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When I am counting in a base greater than ten, I can use the letters of the alphabet. What do I use when I run out of those?

What comes after z?:

0, 1, 2, 3... 9, a, b, c, d... x, y, z, (?)

And how would I find it?

share|improve this question
16  
It doesn't matter what symbols you use; you can use whatever you want. You can make something up; you can use the symbols $d_1, d_2, \ldots $ in place of $a,b,c,\ldots$, or whatever. Or you could use capital letters. Or Greek letters. Or make up arbitrary squiggles. –  MJD Jul 1 at 16:18
1  
take a look at this: en.wikipedia.org/wiki/Base_36 –  Varun Iyer Jul 1 at 16:19
10  
What comes beyond Z? Dr. Seuss wrote a book about that. –  John Jul 1 at 16:26
14  
Base 64 encoding typically uses a-z, A-Z, 0-9, and two others (such as / and +). –  MPW Jul 1 at 16:27
2  
@Questionmark: There is no standard (except for certain special cases). –  Micah Jul 1 at 16:49

8 Answers 8

up vote 24 down vote accepted

You can choose your favorite letter and put an index, e.g. for base $n\in \mathbb{N}$ $$a_1,a_2,a_3,\ldots, a_n.$$

share|improve this answer
6  
Hahahaha, this is hilarious –  Jorge Fernández Jul 1 at 16:58
10  
@banana It's not that hilarious. Suppose you want to write numbers in base 100. It makes perfect sense to denote the 100 digits as 00, 01, …, 99. If you do that, then the base-100 representation of, say, 142857, is the three digits 14 28 57. This kind of system is ubiquitous in computer applications (an example is given by user87690 elsewhere in this thread). The bablyonians used a similar technique for their base-60 system, where the 60 'digits' were effectively 00, 01,… 59, as David K. observed elsewhere in this thread. –  MJD Jul 1 at 18:50
13  
OK, but it's still at least half as hilarious as I claimed originally, which is a non-negligible amount of hilarity –  Jorge Fernández Jul 1 at 18:59
4  
I suppose that if a joke is funny, it needs no explanation. The answer looks good. Not sure how it's hilarious, not even a bit. –  JoeTaxpayer Jul 1 at 20:20
15  
@JoeTaxpayer I believe Bananarama finds it funny because you're effectively giving up on the idea of having a symbol for each possible digit. Instead, you're just recycling base 10, which somewhat defeats the purpose if you're actually hoping to write the number down or print it out in your chosen base. I actually tend to agree there's a high degree of irony there. –  jpmc26 Jul 2 at 4:13

The Babylonians were able to do math in base $60$. Each digit consisted of $N$ wedge marks in one direction and $M$ in the other direction, and the value of the digit was $10N + M$. They lacked a zero, but that is not hard to provide.

You could also say that time of day is base $60$, so $\mbox{1:23:45} = 1 \cdot 60^2 + 23 \cdot 60 + 45.$ The basic idea of a place-value system is that you can tell what digit is in each place somehow (in this case, by taking the symbols $0$ through $9$ in pairs, and some people insert the symbol ":" beteween pairs as a visual aid). Each digit does not need to be drawn as a single connected region of black paint.

share|improve this answer
    
I like that story about the Babylonians... +1 –  Questionmark Jul 2 at 16:44

Take for example base 64, widely used for encoding binary data.

Once it gets to Z (it starts uppercase) it goes with lowercase letters: WXYZabcd. The careful observer will note it's still missing two letters, because 10 + 2 * 26 = 62.

Those two are usually one of /, + and -, but it's more or less implementation dependent.

A base64 number looks like this cGxlYXN1cmUu and you have probably seen it in a URL before.

share|improve this answer
    
This answer pleases me. ;-} –  corsiKa Jul 2 at 16:51
1  
According to that wikipedia article, +/ is the most common pair of highest digits (those are anecdotally the ones used today) and the rest used various combinations of ., _, -, :, and !. –  Aaron Dufour Jul 2 at 19:13
    
Yeah, and there's also the padding =. –  BoppreH Jul 2 at 21:08
    
For URLs you typically use -_ and drop the padding, as all three otherwise typically used characters +/= are not nice to handle in URLs as they all have their special meaning. –  leemes Jul 2 at 22:28

You can also encode each digit of high-base number system by fixed-width representation of a number in low-base number system. For example each digit of a number in base ≤256 can be encoded by a pair of hexadecimal digits.

