What is the best base to use?
Solution 1:
A while back I tried doing some pencil-and-paper arithmetic in base 16 (extraction of square roots) to see what it was like, and found that it was hard mainly because I don't have the base-16 multiplication table memorized, and there is a lot to memorize. My conclusion at the end was that the easiest way to do base-16 arithmetic is to do it in base 4 and then convert at the end; the conversion is trivial. All the numerals have twice as many digits, but it takes much less than half as long to calculate each digit, so it is much faster. Converting to base 2 goes too far and loses the tradeoff; now the numerals are twice as long again, but you do not gain enough speed to make it worth while.
For base $n$, you have to memorize $\frac12\bigl(n^2-3n+2\bigr)$ multiplication table entries, since $0\times n$ and $1\times n$ are trivial, and $m\times n = n\times m$. This increases rapidly with $n$: in base 10 you have 36 products to memorize; in base 16 there are 105. For base 4, you only have to remember $2\times 2 = 10, 2\times 3 = 12, $ and $3\times 3 = 21$.
I did not try any of the usual methods for speeding up paper-and-pencil multiplication, such as Napier's bones, or Genaille-Lucas rulers. But I don't think these would help enough. If you don't have the multiplication tables down pat, you lose a lot of number sense that you probably take for granted. For example, you can't estimate the answer to division problems. The square root algorithm requires that you guess the answer to a lot of sub-problems of the type "How many times does 2D,409 go into 1C0,000?" and whereas this sort of thing is easy in base 10 because you have years of practice (What's $1,835,000\div 185,353$ to one decimal place?) it is hard in base 16. So knowing the multiplication tables is important, and in base 16 there is a lot to know.
I found the exercise very enlightening, and I recommend it. If nothing else, it may be interesting to re-experience feelings that you have not had since third grade.
Converting to and from base 8 is not so felicitous, but the multiplication table is not too hard to memorize. Still I think the advantage over bases 4 or 16 are minimal, if they exist at all. I think the worst choice for pencil calculation would be a large prime number, since there's no easy conversion to a smaller base as there is from bases 8 or 16 to base 4.
The desiderata for computer calculation are of course completely different, and for abacus calculation are different again. For computers I have seen claims that base 3 is in some sense optimal, but I think 70 years of engineering practice refutes that completely. There is a certain sense in which modern computers operate not in base 2 but in base 256, but it seems to be something of a philosophical question which it is; it depends on what level of operation you are looking at.
Solution 2:
We could say that there are two desirable goals for a base system:
- A minimized number of symbols (e.g. there are two symbols in binary, 0 and 1).
- Minimal digital lengths for each number (e.g. the number 10101 has a length of 5 digits in binary).
Let us say our base is $b$, and hence we have exactly $b$ symbols to represent numbers with. The digital length of a number is then represented as $\log_bN$ for a positive number $N.$
There are many ways to attempt to minimize both of these values simultaneously. One way is to minimize $||\langle b,\space \log_bN \rangle||_p$, where $||\small \vec x||_p$ is the p-norm of $\vec x$, and where $N$ can be any arbitrary number (hence $N$ becomes a weighting parameter of sorts).
The results you obtain with the above algorithm are completely dependent on what values you choose for $N$ and $p$. So while there is no absolute answer, you can obtain one by assigning fixed values to these two parameters.
Solution 3:
To have the largest readership possible, along with the greatest probability of your readers reading your work until the end, and the probability of your readers also finding your work easy to read, at present, it seems you want base ten.
There doesn't exist any mathematical argument as to which base system comes as the best until we have criteria to determine what "the best" would mean. Who do we want to talk to? For what purpose do we talk to them? What sort of background do they have? What sort of tools do they have? What can those tools do?
Solution 4:
It depends what you find 'best'. Personally I don't care whether it's easy to convert to binary, since computers do that just fine for any base, so I never need to.
What is a little bit more annoying for humans is fractions that don't terminate. So I think to get the best base, you just multiply prime numbers from low to high until you feel the base gets too big. E.g.
1 * 2 = 2
1 * 2 * 3 = 6
1 * 2 * 3 * 5 = 30
1 * 2 * 3 * 5 * 7 = 210
....
Using base 30, fractions like 1/2, 1/3, 1/5, 1/6, 1/8, 1/9, 1/10 or 1/15 factorizes nicely because the denominator's prime factors are also prime factors of your base. In that respect, base 12, or any $2^N$ base, waste numbers because they have double prime factors. The same fractions that terminate in base 12 also terminate in base 6.
Base 210 is definitely too big since you need to remember like 21.000 multiplications. Base 30 isn't really fun at around 400 either, but that's doable. The best bet may be 6. A quarter is not ideal though (3/20).