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Date:
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Algebra 1 G/T
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Recurring Decimals
A rational number is a real number that can be expressed in the form ab , such that a and b are
integers and b is not zero. Such a fraction is called a vulgar fraction. A recurring decimal (or
repeating decimal) is a decimal in which there is a finite sequence of digits that repeat indefinitely.
We wish to determine whether all recurring decimals are rational numbers.
Let’s convert some commonly known recurring decimals to fractions.
Example 1:
x = 0.333333...
(multiply each side of this equation by 10)
10x = 3.333333...
(subtract the first equation from the second)
9x = 3
3
1
x= =
9
3
(divide each side of this equation by 9)
Example 2:
1
3
1
1
0.111111... = ÷ 3 =
3
9
0.333333... =
(divide each side of this equation by 3)
Example 3:
1
9
1
2
0.222222... = × 2 =
9
9
0.111111... =
(multiply each side of this equation by 2)
1. Convert the following recurring decimals to fractions. Then write each fraction in reduced
form.
(f) 0.666666... = 0.6 =
(a) 0.111111... = 0.1 =
(b) 0.222222... = 0.2 =
(g) 0.777777... = 0.7 =
(c) 0.333333... = 0.3 =
(h) 0.888888... = 0.8 =
(d) 0.444444... = 0.4 =
(e) 0.555555... = 0.5 =
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(i) 0.999999... = 0.9 =
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2. What do you notice about recurring decimals that begin to repeat immediately following the
decimal point and have a recurring sequence of length 1?
3. Convert the following recurring decimals to fractions. Then write each fraction in reduced
form.
(a) 0.232323... = 0.23 =
(b) 0.161616... = 0.16 =
(c) 0.787878... = 0.78 =
4. What do you notice about recurring decimals that begin to repeat immediately following the
decimal point and have a recurring sequence of length 2?
5. Without doing any calculations, convert the following recurring decimals to fractions (you
do not need to write your answers in reduced form). Then use a calculator to verify your
answers.
(a) 0.454545... = 0.45 =
(b) 0.183183... = 0.183 =
(c) 0.62846284... = 0.6284 =
(d) 0.328328... = 0.328 =
(e) 0.0012900129... = 0.00129 =
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Not all recurring decimals begin to repeat immediately following the decimal point. Let’s look for
a pattern in recurring decimals that have zeros after the decimal point leading up to the recurring
sequence.
Example 4:
x = 0.00712712...
100x = 0.712712...
712
100x =
999
712
x=
99900
(multiply each side of this equation by 100)
(apply the shortcut from the previous set of problems)
1
(multiply each side of this equation by
)
100
(divide each side of this equation by 99900)
6. Without doing any calculations, convert the following recurring decimals to fractions (you
do not need to write your answers in reduced form). Then use a calculator to verify your
answers.
(a) 0.004242... = 0.0042 =
(b) 0.000333... = 0.0003 =
(c) 0.000743743... = 0.000743 =
(d) 0.091249124... = 0.09124 =
Finally, we will consider recurring decimals such as 0.25632, where the recurring sequence does not
begin immediately following the decimal point, and whose preceding digits are not all zeros.
Example 5:
x = 000.25632632632...
(multiply each side of this equation by 1000)
1000x = 256.32632632632...
(subtract the first equation from the second)
999x = 256.07
(multiply each side of this equation by 100)
99900x = 25607
25607
x=
99900
(divide each side of this equation by 99900)
7. Explain why we multiplied each side of the first equation by 1000.
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Let’s try the previous example another way.
Example 5:
x = 0.25632632632...
x = 0.25 + 0.00632632...
1
632
x= +
4 99900
24975
632
x=
+
99900 99900
25607
x=
99900
(rewrite as the sum of two decimals)
(apply the shortcut from the previous set of problems)
(combine fractions)
(divide each side of this equation by 99900)
8. Convert the following recurring decimals to fractions (you do not need to write your answers
in reduced form). Then use a calculator to verify your answers.
(a) 0.685151... = 0.6851 =
(b) 2.754444... = 2.754 =
(c) 1.03612612... = 1.03612 =
(d) 0.0910202... = 0.09102 =
9. Would the process in Example 5 work for any recurring decimal? Explain.
10. Use your answer to the previous question to explain whether all recurring decimals are rational
numbers.
Michelle Krummel - http://mrskrummel.com
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