So, maybe we should approach this thing logically; what bases [pl. of "basis"] do we have for supporting one number base over another?

There's the theoretical reasons, and the practical reasons. The former are probably more compelling.


Anyway, what are they? Let's make a list.


One is the number of fractions that can be represented by 1dp or in a simplistic fashion.

Here are some. These pertain to pure (vs. mixed) radixes for general human use. Note that other numbers or groups of numbers are useful for specific applications.

These qualities are important, in my opinion, because the goal of a number system should be the enabling of a human being to make practical use of the base on an everyday basis, or for a particular task. Yes, mental computation is important (I'd say paramount). You aren't going to always have digital stuff (what if the power is out and your flying a plane, what if you're on top of a beam on the 34th floor of a new building, what if you left your calculator at home, etc.)

Herein r = the base, e.g. r=12 equals base 12.

1. Number of prime factors. each prime factor is an avenue that helps the human mind break down the quantity of things that he or she is facing, so that the quantity can be manipulated. The primes can all be the same (hexadecimal) or mixed (decimal or dozenal, etc).
2. Type of prime factors. Numbers r that include 2 (which crops up every other integer), thus even bases, are preferable to those that do not account for 2. Numbers r that include 3 (every third) are next most desirable, when they also account for two. Numbers that include 5 (every fifth) are preferable when they account for two and three. Bases r which skip a smaller prime to account for a larger prime are not as efficient as those which have a compact set of the smallest prime numbers. 12, the product of (2^2)*3, is essentially the same as 20 (2^2)*5, or 28 (2^2)*7, or 45 (3^2)*5, in terms of the "shape" of its prime factors. But 12 has all the power of a number that is formed by the square of a smaller prime multiplied by another prime in the smallest possible package. "You get more bang for your buck". (This is where the sequence 2, 6, 12, 60, 120, 360, 2520, etc. comes in; these seem to be the optimum bases for 1, 2, 3, ... n prime factors, when general use is concerned.)
3. A mix of prime factors. Numbers that feature different prime factors have more divisors in their powers, so that "percentages" are more powerful for the mixed prime bases. Decimal thousandths feature more divisors than hexadecimal. Duodecimal "per-great-gross-ths" are even more powerful.
4. Magnitude of base. Small bases less than +/- 7 do not make efficient use of human computational abilities using positional notation. Bases larger than 12, but I suppose we can stretch it to say maybe 16 or 20 will require a different way to multiply or add, or to employ a "sub-base" or mixed radix so that people can compute with it. Bases between 7 and 12 ("human-scaled" bases) are somewhat aligned with the human ability to deal with groups of objects; these maximize the human ability to mentally compute, while are not so large that their fact tables are too large to memorize in a reasonable period of time. (A subjective but crucial quality).
5. Number of totatives i.e. digits that are relatively prime to r. These digits tend to look "strange" to the base because numbers ending in these digits are not divisible by the factors of the base. 12 has 4 totatives (1,5,7,11); same as decimal (1,3,7,9)and octal (1,3,5,7). We can compare the percent chance we will encounter a "strange" number in each base by examining the ratio of number of totatives divided by r. So 12 gives us 4/12 or 33%, 10: 4/10 or 40%, 8: 4/8 or half. In fact it is interesting to see that the "binary" bases will always yield the 50% ratio.
6. Divisibility Rules. The nature of the number r-1 affects whether or not an important divisibility rule can give us a quick method of testing for divisibility by primes that are not a divisor of r. Because 10 is 1 greater than nine, decimal users can easily determine whether a number is divisible by three, even though 10 is not evenly divisible by three. Simply add up digits; if the sum is divisible by three, the number is divisible by three. Hexadecimal is awesome in that it allows this test for primes 3 and 5 even though it is the 4th power of 2.
7. History of usage.

By these aspects, we have bases 6, 8 through 16, maybe 20 to consider. There may be reasons or developments which make other bases (60) also a good consideration.

Base 8 is lovely and compact but man she's not to friendly to odd numbers. You get a power of three in there and you'll be hard pressed to detect it. At least with 16 or even decimal, you can see the three and the five. Base eight, it's like walking around colorblind. I prefer 16 to 8, but 12 to sixteen. I am recently in awe of 60, and continue to be captured in its thrall. I realize most people won't be using base 60, but I am beginning to think that this is an even more beneficial system than dozenal. It's just hard to reach for a variety of reasons, and therefore is not the optimum base for mankind's general use.

You know, I think it would be grand if there was a general discussion of any base, and that there would be a place where we could be "ecumenical" about the bases we are discussing.

Some aspects to compare, off the top of my head:

- Compactness
- Arithmetic tables
- Finitely representable numbers
- Compatible numbers
- Commonly needed fractions
- Important irrationals
- Powers
- Primes
- Divisibility tests
- Geometric patterns
- Angles
etc.