share|improve this answer
    
Yes, this is seen quite often in computer science (for example, binary-coded decimal encoding). –  augurar Jul 1 at 19:13
3  
IPv4 for example is 4 digits of base 255, traditionally denoted as dot separated base 10 numbers (e.g. 255.255.255.255). IPv6 took a similar approach, though the notation is slightly different, with 6 digits of colon separated base 16 numbers (e.g. ff:ee:dd:cc:bb:aa). This is a practical system that can scale to arbitrarily large bases without having to memorize/invent new symbols. –  Lie Ryan Jul 2 at 11:42
2  
@LieRyan Isnt't IPv4 of base 256 since 0 can occur? –  user87690 Jul 2 at 12:30
1  
@user87690: ah, my mistake, yes you're correct, it should actually be base 256. –  Lie Ryan Jul 2 at 12:43

We have tons of symbols available, Take any number of languages which have pairways disjoint letters. Index each of those languages and use the pre-ezisting indexes inside the language and you shall have a lot of letters, just by using the symbols for base $10$ numbers ,Japanese Hiragana, katakana, Arab, Hebrew and Spanish we get $179$ symbols.

share|improve this answer
    
If we take 5 different styles of writing, wouldn't we get 50 symbols? –  TZakrevskiy Jul 2 at 9:30
    
@TZakrevskiy No. Not all languages reuse the first 10 digits. Hebrew, for example, has unique characters for 10, 20, ... etc. –  AJ Stas Jul 2 at 13:03
    
@ajstas I see. Here's to my education)) –  TZakrevskiy Jul 2 at 13:06
1  
I meant you can also use the letters for each language –  Jorge Fernández Jul 2 at 14:00

+, -, /, { ,], ä, ö, ü, ß, ă, î, ț, ﺞ, 덖, 哶, ... There are millions of characters you could use. There are standards mostly used for base 64 http://en.wikipedia.org/wiki/Base64 or 32 http://en.wikipedia.org/wiki/Base32 but I'd say that it's mostly up to you what you choose.

share|improve this answer
    
Seems too complicated to write,use excel notation, as pointed already. –  lavkush Jul 2 at 13:50
1  
@lavkush I disagree. I would use up all the characters available on my keyboard before adopting an excel notation. Of course in some cases, like base 256, grouping may be easier, but for base 71 for example I would use 71 distinct characters. –  memory of a dream Jul 2 at 14:28

This example shows the first 1,600 values from base 40 represented in a particular fashion:

