§ 7-3 Nonterminating Decimals
Nonterminating Decimals
How can we define nonterminating decimals?
Nonterminating Decimals
How can we define nonterminating decimals?
Definition
A nonterminating decimal is a decimal that does not end in an infinite
sequence of zeros.
Nonterminating Decimals
How can we define nonterminating decimals?
Definition
A nonterminating decimal is a decimal that does not end in an infinite
sequence of zeros.
Of interest to us here will primarily be repeating decimals.
Converting fractions to repeating decimals
Example
Convert
2
9
to a repeating decimal.
Converting fractions to repeating decimals
Example
Convert
2
9
to a repeating decimal.
2
=
9
Converting fractions to repeating decimals
Example
Convert
2
9
to a repeating decimal.
2
= .22222222...
9
Converting fractions to repeating decimals
Example
Convert
2
9
to a repeating decimal.
2
= .22222222...
9
= .2
Converting fractions to repeating decimals
Example
Convert
2
9
to a repeating decimal.
2
= .22222222...
9
= .2
Example
Convert
25
990
to a repeating decimal.
Converting fractions to repeating decimals
Example
Convert
2
9
to a repeating decimal.
2
= .22222222...
9
= .2
Example
Convert
25
990
to a repeating decimal.
25
=
990
Converting fractions to repeating decimals
Example
Convert
2
9
to a repeating decimal.
2
= .22222222...
9
= .2
Example
Convert
25
990
to a repeating decimal.
25
= .02525252525...
990
Converting fractions to repeating decimals
Example
Convert
2
9
to a repeating decimal.
2
= .22222222...
9
= .2
Example
Convert
25
990
to a repeating decimal.
25
= .02525252525...
990
= .025
Converting repeating decimals to fractions
Example
Convert .4 to a fraction.
Converting repeating decimals to fractions
Example
Convert .4 to a fraction.
n = .4
Converting repeating decimals to fractions
Example
Convert .4 to a fraction.
10n = 4.4
n = .4
Converting repeating decimals to fractions
Example
Convert .4 to a fraction.
10n = 4.4
− n = .4
Converting repeating decimals to fractions
Example
Convert .4 to a fraction.
10n = 4.4
− n = .4
9n = 4
Converting repeating decimals to fractions
Example
Convert .4 to a fraction.
10n = 4.4
− n = .4
9n = 4
n=
4
9
Converting repeating decimals to fractions
Example
Convert .043 to a fraction.
Converting repeating decimals to fractions
Example
Convert .043 to a fraction.
First, we want to get all that is not repeating to the left of the decimal
point. How do we do this?
Converting repeating decimals to fractions
Example
Convert .043 to a fraction.
First, we want to get all that is not repeating to the left of the decimal
point. How do we do this?
n = .043 ⇒ 10n = .43
Converting repeating decimals to fractions
Example
Convert .043 to a fraction.
First, we want to get all that is not repeating to the left of the decimal
point. How do we do this?
n = .043 ⇒ 10n = .43
Now, we want to get one copy of the repeating decimal to the left of
the decimal point, as before.
10n = .43 ⇒ 100(10n) = 43.43
Converting repeating decimals to fractions
Now, we can subtract as we did before.
Converting repeating decimals to fractions
Now, we can subtract as we did before.
1000n = 43.43
−10n =
.43
Converting repeating decimals to fractions
Now, we can subtract as we did before.
1000n = 43.43
−10n =
.43
990n =
43
Converting repeating decimals to fractions
Now, we can subtract as we did before.
1000n = 43.43
−10n =
.43
990n =
43
So, we have
n=
43
990
Converting repeating decimals to fractions
Example
Convert .213 to a fraction.
Converting repeating decimals to fractions
Example
Convert .213 to a fraction.
First, we want to get all that is not repeating to the left of the decimal
point. How do we do this?
Converting repeating decimals to fractions
Example
Convert .213 to a fraction.
First, we want to get all that is not repeating to the left of the decimal
point. How do we do this?
n = .213 ⇒ 100n = 21.3
Converting repeating decimals to fractions
Example
Convert .213 to a fraction.
