Cyclic fractions

Somewhere along the way you may have noticed that the digits in the decimal expansion of multiples of 1/7 are all rotations of the same digits:

1/7 = 0.142857142857…
2/7 = 0.285714285714…
3/7 = 0.428571428571…
4/7 = 0.571428571428…
5/7 = 0.714285714285…
6/7 = 0.857142857142…

We can make the pattern more clear by vertically aligning the sequences of digits:

1/7 = 0.142857142857…
2/7 =   0.2857142857…
3/7 =  0.42857142857…
4/7 =     0.57142857…
5/7 =      0.7142857…
6/7 =    0.857142857…

Are there more cyclic fractions like that? Indeed there are. Another example is 1/17. The following shows that 1/17 is cyclic:

 
 1/17 = 0.05882352941176470588235294117647…
 2/17 =           0.1176470588235294117647…
 3/17 =            0.176470588235294117647…
 4/17 =     0.2352941176470588235294117647…
 5/17 =        0.2941176470588235294117647…
 6/17 =      0.352941176470588235294117647…
 7/17 =          0.41176470588235294117647…
 8/17 =               0.470588235294117647…
 9/17 =       0.52941176470588235294117647…
10/17 =  0.5882352941176470588235294117647…
11/17 =              0.6470588235294117647…
12/17 =                0.70588235294117647…
13/17 =             0.76470588235294117647…
14/17 =    0.82352941176470588235294117647…
15/17 =   0.882352941176470588235294117647…
16/17 =         0.941176470588235294117647…

The next denominator to exhibit this pattern is 19. After finding 17 and 19 by hand, I typed “7, 17, 19” into the Online Encyclopedia of Integer Sequences found a list of denominators of cyclic fractions: OEIS A001913. These numbers are called “full reptend primes” and according to MathWorld “No general method is known for finding full reptend primes.”

Related: Applied number theory

15 thoughts on “Cyclic fractions

  1. @joseph: I’d say it falls under number theory or algebra. It’s a simple problem on the surface, just doing division and looking for repetitions. But it quickly gets into some deeper math when you look closer.

  2. Surely it’s trivial that the decimal expansion is periodic, given that it’s rational. The more interesting aspect about 1/7 is that the first few digits are multiples of 7, 0.[14][28]…, which I was told a good reason for some time ago. Unfortunately, I don’t remember it 🙁 There are more like it too

  3. @MAX: The decimal expansion of any rational number either terminates or is periodic. What’s special about 1/7 is not that the decimal expansion itself is period, it’s that the digits of that expansion rotate when you form 2/7, 3/7, etc.

    For example,
    1/13 = 0.076923076923…
    2/13 = 0.153846153846…

    Both decimal expansions have period 6, but they’re made of different digits. One is not simply a shift of the other.

  4. @MAX: The decimal expansion of any rational number either terminates or is periodic. What’s special about 1/7 is not that the decimal expansion itself is period, it’s that the digits of that expansion rotate when you form 2/7, 3/7, etc.

    For a negative example,
    1/13 = 0.076923076923…
    2/13 = 0.153846153846…

    Both decimal expansions have period 6, but they’re made of different digits. One is not simply a shift of the other.

  5. Is this behavior for such values unique to base 10? Are these same values periodic in any base? Or does each base have its own set of “full reptend primes”? If so, do the sets intersect?

  6. I’ve always been interested by the reciprocal of 7 (in base 10). As you go through the reciprocals of the positive integers in order, each seems to add something. 1/1 = 1. Then 1/2 = 0.5 — OK, we now have two digits involved (2 and 5 (sorry, 0, you don’t count)). 1/3 = 0.3333… one digit, but infinitely repeating. 1/4 = 0.25 — three digits! 1/5 = 0.2 — boring (though we see the roles of the digits permuted when compared to 1/2 = 0.5). Next up: 1/6 = 0.166666… — a new beast, an infinitely repeating digit again, but preceded by the prefix of another digit.

    And then we come to , 1/7 — 0.142857142857… what the hell just happened? Six unique digits! Repeating! And (as noted by Max) there is a pattern: It’s 2*7 followed by 4*7 followed by 8*7 … well, almost, 57, not 56.

    Why is this? 7 is prime, but so are 2, 3, and 5. OK, 2 and 5 divide 10 (we’re using base 10). 3 does not divide 10, but 3 divides 10-1, so maybe it’s part of the family, so to speak. Is this why 1/7 is out there in left field? I have intended to investigate this for some time, but I never seem to get around to it.

  7. When you take 1/7 and multiply by 10, you get 10/7 = 1 3/7:
    1/7 = 0.142857142857… = .142857 repeating.
    10/7 = 1.42857142857…
    so 3/7 = .428571 repeating.
    Clearly, 3/7 MUST have the same digits in the same order as 1/7.

    Doing this again, we have:
    30/7 = 4 2/7 = 4.28571428571…
    so 2/7 = .285714 repeating.
    Again, these clearly have to be the same digits as 3/7, and thus as 1/7.

    If we keep multiplying by 10, we cycle through all six possibilities before getting back to 1/7. The remainders when 10^n is divided by 7 are: (starting from n=0)
    1, 3, 2, 6, 4, 5, 1, 3, 2, etc. So 7 is a “full reptend prime”.

    On the other hand, the remainders when dividing powers of 10 by 13 are:
    1, 10, 9, 12, 3, 4, 1, 10, 9, … This pattern repeats before all 12 numbers show up, so the repeating decimal must only be 6 digits long instead of 12 and thus 13 is not full reptend.

    In order for P to be a “full reptend prime”, the numbers 10^n from n=0 to n = P-1 must span the set of numbers from 1 to P-1 (modulo P). This is definitely number theory!

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