By Graham
Posted by hexnet -

Note: Writing about dozenalism always presents some semantic complications. When discussing the natural numbers up to twelve, I have opted to spell out the numbers in English, since this is a clear and base-neutral way of representing them. After trying several different systems, I have settled on writing larger numbers in decimal. Unless otherwise specified, "10" means ten, not twelve, et cetera. When I use dozenal notation, and for clarification purposes elsewhere, I have prefixed the radix as an abbreviation before the number. Thus, "dec. 360" means decimal 360, and "doz. 260" means dozenal 260. "360" by itself, unless otherwise specified, refers to the former. Larger numbers spelled out, where it is stylistically appropriate to do so, will always be given in decimal. My first inclination, of course, was to put all numbers here in dozenal, but on further reflection I see no value in confusing people needlessly. (Confusing them for a good reason though is fine.)

### WTF is dozenalism

The dozenal or duodecimal system is a positional notation system using twelve as its base, or radix. It is an alternative to the decimal system, using ten as its base, which is the universally-understood system for representing numerical values in the modern world. Whereas in decimal there are ten digits, 0-9, in the dozenal system there are twelve digits, including the ten of decimal notation, plus two more representing the numbers ten and eleven. For the purposes of this article, the two additional digits will be represented as "A" for ten and "B" for eleven. Various systems of notation exist for dozenal glyphs, and it is a topic of much discussion amongst dozenal advocates. There are very good reasons not to use "A" and "B" in the long run, but for now it is convenient and simple to do so.

The number twelve is a highly composite number—that is, it is a natural number with more factors than any integer smaller than itself. Twelve can be divided by two, three, four, and six. Compare this to ten, which can only be divided by two and five. One does not have to think too deeply about it to realize how useful a radix divisible by three or even four would be—in everyday life, we are used to encountering situations where ten or a multiple of ten has to be divided into thirds, and we end up with messy results such as "3.3333." Dividing ten into quarters is a little easier, but you still end up with a fractional value (2.5). The trivial and routine calculations most of us perform every day would be vastly simplified by using a positional notation system that allowed for convenient division into thirds and quarters.

Though we often find ourselves dividing things into fifths, it will be noted that, more often than not, we are only doing so because we are using a base ten system. We are used to quantifying things in terms of groups of tens, and as a result we naturally quantify things in groups of five as well, but there is in general no particular reason why this needs to be the case. Certainly most people would agree that, simply in the realm of practical computation, it is necessary or useful to divide things into thirds and quarters more often than into fifths. Obviously, there are always situations where one needs to divide by particular numbers, and equally obviously not all division is particularly hindered or improved by the radix used, but certainly to the extent that we naturally group things into numbers convenient to base ten (we often round off values to the nearest ten, hundred, or what have you), division by thirds or fourths are often at least slightly inconvenient.

Mathematically, divisibility by three is often more useful than divisibility by five simply because the latter is less common in the prime factorization of numbers. That is, one out of every three integers is divisible by three, while only one out of every five is divisible by five. When breaking down large numbers into their prime factors, five is simply less common, and therefore a base that is divisible by five is less useful.

Compare the divisibility of the dozenal radix to almost any other base that has been used for positional notation (* indicates prime factor, numbers in decimal):

• Octal (8): 4, 2*
• Decimal (10): 5*, 2*
• Dozenal (12): 6, 4, 3*, 2*
• Hexadecimal (16): 8, 4, 2*
• Vigesimal (20): 10, 5*, 4, 2*
• Tetravigesimal (24): 12, 8, 6, 4, 3*, 2*
• Hexavigesimal (26): 13*, 2*
• Sexagesimal (60): 30, 20, 15, 12, 10, 6, 5*, 4, 3*, 2*

It will be noted that, of all the bases listed, only dozenal, tetravigesimal, and sexagesimal offer divisibility by the critical prime factor of three. Sexagesimal (which will be touched on later in this article) is somewhat impractical, due to its size, and tetravigesimal does not add any additional prime factors to dozenal, being merely two times twelve. Overall, in fact, it is striking that not only is dozenal the most obviously useful radix listed here, it is also very close to ten. Given the proper intellectual climate, people could transition from thinking in tens to thinking in twelves without too much difficulty, compared to moving to a system such as like sexagesimal, which is simply harder to conceptualize—though we do use it, in a sense, in our reckoning of minutes and seconds.

The only other base worth noting is binary, which of course is used digital computing. The advantages of binary are quite distinct from those of prime factorization, and it is hard to imagine digital computer technology using any other base for the foreseeable future, giving the nature of Boolean logic and of transistor-based computing circuitry. Though with the advent of optical computing, cellular automata logic, and so on down the road, perhaps our technology will at some point transition away from binary. At any rate, it will be noted that digital binary and "human" decimal have coexisted for over 60 years at this point, and binary values will be no harder to work with under the dozenal regime. One slight complication is that, while 2^10 (1,024) is very close in value to 10^3 (1,000), providing the basis for the Greek prefixes used terms like "megabyte," et cetera, there is no similar power of twelve that meshes well with binary magnitudes. This is obviously a fairly trivial concern, but it is worth mentioning.

