6.7 Nonterminating divisions

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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. $9\text{.}\text{8}÷\text{3}\text{.}5$
2. $4÷\text{3}$

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 ).

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 .

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.

$\text{4}÷\text{3}=\text{1}\text{.}\text{333}\dots$

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

$\text{4}÷\text{3}=\text{1}\text{.}\overline{3}$

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 [link] of [link] .
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 [link] and 4 of [link] .

Sample set a

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

$\text{100}÷\text{27}$

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

$\text{100}÷\text{27}=3\text{.}\text{70370370}\dots$ The repeating block is 703.

$\text{100}÷\text{27}=3\text{.}\overline{\text{703}}$

$\text{1}÷\text{9}$

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

$1÷9=0\text{.}\text{111}\dots$ The repeating block is 1.

$1÷9=0\text{.}\overline{1}$

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.

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

$\text{2}÷\text{11}=\text{0}\text{.}\text{182}$

Divide 1 by 6.

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

$1÷6=0\text{.}1\overline{6}$

Practice set a

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

$1÷3$

$0\text{.}\overline{3}$

$5÷6$

$0\text{.}8\overline{3}$

$\text{11}÷9$

$1\text{.}\overline{2}$

$\text{17}÷9$

$1\text{.}\overline{8}$

Divide 7 by 6 and round to 2 decimal places.

1.17

Divide 400 by 11 and round to 4 decimal places.

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.

$4÷9$

$0\text{.}\overline{4}$

$8÷\text{11}$

$4÷\text{25}$

0.16

$5÷6$

$1÷7$

$0\text{.}\overline{\text{142857}}$

$3÷1\text{.}1$

$\text{20}÷1\text{.}9$

10.526

$\text{10}÷2\text{.}7$

$1\text{.}\text{11}÷9\text{.}9$

$0\text{.}1\overline{\text{12}}$

$8\text{.}\text{08}÷3\text{.}1$

$\text{51}÷8\text{.}2$

$6\text{.}\overline{\text{21951}}$

$0\text{.}\text{213}÷0\text{.}\text{31}$

$0\text{.}\text{009}÷1\text{.}1$

$0\text{.}\text{00}\overline{\text{81}}$

$6\text{.}\text{03}÷1\text{.}9$

$0\text{.}\text{518}÷0\text{.}\text{62}$

0.835

$1\text{.}\text{55}÷0\text{.}\text{27}$

$0\text{.}\text{333}÷0\text{.}\text{999}$

$0\text{.}\overline{3}$

$0\text{.}\text{444}÷0\text{.}\text{999}$

$0\text{.}\text{555}÷0\text{.}\text{27}$

$2\text{.}0\overline{5}$

$3\text{.}8÷0\text{.}\text{99}$

Calculator problems

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

$7÷9$

$0\text{.}\overline{7}$

$8÷\text{11}$

$\text{14}÷\text{27}$

$0\text{.}\overline{\text{518}}$

$1÷\text{44}$

$2÷\text{44}$

$0\text{.}0\overline{\text{45}}$

$0\text{.}7÷0\text{.}9$ (Compare this with [link] .)

$\text{80}÷\text{110}$ (Compare this with [link] .)

$0\text{.}\overline{\text{72}}$

$0\text{.}\text{0707}÷0\text{.}\text{7070}$

$0\text{.}\text{1414}÷0\text{.}\text{2020}$

0.7

$1÷0\text{.}\text{9999999}$

Exercise for review

( [link] ) In the number 411,105, how many ten thousands are there?

1

( [link] ) Find the quotient, if it exists. $\text{17}÷0$ .

( [link] ) Find the least common multiple of 45, 63, and 98.

4410

( [link] ) Subtract 8.01629 from 9.00187 and round the result to three decimal places.

( [link] ) Find the quotient. $\text{104}\text{.}\text{06}÷\text{12}\text{.}1$ .

8.6

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