jaap wrote:I vaguely remember reading an argument somewhere that went something like this:

1) The two main factors determining the complexity of the number notation system are the number of symbols the system has (equals the base) and the average length of commonly used numbers when written in that system (average number of digits).

2) These two factors were combined in some way to give a complexity score to using any particular base.

3) Optimising for this score gave e as the best answer.

4) Therefore base 3 was the best base to use.

Looking at this now there is an obvious fudge factor in step 2, where you can weigh the importance of the two aspects differently to get whatever outcome you like. If you really hate using many kinds of symbols but don't mind writing a lot, use binary (or even 'unary'), if you want shorter numbers use a higher base.

At the time I read it, this fudge factor was not obvious, so it was probably implicit due to some assumption somewhere.

Does this ring a bell with anyone?

I'd say that number of symbols is not really that important: we already learn 27*2 symbols(plus punctuation) for writing and nobody complains. Maybe for numbers that would be too much, but 15 symbols are not much harder to learn that 5, specially if we are drawing from symbols we already know. Like arbitreroftruth said, it's easier to memorize short strings with many possible symbols than long strings with few possible symbols: just compare "∂e5@" and "1001101011101001".

And a factor your reasoning does not really include is the utility that comes from having easy multiplication tables and terminating fractions: things that depend on the prime factorization of the base. Small numbers like 3 are bound to be bad both at this and at number of digits(2 is different because multiplication becomes sums and shifts, but that's also the worst number possible when it comes to length).

In my opinion, the base that has risen naturally should be at least close to the optimal one, in terms of memory efficiency: so I'd go with 12, for that reason and the prime factors.