memes

The number to describe a base is always the number 10 in that base

For example binary is base 2, it has only 1 and 0 as digits, and 2 in binary is 10.

Similarly for 4, and base 10.

So no matter what your numbering system is, with that system it is always base “10”

Every base has a 10. Base 4 numbering goes 1, 2, 3, 10, 11, 12, 20 etc...

"Base" is the number of distinct integers you have in play. In Base 10, there are ten of them. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. You can think of the numeric representation 10 as "1 ten, and 0 ones."

In Base 2 (binary) the only two digits available are 0 and 1. The first four binary numbers are 0, 1, 10, 11, which represent zero, one, two, and three. In Base 2, "10" means "1 two, and 0 ones." But, "Base 2" can't be written in binary, there's no concept of 2! Indeed, the way we reflect two in binary is 10. Which means, when we're talking in binary, "Base 2" is written as "Base 10."

This holds true for EVERY base. In Base 4, we have the digits 0, 1, 2, and 3. So if we want a value of four, we need to write it as 10. "1 four, 0 ones". So, when we're talking in Base 4, the way to say "Base 4" is ALSO by saying "Base 10"!

The trick behind it is that numbers written don't have context-free meaning. You can't communicate what "10" means without knowing how many distinct digits your conversational partner is working with. Most people have centralized on base 10, but there's no inherent advantage to doing things that way. Indeed, it's kind of awkward in lots of ways. Consider Base 12 (the digits of which are most often 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, as an aside). In Base 12, you can easily divide your base numbers by 1, 2, 3, 4. That's SUPER handy, since we obviously break things up into groups of 3 and 4 pretty often in our daily lives, but that's pretty painful in Base 10 because you immediately run into the need for fractions.

If you count in base 10 (from 0 to 19):

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

Base 4 (from 0 to 7):

0 1 2 3

10 11 12 13

Base 16 (from 0 to 31):

0 1 2 3 4 5 6 7 8 9 a b c d e f

10 11 12 13 14 15 16 17 18 19 1a 1b 1c 1d 1e 1f

Base 10 means when you count it goes: 1 2 3 4 5 6 7 8 9 10

Base 4 means when you count it goes: 1 2 3 10. 10 would still be equivalent to 4, 11 would be 5, 12 would be 6, and 20 would be 8.

To an alien that counted in base 4, base 4 would be base 10, because 4 is where they start adding 0s to numbers and they don’t have a concept of what 4 is. Probably not really if they were a mathematician alien, but it made me laugh.

All your base

Ah, you ssh into a Unix like system and use $(( )) or maybe bc?