[]
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[37]
[38]
[39]
[1, 0]
[1, 1]
[1, 2]
[1, 3]
[1, 4]
[1, 5]
[1, 6]
[1, 7]
[1, 8]
[1, 9]
[1, 10]
[1, 11]
[1, 12]
[1, 13]
[1, 14]
[1, 15]
[1, 16]
[1, 17]
[1, 18]
[1, 19]
[1, 20]
[1, 21]
[1, 22]
[1, 23]
[1, 24]
[1, 25]
[1, 26]
[1, 27]
[1, 28]
[1, 29]
[1, 30]
[1, 31]
[1, 32]
[1, 33]
[1, 34]
[1, 35]
[1, 36]
[1, 37]
[1, 38]
[1, 39]
[2, 0]
[2, 1]
[2, 2]
[2, 3]
[2, 4]
[2, 5]
[2, 6]
[2, 7]
[2, 8]
[2, 9]
[2, 10]
[2, 11]
[2, 12]
[2, 13]
[2, 14]
[2, 15]
[2, 16]
[2, 17]
[2, 18]
[2, 19]
[2, 20]
[2, 21]
[2, 22]
[2, 23]
[2, 24]
[2, 25]
[2, 26]
[2, 27]
[2, 28]
[2, 29]
[2, 30]
[2, 31]
[2, 32]
[2, 33]
[2, 34]
[2, 35]
[2, 36]
[2, 37]
[2, 38]
[2, 39]
[3, 0]
[3, 1]
[3, 2]
[3, 3]
[3, 4]
[3, 5]
[3, 6]
[3, 7]
[3, 8]
[3, 9]
[3, 10]
[3, 11]
[3, 12]
[3, 13]
[3, 14]
[3, 15]
[3, 16]
[3, 17]
[3, 18]
[3, 19]
[3, 20]
[3, 21]
[3, 22]
[3, 23]
[3, 24]
[3, 25]
[3, 26]
[3, 27]
[3, 28]
[3, 29]
[3, 30]
[3, 31]
[3, 32]
[3, 33]
[3, 34]
[3, 35]
[3, 36]
[3, 37]
[3, 38]
[3, 39]
[4, 0]
[4, 1]
[4, 2]
[4, 3]
[4, 4]
[4, 5]
[4, 6]
[4, 7]
[4, 8]
[4, 9]
[4, 10]
[4, 11]
[4, 12]
[4, 13]
[4, 14]
[4, 15]
[4, 16]
[4, 17]
[4, 18]
[4, 19]
[4, 20]
[4, 21]
[4, 22]
[4, 23]
[4, 24]
[4, 25]
[4, 26]
[4, 27]
[4, 28]
[4, 29]
[4, 30]
[4, 31]
[4, 32]
[4, 33]
[4, 34]
[4, 35]
[4, 36]
[4, 37]
[4, 38]
[4, 39]
[5, 0]
[5, 1]
[5, 2]
[5, 3]
[5, 4]
[5, 5]
[5, 6]
[5, 7]
[5, 8]
[5, 9]
[5, 10]
[5, 11]
[5, 12]
[5, 13]
[5, 14]
[5, 15]
[5, 16]
[5, 17]
[5, 18]
[5, 19]
[5, 20]
[5, 21]
[5, 22]
[5, 23]
[5, 24]
[5, 25]
[5, 26]
[5, 27]
[5, 28]
[5, 29]
[5, 30]
[5, 31]
[5, 32]
[5, 33]
[5, 34]
[5, 35]
[5, 36]
[5, 37]
[5, 38]
[5, 39]
[6, 0]
[6, 1]
[6, 2]
[6, 3]
[6, 4]
[6, 5]
[6, 6]
[6, 7]
[6, 8]
[6, 9]
[6, 10]
[6, 11]
[6, 12]
[6, 13]
[6, 14]
[6, 15]
[6, 16]
[6, 17]
[6, 18]
[6, 19]
[6, 20]
[6, 21]
[6, 22]
[6, 23]
[6, 24]
[6, 25]
[6, 26]
[6, 27]
[6, 28]
[6, 29]
[6, 30]
[6, 31]
[6, 32]
[6, 33]
[6, 34]
[6, 35]
[6, 36]
[6, 37]
[6, 38]
[6, 39]
[7, 0]
[7, 1]
[7, 2]
[7, 3]
[7, 4]
[7, 5]
[7, 6]
[7, 7]
[7, 8]
[7, 9]
[7, 10]
[7, 11]
[7, 12]
[7, 13]
[7, 14]
[7, 15]
[7, 16]
[7, 17]
[7, 18]
[7, 19]
[7, 20]
[7, 21]
[7, 22]
[7, 23]
[7, 24]
[7, 25]
[7, 26]
[7, 27]
[7, 28]
[7, 29]
[7, 30]
[7, 31]
[7, 32]
[7, 33]
[7, 34]
[7, 35]
[7, 36]
[7, 37]
[7, 38]
[7, 39]
[8, 0]
[8, 1]
[8, 2]
[8, 3]
[8, 4]
[8, 5]
[8, 6]
[8, 7]
[8, 8]
[8, 9]
[8, 10]
[8, 11]
[8, 12]
[8, 13]
[8, 14]
[8, 15]
[8, 16]
[8, 17]