### Dozenalism and cryptodozenalism throughout history

Despite the decimal tendencies of modern society, our culture is replete with relics of dozenal thinking from days of yore. The persistence in the English-speaking world of the word "dozen" itself, though originally derived from some variant of the Latin "duodecimal," is a testament to the ongoing usefulness of twelve as a grouping unit. The 24-hour day, divided into two groups of twelve, was an ancient custom of dividing the periods of light and darkness into twelve hours each. Thus, the ancients would not divide the day into ante meridiem or post meridiem as we do, but they would speak of the "first hour" of sunlight, "fourth hour of night," et cetera—this is the sense of "hour" originally meant by the term "eleventh hour." Likewise, we divide the year into twelve months. This was certainly convenient to the length of the lunar cycle, but it is also notably useful for dividing the year into four seasons, et cetera. Even the French revolutionaries, probably the most deranged decimalist thinkers in history, did not eliminate the twelve-month year.

A related matter to the length of the Earthly year is of course the 360 degrees of arc in a circle. The use of this value goes back at least to the Babylonians, and probably has at least something to do with the length of the year. The fact that degrees continue to be used to this day suggests that people throughout history have found it to be a useful way of dividing circles and measuring angles. Like twelve, 360 is a highly composite number, meaning again that it has more factors than any number below it. Thus it is possible to divide a circle of 360 degrees by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180. Which is all very useful.

The Babylonians not only gave us an excellent system for measuring angles (mathematically not as useful as the radian, but certainly more convenient for everyday use), they also gave us something resembling the earliest positional notation system, using base sixty. The Babylonian sexagesimal system was not an exact analogue to modern positional notation, and it included some pseudo-decimal elements, but it provided the framework for some fairly advanced mathematics at the time. Sixty is, of course, a somewhat dozenal number itself, being a multiple of twelve. It is also a highly composite number, and is therefore more convenient for mathematical operations, particularly division, than any number below it. A particularly useful property of sixty is that it integrates divisibility by five with divisibility by twelve. A sexagesimal system is probably impractical for modern purposes, since we are used to conceptualizing numbers in smaller groups, and it would be difficult to come up with sixty truly unique glyphs to construct a strict, glyph-based positional notation system, but it was an interesting and worthy contribution to the early development of mathematics. The Babylonian sexagesimal system is the origin of not only the 360 degree circle, but also of the 60 minute hour, and by extension the 60 second minute, et cetera. Every time you conveniently divide an hour into halves, thirds, quarters, fifths, sixths, tenths, or twelfths, remember the sexagesimal system and how helpful it has been to our ability to conceptualize and divide time.

Beyond this, the twelve inches of the imperial foot is another obvious example of dozenal thinking in earlier times. There were twelve ounces in a Roman pound (libra, lb.), and the term "ounce" itself is derived from the Latin "uncia," meaning twelfth. If you look closely, history is full of examples of subtle dozenal thinking, which all serve to demonstrate that, despite the ubiquity and hegemony of decimalist thinking, the usefulness of dozenalism has had a consistent appeal through the ages. Most people throughout history probably could not have precisely defined what a highly composite number is or why it might be useful, but they could clearly see that dividing things into groups of twelve was more useful for everyday purposes than dividing them into groups of ten. We would do well to remember their example, and learn from it.

### Why it matters

Culturally, humanity is very entrenched with decimalism. Outside of a handful of indigenous cultures (see: Piraha Language on Wikipedia), everyone in the world uses base ten notation, everyone understands Indo-Arabic numerals, everyone learned basic arithmetic using base ten, and nobody wants to learn a whole new way of counting. This is understandable. But consider our situation: Humanity is, possibly, on the verge of a glorious and infinite future. The rise of the metric system is institutionalizing the decimal radix not just for counting, but for measuring and quantifying the world itself. With the advent of modern computers and the internet, we are potentially setting in place structures and systems that will define our relationships with information and information technology for countless generations to come. It would behoove us to pause and reflect on whether or not this is really a system worth further embedding into the fundamental fabric of our civilization.

The metric system in particular is a tragic case. In terms of its intent, it makes perfect sense, and pursues an admirable goal—the systematization and regularization all units of measurement into a consistent and easily-computable system. But in practice, we are replacing some very dozenal systems of measurement—for instance, again, the twelve inches in the imperial foot—with misguided and mathematically limiting decimalism. In my view, the metric system presents the most compelling and urgent reason why dozenalism must be taken seriously and addressed quickly in our society, because it is rapidly becoming so entrenched in technology and science that it will be difficult to extract ourselves from it if we go on too much longer. I am not in any way defending traditional imperial measurements—twelve inches in a foot makes a lot of sense, but 5,280 feet in a mile, less so. But if we as a civilization are going to consciously adopt a new and mathematically consistent system of measurements, we ought to at least put some thought into it.

