text: # 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
explanation: The text extract is from a mathematics module that discusses nonterminating divisions. It explains the concept of nonterminating divisions, where the quotient does not terminate and has a repeating pattern. The module also covers how to denote nonterminating quotients using ellipsis (three dots) or a bar over the repeating digits. The text includes examples and exercises to practice identifying and working with nonterminating divisions.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # 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
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract is from a mathematics module that discusses nonterminating divisions. It explains the concept of nonterminating divisions, where the quotient does not terminate and has a repeating pattern. The module also covers how to denote nonterminating quotients using ellipsis (three dots) or a bar over the repeating digits. The text includes examples and exercises to practice identifying and working with nonterminating divisions.
  
  Label: no-question