Download Lesson 7: Infinite Decimals

Lesson 7
A STORY OF RATIOS
8•7
Lesson 7: Infinite Decimals
Classwork
Opening Exercise
a.
Write the expanded form of the decimal 0.3765 using powers of 10.
b.
Write the expanded form of the decimal 0.3333333… using powers of 10.
c.
Have you ever wondered about the value of 0.99999…? Some people say this infinite decimal has value 1.
Some disagree. What do you think?
Example 1
The number 0.253 is represented on the number line below.
Lesson 7:
Infinite Decimals
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Lesson 7
A STORY OF RATIOS
8•7
Example 2
The number
5
6
which is equal to 0.833333… or 0.83� is partially represented on the number line below.
Lesson 7:
Infinite Decimals
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Lesson 7
A STORY OF RATIOS
8•7
Exercises 1–5
1.
a.
Write the expanded form of the decimal 0.125 using powers of 10.
b.
Show on the number line the placement of the decimal 0.125.
a.
Write the expanded form of the decimal 0.3875 using powers of 10.
b.
Show on the number line the placement of the decimal 0.3875.
2.
Lesson 7:
Infinite Decimals
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Lesson 7
A STORY OF RATIOS
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3.
a.
Write the expanded form of the decimal 0.777777… using powers of 10.
b.
Show the first few stages of placing the decimal 0.777777… on the number line.
Lesson 7:
Infinite Decimals
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Lesson 7
A STORY OF RATIOS
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4.
a.
���� using powers of 10.
Write the expanded form of the decimal 0. 45
b.
���� on the number line.
Show the first few stages of placing the decimal 0. 45
a.
����.
Order the following numbers from least to greatest: 2.121212, 2.1, 2.2, and 2. 12
b.
Explain how you knew which order to put the numbers in.
5.
Lesson 7:
Infinite Decimals
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Lesson 7
A STORY OF RATIOS
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Lesson Summary
An infinite decimal is a decimal whose expanded form is infinite.
Example:
The expanded form of the decimal 0.83� = 0.83333… is
8
10
+
3
102
+
3
103
+
3
104
+ ⋯.
To pin down the placement of an infinite decimal on the number line, we first identify within which tenth it lies,
then within which hundredth it lies, then within which thousandth, and so on. These intervals have widths getting
closer and closer to a width of zero.
This reasoning allows us to deduce that the infinite decimal 0.9999… and 1 have the same location on the number
line. Consequently, 0. 9� = 1.
Problem Set
1.
a.
Write the expanded form of the decimal 0.625 using
powers of 10.
b.
Place the decimal 0.625 on the number line.
Lesson 7:
Infinite Decimals
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Lesson 7
A STORY OF RATIOS
8•7
2.
3.
a.
����� using
Write the expanded form of the decimal 0. 370
powers of 10.
b.
Show the first few stages of placing the decimal 0.370370… on
the number line.
2
Which is a more accurate representation of the fraction : 0.6666 or 0. 6� ? Explain. Which would you prefer to
3
compute with?
4.
Explain why we shorten infinite decimals to finite decimals to perform operations. Explain the effect of shortening
an infinite decimal on our answers.
5.
A classmate missed the discussion about why 0. 9� = 1. Convince your classmate that this equality is true.
6.
Explain why 0.3333 < 0.33333.
Lesson 7:
Infinite Decimals
©2015 Great Minds eureka-math.org
G8-M7-SE-1.3.0-10.2015
S.38