# 6.7 Nonterminating divisions

 Page 1 / 1
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

the art of managing the production, distribution and consumption.
what is economics
okk
damfash
marginal utility is the additional satisfaction one derives from consuming additional unit of a good or service.
Fred
It's the allocation of scarce resources.
Fred
Dishan
marginal utility is the additional satisfaction one derives from consuming additional unit of a good or service.
Fred
I know the definition, but I don't understand its meaning.
Dishan
what is the must definition of economic please?
Nurudeen
demand lfs
Alpha
Economics is derived from the word Oikonomia which means management of household things. Thus, Economics is a study of household things with the constrains of allocating scare resources.
Dishan
what is Open Market Operation
dominating middlemen men activities circumstances
what Equilibrium price
what is gap
mirwais
who is good with the indifference curve
Dexter
What is diseconomic
what are the types of goods
WARIDI
how can price determination be the central problem of micro economics
marginal cost formula
you should differentiate the total cost function in order to get marginal cost function then you can get marginal cost from it
boniphace
Foday
ok
Foday
how can price determination be the central problem if micro economics
simon
formula of cross elasticity of demand
what is ceteris paribus
what is ceteris parabus
Priyanka
Ceteris paribus - Literally, "other things being equal"; usually used in economics to indicate that all variables except the ones specified are assumed not to change.
Abdullah
What is broker
scor
land is natural resources that is made by nature
scor
What is broker
scor
what is land
kafui
What is broker
scor
land is natural resources that is made by nature
scor
whats poppina nigga turn it up for a minute get it
what is this?
Philo
am from nigeria@ pilo
Frank
am from nigeria@ pilo
Frank
so
owusu
what is production possibility frontier
owusu
it's a summary of opportunity cost depicted on a curve.
okhiria
please help me solve this question with the aid of appropriate diagrams explain how each of the following changes will affect the market price and quantity of bread 1. A
ok let me know some of the questions please.
Effah
ok am not wit some if den nw buh by tommorow I shall get Dem
Hi guys can I get Adam Smith's WEALTH OF NATIONS fo sale?
Ukpen
hello I'm Babaisa alhaji Mustapha. I'm studying Economics in the university of Maiduguri
Babaisa
okay
Humaira
my name is faisal Yahaya. i studied economics at Kaduna state university before proceeding to West African union university benin republic for masters
Faisal
Mannan
Wat d meaning of management
disaster management cycle
cooperate social responsibility
igwe
Fedric Wilson Taylor also define management as the act of knowing what to do and seeing that it is done in the best and cheapest way
OLANIYI
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Jeannette has $5 and$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
What is the expressiin for seven less than four times the number of nickels
How do i figure this problem out.
how do you translate this in Algebraic Expressions
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Difference between extinct and extici spicies
7hours 36 min - 4hours 50 min
The content structure of the provided text is as follows: It is a module from a mathematics textbook that discusses nonterminating divisions. The text is divided into sections that explain the concept of nonterminating divisions, the notation used to denote them, and provides examples and exercises for practice.
no_question

the text does not contain a specific mathematical question or problem for the reader to solve. instead, it provides explanations and examples related to nonterminating divisions.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
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system
# 6.7 Nonterminating divisions

 Page 1 / 1
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

the art of managing the production, distribution and consumption.
what is economics
okk
damfash
marginal utility is the additional satisfaction one derives from consuming additional unit of a good or service.
Fred
It's the allocation of scarce resources.
Fred
Dishan
marginal utility is the additional satisfaction one derives from consuming additional unit of a good or service.
Fred
I know the definition, but I don't understand its meaning.
Dishan
what is the must definition of economic please?
Nurudeen
demand lfs
Alpha
Economics is derived from the word Oikonomia which means management of household things. Thus, Economics is a study of household things with the constrains of allocating scare resources.
Dishan
what is Open Market Operation
dominating middlemen men activities circumstances
what Equilibrium price
what is gap
mirwais
who is good with the indifference curve
Dexter
What is diseconomic
what are the types of goods
WARIDI
how can price determination be the central problem of micro economics
marginal cost formula
you should differentiate the total cost function in order to get marginal cost function then you can get marginal cost from it
boniphace
Foday
ok
Foday
how can price determination be the central problem if micro economics
simon
formula of cross elasticity of demand
what is ceteris paribus
what is ceteris parabus
Priyanka
Ceteris paribus - Literally, "other things being equal"; usually used in economics to indicate that all variables except the ones specified are assumed not to change.
Abdullah
What is broker
scor
land is natural resources that is made by nature
scor
What is broker
scor
what is land
kafui
What is broker
scor
land is natural resources that is made by nature
scor
whats poppina nigga turn it up for a minute get it
what is this?
Philo
am from nigeria@ pilo
Frank
am from nigeria@ pilo
Frank
so
owusu
what is production possibility frontier
owusu
it's a summary of opportunity cost depicted on a curve.
okhiria
please help me solve this question with the aid of appropriate diagrams explain how each of the following changes will affect the market price and quantity of bread 1. A
ok let me know some of the questions please.
Effah
ok am not wit some if den nw buh by tommorow I shall get Dem
Hi guys can I get Adam Smith's WEALTH OF NATIONS fo sale?
Ukpen
hello I'm Babaisa alhaji Mustapha. I'm studying Economics in the university of Maiduguri
Babaisa
okay
Humaira
my name is faisal Yahaya. i studied economics at Kaduna state university before proceeding to West African union university benin republic for masters
Faisal
Mannan
Wat d meaning of management
disaster management cycle
cooperate social responsibility
igwe
Fedric Wilson Taylor also define management as the act of knowing what to do and seeing that it is done in the best and cheapest way
OLANIYI
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Jeannette has $5 and$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
What is the expressiin for seven less than four times the number of nickels
How do i figure this problem out.
how do you translate this in Algebraic Expressions
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Difference between extinct and extici spicies
7hours 36 min - 4hours 50 min
user
The content structure of the provided text is as follows: It is a module from a mathematics textbook that discusses nonterminating divisions. The text is divided into sections that explain the concept of nonterminating divisions, the notation used to denote them, and provides examples and exercises for practice.

TYPE: NO_QUESTION

The text does not contain a specific mathematical question or problem for the reader to solve. Instead, it provides explanations and examples related to nonterminating divisions.