Convert any repeating decimal to a fraction?

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Discussion Overview

The discussion revolves around the conversion of repeating decimals to fractions, exploring various methods and examples, including specific cases like 0.88118811... and 0.5333.... Participants share their approaches, challenge each other's reasoning, and seek clarification on the processes involved.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that any repeating decimal can be converted to a fraction, providing examples such as 0.555... = 5/9 and 0.5333... = 8/15.
  • One participant claims to have found the fraction for 0.88118811... as 1762376220/2000000000, indicating a lengthy process.
  • Another participant suggests a method involving dividing by the number of 9s corresponding to the length of the repeating portion, leading to the fraction 8811/9999, which reduces to 89/101.
  • Concerns are raised about the accuracy of the method used for 0.5333..., with a participant questioning whether the repeated portion was correctly identified.
  • Some participants discuss the need to separate non-repeating and repeating parts of a decimal for accurate conversion.
  • There are mentions of different methods to derive fractions from repeating decimals, indicating that multiple approaches exist.
  • One participant expresses confusion about the definitions of variables used in the conversion process.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the methods for converting repeating decimals to fractions, with multiple competing views and approaches presented throughout the discussion.

Contextual Notes

Some methods discussed rely on specific assumptions about the structure of repeating decimals, and there are unresolved questions regarding the identification of repeating portions and the accuracy of various conversion techniques.

YesIam
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A Math quicky-Can you convert any repeating decimal to a fraction? For example: 0.555... converts to 5/9, and 0.533333... converts to 8/15. So convert 0.88118811... to a fraction.
 
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1762376220/2000000000=.88118811 Took me forever to find it.
 


YesIam said:
A Math quicky-Can you convert any repeating decimal to a fraction? For example: 0.555... converts to 5/9, and 0.533333... converts to 8/15. So convert 0.88118811... to a fraction.

You simply divide by the number of 9s matching the length of the repeated pattern, then reduce any common factors between the numerator and denominator.

So for example, 0.88118811... = 8811/9999. You can then factorise the top and bottom and cancel any common factors. In this case, both the top and bottom have a factor of 99, so I get 89/101. (Wolfram Alpha is good for finding prime factors).
 
Hi YesIam! :smile:

A repeating decimal is of the form m(∑k=r∞ 10-kn), = m/10r(1 - 10-n) = m10n/10r(10n - 1) :wink:
 


Jonathan Scott said:
You simply divide by the number of 9s matching the length of the repeated pattern, then reduce any common factors between the numerator and denominator.

So for example, 0.88118811... = 8811/9999. You can then factorise the top and bottom and cancel any common factors. In this case, both the top and bottom have a factor of 99, so I get 89/101. (Wolfram Alpha is good for finding prime factors).

I am, having trouble with this - please show me where I may be going wrong. e.g., In 0.5333... the repeated portion is 3, correct...3/9 does not equal 8/15. At first reading I find the above solution works only if 'all' of the digits of the repeating decimal fraction are included in the 'repeated portion'. I have come across three ways to do this but I only remember one.

As for tiny-tim's solution...WOW, does this really work? Some guy must have stayed up for more than one night to come up with that !
 
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Hi tiny-tim,
I don't mean to put you on the spot, but if you happen to know just what all of the factors equal I would like to give it a shot. Thanks.
 
I mean if you have .00000000147147147147…

then m = 147, r = 11, n = 1000 :wink:
 


YesIam said:
I am, having trouble with this - please show me where I may be going wrong. e.g., In 0.5333... the repeated portion is 3, correct...3/9 does not equal 8/15. At first reading I find the above solution works only if 'all' of the digits of the repeating decimal fraction are included in the 'repeated portion'.

That's correct. So you have to split off the non-repeating part from the repeating part.
0.5333...
= 0.5 + 0.0333...
= 5/10 + (0.333...)/10
= 5/10 + (3/9)/10 (Yeah we both know 0.333... it is 1/3, but I'm using the general method)
= 5/10 + 1/30
= 15/30 +1/30
= 16/30
= 8/15
 


Okay tiny-tim...just so I don't have to spin my tires too much.
M = equals the repeated portion
r = 11...does this mean that, 0.00000000147 has 11 decimal places?
n = 1,000...does this mean that, 147 stands for 147/1000, i.e., if the repeating portion was 1478, then 'n' would be equal to 10,000? This is the only thing that I can think of right now.

Thanks,
 
  • #10
Hi YesIam! :wink:

Yes yes and … yes :smile:
 
  • #11


Well thank you tiny-tim...piece of cake now, huh? But this is the only solution that I knew before this...
For 0.5333...subtract 5 from 53...we get 48, use this as a numerator in a fraction where the denominator is 90. The 9 = a 9 for each number of digits in the repeated portion and the 0 = a zero added for each number of digits in the non-repeating portion.
Thusly: 48/90 reduces to 8/15

And thank you AlephZero.
 