The origin of the metric system, of course, lies in the French Revolution. And one cannot discuss French revolutionary contributions to decimalism without noting the appalling invention of "decimal time," which is perhaps the limit case of absurdity by decimalist agitators. I have already noted the utility of the Babylonian sexagesimal system. When coupled with the (possibly also Babylonian) 24-hour day, it produces a highly dozenal—though perhaps somewhat inconsistent—system for measuring time. Days, half days, hours, and minutes, can all be evenly and usefully divided into halves, thirds, fourths, sixths, twelfths, and in the case of hours and minutes, into fifths, tenths, twentieths, et cetera. Why on earth anyone would feel compelled to replace this elegant and convenient system with a decimalized one is truly beyond my comprehension. If the French of the time had spent less time chopping people's heads off and a bit more time putting even the most minimal degree of thought into what they were doing, perhaps they could've come up with a better system. There is room to improve the clock system handed down to us, by making all units of measurement recursively consistent. One could easily imagine a truly dozenal clock, with perhaps twelve hours in a day, twelve such-and-such in an hour, twelve "minutes" in that, et cetera. That would make a lot more sense. The fact that not only the French people but the world at large soundly rejected decimal time is further proof that, however entrenched decimalism may become, humanity will not give up its dozenalist traditions easily. Nor should they.

### What can be done

What is ultimately and urgently needed, then, is a dozenalization of the SI base units of the metric system. Before anything else is accomplished in the dozenal reconquest of positional notation, this issue must be addressed. There are at present seven base units in the SI system: the meter, the kilogram, the second, the ampere, the kelvin, the mole, and the candela. All other units of measurement are derived from these seven, and they are defined in various ways with various degrees of arbitrarity. For instance, the kilogram is defined by the weight of a block of platinum in France, which is just bizarre. At any rate, the solution to our problems is to simply define a new set of measurements based on these units using powers of twelve rather tha powers of ten. For instance, since a kilo-anything is dec. 1,000 of that unit, we need a new system of terminology to define dozenal magnitudes of dec. 144 (doz. 100), dec. 1728 (doz. 1000), et cetera.

This raises an issue pertinent to the dozenalization of both numbers and units of measurement: terminology. We have a system for referring to both numbers and units according to both a decimal base and a cubic superbase—that is, we divide powers of ten into groups of three, or cubes, separated in American English by commas, and elsewhere by periods or other characters: dec. 1,000 is a "thousand," dec. 1,000,000 is a "million," et cetera. Conversely, in the metric system and elsewhere, we use Greek prefixes to denote the same groups of three powers of ten: kilo-, mega-, giga-, milli-, micro, nano-, et cetera. What is needed is a unique and convenient vocabulary for describing powers of twelve. For twelve squared, terminology already exists: a "gross" is equal to doz. 100 (dec. 144). In various European languages terms exist as well for twelve cubed, doz. 1000, or dec. 1728. However, what we really need is a system of vocabulary that can be consistently applied over large scales. The English terms for numerical orders of magnitude could be repurposed for dozenal use—i.e., by defining, say, a dozenal "million" as doz. 1,000,000, as a counterpart to the decimal million of dec. 1,000,000, et cetera—but this scheme is not only potentially quite confusing, it is also complicated by the fact that these terms, ultimately, incorporate decimalist elements. Specifically, above one "decillion," or dec. 10^33, terminology reverts to a composite decimal form, and the next cubic order is named "undecillion," or one-plus-ten-illion, and so on. Additionally, the fact that the numerical element in these terms (bi-, tri-, et cetera) do not kick in until after the "thousands" level is confusing and inconsistent. For example, to be consistent, a "decillion" should be dec. 10^30, not 10^33, yet it is the latter because the numbering does not start until a cubic order of magnitude after a thousand, which is arbitrary and pointless.

In order to address this issue of magnitude terminology, let us digress for a moment into simple ordinal number terminology. Since English already has unique terms for ten, eleven, and twelve, there is no reason to stop using them in a dozenal system. (It should be noted that "eleven" and "twelve," like the word "dozen," ultimately derive from a decimal etymology, but as they have, for all intents and purposes, been divorced from their decimal roots at this point, it is probably of little concern.) When counting verbally or in written words, we would continue to count "nine, ten, eleven, twelve" in a base twelve system. After that, I would propose that we add the numbers "oneteen" and "twoteen" to represent dec. 13 and 14, respectively, and then so on to dozenal "thirteen," et cetera, until you reach dozenal "nineteen," at which point you also add "tenteen" and "eleventeen," bringing you to dozenal "twenty," being equal to decimal 24, and so on. We simply append "twenty-ten" and "twenty-eleven" before "thirty," and so on with every radix unit up to ninety-eleven, then we add "tenty" and "eleventy," and go on to eleventy-eleven (dec. 143).

