Download 4-7 The Real Numbers

4-7
4-7 The
TheReal
RealNumbers
Numbers
Warm Up
Problem of the Day
Lesson Presentation
Course
Course
33
4-7 The Real Numbers
Warm Up
Each square root is between two integers.
Name the two integers.
1.
119
2. –
15
10 and 11
–4 and –3
Use a calculator to find each value.
Round to the nearest tenth.
3.
4. –
Course 3
2
1.4
123
–11.1
4-7 The Real Numbers
Problem of the Day
The circumference of a circle is
approximately 3.14 times its diameter. A
circular path 1 meter wide has an inner
diameter of 100 meters. How much farther
is it around the outer edge of the path than
the inner edge?
6.28 m
Course 3
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
Course 3
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
Course 3
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
Course 3
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.
Course 3
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
Course 3
Irrational numbers
4-7 The Real Numbers
Additional Example 1: Classifying Real Numbers
Write all names that apply to each number.
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
Course 3
4-7 The Real Numbers
Check It Out: Example 1
Write all names that apply to each number.
A.
9
9
=3
whole, integer, rational, real
B.
C.
Course 3
–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
Additional Example 2: Determining the Classification
of All Numbers
State if each number is rational, irrational,
or not a real number.
A.
21
irrational
B.
0
3
rational
Course 3
0
=0
3
4-7 The Real Numbers
Additional Example 2: Determining the Classification
of All Numbers
State if each number is rational, irrational,
or not a real number.
C.
–4
not a real number
D.
4
9
rational
Course 3
2
3
2
4
=
3
9
4-7 The Real Numbers
Check It Out: Example 2
State if each number is rational, irrational,
or not a real number.
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
Course 3
4-7 The Real Numbers
Check It Out: Example 2
State if each number is rational, irrational,
or not a real number.
C.
–7
not a real number
D.
64
81
rational
Course 3
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.
Course 3
4-7 The Real Numbers
Additional Example 3: Applying the Density Property
of Real Numbers
2
3
Find a real number between 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.
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
Course 3
4-7 The Real Numbers
Check It Out: Example 3
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
Course 3
4-7 The Real Numbers
Lesson Quiz
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 .
8
Course 3
4
8