sets of Numbers
• To graph a set of numbers means to draw, or plot, the
points named by those numbers on a number line.
• The number that corresponds to a point on a number line
is called the coordinate of that point.
• The rational numbers and the irrational numbers
complete the number line.
Example 2
• Graph each set of numbers on a number line. Then
order the numbers from least to greatest.
1
3 1 7
,

,
,

1


2
4 4 4
3

25,
4.6,
15,
4


8


7
 9


,

2
,

2.1
,

8


10
 4

-2
-1
0
1
2
 1 1 3 7
1 ,  , , 
 2 4 4 4
3.2
3.6
4.0
4.4
4.8
5.2
5.6
6.0
3


 15, 4 , 4.6, 25
8


-3.0
-2.8
-2.6
7
9


 8,  2 ,  ,  2.1
10 4


-2.4
-2.2
-2.0
Repeating Decimals
• Any repeating decimal can be written as a fraction.
• Write 0.8 as a fraction in simplest form.
– Step 1: Let N represent the repeating decimal. Since only one
digit repeats, multiply each side by 10 (If two digits repeat
multiply by 100, for three multiply by 1000, etc.). Simplify.
N = 0.888. . .
10(N) = 10(0.888. . .)
10N = 8.888. . .
– Step 2: Subtract N from 10N to eliminate the part of the number
that repeats. Divide each side by the number in front of N.
10N = 8.888. . .
Simplify.
-(N = 0.888. . .)
8
N
9N = 8
9
9
9
Example 3
• Write 0.23 as a fraction in simplest form.
N = 0.232323. . .
100(N) = 100(0.232323. . .)
100N = 23.232323. . .
100N = 23.232323. . .
-(N = 0.232323. . .)
99N = 23
99
99
23
N
99