For example:

- Compactness:

binary: overly lengthy numbers
senary: not very optimal
octal: not very optimal
decimal: acceptable
dozenal: acceptable
hexadecimal: quite compact
vigesimal: quite compact
sexagesimal: amazingly compact

- Arithmetic tables:

binary: trivial
senary: easy
octal: medium
decimal: medium
dozenal: easier than decimal
hexadecimal: more difficult than decimal
vigesimal: a bit too large
sexagesimal: totally unwieldy

- Finitely representable numbers:

Only the rationals with lowest-term denominators of the kind...
binary: 2^n
senary: 2^n * 3^m
octal: 2^n
decimal: 2^n * 5^m
dozenal: 2^n * 3^m
hexadecimal: 2^n
vigesimal: 2^n * 5^m
sexagesimal: 2^n * 3^m * 5^p

Dawgg I almost forgot --- Divisors. But divisors are governed by prime factors.

It is interesting to note, seeing Uax's comment above, that many of the regular three dimensional polyhedra feature nodes, edges, and faces in quantities that are multiples of 2, 3, and 5; so that 60 is perhaps the best way to regard these. I can think of no better suited base to use in geometry than 60.

Here is a chart of the factors of the third power of the even bases between 6 and 18, and base 60. This illustrates the power of each base's "per-mil". There was once a scientific study (which I have yet to find on the internet) that stated that the average joe believed (erroneously) that percents were more precise than vulgar fractions. examining the third power is a handy way of seeing what each base offers the user in terms of their digital ability to represent fractions. One could look at an exponent of two or four or seven, but the advantages are clear at the third power. The figures are divisors of the third power of the base at the heading, expressed in terms of that base.

6 8 10 12 14 16 18

1 1000 1 1000 1 1000 1 1000 1 1000 1 1000 1 1000
2 300 2 400 2 500 2 600 2 700 2 800 2 900
3 200 4 200 4 250 3 400 4 370 4 400 3 600
4 130 10 100 5 200 4 300 7 200 8 200 4 490
10 100 20 40 8 125 6 200 8 137 10 100 6 300
12 43 10 100 8 160 10 100 20 80 8 249
13 40 20 50 9 140 20 70 40 9 200
20 30 25 40 10 100 37 40 C 190
14 90 10 100
16 80 16 D9
20 60 19 C0
23 54 20 90
28 46 30 60
30 40 40 49

You can see that there are "cousins", that 6, 10, and 14 resemble one another. This is because they have the same "formula": they are diprimes, (prime 1)*(prime 2). Six is the first diprime, so if we're in the market for diprime bases, six will offer the greatest efficiency. You get the power of a diprime for the least "price" (which is additional digits, additional facts in the mult. table).
Then we have 12 and 18 (and 20, 28, etc.), which have the "formula" (x^2)(y). Because there is another factor involved, we have much more "power". The binary bases lose some luster because they only have the ability to represent powers of 2. So the interplay among different primes is lost on them. They function more like bigger prime bases because they cannot resolve for any other prime factor. This is why prime composition is crucial. The following table is for sexagesimal's third power, expressed in the mixed radix (six on ten) notation we are familiar with on our clocks.
60

01: 10:00:00:00
02: 30:00:00
03: 20:00:00
04: 15:00:00
05: 12:00:00
06: 10:00:00
08: 07:30:00
09: 06:40:00
10: 06:00:00
12: 05:00:00
15: 04:00:00
16: 03:45:00
18: 03:20:00
20: 03:00:00
24: 02:30:00
25: 02:24:00
27: 02:13:20
30: 02:00:00
32: 01:52:30
36: 01:40:00
40: 01:30:00
45: 01:20:00
48: 01:15:00
50: 01:12:30
54: 01:06:40
01:00 01:00:00
01:04 56:15
01:12 50:00
01:15 48:00
01:20 45:00
01:30 40:00
01:36 37:30
01:40 36:00
01:48 33:20
02:00 30:00
02:05 28:48
02:24 25:00
02:30 24:00
03:00 20:00
03:20 18:00
03:36 16:40
03:45 16:00
04:00 15:00
04:10 14:24
05:00 12:00
06:00 10:00
07:12 08:20
07:30 08:00
Because sixty has four prime factors, two of them the same prime 2, we are afforded an astounding number of combinations at the great price of 60 digits and a mammoth multiplication table.

Oh well, my chart didn't turn out too nicely. Too bad.

