The Best Base
Having explored place notation simply as well as in some detail, we can now proceed to the question of which base, if any, is the best for human use. Currently, of course, we almost universally use the decimal base; in computers, we sometimes use "hex," or base sixteen, and "octal," or base eight; computers themselves function entirely in binary, or base two. Which of these, if any, is best for general human use?
Human Use
It should be emphasized that we are here discussing which base is best for human use. This concept bears some explanation.
Computers not only function in binary, they almost certainly must function in binary. That is, while it's possible to construct logical systems for computers functioning in a different base, it's much more difficult to do so; the most sensible solution is to allow computers to function in binary and simply convert that to some other base when it outputs the data for us to read. When we ask what the best base is here, we're not questioning that for computers, binary is the best base.
Nor are we investigating which base it's best to interact with computers in. Unix-style file permissions consist of a series of octal numbers, for example, and most computer graphics programs accept color codes as hexadecimal numbers; changing this would be foolish, and this discussion doesn't imply that we should. Data which is primarily intended for computer reading and interpreting should almost always be in a base which is some multiple of two, due to computers' internal reliance on binary arithmetic.
This does not mean, however, that the best base for human use is either two or some multiple thereof. Humans must deal with numbers in a way that is easiest for humans, not in a way that is easiest for computers, particularly since the conversion of numbers from binary to other bases is almost trivially easy for computers. What we are seeking here is the best base for humans; this may coincide with the best base for computers, or it may not. Either way, it is important not to confuse the issue.
Criteria for a Good Base
So by what criteria ought we to judge the benefits and deteriments of a particular base?
- As we've discussed elsewhere, a number system utilizing place notation requires a number of digits equal to the base itself. Because human beings are creatures of limited memory and limited ability to deal with large numbers of different symbols, we should select our base with a mind to ensuring that we have a manageable number of symbols; specifically, we should select a base small enough that we will not be overwhelmed by the number of necessary digits. Anything higher than sixteen or so is probably higher than desirable; we'd essentially be learning a new alphabet in order to use it. Anything higher than twenty is almost certainly too high.
- The size of the base will determine the size of the numbers we will be dealing with; that is, a small base will tend to produce much longer numbers than a large one. This is because each time we count through the base or a multiple thereof, we must add another digit to the number; in smaller bases, this number gets larger more quickly. As an example, the number "74" in decimal is "1001010" in binary; "7400" is "10010101110011101000." Numbers very quickly become quite unwieldy, even at scales that are quite common in normal, daily use. So we want to select our base not only to be small enough that we have a manageable number of symbols, but large enough that our numbers are reasonably sized for common magnitudes.
- Number of divisors. Human use requires the ability to "eyeball" fractions quickly and easily, and so we want as many fractions of our base to come to even numbers are possible. Most frequently, we deal with halves, thirds, and the half of the half, quarters; we also deal with halves of thirds, or sixths, and halves of quarters, eighths, very often. The half and the third are particularly important, being the first prime numbers, along with their product, the sixth; the quarter, being the first non-prime number, is also quite important, along with its half, the eighth. Fifths and sevenths would be handy, but is not as vital as these others.
- Some biological basis for our base. In decimal, of course, we can count on our fingers; in base twenty, perhaps we could count on our fingers and our toes. This is a relatively minor consideration; however, it shouldn't be ignored in our explorations.
The first two of these considerations more or less speak for themselves; for purposes of this brief article, I will suggest that the smallest reasonable base size is eight, and the highest is sixteen.
As for the third, justifying the choice of these fractions are more important than others is beyond the scope of this little piece. However, the necessity of as great a number of even factors as possible is certainly justifiable even independently of this.
For our explorations, we will use a simple alphabetical processions for "transdecimal" (that is, ten or higher) digits. So, for example, our counting for base sixteen will proceed as follows: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 10, 11, 12... We will use as many or as few of these letters as is necessary to explain a base.
Exploring Factors
The number of easy factors for a base is fairly easily explored; we simply expand the integer fractions of the base in terms of itself and compare the results. The table below does so for all even bases between eight and sixteen. (Note that odd bases have all been excluded, as they do not even offer the half as an easy fraction, and thus do not merit further consideration.)
| Integer Fractions in Five Bases | |||||
|---|---|---|---|---|---|
| Eight | Ten | Twelve | Fourteen | Sixteen | |
| 2 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 |
| 3 | 0.252525... | 0.333... | 0.4 | 0.494949... | 0.555... |
| 4 | 0.2 | 0.25 | 0.3 | 0.37 | 0.4 |
| 5 | 0.14631463... | 0.2 | 0.2497... | 0.2B2B2B... | 0.333... |
| 6 | 0.1252525... | 0.1666... | 0.2 | 0.2494949... | 0.2AAA... |
| 7 | 0.111... | 0.142857... | 0.186A35... | 0.2 | 0.249249... |
| 8 | 0.1 | 0.125 | 0.16 | 0.1A7 | 0.2 |
| 9 | 0.070707... | 0.111... | 0.14 | 0.17AC63... | 0.1C7... |
| 10 | 0.063146314... | 0.1 | 0.124972497... | 0.1585858... | 0.1999... |
| 11 | 0.0564272135... | 0.0909... | 0.111... | 0.13B65... | 0.1745D... |
| 12 | 0.0525252... | 0.08333... | 0.1 | 0.12494949... | 0.1555... |
| 13 | 0.04730473... | 0.076923... | 0.0B0B... | 0.111... | 0.13B13B... |
| 14 | 0.0444... | 0.071428571... | 0.0A35186A... | 0.1 | 0.1249249... |
| 15 | 0.04210421... | 0.0666... | 0.0972497... | 0.0D0D0D... | 0.111... |
| 16 | 0.04 | 0.0625 | 0.09 | 0.0C37 | 0.1 |
This table gives us a great deal of food for though concerning choice of bases. For one, we can notice a few interesting patterns; for example, the fraction for two is always even (given that all these bases are even, and therefore divisible by two). We can also notice that the fraction for the base minus one (what some base scholars have labeled the base's omega, or "Ω") is always 0.111... But the general rule that we can derive is one that seems obvious, but still bears stating: A base will have even fractions for even divisors of it or its powers.
