# Counting in base 13

So, I was counting in base $$8$$:

$$1,2,3,4,5,6,7,10,11,12,13,14,15,16,17,20,21,22,23,24,25,26...$$

Then I tried counting in base $$13$$ and got confused:

$$1,2,3,4,5,6,7,8,9,10,11,12$$ (confused here, maybe:) $$1(01),1(02),1(03),....1(12),2(00).$$

Is there a nice or more standard way list the counting numbers in a base above $$10$$?

• It is perhaps more common to use letters, e.g., a=10, b=11, c=12. For instance the hexadecimal (16) base has 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, f as “digits”. Sep 14 '21 at 21:26
• $10_{13}$ would be $13_{10}$. You need new symbols. You could write, e.g., $a_{13}, b_{13}, c_{13}$ for $10,11,12$ in base $10$.
– lulu
Sep 14 '21 at 21:26
• Base $16$ (used in computing contexts) takes "ABCDEF" as extra digits. Sep 14 '21 at 21:27
• Indeed, in the computer world, where base $16$ is common, we use $$A=9+1,B=9+2,C=9+3,\\D=9+4,E=9+5,F=9+5$$ But in theory, you could use any symbols, as long as everybody with whom you are conversing in a base agree. Sep 14 '21 at 21:42
• For new digits, either use new symbols or (for large number bases) use decimal numbers separated by a separator. For example, $n=12\cdot13^2+12\cdot13+12$ would be $ccc_{13}$ or $12:12:12_{13}$ (or $(12,12,12)_{13}$). Sep 14 '21 at 21:43

In order to work in base $$13$$, you need three extra symbols, which represent $$10$$, $$11$$, and $$12$$. Suppose that these symbols are $$A$$, $$B$$, and $$C$$ respectively. Then you have:$$1,2,3,4,5,6,7,8,9,A,B,C,10,11,12,13,14,15,16,17,18,19,1A,1B,1C,20,\ldots$$
The standard way (at least as it's used conventionally in base 16 in computing contexts) is to start using letters. So you get $$0,1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C,\\ 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B, 1C,\\ 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 2A, 2B, 2C, \\ \vdots$$ Some prefer to use lower case instead, and that's fine too. And of course you're free to invent your own symbols if you'd like, or steal symbols from somewhere other than the Latin alphabet. Just be sure to inform your readers which symbol means what.
The standard way to count this (which is used in hexadecimal: base-$$16$$) is to use the letters of the alphabet like so: $$1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E ...$$ So in base-$$13$$ it would be: \begin{align} &1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C \\ &10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B, 1C \\ &20... \end{align}