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 2019-04-23, 23:41 #1 neskis     Feb 2019 24 Posts What is this called? repitend of 1/95 Forgive me, I'm an amateur. I call this front/back stack. The repeating decimal of 1/95 is 0.01052631578947368421, which can be found by stacking the sum of 5^n from left to right or the sum of 2^n from right to left as shown in the attached image. Thanks. Attached Thumbnails
 2019-04-24, 08:21 #2 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 22×2,281 Posts $\sum_{k=0}^n {20^k} = {{20^{k+1} -1} \over {19}}$ the right lower corner of your summation is = ${20^{21} -1} \over {19}$
 2019-04-24, 11:27 #3 neskis     Feb 2019 24 Posts Thanks! Is it known that all repitends are in this form?
 2019-04-24, 13:09 #4 Dr Sardonicus     Feb 2017 Nowhere 5×691 Posts The following geometric series expression may also be illustrative. 1/95 = 1/(100 - 5) = (1/100)/(1 - 5/100) = (1/100)[1 + 5/100 + (5/100)^2 + (5/100)^3 + ...] = .01 + .0005 + .000025 + .00000125 + ... Obviously, this sort of expression depends on being close to a power of 10. Of course, repeating decimals are closely related to geometric series. You might not be able to get a small common ratio in a geometric series for a given fraction, but in some cases you can, with some manipulation. A well-known example is 1/7 = 14/98 = (14/100) * [1 + 2/100 + (2/100)^2 + ...] = 14/100 * [1 + .02 + .0004 + .000008 + .00000016 + ...] = .14 28 57... (the "7" in "57" is due to a carry from 7 x 16.) Last fiddled with by Dr Sardonicus on 2019-04-24 at 13:10 Reason: fixing typos
 2019-04-24, 19:22 #5 neskis     Feb 2019 100002 Posts Fascinating. I’ll look into it more. You are right about it being close to a power of 10. Other front/back combinations I have found are more complicated. For example 1/93 is a front stack of 7^n and a back stack of 143^n (again, please excuse my jargon). What is this area of study called?
2019-04-25, 15:05   #6
Dr Sardonicus

Feb 2017
Nowhere

65778 Posts

Quote:
 Originally Posted by neskis Fascinating. I’ll look into it more. You are right about it being close to a power of 10. Other front/back combinations I have found are more complicated. For example 1/93 is a front stack of 7^n and a back stack of 143^n (again, please excuse my jargon). What is this area of study called?
The short answer would be "recreational mathematics." I have a vague recollection of reading the powers-of-two thing for 1/7 somewhere, possibly a Martin Garner "Mathematical Games" column in Scientific American many, many years ago. It seems clear that your "front stack" is related to geometric series whose common ratio is a fraction with denominator a power of 10. Your example 1/93, for instance, may be written

1/93 = (1/100)/(1 - 7/100) = (1/100)* [1 + (7/100) + (7/100)^2 + ...]

The "back stacking" also appears to be related to summation formulas for a geometric series like the one in Batalov's post above, although I confess I don't see 143 jumping off the screen as being related to 1/93.

An unusual sort of "front stacking" involving the Fibonacci sequence instead of the powers of an integer is described here.

2019-04-28, 14:08   #7
neskis

Feb 2019

24 Posts

Quote:
 Originally Posted by Dr Sardonicus The short answer would be "recreational mathematics." I have a vague recollection of reading the powers-of-two thing for 1/7 somewhere, possibly a Martin Garner "Mathematical Games" column in Scientific American many, many years ago. It seems clear that your "front stack" is related to geometric series whose common ratio is a fraction with denominator a power of 10. Your example 1/93, for instance, may be written 1/93 = (1/100)/(1 - 7/100) = (1/100)* [1 + (7/100) + (7/100)^2 + ...] The "back stacking" also appears to be related to summation formulas for a geometric series like the one in Batalov's post above, although I confess I don't see 143 jumping off the screen as being related to 1/93. An unusual sort of "front stacking" involving the Fibonacci sequence instead of the powers of an integer is described here.
You’re right! 1/89 and Fibonacci? Mind blown. When I first started studying repitends, I was hopeful that there would be some application for the patterns. For now, I’ll continue to enjoy it recreationally. I’ll also study up on harmonic series and geometric series. Thanks for the insight.