For example, in decimal, "2" makes an even fraction ("0.5") because ten is divible by two; "4" makes an even fraction ("0.25") because four is divisible by ten to the power of two, or one hundred. But "3" does not make an even fraction because neither ten nor any of its powers are divisible by 3. This can lead us to a corollary of the above general rule: The even fractions of a base will have a number of digits equal to the power of the base by which it is divisible.
Again using decimal as an example, "2" makes an even one-digit fraction because "2" is divisible by the base to the power of one (that is, the base itself), while "4" makes an even two-digit fraction because "4" is divisible by the base to the power of two (that is, one hundred), and "8" makes an even three-digit fraction because "8" is not divisible by ten to the first or ten to the second, but only by ten to the third power (one thousand). Three, on the other hand, makes a fraction with an infinite number of digits because it is not divisible by any power of the base, no matter how high.
The upshot of this is that a base will have a number of even fractions commensurate with the numbers by which it and its powers is divisible. So that base which is divisible by the most numbers, along with its powers, will have the most even fractions; and the more numbers which are divisible by its lower powers, the shorter and therefore easier to deal with will those fractions be.
In other words, that base with the greatest number of factors for its lower-order divisors will have the most and the shortest (and therefore the easiest to deal with) fractions. The table below explores summarizes the results of the table above regarding its number of even fractions and their length. Numbers are in decimal for familiarity's sake.
| Number of Digits | ||||
|---|---|---|---|---|
| Base | One | Two | Three | Infinite |
| Eight | 3 | 1 | 0 | 11 |
| Ten | 3 | 1 | 1 | 9 |
| Twelve | 5 | 3 | 0 | 7 |
| Fourteen | 3 | 1 | 1 | 9 |
| Sixteen | 4 | 0 | 0 | 11 |
This table shows that eight and sixteen, although they are powers of two, are otherwise rather poor choices for a base. Eight contains only three one-digit fractions and one two-digit fraction; the remainder of those less than sixteen are infinite, for a wretched total of eleven. This would result in very frequently dealing with non-terminating fractions, a definitely suboptimal base. Sixteen, while it contains four fractions terminating in a single digit, these are all the powers of two; every single one of the rest of its fractions are infinite, which is also suboptimal.
Fourteen gives us a little more flexibility, but not much more; it has nine nonterminating fractions, more than half of those examined, and only five of its six terminating fractions terminate in three digits or less. (The last, the sixteenth, terminates in four.) Furthermore, its half is seven, a prime number; these means that taking the third or the half, or indeed, any fraction, of the half is impossible. This is suboptimal for a base.
Ten, again, is lackluster, performing almost identically to fourteen. This is due to ten's shared property with fourteen: its half (in this case, five) is a prime, so it's impossible to take its third or half in a terminating number of digits. It shares fourteen's faults that over half of the fractions examined are infinite, and that of the finite fractions only five of the six terminate in three digits or less. (The sixteenth, just as in fourteen, terminates in four.) Ten is then also suboptimal for a base.
Standing out like a shining beacon of light within this morass of infinite fractional expansions and lengthy terminating expressions is twelve. Twelve is the lowest of the abundant numbers, which means that the sum of its factors is greater than the number itself; the next-lowest abundant number is eighteen. Its half, six, while not quite abundant, can itself be evenly divided into thirds and halves, which gives it terminating expansions for the sixth and the third; furthermore, it is divisible by the lowest primes, two and three, as well as the lowest non-prime, four, even in its first power, which gives it single-digit expansions of these vital fractions. Alone among all the bases we've examined, less than half of its fractions are infinite; it has five fractions terminating in only one digit, and three terminating in two.
In terms of factors, twelve clearly stands out as the best base.
Biological Basis
We have ten fingers; this, and only this, is responsible for the dominance of base-ten counting the world today. While it is not fundamentally necessary for a good base, therefore, it would be helpful if our ideal base also had some biological basis for its adoption. For example, we have eight fingers not including thumbs, which would provide a basis for an octal counting system. Is there such a basis for any other bases?
It's hard to imagine such a base for fourteen or sixteen; however, for twelve there is such a basis, and quite simply. As shown in the image above, there are twelve segments on the fingers, not including the thumb, on each hand. This allows easy finger-counting to twice the base, rather than simply to the base itself, in base twelve. So eight, ten, and twelve also have a biological basis in addition to whatever other virtues they might have.
Conclusion
We've examined four criteria: that the base must not be too large; that it must not be too small; that it must have a great abundance of even fractions, especially two, three, four, six, and eight; and that it should have some biological basis in the human form. In all these categories, one base stands out before all the others: the base of twelve.
Twelve, the dozen (from which we derive the name "dozenal"), is clearly the best base for human use. Its use presents some problems, of course, including how to count in words and what digits to use (here, we've only used "A" and "B," good as placeholders but probably not adequate for permanent adoption). But these problems can be surmounted without much difficulty. For this article, it is enough to say that twelve, dozenal, is the best base for man. Other problems are addressed elsewhere on this site.