







The decimal form of a fraction 
20180112 

From Tom: Prove that the decimal representation of the quotient of 2 integers must repeat (if it does not terminate). Answered by Penny Nom. 





4 digit codes with repeating digits 
20170601 

From Morgan: what are all possible 4 digit code repeating numbers? cuz i know that you have one website on it but, it doesn't repeat 2 numbers in a code. Answered by Harley. 





The sum of two repeating decimals 
20151022 

From Kaitlin: Here is the question I am struggling with:
Is the sum of two repeating decimals always repeating decimal? Explain your answer and give an example.
I answered this question thinking that no you cannot but only when adding 0.9 repeating, but my professor said this was incorrect.
Thanks for the help!
Kaitlin Answered by Penny Nom. 





Repeated decimals 
20150509 

From Vir: Years ago I (re?)discovered 'cyclic division'. For example: if you arrange the number along a circle and put the number 142857 at the centre
all the numbers taken cyclically, starting with 1, are fully divisible by 37. Whatever the starting point of this number, it remains fully divisible by 37.
what is more, the number can be formed by taking the digits clockwise as well as anticlockwise..This I call "full cyclic divisibility". In many cases, only clockwise cyclic divisibility is possible. But I have not come across a case where ONLY anticlockwise divisibility occurs. Thus clockwise cyclic divisibility seems to be favoured. Could this be construed as a sign of chirality in mathematics?.. Answered by Chris Fisher. 





Repeating Decimal 
20100314 

From Gerald: Find the 1987th digit in the decimal equivalent to 1785/9999 starting from decimal point. Can you give us a short but powerful technique in solving this problem? thanks so much.. Answered by Chris Fisher. 





Repeating decimal to fraction 
20090918 

From joan: how can i answer it easily to convert terminating decimal to fraction?
example of this is that, convert ...143143143... to fractional form Answered by Robert Dawson. 





A repeating decimal 
20081107 

From mike: what is 0.0028282828 recurring as a fraction? Answered by Penny Nom. 





The part before the repetend 
20081102 

From Puerto: Hello,
I teach mathematics in a bilingual programme using English. I have learned that the group of digits that repeat in a repeating decimal is called repetend but i need to know how are called the previous ones when the decimal is a delayedrepeating one. I mean, in 0.34555555 5 is the repetend, the period is one and...how are called 34?
Thanks in advance. Answered by Stephen La Rocque and Penny Nom. 





0.99999.... 
20080923 

From Eve: Hi, i had a problem with change 0.99999... this recurring decimal to a fraction. I know the method, but the answer I got is 1 as you can see below.
Where have i done wrong? Answered by Harley Weston. 





Repeating fractions 
20080909 

From Juli: My teacher recently put my math class to the test...
We were deiscussing repeating fractions and she asked
us to find out what the bar over a repeating decimal is called. I found out
it was called the vinculum. But she also said to find out what the number
under the vinculum was called. I can't seem to find it anywhere. Answered by Penny Nom. 





How would put .12 with 2 repeating into a fraction? 
20080909 

From Savanna: How would put .12 with 2 repeating in fraction?
Savanna! Answered by Penny Nom. 





0.151515...=15/99 
20080908 

From Emma: This week, my Algebra teacher told us about the pattern between infinitely repeating
decimals and their corresponding fractions.
(ex. .2222222...= 2/9, .151515...=15/99, 456456456...=456/999, etc.)
I was just wondering the reason why this pattern occurs.
Is there a certain element that causes this pattern to occur?
Thanks
Emma Answered by Penny Nom. 





Nonterminating, nonrepeating decimals 
20080803 

From Peter: How do you take a random, nonterminating, nonrepeating decimal into a fraction? Answered by Stephen La Rocque. 





.9 repeating plus .2 repeating 
20080610 

From megan: How do you express the addition of .9 repeating plus .2 repeating? Answered by Penny Nom. 





Repeating decimals 
20080310 

From Blaine: Is it possible to put a repeating decimal number into a calculator to solve a problem?
EX: Write 39.3939... as a fraction. Answered by Penny Nom. 





Repeating decimals 
20080129 

From Malise: Write each repeating decimal using bar notation.
0.428571428... Answered by Penny Nom. 





Permuting the letters in ELEVEN 
20080123 

From Beth: (i) Find the number of distinct permutations that can be formed from all
of the letters of the word ELEVEN.
(ii) How many of them begin and end
with E?
(iii) How many of them have the 3 E's together?
(iv) How many
begin with E and end with N? Answered by Stephen La Rocque. 





Repeating decimals 
20071202 

From Jack: I've been working on an equation to support my theory of repeating decimals. Specifically the correlation between the 9 and 11 denominators. I wanted to know if there was already an equation to describe this correlation? Mine is x/11=.b repeating and b=9x, so if you use any single digit number for x (say 4) if 9 x 4 = 36 then 4/11 = .36363636... I just want to know if this theory already exists and if so, what is it called? Answered by Victoria West. 





