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You are provided with the 8-segment array in the picture together with all the necessary electronics to make any combination of lit segments you want.

enter image description here

Fillable version for graphics. https://i.sstatic.net/K1hnN.png

enter image description here

What is the smallest positive number that you can display without any two boxes being identical? (A box is a grey rectangle containing 8 segments)

Any combination of segments is allowed within a box provided it is accepted by the community as a valid symbol. ('accepted' is decided by net up-votes for your answer)

Any recognised numbering system that is commonly used is allowed. If the one you use is common but little-known (perhaps because it is specialized) then you must back up your claim with a link to an authoritative source. You must also explain it precisely so that it can be understood by others.

You cannot mix numbering systems unless you can show that this is routinely done by some substantial section of the populace.

You cannot invent your own symbols or language. All symbols must be understandable and approved by the community.

The competition remains open until there has been no answer for one week. At that point (or as soon after as I am able) I shall accept the most up-voted answer (net votes).

Update

  1. The question has been put on hold--therefore I'll make an adjustment about the timescale for accepting an answer (if you want to reopen it please vote to do so).

  2. If I could wind back the clock, I would require the answers to be finite. This was an oversight because I did not think an infinite could be considered a number.

  3. Because there are answers and they have been voted on, it is too late to change the criteria. My suggestion is that people who are interested submit answers in the categories of finite or infinite numbers according to their preference. I will stick to my original criterion for acceptance. Note

You can display your answer using ASCII provided it is clear how it would look when translated into segments. Otherwise please use an actual 8-segment representation to clarify the layout of any symbols that aren't obvious.

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  • 1
    $\begingroup$ Most upvoted answer is almost certainly not going to be the "correct" answer. Your scoring condition makes this not a puzzle. $\endgroup$
    – Deusovi
    Commented Oct 30, 2015 at 13:14
  • $\begingroup$ @Deusovi, What scoring system do you suggest? $\endgroup$ Commented Oct 30, 2015 at 13:17
  • $\begingroup$ Smallest valid answer, where validity is determined either by strict rules or, if necessary, positive score. $\endgroup$
    – Deusovi
    Commented Oct 30, 2015 at 13:25
  • $\begingroup$ It's too late for me to change it now. However, if you have an idea for a similar question but with your rules please feel free to post it. P.S. My motivation is to find numerical problems that can't be tackled by computer or brute force.. $\endgroup$ Commented Oct 30, 2015 at 13:27
  • $\begingroup$ It's an interesting problem, but "satisfy other people" is what the scoring system essentially boils down to, and the problem itself seems poorly defined. $\endgroup$
    – Deusovi
    Commented Oct 30, 2015 at 13:44

7 Answers 7

22
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I don't know if you can get smaller than

enter image description here

epsilon - a small positive infinitesimal quantity

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3
  • $\begingroup$ Also possible is: "DEN" for "denormalized number". Not seen on any calculators that I know of, but I've never seen one that displays "EPSILON" either. Commonly used in some programming languages to represent a number too small to be programmatically distinguishable from zero, but specifically NOT zero. $\endgroup$ Commented Oct 30, 2015 at 13:44
  • $\begingroup$ @DarrelHoffman DEN is too short though $\endgroup$
    – DrunkWolf
    Commented Oct 30, 2015 at 13:46
  • $\begingroup$ VS C++ (don't know about other languages/compilers) usually shows DEN with a couple of digits after it, so "DEN0.1234" maybe? $\endgroup$ Commented Oct 30, 2015 at 14:00
10
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Here's one to start off: 1 E - 9 8 7 6 5 $=10^{-98765}$

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  • $\begingroup$ You may want to drop the 1 in front of the E, and instead improve the exponent. See for instance stackoverflow.com/questions/10687016/…. $\endgroup$
    – Gamow
    Commented Oct 30, 2015 at 8:33
  • 3
    $\begingroup$ @Gamow In my opinion, the E[number] is shorthand for a multiplier, and it's not a well-formed number without something in front of it. $\endgroup$
    – f''
    Commented Oct 30, 2015 at 12:22
  • $\begingroup$ I am not sure if the order of operations will be executed correctly, but I think this '1E-7^998' (using different ways of representing a 9) will be smaller. $\endgroup$
    – fibonatic
    Commented Nov 2, 2015 at 15:27
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Building up on f'' s answer:

        ###           ###    ###    ###    ###    ###
    #  #             #   #  #   #  #   #      #  #
    #  #             #   #  #   #  #   #      #  #
    #  #             #   #  #   #  #   #      #  #
        ###    ###    ###    ###    ###           ###
    #  #                 #      #  #   #      #  #   #
    #  #                 #      #  #   #      #  #   #
    #  #                 #      #  #   #      #  #   #
        ###           ###           ###           ###

$=10^{-99876}$

Using the possibility to write the digit 9 in two different ways on this kind of display.