  • #12


Hi tiny-tim...I see in your profile that you are into Math-good. Now the PF may 86 me forever because this may be a back door flagrant violation-I don't know. But this Evo locked me out on the 2+2 posting I made. char.limit and Evo had no idea...I just don't like hard fast statements like I don't know what I am talking about and I wasn't successful in convincing char.limit of this-I was trying to be helpful but...it's their railroad...anyway would you look at my 2+2 posting and tell me if you know of any other numbers. And please don't say that that there are only 'two'. Fact is there are an infinite number of other numbers. Have a nice year if I am forever banned from the PF.
 
  • #13


Define domain.
 
  • #14


YesIam said:
A Math quicky-Can you convert any repeating decimal to a fraction? For example: 0.555... converts to 5/9, and 0.533333... converts to 8/15. So convert 0.88118811... to a fraction.

This is how I do this:

For -.555... ,
Let x = 0.555…, This is the number we want to represent as a fraction.
Then 10x = 5.555… . Now do the subtraction: 10x - x = 9x.

10x = 5.555…
 -1x =-0.555…
  9x = 5.000…

So, we have 9x=5 . Solving for x gives: x=5/9 . Remembering that we started with x = 0.555…, we now have:

0.555… = 5/9 .
​
Now for 0.5333…, without all the words.

Let x = 0.5333… then 10x = 5.333… and 100x = 53.333…

100x = 53.333…
 -10x = -5.333…
  90x = 48.000…  →  x = 48/90 (8×6)/(15×6) = 8/15 .

So, 0.5333… = 8/15 .
​
Another repeating decimal which is in this thread is 0.88118811…

Let x=0.88118811…, then 10000x = 8811.88118811…

10000x = 8811.88118811…
           -x =       -0.88118811…
 9999x = 8811
So x = 0.88118811… = 8811/9999 = 89/101

0.88118811… = 89/101​

I hope that helps.
 
  • #15


YesIam said:
2+2.
A two by two matrix A with a 1 in each of the 4 positions has this property (the product A*A equals the sum A+A), but is not a number.
 
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  • #16


Hi SammyS - Hey, this at first glance looks like the shortest solution so far. I am sure it is helpful to everyone. I did not realize that there were so many ways to do this. I had just remembered the one way from years ago. I am glad I made the post. Yes, thanks.
 
  • #17


Hi Jimmy Snyder - You may be correct about that, I don't know. I was just asking...if someone could think of some other numbers. I sure won't hold it against you in any way if you can't - believe me.
 
  • #18


Re: 2+2, I also concluded "infinity", but that's arguably not a number either-- I guess it depends who you ask?

DaveE
 
  • #19


Hi Jimmy Snyder and others - You can probably learn the answer as soon as any other, as soon as someone thinks about it and comes up with the answers. Being a reasonable person, I can not think of any reason to just tell everyone nor give them even a clue, so I won't, okay. But I will say this...I think that this thread would go much smoother if they would refrain from posting their reasons why they 'know' that there are no other answers to this problem.
 
  • #20


davee123 - Here we go again. I didn't say that infinity 'was' a number..I said that there were an infinite number of possible answers. So just give one or two others, If you can do that, you can easily come up with 'all' of the rest.
 
  • #21


I'll put it differently. Define number.

As it should be obvious after Jimmy's post, solution depends on what x is allowed to be.
 
  • #22


Hi everyone - Let me make a further statement. Since it appears to me that no one on these forums really wants to try to solve anything more complicated than THEY can solve, AND...they don't want to hear about it, and therefore I have learned that I am not the person that belongs here. So, as I stated above, since I believe myself to be a reasonable person, I can see no reason to bring up any more questions on these forums. It appears to me that everyone is perfectly happy with the past pervading attitude.
 
  • #23


YesIam said:
I have learned that I am not the person that belongs here.
If you mean to leave, then would it hurt to publish the answer before you go? After all, that author of that 1943 book told you.
 
  • #24


Jimmy Snyder said:
If you mean to leave, then would it hurt to publish the answer before you go? After all, that author of that 1943 book told you.

Hi Jimmy - That gave me the best smile I have had all day...and you are right and I had to buy two of his books..the same book! I said that I wouldn't be asking anymore question. But hey, there is a lot of information on this forum that I can read after I filter out all the crying. But no, that would be too easy - someone will figure it out. I'll give you a clue. If you will figure it out, you will probably do better than I. I just read it and then forgot it after 20 years.

Cheers,
 
  • #25


YesIam said:
I'll give you a clue. If you will figure it out, you will probably do better than I.

That would seem to imply some sort of complex number? Although I believe anything on the standard complex number plane wouldn't work (other than 0+0i and 2+0i). In theory, I guess there's things like root i and number spaces-- though I'm not about to start going in that direction.