First, we want to get all that is not repeating to the left of the decimal
point. How do we do this?
n = .213 ⇒ 100n = 21.3
Now, we want to get one copy of the repeating decimal to the left of
the decimal point, as before.
100n = 21.3 ⇒ 10(100n) = 213.3
Converting repeating decimals to fractions
Now, we can subtract as we did before.
Converting repeating decimals to fractions
Now, we can subtract as we did before.
1000n = 213.3
−100n = 21.3
Converting repeating decimals to fractions
Now, we can subtract as we did before.
1000n = 213.3
−100n = 21.3
900n =
192
Converting repeating decimals to fractions
Now, we can subtract as we did before.
1000n = 213.3
−100n = 21.3
900n =
192
So, we have
n=
192
16
=
900
75
Ordering Repeating Decimals
Example
Put the following in order from largest to smallest and be ready to
explain why you ordered as you did.
.53
.535
.53
.5
.53
What digit?
Example
What is the 12th digit of .213?
What digit?
Example
What is the 12th digit of .213?
Example
What is the 37th digit of .15274?
Addition of Repeating Decimals
Example
Find ..23 + .154.
Addition of Repeating Decimals
Example
Find ..23 + .154.
Thoughts about how we can do this?
Addition of Repeating Decimals
Example
Find ..23 + .154.
Thoughts about how we can do this?
How many digits should repeat in our answer?
.232323...
+.154154...
.386477...
Addition of Repeating Decimals
Example
Find ..23 + .154.
Thoughts about how we can do this?
How many digits should repeat in our answer?
.232323...
+.154154...
.386477...
So, our answer is .386477.
Fractions from given representations
Example
Given that
1
9
= .1 and
1
99
= .01, find a rational representation for 34.
Fractions from given representations
Example
Given that
1
9
= .1 and
1
99
= .01, find a rational representation for 34.
34 =
Fractions from given representations
Example
Given that
1
9
= .1 and
1
99
= .01, find a rational representation for 34.
1
1
34 = 3
+
9
90
Fractions from given representations
Example
Given that
1
9
= .1 and
1
99
= .01, find a rational representation for 34.
1
34 = 3
+
9
3
= +
9
1
90
1
99
Fractions from given representations
Example
Given that
1
9
= .1 and
1
99
= .01, find a rational representation for 34.
1
34 = 3
+
9
3
= +
9
1
90
1
99
34
=
99
Using an older idea
We can use an idea we talked about before the midterm to find the
rational expression that equals a repeating decimal as well.
Using an older idea
We can use an idea we talked about before the midterm to find the
rational expression that equals a repeating decimal as well.
Example
Write .7 as a fraction.
Using an older idea
We can use an idea we talked about before the midterm to find the
rational expression that equals a repeating decimal as well.
Example
Write .7 as a fraction.
We can begin by writing this in expanded fraction form.
Using an older idea
We can use an idea we talked about before the midterm to find the
rational expression that equals a repeating decimal as well.
Example
Write .7 as a fraction.
We can begin by writing this in expanded fraction form.
.7 =
Using an older idea
We can use an idea we talked about before the midterm to find the
rational expression that equals a repeating decimal as well.
Example
Write .7 as a fraction.
We can begin by writing this in expanded fraction form.
.7 =
7
7
7
+ 2 + 3 + ...
10 10
10
Using an older idea
We can use an idea we talked about before the midterm to find the
rational expression that equals a repeating decimal as well.
Example
Write .7 as a fraction.
We can begin by writing this in expanded fraction form.
7
7
7
+ 2 + 3 + ...
10
10 10
7
1
1
+
+ ...
=
1+
10
10 102
.7 =
Using an older idea
When looking at this representation, what do we have?
Using an older idea
When looking at this representation, what do we have?
7
.7 =
10
1
1
1+
+
+ ...
10 102
Using an older idea
When looking at this representation, what do we have?
7
.7 =
10
1
1
1+
+
+ ...
10 102
This is an infinite geometric series.
Using an older idea
When looking at this representation, what do we have?
