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Comment by chrismorgan | Hacker News Reader

On the decimal expansion part, 1⁄7 has always fascinated me, having something very similar going on. Doubling from 7, you get 14, 28, 56; and 1⁄7 is 0.1̅4̅2̅8̅5̅7̅, 2⁄7 is 0.2̅8̅5̅7̅1̅4̅, 3⁄7 is 0.4̅2̅8̅5̅7̅1̅, &c. (just changing which digit you start the recurring sequence with). https://en.wikipedia.org/wiki/142,857 talks about it a bit more; the doubling sequence thing is covered in the section 1⁄7 as an infinite sequence (including the reason the recurring decimal has 57 instead of 56—that 56 doubled is 112, so the hundred there overlaps with the six, much as the ten of the 13 adds to the 8 in the 1⁄89 expansion of this article).

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I've always liked 1/7 as well, and I never realized that thing about doubling from 7 giving 14, 28, 56. People are always impressed when I can rattle off the digits of x/7.

By the way, I appreciate your use of U+0305 combining overline. Did you enter those manually or do you have some neat way of doing it?

To me, the fact that you’re in an environment where knowing the digits of x/7 is appreciated is the most-impressive thing of all!

Heh, last time I spouted about 1⁄7 here someone asked about the overline too: https://news.ycombinator.com/item?id=24302104. I love my Compose key.

How do you get one of those. Track down an old Sun keyboard? :)

On my last laptop I used the Menu key as Compose. On this one I use Right Alt.

I use CapsLock; lowest usefulness to size ratio possible.

Caps lock is much more useful as a home row ctrl key

Why stop there though? Mine is configured to be ESC when tapped, CTRL when held down. Took a while to get used to, but it's a big improvement compared to other ways of handling ESC on a 60% keyboard.

I find it most useful as a Super key, so that I can attach all my custom keybindings and shortcuts to it.

I've had my CapsLock bound to Compose for ages; it's a great use of that piece of keyboard realestate.

I still don't have a good way to discover compose sequences other than by groveling through xkb and compose files. I really wish there were a character-palette tool that would tell me how to type the characters by introspecting the current input settings.

Cool thing is, this is not a special property of 1/7. 100 / 7 is 14, with a remainder of 2, therefore the series starts with 14, multiplies by 2, and divides with 100 in each iteration. For instance 10 / 7 is 1, with a remainder of 3, therefore 1/7 is also equal to 0.1+0.03+0.009 etc. And 1/8 is 0.1+0.02+0.004 etc.

Then how does 1/8 become 0.125?

If you do the calculation, with each step it goes towards 0.124999999999

> with each step it goes towards 0.124999999999

Also known as .125.

https://en.wikipedia.org/wiki/0.999...

Coming from video, I've always been a fan of 1/1001. 30000/1001 = 29.970029797002997 and 24000/1001 = 23.97600239760023976. There's something about it's clean repeating that I liked. I hear people confusing frame rates by saying something like 29.976. I also don't like 23.98 as that rounding is going to cause problems later.

However, you have to be a special math something to have any of these kind of number "oddities" be anything meaningful. I wear mine like a badge of honour

I had forgotten why the number 1001 mattered in video (it's been too long since I worked with NTSC circuits), so I looked it up. It has to do with avoiding dot crawl in color analog video.

https://en.wikipedia.org/wiki/Frame_rate

It also had to do with allowing the additional of the color information to not break compatibility with the existing B&W TVs in existence. Had the decided to not make 1 video signal that could be broadcast to both color and B&W TVs, they could have just broadcast color at 30fps (and man would my life had been so much easier).

Indeed. NTSC is such a kludgy hack. An incredibly clever kludgy hack, but it's still kludgy.

Kludgy hacks are interesting when their value highly outweighs the lack of careful design or effort put into it. Or, I work in the video games industry and quick work can end up being charming or valuable to your audiences, even if they're a difficult thing to continue developing or maintain. The wide compatibility of NTSC likely has been very valuable to the public, but this public is also unaware of the difficult work it implies.

