# Mathematics: Fraction \> decimal = repeated number sequence **URL:** \ **Category:** Factual Questions\ **Created:** [August 24, 2019, 1:50pm UTC](https://boards.straightdope.com/t/mathematics-fraction-decimal-repeated-number-sequence/839156 "2019-08-24T13:50:33Z")\ **Posts on this page:** 7\ **Page:** 2
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [August 24, 2019, 9:44pm UTC](https://boards.straightdope.com/t/mathematics-fraction-decimal-repeated-number-sequence/839156/21 "2019-08-24T21:44:18Z")
> [@ftg](#): > > 1/27 = 0.037037037 … > > but no one points out that > > 1/37 = 0.027027027 … ? It’s not too hard to come up with these: 1/271 = 0.003690036900369003690036900369 1/369 = 0.002710027100271002710027100271 1/4649 = 0.00021510002151000215100021510002151 1/2151 = 0.00046490004649000464900046490004649 ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [August 24, 2019, 10:52pm UTC](https://boards.straightdope.com/t/mathematics-fraction-decimal-repeated-number-sequence/839156/22 "2019-08-24T22:52:12Z")
All you need is a pair of numbers that multiply to 999…999, for some number of 9s. ---
**Author:** ![k9bfriender](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/k9bfriender/32/3283_2.png) [@k9bfriender](https://boards.straightdope.com/u/k9bfriender)\ **Post date:** [August 24, 2019, 10:57pm UTC](https://boards.straightdope.com/t/mathematics-fraction-decimal-repeated-number-sequence/839156/23 "2019-08-24T22:57:16Z")
> [@Chronos](#): > > All you need is a pair of numbers that multiply to 999…999, for some number of 9s. Ah good, I was worried it would be something hard, like factoring large numbers… 😉 ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [August 25, 2019, 12:38pm UTC](https://boards.straightdope.com/t/mathematics-fraction-decimal-repeated-number-sequence/839156/24 "2019-08-25T12:38:19Z")
UIAM, the repeating period (in any base) of a rational number p/q will be a divisor of [Euler’s totient function φ(q)](https://en.wikipedia.org/wiki/Euler%27s_totient_function). For example φ(49) = 42 and, in base 10, the repeating string attains its maximal length of 42: 1/49 = .020408163265306122448979591836734693877551 020408163265306122448979591836734693877551 020408163265306122448979591836734693877551 020408163265306122448979591836734693877551 … > [@k9bfriender](#): > > Ah good, I was worried it would be something hard, like factoring large numbers… 😉 There is a vague roundabout connection! From the above-linked article: > [@](#): > > Euler’s totient function … is also used for defining the RSA encryption system. ---
**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\ **Post date:** [August 25, 2019, 12:57pm UTC](https://boards.straightdope.com/t/mathematics-fraction-decimal-repeated-number-sequence/839156/25 "2019-08-25T12:57:15Z")
> [@Chronos](#): > > All you need is a pair of numbers that multiply to 999…999, for some number of 9s. But not all are “pretty”. Take another pair of factors of 999: 9 and 111. 1/111 = 0.009009009 … 1/9 = 0.111111111 … which you have to “see” as .111 111 111 … . ---
**Author:** ![Walken\_After\_Midnight](https://avatars.discourse-cdn.com/v4/letter/w/b487fb/32.png) [@Walken\_After\_Midnight](https://boards.straightdope.com/u/Walken_After_Midnight)\ **Post date:** [August 25, 2019, 2:55pm UTC](https://boards.straightdope.com/t/mathematics-fraction-decimal-repeated-number-sequence/839156/26 "2019-08-25T14:55:18Z")
> [@ftg](#): > > An OP that mentions that > > 1/27 = 0.037037037 … > > but no one points out that > > 1/37 = 0.027027027 … ? > > Charles Fleischer is not going to be happy when he hears this. > > [Moleeds](https://www.ted.com/talks/charles_fleischer_insists_all_things_are_moleeds?language=en#t-846206). (If you don’t want to watch the whole thing, jump to about 3:50 left.) That was a really good video. I’m going to watch it again later in detail in order to fully absorb the moleeds theory into my thinking. I find the area of mathematical ratios and rational numbers fascinating, especially since through sound/music we are able to [hear them](https://en.wikipedia.org/wiki/List_of_pitch_intervals). ---
**Author:** ![JWT\_Kottekoe](https://avatars.discourse-cdn.com/v4/letter/j/3ec8ea/32.png) [@JWT\_Kottekoe](https://boards.straightdope.com/u/JWT_Kottekoe)\ **Post date:** [August 25, 2019, 4:18pm UTC](https://boards.straightdope.com/t/mathematics-fraction-decimal-repeated-number-sequence/839156/27 "2019-08-25T16:18:15Z")
Many years ago, Martin Gardner had a great [Mathematical Games column](https://www.jstor.org/stable/24925760?seq=1#page_scan_tab_contents) about the related topic of “[cyclic numbers](https://en.wikipedia.org/wiki/Cyclic_number)”, for example the integer 142857, which is the repeating part of 1/7 (see the post by Chronos above). If you multiply it by any integer less than 7, you get a cyclic permutation of the digits. 142857 × 1 = 142857 142857 × 2 = 285714 142857 × 3 = 428571 142857 × 4 = 571428 142857 × 5 = 714285 142857 × 6 = 857142 You can find more of these by looking at the decimal expansion of reciprocals of prime numbers. The example of 1/17 was given above by Dr. Strangelove. [Previous page](https://boards.straightdope.com/t/mathematics-fraction-decimal-repeated-number-sequence/839156.md?page=1)