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4-7 The Real Numbers
Warm Up
1-28-09
Each square root is between two integers.
Name the two integers.
1.
119
2. –
15
10 and 11
–4 and –3
Estimate each value.
Round to the nearest tenth.
3.
4. –
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2
1.4
123
–11.1
4-7 The Real Numbers
Learn to determine if a number is
rational or irrational.
Course 3
4-7 The Real Numbers
Vocabulary
irrational number
real number
Density Property
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4-7 The Real Numbers
Biologists classify animals based on shared
characteristics. The horned lizard is an animal, a
reptile, a lizard, and a gecko.
Animal
Reptile
Lizard
Gecko
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You already know that some
numbers can be classified as
whole numbers, integers, or
rational numbers. The number
2 is a whole number, an
integer, and a rational number.
It is also a real number.
4-7 The Real Numbers
Recall that rational numbers can be written as
fractions. Rational numbers can also be written
as decimals that either terminate or repeat.
3
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4
= 3.8
5
2
= 0.6
3
1.44 = 1.2
4-7 The Real Numbers
Irrational numbers can only be written as
decimals that do not terminate or repeat. If a
whole number is not a perfect square, then its
square root is an irrational number.
2 ≈1.4142135623730950488016…
Caution!
A repeating decimal may not appear to
repeat on a calculator, because
calculators show a finite number of digits.
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4-7 The Real Numbers
The set of real numbers consists of the set of
rational numbers and the set of irrational numbers.
Real Numbers
Rational numbers
Integers
Whole
numbers
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Irrational numbers
1
3
4-7 The Real Numbers
What are the different types of numbers?
Real Numbers
Rationals
Naturals
Wholes
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Irrationals
Integers
4-7 The Real Numbers
Fill In Your Real Number Chart
Counting “Natural” Numbers: 1, 2, 3, 4, 5, 6, . . .
Whole Numbers: 0, 1, 2, 3, 4, . . .
Integers: . . . -3, -2, -1, 0, 1, 2, 3, 4. . .
Rational Numbers: 0, …1/10, …1/5, …1/4, ... 33,
…1/2, …1, perfect squares
Real Numbers: all numbers
Irrationals: π, non-repeating decimal, nonperfect
squares
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4-7 The Real Numbers
Classifying Real
Numbers
Write all names that apply to each
number (whole, integer, rational,
irrational, real)
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4-7 The Real Numbers
Example 1
A.
5 is a whole number that is
not a perfect square.
irrational, real
5
B. –12.75 –12.75 is a terminating decimal.
rational, real
C.
16
2
16
4
=
=2
2
2
whole, integer, rational, real
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4-7 The Real Numbers
Example 2
A.
9
9
=3
whole, integer, rational, real
B.
C.
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–35.9
–35.9 is a terminating decimal.
rational, real
81
81
9
=
=3
3
3
3
whole, integer, rational, real
4-7 The Real Numbers
Determining the
Classification of All
Numbers
State if each number is rational, irrational,
or not a real number.
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4-7 The Real Numbers
Example 3
A.
21
irrational
B.
0
3
rational
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0
=0
3
4-7 The Real Numbers
Example 3 continued..
C.
–4
not a real number
D.
4
9
rational
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2
3
2
4
=
3
9
4-7 The Real Numbers
Example 4
A.
23
23 is a whole number that
is not a perfect square.
irrational
B.
9
0
not a number, so not a real number
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4-7 The Real Numbers
Example 4 Continued…
C.
–7
not a real number
D.
64
81
rational
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8
9
8
64
=
9
81
4-7 The Real Numbers
The Density Property of real numbers
states that between any two real
numbers is another real number. This
property is also true for rational
numbers, but not for whole numbers or
integers. For instance, there is no integer
between –2 and –3.
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4-7 The Real Numbers
Find a real number between a set
of numbers
There are many solutions. Let’s try to
find the solution that is halfway
between the two numbers
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4-7 The Real Numbers
Example 5
2
3
and 3 .
5
5
There are many solutions. One solution is
halfway between the two numbers. To find it,
add the numbers and divide by 2.
Find a real number between 3
2
3
5
1
3
+3
÷2 =6
÷2 =7÷2=3
5
5
5
2
3
1
2
3
4
3 5 3 5 13 5 35
4
32
3
2
1
A real number between 3
and 3
is 3 .
5
5
2
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4-7 The Real Numbers
Example 6
3
4
Find a real number between 4
and 4 .
7
7
There are many solutions. One solution is halfway
between the two numbers. To find it, add the
numbers and divide by 2.
3
4
4
+4
7
7
÷2
1
2
7
=8
÷2
7
3
4
47 47 4 7 14 7
42
5
1
=9÷2=4
2
6
4 7 47
4
1
A real number between 4 3 and 4
is 4 .
7
2
7
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4-7 The Real Numbers
Lesson Summary
Write all names that apply to each number.
1.
2. – 16
2
2
real, integer, rational
real, irrational
State if each number is rational, irrational, or
not a real number.
3. 25
4.
0
not a real number
4 •
9
rational
5. Find a real number between –2 3 and –2 3 .
Possible answer –25 .
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