[8, 18]
[8, 19]
[8, 20]
[8, 21]
[8, 22]
[8, 23]
[8, 24]
[8, 25]
[8, 26]
[8, 27]
[8, 28]
[8, 29]
[8, 30]
[8, 31]
[8, 32]
[8, 33]
[8, 34]
[8, 35]
[8, 36]
[8, 37]
[8, 38]
[8, 39]
[9, 0]
[9, 1]
[9, 2]
[9, 3]
[9, 4]
[9, 5]
[9, 6]
[9, 7]
[9, 8]
[9, 9]
[9, 10]
[9, 11]
[9, 12]
[9, 13]
[9, 14]
[9, 15]
[9, 16]
[9, 17]
[9, 18]
[9, 19]
[9, 20]
[9, 21]
[9, 22]
[9, 23]
[9, 24]
[9, 25]
[9, 26]
[9, 27]
[9, 28]
[9, 29]
[9, 30]
[9, 31]
[9, 32]
[9, 33]
[9, 34]
[9, 35]
[9, 36]
[9, 37]
[9, 38]
[9, 39]
[10, 0]
[10, 1]
[10, 2]
[10, 3]
[10, 4]
[10, 5]
[10, 6]
[10, 7]
[10, 8]
[10, 9]
[10, 10]
[10, 11]
[10, 12]
[10, 13]
[10, 14]
[10, 15]
[10, 16]
[10, 17]
[10, 18]
[10, 19]
[10, 20]
[10, 21]
[10, 22]
[10, 23]
[10, 24]
[10, 25]
[10, 26]
[10, 27]
[10, 28]
[10, 29]
[10, 30]
[10, 31]
[10, 32]
[10, 33]
[10, 34]
[10, 35]
[10, 36]
[10, 37]
[10, 38]
[10, 39]
[11, 0]
[11, 1]
[11, 2]
[11, 3]
[11, 4]
[11, 5]
[11, 6]
[11, 7]
[11, 8]
[11, 9]
[11, 10]
[11, 11]
[11, 12]
[11, 13]
[11, 14]
[11, 15]
[11, 16]
[11, 17]
[11, 18]
[11, 19]
[11, 20]
[11, 21]
[11, 22]
[11, 23]
[11, 24]
[11, 25]
[11, 26]
[11, 27]
[11, 28]
[11, 29]
[11, 30]
[11, 31]
[11, 32]
[11, 33]
[11, 34]
[11, 35]
[11, 36]
[11, 37]
[11, 38]
[11, 39]
[12, 0]
[12, 1]
[12, 2]
[12, 3]
[12, 4]
[12, 5]
[12, 6]
[12, 7]
[12, 8]
[12, 9]
[12, 10]
[12, 11]
[12, 12]
[12, 13]
[12, 14]
[12, 15]
[12, 16]
[12, 17]
[12, 18]
[12, 19]
[12, 20]
[12, 21]
[12, 22]
[12, 23]
[12, 24]
[12, 25]
[12, 26]
[12, 27]
[12, 28]
[12, 29]
[12, 30]
[12, 31]
[12, 32]
[12, 33]
[12, 34]
[12, 35]
[12, 36]
[12, 37]
[12, 38]
[12, 39]
[13, 0]
[13, 1]
[13, 2]
[13, 3]
[13, 4]
[13, 5]
[13, 6]
[13, 7]
[13, 8]
[13, 9]
[13, 10]
[13, 11]
[13, 12]
[13, 13]
[13, 14]
[13, 15]
[13, 16]
[13, 17]
[13, 18]
[13, 19]
[13, 20]
[13, 21]
[13, 22]
[13, 23]
[13, 24]
[13, 25]
[13, 26]
[13, 27]
[13, 28]
[13, 29]
[13, 30]
[13, 31]
[13, 32]
[13, 33]
[13, 34]
[13, 35]
[13, 36]
[13, 37]
[13, 38]
[13, 39]
[14, 0]
[14, 1]
[14, 2]
[14, 3]
[14, 4]
[14, 5]
[14, 6]
[14, 7]
[14, 8]
[14, 9]
[14, 10]
[14, 11]
[14, 12]
[14, 13]
[14, 14]
[14, 15]
[14, 16]
[14, 17]
[14, 18]
[14, 19]
[14, 20]
[14, 21]
[14, 22]
[14, 23]
[14, 24]
[14, 25]
[14, 26]
[14, 27]
[14, 28]
[14, 29]
[14, 30]
[14, 31]
[14, 32]
[14, 33]
[14, 34]
[14, 35]
[14, 36]
[14, 37]
[14, 38]
[14, 39]
[15, 0]
[15, 1]
[15, 2]
[15, 3]
[15, 4]
[15, 5]
[15, 6]
[15, 7]
[15, 8]
[15, 9]
[15, 10]
[15, 11]
[15, 12]
[15, 13]
[15, 14]
[15, 15]
[15, 16]