Converting a repeating decimal to a common fraction 
20070822 

From isabelle: how do you turn 6.333... into a fraction in simplest form? Answered by Stephen La Rocque and Penny Nom. 





Repeating decimals and rational numbers 
20070531 

From lil: Why are repeating decimals considered rational numbers? Answered by Penny Nom and Gabriel Potter. 





0.999..., asymptotes and infinity 
20041217 

From Mike: My Name is Mike and I teach high school. I had a student ask me to explain why .9 repeating is equal to 1. Then he asked me about an asymptote, or why a parabola or any other curve for that matter can continually approach a value (like 1) and yet never attain a value of 1. He is thinking that these two should represent the same concept and yet they contradict each other. Do you have a solid explanation for him? Of by the way he is a 7th grader. Great little thinker!!!!! Answered by Claude Tardif and Harley Weston. 





The bar over repeating digits. 
20040907 

From Debbie: I would like to know the name given to the bar that is written over the repeating digits of a decimal. Answered by Penny Nom. 





7,473,000,000 divided by 52000 
20030906 

From A student: I am having trouble with a question. 7,473,000,000 divided by 52000. Our calculater gets 147311.5385. I come up with 143711.53846, and then the number starts to repeat itself Answered by Penny Nom. 





X.9999... and X+1 
20030823 

From David: I have read your answers to the questions on rational numbers, esp. 6.9999... = ? and still have a question: The simple algebraic stunt of converting repeating decimals to rational numbers seems to work for all numbers except X.999999.... where X is any integer. The fact that the method yields the integer X+1 in each case seems to violate the completeness axiom of the real numbers, namely that there is no space on the number line which does not have an number and conversely that every geometric point on the number line is associated with a unique real number. In the case of 3.999... for example, it seems that both the number 4 and the number 3.9999.... occupy the same point on the number line. How is this possible??? Answered by Penny Nom. 





AABBBCCCCAABBBCCCC... 
20030209 

From Patty: The pattern AABBBCCCCAABBBCCCC continuously repeats. What is the 2003rd letter in the pattern? Please help, I am trying to figure out. Answered by Penny Nom. 





Repeating decimals 
20030108 

From A student: If k=.9repeating, and 10k=9.9repeating then 10kk=9k, k=1 therefore .9repeating=1 and 1/3=.3repeating 3x1/3=.3repeatingx3, 3/3=.9repeating, therefore 1=.9repeating It would seem to me that .9repeating approaches one but never quite makes it. Can you clarify? Answered by Penny Nom. 





0.99999... 
20020926 

From Erica: Yesterday in my 8th grade math class we were being taught how to convert a Repeating Decimal into a fraction. Since I, for some odd reason, seem to understand math better than the rest of my classmates, i began to drown out my teachers explaination for the rule. While she was about half way through with explaining mixed decimals i came up with an unsolvable question. Like I said before, I understand how to turn a repeating decimal into a fraction, but how would I turn a repeating .9 into a fraction? We all know it would equal 9/9, but doesn't 9 over 9 also equal 1? Even though it comes very close to one, it never really equals one. I'm very confused about this and i would love it if you could clear this up for me. Answered by Penny Nom. 





The square root of 2 
20020305 

From Roger: Does two (2) have a square root or do the numbers just keep going? Are there any other numbers that behave like two when it comes to extracting the square root? Answered by Penny Nom. 





0.999999=1? 
20010906 

From Catherine: Hi! My teacher told us that 0.9 repeating equals one. We discussed how this is true. But, I was wondering if there is a proof that this is true. If so what is this called? I was trying to find information, but, it's hard when you don't know the name. Answered by Walter Whiteley. 





Repeating decimals 
20010421 

From Sarah: Hi, I'm working on a project for school. The theory I choose was "When turned into a fraction, a repeating decimal has a denometor that is a multiple of three." I have a couple of questions about this topic. My first question is, have you ever heard of this, and what can you tell me about it? My second question is, when I was testing this theory I came across .999... now, when this is a fraction it is 9/9 which is equal to one. The denometor is a multiple of 3, but it's a whole number. I don't understand how a decimal can be equal to a whole number since a decimal is a piece of a whole number. Please don't just show me a math problem, I don't want to see a math problem. I want to see an explanation of this theory and the decimal .999... Answered by Penny Nom. 





1 = 0.999... 
20010413 

From Joan: I have a middle grade math question for you. I would like to know why .9999... = 1 ? I can not use algebra to show this or the following: We agree that 2 = 2 and that 22 = 0, so
1.00000...... 0.99999.....  0.000000...... and 0.000... = 0 therefore 0.9 = 1 OR 1/3 = 0.333333 and 3 X 1/3 = 1, so if 3 X 0.333... = 0.999... then 0.999... = 1 My teacher says that I can not use the above example to show why this is true, and that I must use a couple different examples. He says that there are several other ways. Do you know any? I could really use the help because I can't think of any other ways to show this is true. Thanks for any help you can give. Answered by Penny Nom. 





