Transcript Real Numbers

Page 152 #19-38 ANSWERS
3-10 The Real Numbers
Student Learning Goal Chart
Lesson Reflections
3-10 LAST ONE! 
Pre-Algebra
Pre-Algebra Learning Goal
Students will
understand rational
and real numbers.
Students will understand rational and real numbers
by being able to do the following:
• Learn to write rational numbers in equivalent forms (3.1)
• Learn to add and subtract decimals and rational numbers with like denominators
(3.2)
• Learn to add and subtract fractions with unlike denominators (3.5)
• Learn to multiply fractions, decimals, and mixed numbers (3.3)
• Learn to divide fractions and decimals (3.4)
• Learn to solve equations with rational numbers (3.6)
• Learn to solve inequalities with rational numbers (3-7)
• Learn to find square roots (3-8)
• Learn to estimate square roots to a given number of decimal places and solve
problems using square roots (3-9)
• Learn to determine if a number is rational or
irrational (3-10)
3-10 The Real Numbers
Today’s Learning Goal Assignment
Learn to
determine if a
number is rational
or irrational.
Pre-Algebra
3-10 The Real Numbers
Pre-Algebra HW
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Pre-Algebra
3-10
Real
Numbers
3-10The
The
Real
Numbers
Warm Up
Problem of the Day
Lesson Presentation
Pre-Algebra
Pre-Algebra
3-10 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. –
Pre-Algebra
2
1.4
123
–11.1
3-10 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
Pre-Algebra
3-10 The Real Numbers
Today’s Learning Goal Assignment
Learn to
determine if a
number is rational
or irrational.
Pre-Algebra
3-10 The Real Numbers
Vocabulary
irrational number
real number
Density Property
Pre-Algebra
3-10 The Real Numbers
Biologists classify animals based on shared
characteristics. The gray lesser mouse lemur is
an animal, a mammal, a primate, and a lemur.
Animals
Mammals
Primates
Lemurs
Pre-Algebra
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.
3-10 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
Pre-Algebra
2
= 0.6
3
1.44 = 1.2
3-10 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…
Helpful Hint
A repeating decimal may not appear to repeat
on a calculator, because calculators show a
finite number of digits.
Pre-Algebra
3-10 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
Pre-Algebra
Irrational numbers
3-10 The Real Numbers
Additional Examples 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
Pre-Algebra
3-10 The Real Numbers
Try This: Example 1
Write all names that apply to each number.
A.
9
9
=3
whole, integer, rational, real
B.
C.
Pre-Algebra
–35.9
–35.9 is a terminating decimal.
rational, real
81
81
9
=
=3
3
3
3
whole, integer, rational, real
3-10 The Real Numbers
Additional Examples 2: Determining the
Classification of All Numbers
State if the number is rational, irrational,
or not a real number.
A.
15
15 is a whole number that
is not a perfect square.
irrational
B.
0
3
rational
Pre-Algebra
0
=0
3
3-10 The Real Numbers
Additional Examples 2: Determining the
Classification of All Numbers
State if the number is rational, irrational,
or not a real number.
C.
–9
not a real number
D.
4
9
rational
Pre-Algebra
2
3
2
4
=
3
9
3-10 The Real Numbers
Try This: Examples 2
State if the 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
Pre-Algebra
3-10 The Real Numbers
Try This: Examples 2
State if the number is rational, irrational,
or not a real number.
C.
–7
not a real number
D.
64
81
rational
Pre-Algebra
8
9
8
64
=
9
81
3-10 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.
Pre-Algebra
3-10 The Real Numbers
Additional Examples 3: Applying the Density Property
of Real Numbers
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
Pre-Algebra
3-10 The Real Numbers
Try This: 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
Pre-Algebra
3-10 The Real Numbers
Lesson Quiz
Write all names that apply to each number.
2. – 16
1. 2
2
real, integer, rational
real, irrational
State if the number is rational, irrational, or not
a real number.
3. 25
4. 4 • 9
0
rational
not a real number
5. Find a real number between –2 34 and –2 38 .
5
Possible answer –2 8 .
Pre-Algebra