Transcript 4-7 PP

4-7 The Real Numbers
Learn to determine if a number is
rational or irrational.
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
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.
setReal
of realNumbers
numbers consists of the set of
4-7 The
The
rational numbers and the set of irrational numbers.
4-7 The Real Numbers
Example 1: Classifying Real Numbers
Identify as rational or irratioanl.
A.
5
irrational
B. –12.75
rational
C.
16
2
–12.75 is a terminating decimal.
16
4
=
=2
2
2
rational
4-7 The Real Numbers
Example 2
Write all names that apply to each number.
A.
9
9
=3
rational
B.
C.
–35.9
–35.9 is a terminating decimal.
rational
81
3
rational
81
9
=
=3
3
3
4-7 The Real Numbers
Example 3: 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
0
=0
3
4-7 The Real Numbers
Example 4: 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
4-7 The Real Numbers
Example 5
State if each number is rational, irrational,
or not a real number.
A.
23
irrational
B.
9
0
undefined, so not a real number
4-7 The Real Numbers
Example 6
State if each number is rational, irrational,
or not a real number.
C.
–7
not a real number
D.
64
81
rational
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.
4-7 The Real Numbers
Example 7: 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.
32 +33 ÷2 =6 5 ÷2
5
5
5
3
1
2
3
4
=7÷2=3 1
2
3 5 3 5 13 5 35
4
32
3
2
1
A real number between 3
and 3
is 3 .
5
5
2
4-7 The Real Numbers
Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems
4-7 The Real Numbers
Lesson Quiz
Write all names that apply to each number.
1.
2. – 16
2
2
rational
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
4
8
4-7 The Real Numbers
Lesson Quiz for Student Response Systems
1. Identify rational, irrational, not real.
A. irrational
B. rational
C. Not real
D. All of the above
4-7 The Real Numbers
Lesson Quiz for Student Response Systems
2. Identify the name that applies to
A. irrational
B. rational
C. not a real number
D. none
.
4-7 The Real Numbers
Lesson Quiz for Student Response Systems
3. Identify a real number between
A. –4
B.
C.
D.
.