DaveE
 
  • #26


I got it.
In arithmetic modulo 8, 4 + 4 = 4 x 4 = 0. There are an infinite number of such examples.
 
  • #27


Because the first few examples I found for a + a = a x a = b all had b = 0 or b = 4 and since these are the values for 0 + 0 and 2 + 2, I conjectured that in all cases b = 0 or b = 4. However, the famous Snyder conjecture fell flat.

In arithmetic modulo 15, 5 + 5 = 5 x 5 = 10.

Do I get a fields medal for solving this problem which had remained unsolved for so many hours?
 
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  • #28
Hi Jimmy! :smile:

Since the equation is x(x-2) = 0,

the solutions are 0, 2, and any n mod k, where
k divides n(n-2)

eg 5 mod 15, 6 mod 8 12 or 24, 10 mod 16 20 40 or 80 … :wink:
 
  • #29


Good morning - I swear, some people that hang around on this forum have to be heavy into self penance or at least clinically masochistic and I do not think that I should be associated with either, but unfortunately, so far I have been guilty . It has been a perfect example of, "Crowd mentality," where some feed on the 'pushed' and 'provoked' thoughts of others. The darn numbers are plain ole garden variety numbers that the typical forth grader should be familiar with. And yes, a forth grader would have a better chance of solving this than ANYONE that I have come across on this forum. I wasn't trying to pull some kind of 'magic' as one implied. So far everyone has just plain been over thinking the problem...possibly due to inexperience, I don't know.
And Jimmy Snyder, I don't want to belittle your possible Fields Medal but that equation has been around for maybe a couple of thousand years and possibly longer. This is what 'char-limit' came up with and told me in his way 'that I didn't know what I was talking about'. His math told him that the only possible answers could be 2 and 0 right after I told him and everyone that there were others. His math is something that a first year algebra student should have been familiar with. This is a common misconception about schools. Schools are where one is familiarized with the TOOLS to help START their real education. It just takes some longer than others to realize this. And then BOREK 'DEMANDS' that I define 'domain' for him. I told him that he has a dictionary like anyone else, so he changes it to demanding that I define 'number' for him. If one doesn't have a dictionary, there is good one on the web. Google 'dictionary Webster's'.
And Davee123...That part about, "That would seem to imply..." may be true but I am not implying any such thing...You have 'implied' by saying that.
And I don't want to hear about sentences should not start with 'and'. I wouldn't do this in formal writing.
 
  • #30
YesIam said:
Good morning - I swear, some people that hang around on this forum have to be heavy into self penance or at least clinically masochistic and I do not think that I should be associated with either, but unfortunately, so far I have been guilty . It has been a perfect example of, "Crowd mentality," where some feed on the 'pushed' and 'provoked' thoughts of others. The darn numbers are plain ole garden variety numbers that the typical forth grader should be familiar with. And yes, a forth grader would have a better chance of solving this than ANYONE that I have come across on this forum. I wasn't trying to pull some kind of 'magic' as one implied. So far everyone has just plain been over thinking the problem...possibly due to inexperience, I don't know.
And Jimmy Snyder, I don't want to belittle your possible Fields Medal but that equation has been around for maybe a couple of thousand years and possibly longer. This is what 'char-limit' came up with and told me in his way 'that I didn't know what I was talking about'. His math told him that the only possible answers could be 2 and 0 right after I told him and everyone that there were others. His math is something that a first year algebra student should have been familiar with. This is a common misconception about schools. Schools are where one is familiarized with the TOOLS to help START their real education. It just takes some longer than others to realize this. And then BOREK 'DEMANDS' that I define 'domain' for him. I told him that he has a dictionary like anyone else, so he changes it to demanding that I define 'number' for him. If one doesn't have a dictionary, there is good one on the web. Google 'dictionary Webster's'.
And Davee123...That part about, "That would seem to imply..." may be true but I am not implying any such thing...You have 'implied' by saying that.
And I don't want to hear about sentences should not start with 'and'. I wouldn't do this in formal writing.

https://www.physicsforums.com/showpost.php?p=3060989&postcount=1" …
YesIam said:
It may be in here, I don't know...but how many other numbers can you think of that can be added to or multiplied by itself and get the same answer such as 2 and 0? There are others.

… had (as Evo pointed out) missing information.

"Numbers" without more is not a mathematically correct way to describe elements of a modulo arithemetic.

As Borek gently pointed out, you needed to correct your question by correctly specifying the domain.

YesIam, this is a forum where members try to help each other.

Wasting members' time by posting a question with inaccurate or missing information is not helpful …

and criticising them when they try to help you is ungrateful and unfriendly. :redface:

Try playing nice. :smile:​
 
Last edited by a moderator:

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