which is the best base?

twocubes:

In base 10, it’s easy to tell if things are multiples of 2 or 5. You just, look at the last digit, and if that’s a multiple of 2 (or 5) then the whole number is a multiple of 2 (or 5). There’s similarly easy ways to tell for any product of a power of 2 and a power of 5.

People sometimes argue that base 12 would be better than base 10 because in base 12, it’s easy to tell when things are divisible by 2 or 3 (or products of powers thereof) and multiples of 3 are more common than multiples of 5.

Ignoring briefly that you could say the same thing about base 6 this is all completely reasonable. It’s just that it misses something.

See, in base 10, there’s this trick that you can use to tell if something is a multiple of 3; what you do is you add up the digits, and if the thing at the end is a multiple of 3, then the whole thing was a multiple of 3 all along.

Is 2018 a multiple of 3? Well, 2+0+1+8 = 11, which isn’t a multiple of 3, so it isn’t. What about 20181201, today’s date but written as a big number? well, we have 2+0+1+8+1+2+0+1 = 15, which is a multiple of 3, so we indeed have that 20181201 is a multiple of 3.

There’s a more obscure trick to tell if a number is a multiple of 11. What you do is you alternatingly add and subtract the digits and check whether the thing at the end is a multiple of 11.

Is 2018 a multiple of 11? Well, 2-0+1-8 = -5 isn’t, so it isn’t. What about, 2321, the time that it is right now? well, 2-3+2-1 = 0 is, so it is.

But, like I said, that’s obscure, and the reason it’s obscure is that no one cares about 11. multiples of 11 are only 1/11ths of all the numbers, while multiples of 3 are 1/3rd, so obviously, people care more about the first than the second. The value of easily telling if a thing is divisible by 3 is 1/3, which is much larger than the value of being able to tell if a thing is divisible by 11, which is only 1/11.

But, ok, so it turns out that in general, in any base b, the 3s trick will work to tell when things are divisible by any factor of b-1, while the 11s trick will work to tell when things are divisible by any factor of b+1.

In fact, this allows us to do another trick: we can group digits! Basically, we take our number (let’s say 12345) in some base b, and then we can pretend that we’re in base b² by just… mushing the numbers together and thinking of it as 1′23′45, if you follow.

So, how do we tell if 34542 is a multiple of 101? well, we notice that 3 – 45 + 42 = 0, and thus we know that it is. (Frankly doing this with groups of digits or more is a pain so let’s just stay with groups of 2 digits) 

So, to find the best base b, what we want is to have as large a sum of the reciprocals of the distinct prime divisors of b, b-1, b+1, b²-1 and b²+1. But, also, since we don’t want to memorize multiplication and addition tables that are too large, let’s say that we don’t want any base that is larger than 20. So, with all that in mind, which is the best base? And the answer is…

13

we get 2 and 3 from 13-1 = 12, we get 7 from 13+1 = 14, we get 13 from 13 itself and we get 5 and 17 from 13²+1 = 170. This is better than any other base under 20.

strictly 21 is slightly better, since you get all those and also 11 from 21+1 = 22, but after that it doesn’t get any better until 34 (which only adds 89 from 34²+1 = 1157 = 13*89) but, I set the rules, so I’ll abide by them.

So, now you know. Have fun!