What could be better than base 10?

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?

Solution 1:

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.

Solution 2:

I think base $6$ would make counting on our hands particularly convenient, we would have a $1$'s hand and a $6$'s hand and would be able to count up to $35$.

Solution 3:

In order to answer this question, it is first necessary to ask: What makes one base “better” than other? Some reasonable things to consider are:

Size

There is a tradeoff between the number of distinct digit characters used in a base (Base $b$ has exactly $b$ of these, from $0$ to $b - 1$, inclusive) and the length of the numeral required to represent a given number (which is $O(1/\log{b})$).

If the base is too small, then numbers explode into cumbersome long strings of digits. For example, in binary, the current year is 111 1101 1101, and the population of China (according to its 2010 census) was 100 1111 1101 1010 1001 0100 0011 0100. Modern computers can easily work with 32-bit or 64-bit binary numbers, but humans can't, which is why programmers have developed more compact encodings of binary, such as hexadecimal.

On the other hand, if we picked a very large base, like 2520, then you would need only 3 characters to represent the population of China, but typing them would be just as challenging as typing Chinese. And forget about learning the mulitplication table, whose size is $O(b^2)$. The only practical way to use such a large base is to split it into sub-bases, the way base-60 is represented as a mixture of base-6 and base-10.

So, what we want is a happy medium.

Fraction-friendliness

This is the main argument advanced in favor of base-12 or other highly composite bases (2, 4, 6, 12, 24, 36, 48, 60, 120, ...).

If a base has a lot of factors, it makes fractions easier to work with. For example, in base ten, 1/3 is represented as the infinitely repeating 0.333 333 333... (often rounded to 0.33 or 0.333), and this awkwardness crops up in deals like “3 for $5” or +/- grading systems. But in base-12, 1/3 is a nice simple 0.4.

Of course, because there are an infinite number of primes, it's impossible to completely avoid repeating “decimals”. And base-12's simplicity for the fractions 1/3 (0.4), 1/4 (0.3), 1/6 (0.2), 1/8 (0.16) and 1/9 (0.14) comes at the price of making 1/5 (0.24972497...) and 1/10 (0.124972497...) recurring dozenal fractions. But 1/3 is more common than 1/5.

Solution 4:

Brian Hayes in his American Scientist article Third Base argues that "When base 2 is too small and base 10 is too big, base 3 is just right."

Figure 1 has the caption

Most economical radix for a numbering system is $e$ (about $2.718$) when economy is measured as the product of the radix and the width, or number of digits, needed to express a given range of values. Here both the radix and the width are treated as continuous variables.

Figure 2 has the caption

Most economical integer radix is almost always 3, the integer closest to $e$. If the capacity of a numbering system is $r^w$, and the cost of a representation is $rw$, then $r=3$ is the best integer radix for all but a finite set of capacities. Specifically, ternary is inferior to binary only for 8,487 values of $r^w$; ternary is superior for infinitely many values.

Figure 3 has the caption

Ternary structure may offer the quickest path through a telephone menu system. Putting eight choices (assumed to be equally likely) in a single octonary menu (left) forces the caller to listen to 4.5 menu items on average. A binary structure (middle) has the same performance, but the ternary tree (right) reduces the average to 3.75.

Solution 5:

Balanced Nonary (base 9) would probably be really good. The digits go from -4 to 4, so taking the negative of a number would just be taking the negative of each digit, so subtraction is easy. Multiplication and division are particularly easy too if you make the easy conversion to balanced ternary first. Then there's no carrying when multiplying single digits (like in binary), and division is just testing inequalities (if you can divide by 2). Of course, if you want to do things faster, learning a balanced nonary times table would be easier than learning a regular nonary times table since you only really need to know the table for 1,2,3,4 and then handle negatives (and zero) appropriately.

There have even been computers based on balanced ternary.