That said, games traditionally have a point where development stops and doesn't resume (not counting from more live-ops-style games today), so the calculus of that sort of thing changes to management.

This is especially true to the people building the ROM emulators. There are so many tricks/hacks/kludges that were at the heart of some games even being usable. Timing for interlacing that was needed for the game to work in NTSC has to be accounted for on today's faster hardware and progressive scanning. Reproducing color accurately from NTSC seems to also be another thing I've seen. I'm sure there are plenty more that have been posted here before, but they are always a fun reminder that porting code can be a nightmare.

I discovered this in my late teens and thought it was super cool and made me want to understand more about repeating decimals. After playing around for a bit I realized you could form arbitrary repeating decimals by dividing by 9, 99, 999 etc. So for example, 1/7 = 142857/999999. Or written another way, 999999/7=142857.

Which also makes it easier to find the patterns in its multiple(or just divide by the extra multiple) say 1/14 starts .07142857142857. Multiples with 3 don't give us the common repeating 1428 but still repeats in its own way.. but 1/49 is pretty cool. 1/49 looks to do what 1/89 is doing but with the powers of 2. Nice!

edit: Dont know the format for proofs but heres a try.

1/49 = i=1 towards inf

sum 2^i*100^-i

I learned about 1/7 back in my youth, and it's just been one of those things that I enjoyed knowing as I went on in life.

Imagine my amusement when I ran across a Project Euler problem where those digits were the answer. I recall just looking at it and thinking I __know__ this one, there's no need to code anything. An easy point, but I didn't feel like I cheated on it.

Does this work or other bases too? Other than 10?

No. A trivial counterexample would be base 7.

That you can find other special properties on various numbers is true in all bases, but 1/7 in base 10 is pretty special.

So the property it has is:

    1/n = n (2/b^2 + 4/b^4 + 8/b^6 + ...)

which by geometric series sums to

    1/n = n / (b²/2 — 1)

    n² = b²/2 – 1

So this works precisely because 7² = 49 = 50 – 1 = 100/2 — 1.

Calculating some of these out these appear to be the Newman-Shanks-Williams numbers [1], the next one is 41 in base 3364, where

    1/41 = {0}.{82}{164}{328}{656}{1312}{2625}...

notice the 5 finally coming from some overflow.

But, supposing that we just like the idea of starting with some digit d and then the next digit being k times that and the next digit being k times that, we get a more general set of numbers,

    d/b + dk/b² + dk²/b³ + ...
      = d/(b - k)

Given that, this becomes much more boring. So for example for doubling in base-100 we think about 1/98 (b=100, k=2) and we find

   1/98 = 0.01020408163265...

and factors of that 98 also may have similar patterns, so 7 has this strength because it is a factor of 98.

So for example we want to think about 1/7 in base-12, this suggests that maybe we should look for things that quintuple base 12, but that rapidly overflows base 12. So we do the same trick as 1/7 where we take pairs of digits, and maybe things quadruple base-144 (since 144 - 4 is 140 which is divisible by 7), and so we find that

    1/7 = 0.{20}{82}{41} repeating

and if you squint closely you can see starting with 20, quadrupling to 80, quadrupling to 320 but then getting a bit unwieldy. Of course even on single digits 12 - 2 = 10 which has 5 as a factor so you can expect to see a pattern in base-12 on

    1/5 = 0.{2}{4}{9}{7} [repeating]

which you can see a sort of "2, 4, 8, 16," pattern happening.

The other base that I really like is nonnary, if we met aliens we might find that they count in balanced nonnary with digits -4, -3, -2, -1, 0, 1, 2, 3, 4, (so like 7 is actually {1, -2}, 7 = 9 - 2), but it's harder to search for patterns in that because you really feel the cap of having only half the base to count up to before you carry.

That really is rather fascinating.

Wow, it keeps going also with the further digits, e.g.

=14285712

+0000000224

=1428571424

+000000000448

=142857142848

+00000000000896

=14285714285696

+0000000000001792

=142857142857....

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