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Nonterminating Divisions

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses nonterminating divisions. By the end of the module students should understand the meaning of a nonterminating division and be able to recognize a nonterminating number by its notation.

Section Overview

  • Nonterminating Divisions
  • Denoting Nonterminating Quotients

Nonterminating Divisions

Let's consider two divisions:

  1. 9.8 ÷ 3.59.8 ÷ 3.5 size 12{9 "." "8 " div " 3" "." 5} {}
  2. 4÷ 34÷ 3 size 12{4 div " 3"} {}

Terminating Divisions

Previously, we have considered divisions like example 1, 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

Exact Divisions

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

Nonterminating Division

The division in example 2 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.

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 … 4 ÷ 3 = 1 . 333 … size 12{"4 " div " 3 "=" 1" "." "333 " dotslow } {}

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 . 3 ¯ 4 ÷ 3 = 1 . 3 ¯ size 12{"4 " div " 3 "=" 1" "." {overline {3}} } {}

The bar indicates the repeated pattern of 3.

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

  1. 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 1 of Section 4.
  2. 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 2 and 4 of Section 4.

Sample Set A

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

Example 1

100 ÷ 27100 ÷ 27 size 12{"100 " div " 27"} {}

3.70370 27 100.00000 81            ̲ 19 0         18 9         ̲ 100     81     ̲ 190   189   3.70370 27 100.00000 81            ̲ 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…100÷27=3.70370370… size 12{"100" div "27"=3 "." "70370370" dotslow } {} The repeating block is 703.

100 ÷ 27 = 3 . 703 ¯ 100 ÷ 27 = 3 . 703 ¯ size 12{"100" div "27"=3 "." {overline {"703"}} } {}

Example 2

1 ÷ 91 ÷ 9 size 12{"1 " div " 9"} {}

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…1÷9=0.111… size 12{1 div 9=0 "." "111" dotslow } {} The repeating block is 1.

1 ÷ 9 = 0 . 1 ¯ 1 ÷ 9 = 0 . 1 ¯ size 12{1 div 9=0 "." {overline {1}} } {}

Example 3

Divide 2 by 11 and round to 3 decimal places.

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.

.1818 11 2.0000 1 1       ̲ 90     88     ̲ 20   11   ̲ 90 .1818 11 2.0000 1 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 2 ÷ 11 = 0 . 182 size 12{"2 " div " 11 "=" 0" "." "182"} {}

Example 4

Divide 1 by 6.

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 . 1 6 ¯ 1 ÷ 6 = 0 . 1 6 ¯ size 12{1 div 6=0 "." 1 {overline {6}} } {}

Practice Set A

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

Exercise 1

1÷31÷3 size 12{1 div 3} {}

Solution

0.3¯0.3¯ size 12{0 "." {overline {3}} } {}

Exercise 2

5÷65÷6 size 12{5 div 6} {}

Solution

0.83¯0.83¯ size 12{0 "." 8 {overline {3}} } {}

Exercise 3

11÷911÷9 size 12{"11" div 9} {}

Solution

1.2¯1.2¯ size 12{1 "." {overline {2}} } {}

Exercise 4

17÷917÷9 size 12{"17" div 9} {}

Solution

1.8¯1.8¯ size 12{1 "." {overline {8}} } {}

Exercise 5

Divide 7 by 6 and round to 2 decimal places.

Solution

1.17

Exercise 6

Divide 400 by 11 and round to 4 decimal places.