This is where it gets complicated. As mentioned, "gross" is already an accepted and conventional term for groups of a dozen dozen, or dec. 144, so it would certainly make sense to use this term, or some variant of it, as the dozenal equivalent of a "hundred." My particular proposal is that we should use Greek and/or Latin numerical prefixes for all superbases beyond this, up to "deca" for doz. 10^26 (dec. 12^30). At that point we should add two new terms, based on English "eleven" and "twelve," instead of for instance the Greek "hendeca" and "dodeca." For the sake of argument, I will use "eli-" and "dozi-," but I will be the first to admit these sound a bit silly, and particular terms can obviously be worked out later. Thus, we could name all dozenal superbases according the following convention, using a tasteful combination of Latin and Greek etymology (my prefence is for the latter, but I have made adjustments to make it sound less retarded where appropriate):

• . . . .

I don't know if anyone is following me here—I more or less made most of this up on the spur of the moment. This is simply an example of how a grammatically consistent ordinal counting system can be constructed for dozenal notation. Note that I have restructured the syllables of Greek numerical prefixes and eliminated duplicate consonants, et cetera—again, this is just something I made up literally right now as I am writing this, but I think it is pretty sensible. As with our current decimal naming conventions, one would expect that astronomically large numbers, at a certain point, would simply be referred to mathematically, without the need for "common" names, so there is probably no need to extend the system indefinitely for absurdly large values (as opposed to doz. 10^63, which of course is a number we all use on a daily basis).

What is important here, among other things, is that such a system can be easily adopted for units of measurement as well. That is, we do not need to use terms like "mega-," "giga-," et cetera, we can simply use the same prefixes and terms we use for ordinal counting. Thus, the dozenal equivalent of a "gigagram" might be a "triligram" or some such thing. (Though by the same logic, a kilogram might be a "miligram," which admittedly is sort of confusing—again, the details can be worked out down the road.) Ultimately it would of course probably make sense to completely rename the SI units themselves to avoid confusion, but that is another issue altogether. Presumably, a system of terminology such as that outlined above can be adapted to negative exponential values with equal consistency—whether by adding a negative prefix or changing the "-ad" suffix to something else.

At any rate, the whole issue of terminology is one that should be defined by community consensus. And it is certainly a topic of rather heated debate amongst dozenal advocates. Many useful proposals have been put forward, and I am indebted to those who have contributed to this debate in the past for helping to shape my views on the matter. It may seem like a trivial issue at first glance, but the need for a useful and intuitive semantic system for conceptualizing dozenal notation cannot be overstated. We are asking people to make very fundamental changes in the way they conceptualize and organize numbers, both mentally and expressively. This cannot be done until there are tools in place to facilitate it. We cannot simply ask people to conceptualize numbers as doz. "10^6," or what have you, and expect this to make any sense to them. The conversion of human civilization to dozenalism will only be successful if and when people are given both reason and ability to mentally count in dozenal, conceptualize numbers in dozenal, et cetera.

The other great challenge facing dozenalism in somewhat the same vein is that of glyph notation. As noted at the beginning of this article, I have chosen to express dec. 10 and dec. 11 as "A" and "B" respectively. (Though as you have perhaps noticed, these two digits have not come up often, as I have been largely concerned with powers of twelve or multiples of twelve, which, in the former case at least, do not require these digits at all.) This is perhaps the most widely-discussed issue in the dozenal community, as far as I can tell. Several proposals put forward in the past no longer seem as compelling as they may once have. For example, the Dozenal Society of America used to recommend using the asterisk (*) and pound sign (#) for ten and eleven, due largely to their inclusion on North American telephone keypads as far as I can tell. At this point, this makes no sense whatsover, for the simple reason that both the asterisk and the pound sign can be—and often are—used as mathematical operators in computer programming languages. It would be confusing in the extreme to take a symbol that has conventionally implied either a mathematical operation, or at least some sort of syntax function, and start using it as a simple positional value glyph. Alphanumeric characters are rarely if ever used for such syntatical purposes, which is presumably why letters have traditionally been used for transdecimal radix notation (i.e., A-F in hexadecimal, et cetera).

Various other exotic characters have been proposed for ten and eleven, which are beyond my present scope to review fully. For instance, a dec. 90-degree rotation of the glyphs for "2" and "3"—I have never entirely understood the point of this, but apparently it is popular in Great Britain. Suffice to say, I believe the only long-term solution is to develop an entirely new set of glyphs for dozenal counting—replacing not just ten and eleven, but all digits. "0" and "1" make a certain degree of intuitive sense, but everything beyond that in the Indo-Arabic glyph system is complete crap as far as I'm concerned. We need a consistent, logical, easy to write set of glyphs for the new dozenal age.

