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 Post subject: fractions puzzle
PostPosted: Thu Jul 07, 2011 1:09 pm 
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I was thinking about this the other day after seeing 0.142857142857...
given the fractions 1/1 1/2 1/3 ... 1/99 1/100 (all in base 10)

Is there a way to intuitively or logically (without using brute force) determine which fraction has the longest repeating sequence? If so, I'd be curious to see this. I have a hunch that it would be a prime number, but have no proof or anything.

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 Post subject: Re: fractions puzzle
PostPosted: Thu Jul 07, 2011 1:37 pm 
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I probably wouldn't be the person to ask about this, but doesn't 1/3 have an infinitely repeating sequence? 0.333333333...

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 Post subject: Re: fractions puzzle
PostPosted: Thu Jul 07, 2011 2:08 pm 
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Yes it does, but it's just one 3 repeating. I'm looking for the longest sequence. one 3 repeating would just be a sequence of 1 number. 1/7 is a sequence of six numbers repeated (.142857...)

Obviously numbers like 1/8 wouldn't count since the sequence ends and it's only three numbers (0.125)

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 Post subject: Re: fractions puzzle
PostPosted: Thu Jul 07, 2011 2:48 pm 
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Last edited by bish on Wed Feb 26, 2014 7:23 am, edited 2 times in total.

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 Post subject: Re: fractions puzzle
PostPosted: Thu Jul 07, 2011 2:50 pm 
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No, he's looking for a fraction where the repeating part is as long as possible in 1/3=0.33333333... the length of the repeating part is only 1, for 1/7=0.142857142857 the repeating part is 142857, which has length 6.

The period of the decimal expansion of a fraction is unbounded. Consider any real number with an infinite expansion that is repeating. It is clear that it's possible to find a real number that has an arbitrarily large period in it's decimal expansion, since we can just define it by taking some suitable (1) natural number n and writing x = 0.nnnnnnn..., which is a converging series and by the completeness of R, must be in it.
Image
So the number we just constructed of the form 0.nnnnnnn... is a fraction and since we can construct such numbers with arbitrarily long period, there is no bound on the period of the decimal expansion of a fraction.

(1) By suitable, the natural number should not be repeating, ie 1212 is not suitable but 1221 is. We can easily find a suitable number of any length as 12222...222221.

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 Post subject: Re: fractions puzzle
PostPosted: Thu Jul 07, 2011 3:40 pm 
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TomZ wrote:
No, he's looking for a fraction where the repeating part is as long as possible in 1/3=0.33333333... the length of the repeating part is only 1, for 1/7=0.142857142857 the repeating part is 142857, which has length 6.

The period of the decimal expansion of a fraction is unbounded. Consider any real number with an infinite expansion that is repeating. It is clear that it's possible to find a real number that has an arbitrarily large period in it's decimal expansion, since we can just define it by taking some suitable (1) natural number n and writing x = 0.nnnnnnn..., which is a converging series and by the completeness of R, must be in it.
Image
So the number we just constructed of the form 0.nnnnnnn... is a fraction and since we can construct such numbers with arbitrarily long period, there is no bound on the period of the decimal expansion of a fraction.

(1) By suitable, the natural number should not be repeating, ie 1212 is not suitable but 1221 is. We can easily find a suitable number of any length as 12222...222221.


For an even easier to see example, the reciprocal of a string of n nines has decimal expansion 0.[[n-1 zeros]] 1 [[n-1 zeros]] 1 [[n-1 zeros]] 1...


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 Post subject: Re: fractions puzzle
PostPosted: Thu Jul 07, 2011 4:01 pm 
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Well, there you go, the simple proofs are always the best. But at least mine looks impressive :lol:

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 Post subject: Re: fractions puzzle
PostPosted: Thu Jul 07, 2011 10:55 pm 
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I think I'm missing something. So out of all the fractions 1/X where X is an Integer between 1 and 100 which X has the longest sequence prior to the sequence repeating?

-d


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 Post subject: Re: fractions puzzle
PostPosted: Fri Jul 08, 2011 1:46 am 
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Oh sorry, I missed the part of you wanting to look just at the fractions 1/1...1/100 and assumed that you wanted to consider all fractions.

Just taking some theorems from Wikipedia, it's easy to see that 1/97 with period 96 has the decimal expansion with the longest period of the fractions in 1/1....1/100. First off, we do not need to consider any fraction greater than 1/97 because the period of 1/n is strictly less than n, so the period of 1/96 would be at most 95 which is less than the 96 we have. So we just need to check 98, 99, 100. 1/98=1/2*1/49, we can see that the period of the expansion must be less than that of 1/49, so less than 49 - this does not beat the 96 we have. As bish stated, 1/99 would have period 2. Finally 1/100 = 1/2*1/50 must have period less than 50.
So 1/97 is the fraction in 1/1....1/100 that has the longest repeating decimal expansion with period 96.

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 Post subject: Re: fractions puzzle
PostPosted: Fri Jul 08, 2011 11:19 am 
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That works, thanks!

-d


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 Post subject: Re: fractions puzzle
PostPosted: Fri Jul 08, 2011 6:25 pm 
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What about transcendental numbers?

EDIT: Nevermind. I was thinking something else.

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Last edited by Jorbs3210 on Sat Jul 09, 2011 9:45 pm, edited 1 time in total.