[15, 17]
[15, 18]
[15, 19]
[15, 20]
[15, 21]
[15, 22]
[15, 23]
[15, 24]
[15, 25]
[15, 26]
[15, 27]
[15, 28]
[15, 29]
[15, 30]
[15, 31]
[15, 32]
[15, 33]
[15, 34]
[15, 35]
[15, 36]
[15, 37]
[15, 38]
[15, 39]
[16, 0]
[16, 1]
[16, 2]
[16, 3]
[16, 4]
[16, 5]
[16, 6]
[16, 7]
[16, 8]
[16, 9]
[16, 10]
[16, 11]
[16, 12]
[16, 13]
[16, 14]
[16, 15]
[16, 16]
[16, 17]
[16, 18]
[16, 19]
[16, 20]
[16, 21]
[16, 22]
[16, 23]
[16, 24]
[16, 25]
[16, 26]
[16, 27]
[16, 28]
[16, 29]
[16, 30]
[16, 31]
[16, 32]
[16, 33]
[16, 34]
[16, 35]
[16, 36]
[16, 37]
[16, 38]
[16, 39]
[17, 0]
[17, 1]
[17, 2]
[17, 3]
[17, 4]
[17, 5]
[17, 6]
[17, 7]
[17, 8]
[17, 9]
[17, 10]
[17, 11]
[17, 12]
[17, 13]
[17, 14]
[17, 15]
[17, 16]
[17, 17]
[17, 18]
[17, 19]
[17, 20]
[17, 21]
[17, 22]
[17, 23]
[17, 24]
[17, 25]
[17, 26]
[17, 27]
[17, 28]
[17, 29]
[17, 30]
[17, 31]
[17, 32]
[17, 33]
[17, 34]
[17, 35]
[17, 36]
[17, 37]
[17, 38]
[17, 39]
[18, 0]
[18, 1]
[18, 2]
[18, 3]
[18, 4]
[18, 5]
[18, 6]
[18, 7]
[18, 8]
[18, 9]
[18, 10]
[18, 11]
[18, 12]
[18, 13]
[18, 14]
[18, 15]
[18, 16]
[18, 17]
[18, 18]
[18, 19]
[18, 20]
[18, 21]
[18, 22]
[18, 23]
[18, 24]
[18, 25]
[18, 26]
[18, 27]
[18, 28]
[18, 29]
[18, 30]
[18, 31]
[18, 32]
[18, 33]
[18, 34]
[18, 35]
[18, 36]
[18, 37]
[18, 38]
[18, 39]
[19, 0]
[19, 1]
[19, 2]
[19, 3]
[19, 4]
[19, 5]
[19, 6]
[19, 7]
[19, 8]
[19, 9]
[19, 10]
[19, 11]
[19, 12]
[19, 13]
[19, 14]
[19, 15]
[19, 16]
[19, 17]
[19, 18]
[19, 19]
[19, 20]
[19, 21]
[19, 22]
[19, 23]
[19, 24]
[19, 25]
[19, 26]
[19, 27]
[19, 28]
[19, 29]
[19, 30]
[19, 31]
[19, 32]
[19, 33]
[19, 34]
[19, 35]
[19, 36]
[19, 37]
[19, 38]
[19, 39]
[20, 0]
[20, 1]
[20, 2]
[20, 3]
[20, 4]
[20, 5]
[20, 6]
[20, 7]
[20, 8]
[20, 9]
[20, 10]
[20, 11]
[20, 12]
[20, 13]
[20, 14]
[20, 15]
[20, 16]
[20, 17]
[20, 18]
[20, 19]
[20, 20]
[20, 21]
[20, 22]
[20, 23]
[20, 24]
[20, 25]
[20, 26]
[20, 27]
[20, 28]
[20, 29]
[20, 30]
[20, 31]
[20, 32]
[20, 33]
[20, 34]
[20, 35]
[20, 36]
[20, 37]
[20, 38]
[20, 39]
[21, 0]
[21, 1]
[21, 2]
[21, 3]
[21, 4]
[21, 5]
[21, 6]
[21, 7]
[21, 8]
[21, 9]
[21, 10]
[21, 11]
[21, 12]
[21, 13]
[21, 14]
[21, 15]
[21, 16]
[21, 17]
[21, 18]
[21, 19]
[21, 20]
[21, 21]
[21, 22]
[21, 23]
[21, 24]
[21, 25]
[21, 26]
[21, 27]
[21, 28]
[21, 29]
[21, 30]
[21, 31]
[21, 32]
[21, 33]
[21, 34]
[21, 35]
[21, 36]
[21, 37]
[21, 38]
[21, 39]
[22, 0]
[22, 1]
[22, 2]
[22, 3]
[22, 4]
[22, 5]
[22, 6]
[22, 7]
[22, 8]