7
.7 =
10
1
1
1+
+
+ ...
10 102
This is an infinite geometric series.
What is a?
Using an older idea
When looking at this representation, what do we have?
7
.7 =
10
1
1
1+
+
+ ...
10 102
This is an infinite geometric series.
What is a? a =
7
10
Using an older idea
When looking at this representation, what do we have?
7
.7 =
10
1
1
1+
+
+ ...
10 102
This is an infinite geometric series.
What is a? a =
What is r?
7
10
Using an older idea
When looking at this representation, what do we have?
7
.7 =
10
1
1
1+
+
+ ...
10 102
This is an infinite geometric series.
What is a? a =
What is r? r =
7
10
1
10
Using an older idea
When looking at this representation, what do we have?
7
.7 =
10
1
1
1+
+
+ ...
10 102
This is an infinite geometric series.
What is a? a =
What is r? r =
7
10
1
10
What we didn’t do before is look at a closed form for the sum of such
a series.
Using an older idea
Sum of an infinite geometric series
If we have a convergent infinite geometric series with initial value a
and ratio r, when the sum of the series is given by
S=
a
1−r
Using an older idea
Sum of an infinite geometric series
If we have a convergent infinite geometric series with initial value a
and ratio r, when the sum of the series is given by
S=
a
1−r
In our example, we have
S=
a
1−r
Using an older idea
Sum of an infinite geometric series
If we have a convergent infinite geometric series with initial value a
and ratio r, when the sum of the series is given by
S=
a
1−r
In our example, we have
S=
=
a
1−r
7
10
1−
1
10
Using an older idea
Sum of an infinite geometric series
If we have a convergent infinite geometric series with initial value a
and ratio r, when the sum of the series is given by
S=
a
1−r
In our example, we have
S=
=
a
1−r
7
10
1−
=
1
10
7
10
9
10
Using an older idea
Sum of an infinite geometric series
If we have a convergent infinite geometric series with initial value a
and ratio r, when the sum of the series is given by
S=
a
1−r
In our example, we have
S=
=
a
1−r
7
10
1−
=
=
1
10
7
10
9
10
7
9
Using the Geometric Series
Example
Write .34 as an infinite geometric series and find the sum.
Using the Geometric Series
Example
Write .34 as an infinite geometric series and find the sum.
.34 =
34
34
34
+
+
+ ...
100 1002 1003
Using the Geometric Series
Example
Write .34 as an infinite geometric series and find the sum.
34
34
34
+
+
+ ...
2
3
100
100
100
34
1
1
=
1+
+
+ ...
100
100 1002
.34 =
Using the Geometric Series
Example
Write .34 as an infinite geometric series and find the sum.
34
34
34
+
+
+ ...
2
3
100
100
100
34
1
1
=
1+
+
+ ...
100
100 1002
.34 =
What is a?
Using the Geometric Series
Example
Write .34 as an infinite geometric series and find the sum.
34
34
34
+
+
+ ...
2
3
100
100
100
34
1
1
=
1+
+
+ ...
100
100 1002
.34 =
What is a? a =
34
100
Using the Geometric Series
Example
Write .34 as an infinite geometric series and find the sum.
34
34
34
+
+
+ ...
2
3
100
100
100
34
1
1
=
1+
+
+ ...
100
100 1002
.34 =
What is a? a =
What is r?
34
100
Using the Geometric Series
Example
Write .34 as an infinite geometric series and find the sum.
34
34
34
+
+
+ ...
2
3
100
100
100
34
1
1
=
1+
+
+ ...
100
100 1002
.34 =
What is a? a =
What is r? r =
34
100
1
100
Using the Geometric Series
So, the sum is
S=
a
1−r
Using the Geometric Series
So, the sum is
S=
=
a
1−r
34
100
1−
1
100
Using the Geometric Series
So, the sum is
S=
=
a
1−r
34
100
1−
=
1
100
34
100
99
100
Using the Geometric Series
So, the sum is
S=
=
a
1−r
34
100
1−
=
=
1
100
34
100
99
100
34
99