 2019-04-29, 11:41 #8 BudgieJane     "Jane Sullivan" Jan 2011 Beckenham, UK EA16 Posts Have you read Samuel Yates, Repunits and Repetends, Star Publishing, 1982?
2019-04-29, 17:09   #9
neskis

Feb 2019

100002 Posts

Quote:
 Originally Posted by BudgieJane Have you read Samuel Yates, Repunits and Repetends, Star Publishing, 1982?
No, I haven’t. Thanks for the recommendation!

2019-05-04, 20:42   #10
neskis

Feb 2019

24 Posts

Quote:
 Originally Posted by Dr Sardonicus The short answer would be "recreational mathematics." I have a vague recollection of reading the powers-of-two thing for 1/7 somewhere, possibly a Martin Garner "Mathematical Games" column in Scientific American many, many years ago. It seems clear that your "front stack" is related to geometric series whose common ratio is a fraction with denominator a power of 10. Your example 1/93, for instance, may be written 1/93 = (1/100)/(1 - 7/100) = (1/100)* [1 + (7/100) + (7/100)^2 + ...] The "back stacking" also appears to be related to summation formulas for a geometric series like the one in Batalov's post above, although I confess I don't see 143 jumping off the screen as being related to 1/93. An unusual sort of "front stacking" involving the Fibonacci sequence instead of the powers of an integer is described here.
In trying to show how 143 is related, I actually discovered something more advanced this week. Thanks for pushing me even if you didn't know it. The repetend (I've even learned how to spell it correctly!) of 1/93 is 010752688172043 which is a front stack of 7^n and a back stack of 142,857,142,857,143^n as shown in the attached image (again, please excuse the crude presentation). Accuracy is improved with the repetition of 142,857 before 143. The same phenomenon can be seen with the front and back stacks of many other fractions I have analyzed. The layering depends on how many digits long the divisor is. For example, to see this clearly, I had to use 1/999993 and manually do the rest. I only have access to a one million digit calculator, so I am limited to six digit divisors most of the time. Do you think there's a way to build a formula to describe the rules of front and back stacking?
Attached Thumbnails

2019-05-07, 13:04   #11
Dr Sardonicus

Feb 2017
Nowhere

1101011111112 Posts

Quote:
 Originally Posted by neskis In trying to show how 143 is related, I actually discovered something more advanced this week. Thanks for pushing me even if you didn't know it. The repetend (I've even learned how to spell it correctly!) of 1/93 is 010752688172043 which is a front stack of 7^n and a back stack of 142,857,142,857,143^n as shown in the attached image (again, please excuse the crude presentation). Accuracy is improved with the repetition of 142,857 before 143. The same phenomenon can be seen with the front and back stacks of many other fractions I have analyzed. The layering depends on how many digits long the divisor is. For example, to see this clearly, I had to use 1/999993 and manually do the rest. I only have access to a one million digit calculator, so I am limited to six digit divisors most of the time. Do you think there's a way to build a formula to describe the rules of front and back stacking?
Note that 142857 is the decimal period of 1/7. The fact that prepending this period repeatedly improves the "back stack" result, leads me to conclude that your "back stack" here actually corresponds to the geometric series identity

1/7 + 100/7^2 + 100^2/7^3 + ... + 100^(k-1)/7^k = ((100/7)^k - 1)/93.

The numerators of the fractions for the partial sums are what you're seeing.

? t=1/7;s=0;for(i=1,20,s+=t;t*=100/7;print(s))
1/7
107/49
10749/343
1075243/2401
107526701/16807
10752686907/117649
1075268808349/823543
107526881658443/5764801
10752688171609101/40353607
1075268817201263707/282475249
107526881720408845949/1977326743
10752688172042861921643/13841287201
1075268817204300033451501/96889010407
107526881720430100234160507/678223072849
10752688172043010701639123549/4747561509943
1075268817204301074911473864843/33232930569601
107526881720430107524380317053901/232630513987207
10752688172043010752670662219377307/1628413597910449
1075268817204301075268694635535641149/11398895185373143
107526881720430107526880862448749488043/79792266297612001

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