That, ultimately, is a battle for another time. The glyph issue is, in my opinion, far less important than the magnitude terminology issue addressed above. As I've noted before, and as is obvious, English already has verbal terms for "eleven" and "twelve" that work quite well, and how we actually write the numbers is probably less important than how we verbally and linguistically conceptualize them. In the interim, I am prepared to use "A" and "B," in the manner that hexadecimal notation uses A-F, as an easily-recognizable standard for transdecimal radix notation. The major complicating issue with this, of course, is that the capital "B" looks an awful lot like "8"—indeed, the two are identical on 7-segment displays. But as long as we are confining ourselves to computer fonts this will probably not be too much of an issue. In written form, perhaps the lowercase "b" would be more appropriate. At any rate, I do not consider this a major stumbling block to the implementation of dozenal notation—any number of options are available. The only thing I would urge my fellow dozenalists to consider is, again, to avoid using special characters that are either overly obscure or that could be confused with syntactical elements in computer languages.

If the world is to be brought around to dozenalism, it must be done with clarity. It will do nobody any good to just start talking in dozenal numbers and assume people will understand them. This can partly be addressed by adopting a unique vocabulary and glyph system, yes. But beyond that, it would behoove us to simply prefix all dozenally-expressed values with an appropriate identifier. For instance, if someone asks you what year it is (for whatever reason), instead of replying "oneteen eleventy six" (though that might be amusing), it might be advisable to say "dozenal oneteen eleventy six." (Which will of course be perfectly clear to them.) As with my use of the prefixes "dec." and "doz." in this article, this will be a time-consuming and arguably completely inane exercise. Yet it may be the only way we will ever make progress in this matter. We must teach people to think in dozenal—starting with ourselves.

I find it an amusing and constructive intellectual exercise to count in dozenal to myself whenever possible. Whether at home or work or where have you, menial counting is a part of everyday life, and there is in my view no better place to start one's personal adaptation to dozenal thinking than on these occasions. It is also possible to adopt various systems for personal number-keeping that reflect dozenal values, without committing to any large-scale lifestyle changes. For instance, conventional Western tally marks are arranged in a base five glyph system—four lines plus a fifth crossing line. A simple quasi-dozenal version of this would be to use a six-lined version of the hash symbol, or number sign. One could start by drawing three vertical lines, then three crossing horizontal lines, for a total of six lines representing six tally figures. This provides an eminently usable equivalent to the decimally-skewed base five tally system, which should fit nicely into our dozenal future.

Finally, I think it is important to simply question decimalism at every opportunity. One of the most disturbing manifestations of the decimal hegemony is the degree to which decimal notation is considered by the general public to represent something "intrinsic" about numbers. It of course does not. One often hears of specific attributes of certain numbers such as, "If you add all the digits together, you get such-and-such." Or, "If you reverse the digits, you get such-and-such." These are of course trivial and superficial properties that do not deserve to be held up as some sort of fundamentally interesting aspect of the number in question. This tendency also arises in many occult numerological traditions, such as gematria, where certain numbers are assigned symbolic values based on their decimal digits, and in various other fields of human activity too numerous to list here. The dozenal community would do well to aggressively and publicly challenge those who would assign some sort of praeternatural significance to the decimal radix. Indeed, if geometry and elemental number theory teach us anything on this matter, it's that it is the dozenal radix, not the decimal, that is favored by the laws of nature and mathematics.

### Conclusions

If aliens landed on earth tomorrow—proper aliens, in ships and crap—they would, I believe, be astonished and dismayed to see us using a decimal radix as the basis of our mathematics and of our civilization. They would wonder how we got as far as we did using a counting system derived, ultimately, from the rather incidental number of prehensile outcroppings on our hands. And I have to wonder myself. I am at least mildly embarrassed to be associated with a species so dependent on such a nonsensical and counterproductive system. Dozenalism, in various forms, has had its triumphs throughout history. Next to decimalism, it is by far the most prominent and well-represented grouping value across all fields of human endeavor. We are therefore not starting from nothing—momentum and history are on the side of the dozenalist cause.

It is true that decimalism has never been as entrenched in human affairs as it presently is. Converting to dozenal at the dawn of the modern era, when Indo-Arabic numbers were first gaining a foothold in the West, would have been a relatively simple affair. That window of opportunity, however small it may have been, is now long closed, and we are left with a system ingrained into the fabric of our very thoughts themselves.

Yet it is equally clear that the decimal hegemony will only continue to gain ground unless and until it is challenged rigorously, rationally, and openly by those of us who see a better way. And in our modern age of internet publishing and what have you, we have an opportunity unique in the history of human affairs since the dawn of mathematics to get our message out to a wider audience. I believe the case for dozenalism is compelling and overwhelming. If presented with a reasonable argument, I think people are capable of changing. With the aid of computer software, programming libraries, conversion tools, et cetera, the change need not be as onerous as it may have seemed to previous generations. As noted previously in this article, computers do not care, architecturally, what radix human-readable numbers are presented in. It does not change their underlying binary arithmetic at all. It is true that many low-level functions in modern computer systems would have to be rewritten, but to do so would not be earth-shattering, nor would it challenge the integrity of our technological infrastructure.

I believe that, with a focused and sustained effort, we can make the dozenal radix coequal to decimal in popular culture as well as in scientific and mathematical use by the end of this century. (Or grossury?) It is incumbent upon those of us who understand the argument for dozenalism to present it forcefully, enthusiastically, and clearly at every reasonable opportunity. If this is done effectively, I have no doubt the natural course of human events will trend more and more towards a dozenal future.

