How would you count a base $> 36$ system?

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 $z$?:

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

And how would I find it?

• 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, 2014 at 16:18
• What comes beyond Z? Dr. Seuss wrote a book about that.
– John
Jul 1, 2014 at 16:26
• Base 64 encoding typically uses a-z, A-Z, 0-9, and two others (such as / and +).
– MPW
Jul 1, 2014 at 16:27
• @Questionmark: There is no standard (except for certain special cases). Jul 1, 2014 at 16:49
• @awashburn There are potentially 1,111,998 characters in unicode, although only 109,384 are assigned in unicode 6.0. Also, "up to" is two words. Jul 2, 2014 at 19:52

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

• Hahahaha, this is hilarious Jul 1, 2014 at 16:58
• @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, 2014 at 18:50
• OK, but it's still at least half as hilarious as I claimed originally, which is a non-negligible amount of hilarity Jul 1, 2014 at 18:59
• 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. Jul 1, 2014 at 20:20
• @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. Jul 2, 2014 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.

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 $10 + 2 \cdot 26 = 62.$

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

A base $64$ number looks like this cGxlYXN1cmUu and you have probably seen it in a URL before.

• 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 !. Jul 2, 2014 at 19:13
• Yeah, and there's also the padding =. Jul 2, 2014 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. Jul 2, 2014 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 $\leq 256$ can be encoded by a pair of hexadecimal digits.

• Yes, this is seen quite often in computer science (for example, binary-coded decimal encoding). Jul 1, 2014 at 19:13
• 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. Jul 2, 2014 at 11:42
• @LieRyan Isnt't IPv4 of base 256 since 0 can occur? Jul 2, 2014 at 12:30
• @user87690: ah, my mistake, yes you're correct, it should actually be base 256. Jul 2, 2014 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.

• If we take 5 different styles of writing, wouldn't we get 50 symbols? Jul 2, 2014 at 9:30
• @TZakrevskiy No. Not all languages reuse the first 10 digits. Hebrew, for example, has unique characters for 10, 20, ... etc. Jul 2, 2014 at 13:03
• @ajstas I see. Here's to my education)) Jul 2, 2014 at 13:06
• I meant you can also use the letters for each language Jul 2, 2014 at 14:00

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.

• A little on the lengthy side :D Jul 2, 2014 at 16:45

+, -, /, { ,], 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 $64$ or base $32$ but I'd say that it's mostly up to you what you choose.

• Seems too complicated to write,use excel notation, as pointed already.
– L.K.
Jul 2, 2014 at 13:50
• @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. Jul 2, 2014 at 14:28

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.