Edit:

Dropping the 1 in front we could get as low as $10^{-998766}$:

 ###           ###    ###    ###    ###    ###
#             #   #  #   #  #   #      #  #      #
#             #   #  #   #  #   #      #  #      #
#             #   #  #   #  #   #      #  #      #
 ###    ###    ###    ###    ###           ###    ###
#                 #      #  #   #      #  #   #  #   #
#                 #      #  #   #      #  #   #  #   #
#                 #      #  #   #      #  #   #  #   #
 ###           ###           ###           ###    ###
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  • $\begingroup$ As with Gamow's comment to f", you can use this to get E-998765, which I would accept $\endgroup$ Commented Oct 30, 2015 at 8:42
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    $\begingroup$ I don't think a leading E is widely interpreted as a number. It's certainly not one in C syntax that's widely used as-is or slightly adapted for other languages. $\endgroup$ Commented Oct 30, 2015 at 15:55
  • $\begingroup$ You could also use an alternative for 7. $\endgroup$
    – fibonatic
    Commented Nov 2, 2015 at 15:12
3
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In non-standard analysis, a halo (or monad) is described as "the collection hal(x) of numbers which are infinitely close to x " (Lobry et al). Thus, the numbers close to 0 would be hal(0) or monad(0). Hyperreal numbers are described further in "Foundations of Infinitesimal Calculus" (Keisler)

       _            _   _      _
| |   | |   |      |   | |      |
|-|   |-|   |      |   | |      |
| |   | |   |_     |_  |_|   . _|

References:

Keisler, H. Jerome. Foundations of infinitesimal calculus. Vol. 20. Boston: Prindle, Weber & Schmidt, 1976.

Lobry, Claude, and Tewfik Sari. "Non-standard analysis and representation of reality." International Journal of Control 81.3 (2008): 519-536.

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  • $\begingroup$ This has 2 boxes identical (empty) $\endgroup$
    – DrunkWolf
    Commented Oct 30, 2015 at 13:45
  • $\begingroup$ @DrunkWolf fixed $\endgroup$
    – user2674
    Commented Oct 30, 2015 at 14:17
  • $\begingroup$ Is that really a number or is it a set of numbers? $\endgroup$ Commented Oct 30, 2015 at 17:16
3
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What about

this.
Using Hex, that's A^-FEDCB

Equivalent to 10^-1043915

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    $\begingroup$ I think you need lower-case 'd' or call it zero. $\endgroup$ Commented Oct 30, 2015 at 18:57
  • 1
    $\begingroup$ Also if that is B, then what would 8 be? $\endgroup$ Commented Oct 30, 2015 at 20:07
0
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Here's my attempt:

enter image description here

Probably not a valid solution though since it is actually not positive.

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7
  • 1
    $\begingroup$ That's exactly 0. $\endgroup$
    – Deusovi
    Commented Oct 30, 2015 at 10:24
  • $\begingroup$ Yes, that's why I wrote it's not positive. And the more I thought about it after posting it, the sillier it seemed, since it could just as well have said 1-1! $\endgroup$
    – Petter
    Commented Oct 30, 2015 at 10:30
  • $\begingroup$ What about 1 - 0.988... or 1 - 0.911... Do all repeating decimal digits round up? $\endgroup$ Commented Oct 30, 2015 at 12:11
  • 1
    $\begingroup$ 1 - 0.988... would be equal to 0.111..., and 1 - 0.9111... would equal 0.088888, they would be positive, but not very small :) $\endgroup$
    – Petter
    Commented Oct 30, 2015 at 12:49
  • 8
    $\begingroup$ Lots of duplicate boxes as well in this one $\endgroup$
    – DrunkWolf
    Commented Oct 30, 2015 at 13:47
0
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What about...

1 / treE(3)

I think you can do a divide sign...

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3
  • 1
    $\begingroup$ Can you show how this can be done with the LEDs? $\endgroup$
    – Ric
    Commented Oct 30, 2015 at 18:45
  • $\begingroup$ center, top right, and bottom left segment could be a passable slash. Not sure what t looks like, though. $\endgroup$
    – Random832
    Commented Oct 30, 2015 at 18:56
  • $\begingroup$ Top-left, center, bottom-left and bottom $\endgroup$ Commented Oct 30, 2015 at 20:36

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