In this article I have provided only the most cursory summary of the arguments for dozenalism. The corpus of dozenal literature is not vast—this is one of the primary reasons I felt compelled to contribute to it—but it is substantive. Several interesting dozenal-related papers and articles can be found in the Hexnet.org Hexagonal Library. Others can be found on the Dozenal Society of America and Dozenal Society of Great Britain websites. I would also recommend that any serious student of dozenalism read the Duodecimal article on Wikipedia, which is perhaps one of the most mathematically interesting and exhaustive explanations of the principles of dozenalism in the popular literature, at least as of this writing.

A further selection of dozenal links to be found in the Links page.

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Submitted by Greg Campbell )esign( )z (http://twelvish.org) - 2012-12-13 21:32

Great contribution to the discussion and as such I have included it high on the list of links at twelvish.org (note: "twelvish" (a term that may be easier for the public to understand or relate to)

There are two things which I think you have not yet considered. The first is the benefits of the new "twelvish system" to have a starting date. I have selected 12.12.12 as an opportune and elegant option. It is just a handful of days from the next great era as defined by the 5,000 year great cycle in the Mayan calendar. It can also coincide with the notion that the world is finally realising that it needs to seriously work together to globally address the mounting issues of environment, population, pollution and equity. This leads on to the second point that as base 12 is a more efficient system it not only offers easier calculations but often requires less room to display larger numbers. This can be equated to smaller devices and savings in materials for manufacture and therefore also carbon emission reductions. This obviously requires good Life Cycle Analysis to confirm just how much but I believe that this may be just enough to sway public opinion that counting using base 12 is worthy to pursue to make the standard system and not just something to please "mathematician types". I am proposing a global research and design process to examine the optimal results to redefine and better relate our current systems. Naturally we need to seriously explore some funding and organisation to achieve this. Obviously there is a lot required to get this movement growing and your comments are welcome.

Kind Regards Greg Campbell

Submitted by themumm - 2013-03-22 11:26

I'm surprised you neglected to probably the most-widely used base-12 positional system ... the Didot point/pica point/cicero (NA & EU, respectively). 6 Picas to 1 inch; 12 points to a pica.

EVERYONE who has used a computer is familiar with the point system -- it's the same one that they use to set their type in any application, and, moreover, it's been in use in the printing industry for some time now.

Submitted by Chris - 2013-10-11 03:17

It's natural to assume that decimalism results from having ten fingers, but apparently there have been (and perhaps still are) cultures that use the twelve knuckles of the four fingers, sometimes with the thumb on the same hand as a pointer to count them with. The uncertain distinction we retain between the terms thumb and finger may thus also be a "relic of dozenal thinking."

I also suggest the 10 system didn't come from fingers at all, but emerged with an additive place value system, using a grid still with us as the "+" sign. Numbers 1 to 4 could be represented successively by a single dot in each of the different quadrants.

Then 5 by two dots: one in, say, the 3 quadrant and the other in the 2 quadrant, and so on. The grid fully occupied according to this rule would express 10.

### Some evidence/arguments?

Submitted by anonymous - 2014-09-01 13:53

I'd like to see this article make a case for why decimal is such a misguided choice. It's claimed often enough, but never supported by arguments. For instance, why is the metric system implementation such a failure? Seems to work every bit as good as the Imperial system....

My last question is why, on this site especially, is no one considering base 6? Every fraction that terminates in base 12 will also terminate in base 6--the have the same prime factors. You don't improve that until you throw a 5 in there and jump to base 30. That's where the real money is.

### Re: Some evidence/arguments?

Submitted by graham - 2014-09-02 02:32

Well I for one would differ from most of my dozenalist colleagues in not really favoring the imperial system over the metric. Certainly it is profoundly unfortunate that the French of yore chose to adopt decimal as the base of the latter system, and if possible I would love to see a metric-style system built up in dozenal (as outlined at length in this article), but I find it hard to understand how any objective observer could possibly argue that imperial units make more sense than metric at this point (though people do of course make this argument). This has less to do with the base itself of course (at least as far as I'm concerned) than with the sheer inconsistency of the imperial "system."