[22, 9]
[22, 10]
[22, 11]
[22, 12]
[22, 13]
[22, 14]
[22, 15]
[22, 16]
[22, 17]
[22, 18]
[22, 19]
[22, 20]
[22, 21]
[22, 22]
[22, 23]
[22, 24]
[22, 25]
[22, 26]
[22, 27]
[22, 28]
[22, 29]
[22, 30]
[22, 31]
[22, 32]
[22, 33]
[22, 34]
[22, 35]
[22, 36]
[22, 37]
[22, 38]
[22, 39]
[23, 0]
[23, 1]
[23, 2]
[23, 3]
[23, 4]
[23, 5]
[23, 6]
[23, 7]
[23, 8]
[23, 9]
[23, 10]
[23, 11]
[23, 12]
[23, 13]
[23, 14]
[23, 15]
[23, 16]
[23, 17]
[23, 18]
[23, 19]
[23, 20]
[23, 21]
[23, 22]
[23, 23]
[23, 24]
[23, 25]
[23, 26]
[23, 27]
[23, 28]
[23, 29]
[23, 30]
[23, 31]
[23, 32]
[23, 33]
[23, 34]
[23, 35]
[23, 36]
[23, 37]
[23, 38]
[23, 39]
[24, 0]
[24, 1]
[24, 2]
[24, 3]
[24, 4]
[24, 5]
[24, 6]
[24, 7]
[24, 8]
[24, 9]
[24, 10]
[24, 11]
[24, 12]
[24, 13]
[24, 14]
[24, 15]
[24, 16]
[24, 17]
[24, 18]
[24, 19]
[24, 20]
[24, 21]
[24, 22]
[24, 23]
[24, 24]
[24, 25]
[24, 26]
[24, 27]
[24, 28]
[24, 29]
[24, 30]
[24, 31]
[24, 32]
[24, 33]
[24, 34]
[24, 35]
[24, 36]
[24, 37]
[24, 38]
[24, 39]
[25, 0]
[25, 1]
[25, 2]
[25, 3]
[25, 4]
[25, 5]
[25, 6]
[25, 7]
[25, 8]
[25, 9]
[25, 10]
[25, 11]
[25, 12]
[25, 13]
[25, 14]
[25, 15]
[25, 16]
[25, 17]
[25, 18]
[25, 19]
[25, 20]
[25, 21]
[25, 22]
[25, 23]
[25, 24]
[25, 25]
[25, 26]
[25, 27]
[25, 28]
[25, 29]
[25, 30]
[25, 31]
[25, 32]
[25, 33]
[25, 34]
[25, 35]
[25, 36]
[25, 37]
[25, 38]
[25, 39]
[26, 0]
[26, 1]
[26, 2]
[26, 3]
[26, 4]
[26, 5]
[26, 6]
[26, 7]
[26, 8]
[26, 9]
[26, 10]
[26, 11]
[26, 12]
[26, 13]
[26, 14]
[26, 15]
[26, 16]
[26, 17]
[26, 18]
[26, 19]
[26, 20]
[26, 21]
[26, 22]
[26, 23]
[26, 24]
[26, 25]
[26, 26]
[26, 27]
[26, 28]
[26, 29]
[26, 30]
[26, 31]
[26, 32]
[26, 33]
[26, 34]
[26, 35]
[26, 36]
[26, 37]
[26, 38]
[26, 39]
[27, 0]
[27, 1]
[27, 2]
[27, 3]
[27, 4]
[27, 5]
[27, 6]
[27, 7]
[27, 8]
[27, 9]
[27, 10]
[27, 11]
[27, 12]
[27, 13]
[27, 14]
[27, 15]
[27, 16]
[27, 17]
[27, 18]
[27, 19]
[27, 20]
[27, 21]
[27, 22]
[27, 23]
[27, 24]
[27, 25]
[27, 26]
[27, 27]
[27, 28]
[27, 29]
[27, 30]
[27, 31]
[27, 32]
[27, 33]
[27, 34]
[27, 35]
[27, 36]
[27, 37]
[27, 38]
[27, 39]
[28, 0]
[28, 1]
[28, 2]
[28, 3]
[28, 4]
[28, 5]
[28, 6]
[28, 7]
[28, 8]
[28, 9]
[28, 10]
[28, 11]
[28, 12]
[28, 13]
[28, 14]
[28, 15]
[28, 16]
[28, 17]
[28, 18]
[28, 19]
[28, 20]
[28, 21]
[28, 22]
[28, 23]
[28, 24]
[28, 25]
[28, 26]
[28, 27]
[28, 28]
[28, 29]
[28, 30]
[28, 31]
[28, 32]
[28, 33]
[28, 34]
[28, 35]
[28, 36]
[28, 37]
[28, 38]
