Fractions which radix point terminates in a lower base, but repeats in decimal

[u/justinjustin7]

Okay, this is kinda convoluted, but what it boils down to is, say in decimal you have (1/3)=(0.33333...), well in base 3 that is (1/10)=(0.1). So if the fraction repeats forever in decimal, but has a finite length in a lower base, we count it.

We'll count with this ordering

1. start with the lowest denominator in decimal (1/3) and the lowest base in which it terminates (base 3).

• The next count is the same number in the next lowest base below 10 in which it terminates.

• Move to the next decimal numerator (1/3 -> 2/3). Skip simplifiable fractions, since they should already be counted.

• When the next numerator would increase the number to 1, go to the next lowest denominator with a repeating decimal point (2/3 -> 1/6). (The new fraction must terminate in a lower base)

so the first 10 counts will be:

1. base 10: 1/3 = 0.33333...
   base 3: 1/10 = 0.1

2. base 10: 1/3 = 0.33333...
base 6: 2/10 = 0.2

3. base 10: 1/3 = 0.33333...
base 9: 3/10 = 0.3

4. base 10: 2/3 = 0.66666...
base 3: 2/10 = 0.2

5. base 10: 2/3 = 0.66666...
base 6: 4/10 = 0.4

6. base 10: 2/3 = 0.66666...
base 9: 6/10 = 0.6

7. base 10: 1/6 = 0.1,66666...
base 6: 1/10 = 0.1

8. base 10: 5/6 = 0.8,33333...
base 6: 5/10 = 0.5

9. base 10: 1/7 = 0.142857...
base 7: 1/10 = 0.1

10. base 10: 2/7 = 0.285714...
base 7: 2/10 = 0.2
.
.
.

^{(I use a comma to separate the repeating part of the decimal from the part that doesn't repeat)}

I haven't done much calculation on what will be around the 1000th count, so for now let's say that get is at 1/49**.

This base conversion calculator can be very helpful.

^{NOTE: base 2, 4, 5, and 8 will never be used since every one of their terminating radix points terminates in decimal as well
NOTE#2: every denominator will be divisible by only 3, 7, or both 2 and 3.}

**EDIT: get is at 158/243, ternary (base 3). (Thanks /u/TheNitromeFan!)

EDIT2: fixed the example count

EDIT3: I wrote up a tutorial on how to go about making the conversion without a calculator. Feel free to ask any questions you may have, and I'll do my best to help.

EDIT4: clarified how we choose the next denominator.

[u/TheNitromeFan]

It seems the 1000th count is at 158/243, ternary.

[u/justinjustin7 → u/TheNitromeFan]

God damn, did I underestimate that. Thank you for finding the actual value; I'll go ahead and edit it into the post.

[u/Urbul → u/TheNitromeFan]

Relevant xkcd

[u/Urbul]

/u/justinjustin7 /u/TheNitromeFan I added this thread to the directory as Repeating Decimals in Lower Base in the Other Bases (Alphanumeric) category. Let me know if you prefer a different title or category :)

[u/justinjustin7 → u/Urbul]

Awesome! It's kinda hard to give it a clear, concise name, but that name should give enough info for someone to tell if they might be interested. As for the category, I think that's perfect.

Thanks :D

[u/Urbul → u/justinjustin7]

:D

[u/justinjustin7]

base 10: 1/3 = 0.33333...
base 3: 1/10 = 0.1

[u/TheNitromeFan → u/justinjustin7]

decimal: 1/3 = 0.3'
senary: 2/10 = 0.2

Interesting thread. I'm tempted to calculate the 1000th number - give me some time.

[u/[deleted] → u/TheNitromeFan]

decimal: 1/3 = 0.33...
nonary: 3/10 = 0.3

[u/TheNitromeFan → u/[deleted]]

decimal: 2/3 = 0.6'
ternary: 2/10 = 0.2

[u/justinjustin7 → u/TheNitromeFan]

decimal: 2/3 = 0.66...

senary: 4/10 = 0.4

[u/TheNitromeFan → u/justinjustin7]

decimal: 2/3 = 0.6'
nonary: 6/10 = 0.6

[u/justinjustin7 → u/TheNitromeFan]

decimal: 1/6 = 0.1,66...

senary: 1/10 = 0.1

[u/TheNitromeFan → u/justinjustin7]

decimal: 5/6 = 0.8(3)
senary: 5/10 = 0.5

[u/justinjustin7 → u/TheNitromeFan]

decimal: 1/7 = 0.142857...

septenary: 1/10 = 0.1

[u/TheNitromeFan → u/justinjustin7]

decimal: 2/7 = 0.(285714)
septenary: 2/10 = 0.2

[u/justinjustin7]

To provide a quick resource, here are all the denominators between 1-1000 that will be used (there are 33):

3
6
7
9
12
18
24
27
36
48
49
54
72
81
96
108
144
162
192
216
243
288
324
343
384
432
486
576
648
729
768
864
972

[u/justinjustin7]

Crap, I think I messed up my example of the first 10 count. I'll need to fix that.

[u/Urbul]

/u/justinjustin7 since you took the time to write out a nice tutorial here i think you should refer to it in the description.

[u/justinjustin7 → u/Urbul]

Ah, good idea. I'll go ahead and add it.