Yes, sexagesimal would be very nice... if only humans could easily handle such a large base. But, unfortunately, for most of us sexagesimal is unwieldy, which is probably the reason why its use in ancient cultures was discontinued. Just like, on the other end of the spectrum, binary is wonderful for computers and greatly simplifies many things, but is awful for human use, being annoyingly verbose and too prone to reading and copying errors, and doesn't take full advantage of our capabilities. One could argue that, mathematically, base 360 could be even better than 60, not to mention base 5040, but it's very clear such bases are completely prohibitively large as far as humans are concerned, so one has to choose the optimal base taking into account not only the mathematical advantages it offers, but also the limitations of human capabilities. Maybe out there in the universe there exists some intelligent species that, like a few humans, possesses a subitizing range much larger than that of the typical human and can count 20 items in an instant, paired with mental arithmetic abilities which in humans are considered exclusive of a few so-called mental calculators, and thus the members of this alien species would feel as comfortable with a base the size of 60 as we feel with one the size of 12, and so, for them base 60 would be the ideal and base 12 would under-exploit their natural abilities. But for common use by humans, sexagesimal is not the optimal choice because it falls outside the range of what we find comfortable to use.

Totally agreed Uaxactum.

I've found 120 and 360 have so many "strange" numbers interspersed that they aren't really very practical. I think 60 is semipractical. It is interesting to note that these numbers, 12, 60, 120, 360 appear in our traditional systems of measurement. It seems that their usefulness is stronger than the propensity toward decimal conformity.

I own a business; yesterday, as with so many other weeks, i received a catalog for office products and marketing materials. This catalog features five levels of purchase. Now we would expect that the items they are selling in bulk would be 50, 100, 150, 200, 250, 300, etc. They tended to double the volume across the range, but round this out, like 75, 150, 300, 500, 1500. Out of the 100 or so products they offer, I'd say one half of these were offered with the following ranges:

72, 144, 288, 576, 864
96, 144, 216, 432, 864
144, 288, 576, 1008, 1536 (1008 = 7 gross! Why 7? an attempt to make it close to 1 grand? Why not just one grand?)

What is the purpuse of selling 1008 pens when this could easily be called 1000? An item like pens could be bundled in dozens, so could shirts, but these packages are flat or oblong, so the bundles needn't use dozens. Yet these shirts, memo pads, technical pencils, etc, all by the dozen.

So I think that the shape the grouping makes is a consideration. The dozen is so powerful in packaging because of its versatility 2x6, 3x4, etc., as are its multiples, that even though we use decimal, we are compelled to package in dozenal. It might be argued that the dozen is the most versatile small size package grouping. Note that zero of these items in this particular catalg comes in 8's or 256s. This is a testament to the practical strength of grouping items for sale by the dozen.

One day I asked the wood salesman in Tuscany back in 1992 for a third of a meter (that was part of the design, I don't know why the designer chose 1/3). The cut needed to be close because I was in school and didn't have a saw to trim the piece. (ho detto "dev'essere precisamente un terzo, caro signor.") The guy gave me 35 cm. It's strange that a purely decimal system has done two things: 1. guided the woodsman's hand to move the saw to the next decimal "click", i.e. 35 cm. 2. Made me think that 1/3 was a strange and unbefitting fraction to use when operating in meters. This is because 2 and 5 are all we have when operating in pure decimal. I suppose a dozenal measurement system would have the same difficulty with a request to divide a span by five.

I have underlined and boldened the bits I am referrign to.

uax, this is why I like base eight. Out ofthe binary bases, it strikes me as most usable. There really is only 16 and 8 in it, and 8 has one advantage over sixteen that is so big I could not possibly support base sixteen; namely, base eight does not require all those extra digits.

What makes binary good for certain things is its extreme simplicity. Its arithmetic tables are trivial (reduced to the rows for 0 and 1 that are usually left out of the tables for other bases because they are so straightfoward), it can represent boolean values and logic, it can be easily re-codified to include the sign (as in two's complement representation) or to make successive numbers differ in only one bit (as in Gray code), it is very easy to implement in circuitry (since values can be stored in many kinds of two-state devices and operations can be implemented with a series of logic gates), the algorithms for operating in binary are simple and ideally suited to automatic machine processing (e.g., multiplication is reduced to bit-shifting and adding) and can take advantage of certain properties that are unique to binary (e.g., sometimes the leading bit of a number can be left out because it can only take the value 1, as in the implicit bit of IEEE floating point representations), etc. All such advantages are lost both in octal and in hexadecimal, which do not share the properties that make binary an ideal base for certain purposes. Two is a superior highly composite, which places binary in the series of optimal bases relative to their size; whereas 8 and 16 are not. They are useful mainly as a shorthand to facilitate the handling of binary by humans; other than that, they are not a serious competitor to some other bases of comparable size, since even decimal, which can handle powers of five apart from powers of two, offers more advantages than octal and hex. Moreover, as I've already pointed out, octal betrays its very "binariness" with its grouping of bits in threes, and a base should not be chosen over another merely because it doesn't require a set of digits that is larger than the one for decimal (if so, then senary wins the battle hands down). The fact that in computing the use of hexadecimal as a shorthand for binary has prevailed while octal is now seldom used, is a testimony to the lesser usefulness of octal as compared to hex; precisely because of its "ternary" way of dealing with binary which contradicts the halving-or-doubling mindset imposed by all dyadic bases including octal itself, whereas hex, which is two doubly squared, remains faithfully "binary" to the core.