As I may or may not have mentioned somewhere here—I am certainly not rereading this ridiculous article at this hour—my interest in dozenal actually came about because, in the early days of my hexagon infatuation, nigh 20 years ago, I seriously considered using a base six system. However after a few hours or days of playing around with this and seeing how it would work, I quickly realized that base twelve made far more sense, at least for everyday counting and arithmetic purposes. There are various reasons for this, and twelve has a number of interesting properties beyond the additional 4 factor, which I don't really want to go into here, but the primary reason I think why it is a more appropriate replacement for decimal is just that it is so close to decimal. Arithmetic, for most people I think, becomes a very habitual and intuitive thing. We are used to our radix positions having certain quantities. Unless you are like weirdly into number bases or something, even those of us who deal with alternative bases on a regular basis—and I am thinking here mainly of binary and hexadecimal—find it difficult to "think in" these bases the way we think in decimal. And a large part of this I feel is due to the fact that these bases (particularly binary obvs) are just so, if you will, radically different than the decimal we are familiar with that it simply goes against the grain of our acquired arithmetic habit to try and intuitively, unconsciously quantify values in these systems. I don't know if that makes sense, and maybe it isn't true of everyone, but the bottom line I think is that going from base ten to base twelve would be a relatively easier transition for both mathematically-inclined and non-mathematically-inclined people. Indeed, as is often mentioned in the dozenalist literature, there are already de facto dozenalist systems at work across all sectors of society—imperial units themselves are an example of this, as is the customary packaging of eggs, baked goods, etc., in groups of twelve, etc. But I don't want to make it sound like dozenal is some halfassed compromise measure in this respect—I think it stands on its own as the ideal base for doing non-machine-augmented conscious arithmetic, etc. It is a nice, middle-of-the-road value—not uselessly small like 2 or 3, not unreasonably large like sixty, etc. Maybe intelligent agents with radically different mental structures than us would conceptualize this differently, I don't know, but from my experience, twelve is really right in the ideal zone for what the human mind can make sense of.

This is not to say that base six is not I think important. I don't really have time to get into it now but yes, you are right, base six does have its value. But this is just another argument for dozenalism as far as I'm concerned—base six meshes quite well with dozenal. It would not mesh well with decimal, or really any other widely-used base, except I guess ternary (and let's be honest, ternary's not very widely used). And that is part of why I emphasized human cognition and human use of arithmetic above—dozenal may eventually just be like the "high level" universal base that everyone learns and uses in everyday transactions—or whatever passes for everyday transactions come the Singularity, etc. Other bases, congruent with this overarching twelveness, will presumably inhabit other mathematical niches. We could see vast mathematical, informational systems develop in our glorious transhuman future where dozenal, hexal, ternary, binary, sexagesimal even, etc., are all used in different capacities, for different things. But twelve is still I think a sensible base to unite them all, and to serve as the lynchpin of the whole system.

As for the issue of prime factors, I think the frequency with which "2.5"s come up in dividing 10 into quarters is evidence enough of the value of divisibility by four. The fact is, in everyday life, divisibility by twos, threes, and fours are far more prevalent than division by other values. Fives you may think seem to come up a lot in business, etc., but on reflection (as again I think I mention somewhere above, maybe), these fives mainly come up because of the decimalist hegemony—they are not some natural feature of our everyday math that decimalism is helping to facilitate. In general, five's main utility is in being half of ten. Beyond that, it doesn't really do much, at least as far as arithmetic usefulness is concerned. For instance, when was the last time you told someone you would be somewhere at a fifth past the hour (which is of course perfectly feasible under sexagesimal), or something in a similar vein? When was the last time, when reaching for an arbitrary fraction to estimate some value, or some fraction of a thing, you decided to settle on a fifth? I'm sure it's happened, but it happens infrequently compared to quarters, halves, thirds, etc.—even sixths or eighths (and eighths are still better off in dozenal than decimal). We gravitate to the more commonsensical fractions, despite the advantages of divisibility by fifths in our decimal system. Again, this is sort of an intuitive thing, and I would appeal to your honest reflection on your everyday experience to see the truth in it. Humans are thirds-and-quarters-dividing creatures. This is what we do. We deserve a system that accommodates this. I would go so far as to say it is our birthright.

Anyway, I really don't have time to get into the broader nuts-and-bolts of why decimal is lame (I literally cannot believe I am on my fifth paragraph here, but I really need to wrap it up at this point). I have always assumed that to most people, after reflecting on this issue long enough (which you clearly have done), decimal's deficiency would be self-evident. As noted at the conclusion of the above text though, I recommend checking out the other dozenal content in our Library, or the voluminous material available on the Dozenal Society of America's site.

Submitted by graham - 2014-09-02 02:37

Looking over this thread after posting my recent reply to anon's questions from ealier today, I see there are a number of earlier comments that probably deserve a response as well. I apologize for my seemingly inexplicable lack of a reaction to these—this latest comment just came in at a time I could conveniently respond, and it raised some issues I was able to easily rant on. I am not ignoring you, earlier commenters, and I appreciate your input and support, and someday I may even get around to responding to you. Thank you for your forbearance.

Submitted by anonymous - 2014-11-30 19:21

I am no expert in the matter of glyph notations, but maybe something similar to the D'ni numeral system (http://dni.wikia.com/wiki/D%27ni_Numerals) might be a good way to go. I am not suggesting to use the same exact glyphs or even similar ones, just that it seems to have an interesting logic for writing those glyphs.

