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6.7: Nonterminating Divisions

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Learning Objectives
  • understand the meaning of a nonterminating division
  • be able to recognize a nonterminating number by its notation

Nonterminating Divisions

Let's consider two divisions:

9.8÷3.5
4÷3

Definition: Terminating Divisions

Previously, we have considered divisions like example, which is an example of a terminating division. A terminating division is a division in which the quotient terminates after several divisions (the remainder is zero).

Long division. 9.8 divided by 3.5. The decimal place needs to be moved to the right one space, making the problem 98 divided by 35. 35 goes into 98 twice, with a remainder of 28. Bring down a zero to find the quotient in decimal form. 35 goes into 280 exactly 8 times. The quotient is 2.8

Definition: Exact Divisions

The quotient in this problem terminates in the tenths position. Terminating divi­sions are also called exact divisions.

Definition: Nonterminating Division

The division in example is an example of a nonterminating division. A non-terminating division is a division that, regardless of how far we carry it out, always has a remainder.

Long division. 4 divided by 3 equals 1.333, with a repeating unresolved remainder, leading to a division problem that never terminates.

Definition: Repeating Decimal

We can see that the pattern in the brace is repeated endlessly. Such a decimal quotient is called a repeating decimal.

Denoting Nonterminating Quotients

We use three dots at the end of a number to indicate that a pattern repeats itself endlessly.

4÷3=1.333...

Another way, aside from using three dots, of denoting an endlessly repeating pattern is to write a bar ( ¯ ) above the repeating sequence of digits.

4÷3=1.ˉ3

The bar indicates the repeated pattern of 3.

Repeating patterns in a division can be discovered in two ways:

As the division process progresses, should the remainder ever be the same as the dividend, it can be concluded that the division is nonterminating and that the pattern in the quotient repeats. This fact is illustrated in Example of Sample Set A.
As the division process progresses, should the "product, difference" pattern ever repeat two consecutive times, it can be concluded that the division is nonter­minating and that the pattern in the quotient repeats. This fact is illustrated in Example and 4 of Sample Set A.

Sample Set A

Carry out each division until the repeating pattern can be determined.

100÷27

Solution

3.7037027¯)100.0000081           _19 0        18 9        _100    81    _190  189  

When the remainder is identical to the dividend, the division is nonterminating. This implies that the pattern in the quotient repeats.

100÷27=3.70370370... The repeating block is 703.
100÷27=3.¯703

Sample Set A

1÷9

Solution

Long division. 1 divided by 9 a nonterminating division problem with a repeating quotient of .111

We see that this “product, difference”pattern repeats. We can conclude that the division is nonterminating and that the quotient repeats.

1÷9=0.111... The repeating block is 1.
1÷9=0.¯1

Sample Set A

Divide 2 by 11 and round to 3 decimal places.

Solution

Since we wish to round the quotient to three decimal places, we'll carry out the division so that the quotient has four decimal places.

.181811¯)2.00001.1      _90    88    _20  11  _90

The number .1818 rounded to three decimal places is .182. Thus, correct to three decimal places,

2÷11=0.182

Sample Set A

Divide 1 by 6.

Solution

Long division. 1 divided by six equals .166

We see that this “product, difference” pattern repeats. We can conclude that the division is nonterminating and that the quotient repeats at the 6.

1÷6=0.16¯6

Practice Set A

Carry out the following divisions until the repeating pattern can be determined.

1÷3

Answer

0.¯3

Practice Set A

5÷6

Answer

0.8¯3

Practice Set A

11÷9

Answer

1.¯2

Practice Set A

17÷9

Answer

1.¯8

Practice Set A

Divide 7 by 6 and round to 2 decimal places.

Answer

1.17

Practice Set A

Divide 400 by 11 and round to 4 decimal places.

Answer

36.3636

Exercises

For the following 20 problems, carry out each division until the repeating pattern is determined. If a repeating pattern is not apparent, round the quotient to three decimal places.

Exercise 6.7.1

4÷9

Answer

0.¯4

Exercise 6.7.2

8÷11

Exercise 6.7.3

4÷25

Answer

0.16

Exercise 6.7.4

5÷6

Exercise 6.7.5

1÷7

Answer

0.¯142857

Exercise 6.7.6

3÷1.1

Exercise 6.7.7

20÷1.9

Answer

10.526

Exercise 6.7.8

10÷2.7

Exercise 6.7.9

1.11÷9.9

Answer

0.1¯12

Exercise 6.7.10

8.08÷3.1

Exercise 6.7.11

51÷8.2

Answer

6.¯21951

Exercise 6.7.12

0.213÷0.31

Exercise 6.7.13

0.009÷1.1

Answer

0.00¯81

Exercise 6.7.14

6.03÷1.9

Exercise 6.7.15

0.518÷0.62

Answer

0.835

Exercise 6.7.16

1.55÷0.27

Exercise 6.7.17

0.333÷0.999

Answer

0.¯3

Exercise 6.7.18

0.444÷0.999

Exercise 6.7.19

0.555÷0.27

Answer

2.0¯5

Exercise 6.7.20

3.8÷0.99

Calculator Problems

For the following 10 problems, use a calculator to perform each division.

Exercise 6.7.21

7÷9

Answer

0.¯7

Exercise 6.7.22

8÷11

Exercise 6.7.23

14÷27

Answer

0.¯518

Exercise 6.7.24

1÷44

Exercise 6.7.25

2÷44

Answer

0.0¯45

Exercise 6.7.26

0.7÷0.9 (Compare this with Exercise above)

Exercise 6.7.27

80÷110 (Compare this with Exercise above)

Answer

0.¯72

Exercise 6.7.28

0.0707÷0.7070

Exercise 6.7.29

0.1414÷0.2020

Answer

0.7

Exercise 6.7.30

1÷0.9999999

Exercise for Review

Exercise 6.7.31

In the number 411,105, how many ten thousands are there?

Answer

1

Exercise 6.7.32

Find the quotient, if it exists. 17÷0.

Exercise 6.7.33

Find the least common multiple of 45, 63, and 98.

Answer

4410

Exercise 6.7.34

Subtract 8.01629 from 9.00187 and round the result to three decimal places.

Exercise 6.7.35

Find the quotient. 104.06÷12.1.

Answer

8.6


6.7: Nonterminating Divisions is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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