[28, 39]
[29, 0]
[29, 1]
[29, 2]
[29, 3]
[29, 4]
[29, 5]
[29, 6]
[29, 7]
[29, 8]
[29, 9]
[29, 10]
[29, 11]
[29, 12]
[29, 13]
[29, 14]
[29, 15]
[29, 16]
[29, 17]
[29, 18]
[29, 19]
[29, 20]
[29, 21]
[29, 22]
[29, 23]
[29, 24]
[29, 25]
[29, 26]
[29, 27]
[29, 28]
[29, 29]
[29, 30]
[29, 31]
[29, 32]
[29, 33]
[29, 34]
[29, 35]
[29, 36]
[29, 37]
[29, 38]
[29, 39]
[30, 0]
[30, 1]
[30, 2]
[30, 3]
[30, 4]
[30, 5]
[30, 6]
[30, 7]
[30, 8]
[30, 9]
[30, 10]
[30, 11]
[30, 12]
[30, 13]
[30, 14]
[30, 15]
[30, 16]
[30, 17]
[30, 18]
[30, 19]
[30, 20]
[30, 21]
[30, 22]
[30, 23]
[30, 24]
[30, 25]
[30, 26]
[30, 27]
[30, 28]
[30, 29]
[30, 30]
[30, 31]
[30, 32]
[30, 33]
[30, 34]
[30, 35]
[30, 36]
[30, 37]
[30, 38]
[30, 39]
[31, 0]
[31, 1]
[31, 2]
[31, 3]
[31, 4]
[31, 5]
[31, 6]
[31, 7]
[31, 8]
[31, 9]
[31, 10]
[31, 11]
[31, 12]
[31, 13]
[31, 14]
[31, 15]
[31, 16]
[31, 17]
[31, 18]
[31, 19]
[31, 20]
[31, 21]
[31, 22]
[31, 23]
[31, 24]
[31, 25]
[31, 26]
[31, 27]
[31, 28]
[31, 29]
[31, 30]
[31, 31]
[31, 32]
[31, 33]
[31, 34]
[31, 35]
[31, 36]
[31, 37]
[31, 38]
[31, 39]
[32, 0]
[32, 1]
[32, 2]
[32, 3]
[32, 4]
[32, 5]
[32, 6]
[32, 7]
[32, 8]
[32, 9]
[32, 10]
[32, 11]
[32, 12]
[32, 13]
[32, 14]
[32, 15]
[32, 16]
[32, 17]
[32, 18]
[32, 19]
[32, 20]
[32, 21]
[32, 22]
[32, 23]
[32, 24]
[32, 25]
[32, 26]
[32, 27]
[32, 28]
[32, 29]
[32, 30]
[32, 31]
[32, 32]
[32, 33]
[32, 34]
[32, 35]
[32, 36]
[32, 37]
[32, 38]
[32, 39]
[33, 0]
[33, 1]
[33, 2]
[33, 3]
[33, 4]
[33, 5]
[33, 6]
[33, 7]
[33, 8]
[33, 9]
[33, 10]
[33, 11]
[33, 12]
[33, 13]
[33, 14]
[33, 15]
[33, 16]
[33, 17]
[33, 18]
[33, 19]
[33, 20]
[33, 21]
[33, 22]
[33, 23]
[33, 24]
[33, 25]
[33, 26]
[33, 27]
[33, 28]
[33, 29]
[33, 30]
[33, 31]
[33, 32]
[33, 33]
[33, 34]
[33, 35]
[33, 36]
[33, 37]
[33, 38]
[33, 39]
[34, 0]
[34, 1]
[34, 2]
[34, 3]
[34, 4]
[34, 5]
[34, 6]
[34, 7]
[34, 8]
[34, 9]
[34, 10]
[34, 11]
[34, 12]
[34, 13]
[34, 14]
[34, 15]
[34, 16]
[34, 17]
[34, 18]
[34, 19]
[34, 20]
[34, 21]
[34, 22]
[34, 23]
[34, 24]
[34, 25]
[34, 26]
[34, 27]
[34, 28]
[34, 29]
[34, 30]
[34, 31]
[34, 32]
[34, 33]
[34, 34]
[34, 35]
[34, 36]
[34, 37]
[34, 38]
[34, 39]
[35, 0]
[35, 1]
[35, 2]
[35, 3]
[35, 4]
[35, 5]
[35, 6]
[35, 7]
[35, 8]
[35, 9]
[35, 10]
[35, 11]
[35, 12]
[35, 13]
[35, 14]
[35, 15]
[35, 16]
[35, 17]
[35, 18]
[35, 19]
[35, 20]
[35, 21]
[35, 22]
[35, 23]
[35, 24]
[35, 25]
[35, 26]
[35, 27]
[35, 28]
[35, 29]
[35, 30]