Solution

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 7

4÷94÷9 size 12{4 div 9} {}

Solution

0.4¯0.4¯ size 12{0 "." {overline {4}} } {}

Exercise 8

8÷118÷11 size 12{8 div "11"} {}

Exercise 9

4÷254÷25 size 12{4 div "25"} {}

Solution

0.16

Exercise 10

5÷65÷6 size 12{5 div 6} {}

Exercise 11

1÷71÷7 size 12{1 div 7} {}

Solution

0.142857¯0.142857¯ size 12{0 "." {overline {"142857"}} } {}

Exercise 12

3÷1.13÷1.1 size 12{3 div 1 "." 1} {}

Exercise 13

20÷1.920÷1.9 size 12{"20" div 1 "." 9} {}

Solution

10.526

Exercise 14

10÷2.710÷2.7 size 12{"10" div 2 "." 7} {}

Exercise 15

1.11÷9.91.11÷9.9 size 12{1 "." "11" div 9 "." 9} {}

Solution

0.112¯0.112¯ size 12{0 "." 1 {overline {"12"}} } {}

Exercise 16

8.08÷3.18.08÷3.1 size 12{8 "." "08" div 3 "." 1} {}

Exercise 17

51÷8.251÷8.2 size 12{"51" div 8 "." 2} {}

Solution

6.21951¯6.21951¯ size 12{6 "." {overline {"21951"}} } {}

Exercise 18

0.213÷0.310.213÷0.31 size 12{0 "." "213" div 0 "." "31"} {}

Exercise 19

0.009÷1.10.009÷1.1 size 12{0 "." "009" div 1 "." 1} {}

Solution

0.0081¯0.0081¯ size 12{0 "." "00" {overline {"81"}} } {}

Exercise 20

6.03÷1.96.03÷1.9 size 12{6 "." "03" div 1 "." 9} {}

Exercise 21

0.518÷0.620.518÷0.62 size 12{0 "." "518" div 0 "." "62"} {}

Solution

0.835

Exercise 22

1.55÷0.271.55÷0.27 size 12{1 "." "55" div 0 "." "27"} {}

Exercise 23

0.333÷0.9990.333÷0.999 size 12{0 "." "333" div 0 "." "999"} {}

Solution

0.3¯0.3¯ size 12{0 "." {overline {3}} } {}

Exercise 24

0.444÷0.9990.444÷0.999 size 12{0 "." "444" div 0 "." "999"} {}

Exercise 25

0.555÷0.270.555÷0.27 size 12{0 "." "555" div 0 "." "27"} {}

Solution

2.05¯2.05¯ size 12{2 "." 0 {overline {5}} } {}

Exercise 26

3.8÷0.993.8÷0.99 size 12{3 "." 8 div 0 "." "99"} {}

Calculator Problems

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

Exercise 27

7÷97÷9 size 12{7 div 9} {}

Solution

0.7¯0.7¯ size 12{0 "." {overline {7}} } {}

Exercise 28

8÷118÷11 size 12{8 div "11"} {}

Exercise 29

14÷2714÷27 size 12{"14" div "27"} {}

Solution

0.518¯0.518¯ size 12{0 "." {overline {"518"}} } {}

Exercise 30

1÷441÷44 size 12{1 div "44"} {}

Exercise 31

2÷442÷44 size 12{2 div "44"} {}

Solution

0.045¯0.045¯ size 12{0 "." 0 {overline {"45"}} } {}

Exercise 32

0.7÷0.90.7÷0.9 size 12{0 "." 7 div 0 "." 9} {} (Compare this with Exercise 27.)

Exercise 33

80÷11080÷110 size 12{"80" div "110"} {} (Compare this with Exercise 28.)

Solution

0.72¯0.72¯ size 12{0 "." {overline {"72"}} } {}

Exercise 34

0.0707÷0.70700.0707÷0.7070 size 12{0 "." "0707" div 0 "." "7070"} {}

Exercise 35

0.1414÷0.20200.1414÷0.2020 size 12{0 "." "1414" div 0 "." "2020"} {}

Solution

0.7

Exercise 36

1÷0.99999991÷0.9999999 size 12{1 div 0 "." "9999999"} {}

Exercise for Review

Exercise 37

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

Solution

1

Exercise 38

((Reference)) Find the quotient, if it exists. 17÷017÷0 size 12{"17" div 0} {}.

Exercise 39

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

Solution

4410

Exercise 40

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

Exercise 41

((Reference)) Find the quotient. 104.06÷12.1104.06÷12.1 size 12{"104" "." "06" div "12" "." 1} {}.

Solution

8.6

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