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I was browsing the Puzzling.SE site when I came across this little challenge. The idea is to prove that 2016 is a "self-composable" number, i.e. that it can be computed using a given set of operations, and its digits only. Basically, using 2, 0, 1 and 6 (along with the operations allowed in the challenge), the objective is to reach a total of 2016. This answer appears to be a valid one.

Now I don't know whether it's because I'm not a native English speaker (I do my math in another language), or because it's something that is simply "used" on Puzzling.SE, but among the allowed set of extended operations, 2 of them make absolutely no sense to me. Since I appear to be the only one puzzled there (no pun intended), I thought it'd be best to ask about them here.

  • $.x$ ; I know it is common in English to remove the heading zero in decimal numbers (0.5 becomes .5), so I thought it was a just a way to disguise a division by 10. But then I saw...
  • $.\bar{x}$ ; this one, which I assume is related to the first one, yet I have no idea how this negation bar fits into the previous definition... Is it $(1 - .x)$, or some kind of base-2 complement? I'd like to believe there isn't any complex number complement hidden in there, but who knows how far the Puzzling.SE guys can go to solve a puzzle!

Since there's a chance this question will look trivial to many of you, I'll of course be more than happy to delete it, or have it closed as a duplicate if it has been asked already. Same thing of course if you believe the question to be more within the scope of Puzzling.SE's meta. The truth is, I have no idea what to Google or search for on Math.SE, and I'm sure once one of you gives me the actual names of these operators (if they exist), I'll be able to find the information I need.

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  • $\begingroup$ Interesting. I've never seen the underscore to mean repeating. And you are right. 0.x makes things much clearer than just .x $\endgroup$ – fleablood Feb 22 '16 at 21:03
  • $\begingroup$ @fleablood Actually it looked just as strange to me when I learnt it (the three dots at the end were part of the notation too : $.\underline{x}...$). Then again, I don't work with decimals very often, so I just went with it :') $\endgroup$ – John WH Smith Feb 22 '16 at 21:05
  • $\begingroup$ In the US we learn the notation as $.pqr\overline{xyz} = .pqrxyzxyzxyz...$ and the bar takes the three dots as a given. I'm so used to it I never question it. $\endgroup$ – fleablood Feb 22 '16 at 21:15
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The $.x$ is as you described.

As for the bar, it refers to the repeating part of the fraction, so $.\overline x$ means $0.xxx\ldots$. For example, $\frac13 = 0.333\ldots = 0.\overline3$. The bar can be used over several digits: $\frac5{22}=0.2272727\ldots = 0.2\overline{27}$.

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  • $\begingroup$ Dang it, in French I was taught to write these as $0.\underline{x}...$ instead of $.\bar{x}$, that's why! $\endgroup$ – John WH Smith Feb 22 '16 at 21:02

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