[35, 31]
[35, 32]
[35, 33]
[35, 34]
[35, 35]
[35, 36]
[35, 37]
[35, 38]
[35, 39]
[36, 0]
[36, 1]
[36, 2]
[36, 3]
[36, 4]
[36, 5]
[36, 6]
[36, 7]
[36, 8]
[36, 9]
[36, 10]
[36, 11]
[36, 12]
[36, 13]
[36, 14]
[36, 15]
[36, 16]
[36, 17]
[36, 18]
[36, 19]
[36, 20]
[36, 21]
[36, 22]
[36, 23]
[36, 24]
[36, 25]
[36, 26]
[36, 27]
[36, 28]
[36, 29]
[36, 30]
[36, 31]
[36, 32]
[36, 33]
[36, 34]
[36, 35]
[36, 36]
[36, 37]
[36, 38]
[36, 39]
[37, 0]
[37, 1]
[37, 2]
[37, 3]
[37, 4]
[37, 5]
[37, 6]
[37, 7]
[37, 8]
[37, 9]
[37, 10]
[37, 11]
[37, 12]
[37, 13]
[37, 14]
[37, 15]
[37, 16]
[37, 17]
[37, 18]
[37, 19]
[37, 20]
[37, 21]
[37, 22]
[37, 23]
[37, 24]
[37, 25]
[37, 26]
[37, 27]
[37, 28]
[37, 29]
[37, 30]
[37, 31]
[37, 32]
[37, 33]
[37, 34]
[37, 35]
[37, 36]
[37, 37]
[37, 38]
[37, 39]
[38, 0]
[38, 1]
[38, 2]
[38, 3]
[38, 4]
[38, 5]
[38, 6]
[38, 7]
[38, 8]
[38, 9]
[38, 10]
[38, 11]
[38, 12]
[38, 13]
[38, 14]
[38, 15]
[38, 16]
[38, 17]
[38, 18]
[38, 19]
[38, 20]
[38, 21]
[38, 22]
[38, 23]
[38, 24]
[38, 25]
[38, 26]
[38, 27]
[38, 28]
[38, 29]
[38, 30]
[38, 31]
[38, 32]
[38, 33]
[38, 34]
[38, 35]
[38, 36]
[38, 37]
[38, 38]
[38, 39]
[39, 0]
[39, 1]
[39, 2]
[39, 3]
[39, 4]
[39, 5]
[39, 6]
[39, 7]
[39, 8]
[39, 9]
[39, 10]
[39, 11]
[39, 12]
[39, 13]
[39, 14]
[39, 15]
[39, 16]
[39, 17]
[39, 18]
[39, 19]
[39, 20]
[39, 21]
[39, 22]
[39, 23]
[39, 24]
[39, 25]
[39, 26]
[39, 27]
[39, 28]
[39, 29]
[39, 30]
[39, 31]
[39, 32]
[39, 33]
[39, 34]
[39, 35]
[39, 36]
[39, 37]
[39, 38]
[39, 39]
[1, 0, 0]

You can invent any way you want to represent a number.

share|improve this answer
    
A little on the lengthy side :D –  Questionmark Jul 2 at 16:45

I don't mean to be silly, but why not use base 6? If you want join the gitis in groups of two bits

1234123051 can be understood as a 5-digit number: 12 34 12 30 51

In the decimal system multiplying by ten $T: x\mapsto 10x$ plays a very special role.

If we want to learn about a number $\alpha$ we could take the integer part (e.g. $\lfloor 3.5\rfloor = 3$), then multiply by 10, take the integer part again. Doing this over and over would give our decimal system.

  • $10^0\pi \mapsto 3$
  • $10^1\pi \mapsto 1$
  • $10^2\pi \mapsto 4$
  • ...

In your case, using 36 instead of 10. Or just use two base-6 digits. This is also related to channel coding from information theory.

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