Finite nonperiodical numbers 
20010327 

From Wouter: Is there anyone who knows the official name for decimal finite nonperiodical numbers such as 0.4 or 0.25 as opposite of numbers like 0.3333333... or 0.28571428571428...? Answered by Penny Nom. 





The repetend in repeating decimals 
20010321 

From Sharon: What is the name for the bar over the repetend in repeating decimals? Also, what is the name of the long division "house"? Answered by Penny Nom. 





More repeating decimals 
20010117 

From Alan: I am neither a math teacher nor a student, but I hope you will consider my question anyway. I recently was discussing repeated decimals with a friend, and went on the web to find out more about a pattern was looking for years ago. In doing so I came upon your correspondence on repeating decimals. . . . Answered by Chris Fisher and Penny Nom. 





Repeating decimal notation 
20001231 

From Clarende Duby: I have seen single repeating decimals for ex. 1/3 = 0.3 with the dot above the 3 to represent the repeating decimal. Then, there is the more common form of the bar over the top of the repeating number or group of numbers (called the period?). Which notation is correct? Answered by Penny Nom. 





Repeating decimals 
20001006 

From Mary O'Sullivan: I understand that with repeating decimals (those with a pattern), the number of digits repeated is put into fraction form with the same number of 9's ex. 0.4444 = 4/9 0.145145145 = 145/999 How can I explain why this is so? Some inquisitive 8th graders are anxious to find out! Answered by Penny Nom and Walter Whiteley. 





Rational Numbers 
20000914 

From Josh Kuhar: How can you tell a number is rational? Answered by Harley Weston. 





The sum of repeating decimals 
20000215 

From Caitlin Harris: Express 0.111... + 0.121212... + 0.123123123... as a repeating decimal and its equivalent fraction. Show work. Also, are there any extensions to this problem? In other words, are there any questions that we could ask that may be related to this problem? Answered by Penny Nom. 





Repeating decimals 
19991122 

From Andrew: Is 1.9 repeating the same as 2? Answered by Walter Whiteley. 





Repeating decimals 
19990918 

From Kavoos Bybordi: I dont know how to change a reccuring decimal to a fraction please could you tell me the method. Answered by Penny Nom. 





Repeating decimals 
19990521 

From Stan: Hi, I am in Honors Math, and have confronted everyone, including teachers, about repeating decimals. What interests me is the number 0.9... and 1. Everyone says that since there is no number between 0.9...(repeating) and 1, that 0.9... = 1. However, isn't a repeating number a representation of a number, and not a real number? Let's look at it this way. 0.9 is close to 1. 0.99 is closer. 0.99999999999999 is even closer. so, 0.9... is a representation of it's closeness to 1. it's an active number... I don't understand how 0.9... is equal to 1. Please help me prove that 0.9... does NOT = 1. Answered by Penny Nom. 





The square root of two is never supposed to end 
19990306 

From a wondering student: i am algebra II and am in the 9th grade. today we were talking about rational and irrational numbers. When we were talking about square roots my friend and i were talking and we thought of something. if you have a square with sides of length one then the diagonal of the square is the square root of 2. Now the square root of two is never supposed to end. But the diagonal of the square ends so therefore doesn't the square root of 2 end. our math teacher did not really answer our question because it was not in the lesson plan and not to many people would see where we were coming from. the answer is really bugging me and i would like to have your input. Answered by Jack LeSage and Penny Nom. 





6.99999... = ? 
19981205 

From Tom: I have had a rather heated arguement with my students. Please settle this for me. Solve <,>, = 6.99999... __ 7 Thank you. Answered by Penny Nom. 





Terminating decimals 
19981116 

From Debra Karr: A college student studying elementary education asked me a question that I could not think of the correct answer. How can you look at a fraction and tell if is a terminating or non terminating decimal? Answered by Jack LeSage and Penny Nom. 





Repeating Decimals 
19981001 

From Chris Norton: Could you PLEASE give the formula to find out the number of digits in a repeating decimal before it repeats. I have been trying to get it for weeks from Math sites on the Internet. Can you please, please help me ? Chris Norton Answered by Chris Fisher and Penny Nom. 





Rational and Irrational Numbers 
19980919 

From Ri: I am trying to explain rational & irrational numbers to my niece who is grade 7 and am having difficulties. Could you please explain the difference between rational & irrational numbers. Thank you Ri Answered by Penny Nom. 





A Question About Pi. 
19970708 

From Mike Chan: I have read the section Repeating Decimals in your data base. It mention that 1/17 has at most only 16 repeating digits. But, why does "pi" have an infinite number of digits (and not repeating ). Answered by Penny Nom. 





Repeating Decimals 
19970124 

From Grant Reed: Is there a way to tell that the repeating decimal for 1/17 has no more than 16 repeating digits? Answered by Penny Nom. 





Repeating decimals 
19960909 

From Alice: What is the line called that is placed over the decimal to show that it is infinitely repeating? Answered by The Centralizer. 

