Most people use base 10; it's obviously the common notation in the modern world.
However, if we could change what became the common notation, would there be a better choice?
I'm aware that it very well may be that there is no intrinsically superior base, but for the purposes of humans, is there a better one?
I've heard from sources such as this and this that base 12 is better, from here that base 8 is better, and, being into computer science, I would say that base 16 is the most handy.
Base 12 does seem to be the most supported non-base 10 number system, mainly due to the following reason pointed out by George Dvorsky:
First and foremost, 12 is a highly composite number — the smallest
number with exactly four divisors: 2, 3, 4, and 6 (six if you count 1
and 12). As noted, 10 has only two. Consequently, 12 is much more
practical when using fractions — it's easier to divide units of
weights and measures into 12 parts, namely halves, thirds, and
quarters.
And, on top of that, previous societies considered very advanced used other systems, such as the Mayans using base 20, and the Babylonians using base 60.
So, summarized, my question is: Is there an intrinsically superior base? If not, is there one that would be best for society's purposes? Or does the best base depend on the context it is being used in?
Best Answer
I like the factorial base, where the integer part of a real number is written as $\sum_{i=2}^n a_i i!$ where the $a_i$ are integers such that $0 \le a_i < i$ and the fractional part is written as $\sum_{i=2}^{\infty} \frac{b_i}{i!}$ where the $b_i$ are integers such that$0 \le b_i < i$.
The nice thing about this is that the integer part has a unique representation and the fractional part terminates if and only if the number if rational (except for the case corresponding to $\frac1{n!} = \sum_{i=n+1}^{\infty} \frac{i-1}{i!}$, the same as 1 = .99999...).
This is a special case of the following result: If $(B_i)_{i=0}^{\infty}$ is an increasing series of positive integers with $B_0 = 1$, we can represent all positive integers in the form $N=\sum_{i=1}^m a_i B_i$ where $0 \le a_i < B_{i}/B_{i-1}$ and $N < B_m$. This representation is unique if and only if $B_{i}/B_{i-1}$ is an integer for all $i$.
The usual decimal, binary, and hexadecimal bases have $B_i = 2^i, 10^i$, or $16^i$. The factorial base has $B_i = (i+1)!$.
I worked this out over 40 years ago and found it quite interesting. I am sure the result is several hundered years old.