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 Post subject: Re: fractions puzzle
PostPosted: Sat Jul 09, 2011 2:47 am 
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To ask a more generalized question:

N is an integer such that 1/N is an infinite repeating decimal.
PN is the length of the repeating digit sequence in 1/N.
k is an integer such that 1/k has Pk > PN for any N<k

What are the values for k and Pk? Can these sequences be calculated without brute force? Do these sequence happen to be list in the OEIS?

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 Post subject: Re: fractions puzzle
PostPosted: Sun Jul 10, 2011 10:44 am 
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Theoretically shouldn't it be the fraction with the largest prime number in the denominator?

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 Post subject: Re: fractions puzzle
PostPosted: Sun Jul 10, 2011 11:01 am 
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There is no largest prime number.

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 Post subject: Re: fractions puzzle
PostPosted: Sun Jul 10, 2011 12:07 pm 
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There is if you have an upper bound, e.g. from 1/1 to 1/100, which was the question.

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 Post subject: Re: fractions puzzle
PostPosted: Sun Jul 10, 2011 2:04 pm 
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Jerry's problem does not specify an upper bound (since Darryl's question has been answered I assumed your answer to be for that) or one could assume Jerry to be looking for the smallest number that satisfies his properties.

But it's incorrect either way. In the case 6 is the upper bound, 5 would be the largest prime but it has a finite decimal expansion (which means period 0?) while 1/3 has period 1. Or for upper bound 101, 1/101 has period 4 but 1/97 has period 96.

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 Post subject: Re: fractions puzzle
PostPosted: Sun Jul 10, 2011 2:38 pm 
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NType3 wrote:
Theoretically shouldn't it be the fraction with the largest prime number in the denominator?


Limiting N to prime values, the first few values of 1/N are(repeating sequence unlined):
1/2 = .5
1/3 = .3
1/5 = .2
1/7 = .142857
1/11 = .09
1/13 = .076923
1/17 though 1/31 have long decimal expansions
1/37 = .027
1/41 = .02439
1/43 though 1/97 have long decimal expansions

With periods of: {}, 1, {}, 6, 2, 6. ..., 3, 5.

As defined in my previous post, it might be that k has to be prime, but primality is not sufficient.

The first few values of k would be: 3, 7, and 17, but I cannot test any larger values of N without a more precise calculator.

Anyone know a good online calculator that can do really long decimal expansions?

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 Post subject: Re: fractions puzzle
PostPosted: Sun Jul 10, 2011 2:42 pm 
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Wolfram Alpha does a pretty good job at it. It tells you exactly what the period is.

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 Post subject: Re: fractions puzzle
PostPosted: Wed Jul 13, 2011 8:00 am 
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Let's examine 1/7 for example.
Multiply 1/7 by 10, to get 1+3/7. So the first digit in the expansion is 1.
Multiply 3/7 by 10, to get 4+2/7. So the second digit in the expansion is 4.
Multiply 2/7 by 10, to get 2+6/7. So the third digit in the expansion is 2.
Multiply 6/7 by 10, to get 8+4/7. So the fourth digit in the expansion is 8.
Multiply 4/7 by 10, to get 5+5/7. So the fifth digit in the expansion is 5.
Multiply 5/7 by 10, to get 7+1/7. So the sixth digit in the expansion is 7.
We are now back to 1/7, so the whole cycle starts again.

As you can see, all six possible fractions x/7 occur.

Now lets try it with 1/13.
Multiply 1/13 by 10, to get 0+10/13. So the first digit in the expansion is 0.
Multiply 10/13 by 10, to get 7+9/13. So the second digit in the expansion is 7.
Multiply 9/13 by 10, to get 6+12/13. So the third digit in the expansion is 6.
Multiply 12/13 by 10, to get 9+3/13. So the fourth digit in the expansion is 9.
Multiply 3/13 by 10, to get 2+4/13. So the fifth digit in the expansion is 2.
Multiply 4/13 by 10, to get 3+1/13. So the sixth digit in the expansion is 3.
and we're back to 1/13, and the cycle starts again.

So even though there are 12 possible fractions x/13, which could have resulted in 12 repeated digits, we've seen only 6 before it started repeating.

The interesting thing is that if you now start with 2/13, one of the missing fractions, then you also get a sequence of 6 steps. It is essentially the same as the above sequence but everything doubled in some sense (modulo 13 if that means anything to you). You can prove that all the sequences that you could get are the same length since they are all multiples of each other. Therefore, the length of the sequence must divide the number of possible fractions you have.

For example, for any prime p (except 2 or 5), the length of the repeating decimals of 1/p will divide p-1.

For 1/p (p prime) to have repeating decimal of length p-1, we need 10 to be a so called primitive root modulo p. The actual definition of 10 being a primitive root modulo p is that 10, 10^2, ..., 10^(p-1) all have different remainders after division by p. This is simply another way of saying that the procedure I described above, done for p-1 steps, results in p-1 different fractions.

There is no simple way to predict which primes p have 10 as a primitive root as far as I know. The only way is to calculate the period and see, or for large p it is quicker to check that the period is not any of the strict divisors of p-1.

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 Post subject: Re: fractions puzzle
PostPosted: Wed Jul 13, 2011 7:46 pm 
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Assuming primality is required for the period of 1/k being greater than the period of all 1/n for n < k, the values of k less than 100 are:

3, 7, 17, 19, 23, 29, 47, 59, 61, 97.

Also, with the exception of three, all of these have periods equal to k-1.

Removing three gives A001913.
Replacing three with 2 gives A006883

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