Submitted by kodegadulo (primel-metrology.wikia.com) - 2015-03-30 03:05

For a well-developed system of technical prefixes expressing dozenal orders of magnitude, based on familiar Latin and Greek roots (as well as the IUPAC's Systematic Element Names) see Systematic Dozenal Nomenclature, (described at http://z13.invisionfree.com/DozensOnline/index.php?showforum=29 or at http://primel-metrology.wikia.com/wiki/Primel_Metrology_Wiki#Systematic_Dozenal_Nomenclature).

For a well-developed dozenal analog to the metric system, see the Tim-Grafut-Maz (TGM) metrology (described at http://www.dozenal.org/drupal/sites/default/files/tgm_0.pdf). For a somewhat more in-progress effort to create a dozenal metric system based on similar principles, but somewhat different starting conditions, see the Primel Metrology (described at http://primel-metrology.wikia.com/wiki/Primel_Metrology_Wiki).

### Fourbang is best base.

Submitted by autymn_d_c (http://twitter.com/alysdexia/favorites) - 2015-06-04 23:11

Of your essay: You ugly yoked Hellènic prefixes with Latin stems; rather than quattuor you wrote tetra. In your dismissal of bases between 12 and 60 you said they introduce no further prime factors which was why 12 was most useful, allegedly, but in your defense of 12 and 60 you said it was their compositeness that made them useful without stress on their prime factors. Also you said that 5 was not often a divisor. Therefore your dismissal of base 24 fails. http://en.wikipedia.org/wiki/Ternary_computer.

https://plus.google.com/b/115025073359073355980/108416959794948591172/posts/ZvwDGyfUWwK spaceLem commented on a video on YouTube. Shared publicly - Mar 7, 2015 I'm a base 12 supporter! (And a tau supporter). ... Stephanie Currie Apr 30, 2015+ +spaceLem base 60!

alysdexia Apr 30, 2015 +Stephanie Currie What do you want a 5-factor for?

Base 24 is demonstrably the best.

spaceLem Apr 30, 2015 +alysdexia ... Base 24 doesn't really provide much over base 12, other than the ability to divide by 8, and require a whole pile more digits. 12 is fine.

alysdexia May 1, 2015 +spaceLem ... That's the point of 24 over 12; the latter isn't a multiple of 8. My first comment touted its cubic and diametric coordination versatility. Base 12 doesn't evenly conform to three dimensions. There are readily 24 finger segments to thumb with.

spaceLem May 1, 2015 +alysdexia ... In order to conform to 3 dimensions, you'd need some a base divisible by some n^3, which gives you 8, 27, 64. We can discard the latter two, because they're huge, and no one wants to remember that many different digits. 8 would be okay, but being able to divide by 3 is really important, so we need to step up to base 24, which again is not going to be fun to work with. Base 12 has the advantage that it is highly composite, and useful in every day calculations (Base 10 fails because dividing by 5 happens much more rarely than 3, 4, or even 6, and really is only done because it's convenient for Base 10, which is fairly circular reasoning).

alysdexia May 1, 2015 +spaceLem ... "base 24, which again is not going to be fun to work with." Asserted again without support. I use military time whenever I can. In my other comment I laid out the familiar Roman staves correspondent with base 24:

0 = Z 1 = A 2 = B 3 = C 4 = D 5 = E 6 = F 7 = G 8 = H 9 = I 10 = J 11 = K 12 = L 13 = M 14 = N 15 = O 16 = P 17 = Q 18 = R 19 = S 20 = T 21 = U 22 = V 23 = X

To mark these as numerals instead of verbals or initialisms they can be forepadded with Z. Conveniently Thewdish already calls tales Zahlen whose heir is the Z field. A base can be written as Y or as AZ, intuitively.

How is the above not any fun?

Your so-called everyday calculations only work in two dimensions. Divisibility by three in many dimensions isn't as important as in one dimension; therefore base 12 deliberately doesn't lose over base 18 or 54. Divisibility by two however is important for every dimension. The base must hold 8 cloves. Real world wares take up three dimensions which base 24 fulfils for dry and wet measures. Its 8-factor complies with imperial base-2 bulk measures, between ounce and cup or gill and quart or cup and gallon.

Cooper Gates May 15, 2015 +Stephanie Currie Using sub-bases of ten and six? Such as 1/5 = 0.12, 1800 (decimal) = 30;00 in base sixty?

Cooper Gates May 15, 2015 +alysdexia Stephanie wanted a base in which 1/2, 1/3, 1/4, 1/5, and lots of other fractions just have one digit after the floating point. The same fractions terminate in bases six, twelve, eighteen, and twenty-four (numbers whose factorizations contain two and three and no other primes).

Cooper Gates May 15, 2015 +spaceLem 24 is a highly composite number as well (8 factors), although not a superior highly composite number. Maybe alys likes the fact that 1/5 and 1/5^2 only repeat two digits in base 24.

alysdexia May 15, 2015 +Cooper Gates I don't care about 1/5's representation. When does it arise?

Cooper Gates May 17, 2015 +alysdexia In a hypothetical world you would only have to divide by three-smooth numbers, do you mean you can't have a denominator above 4? For large numbers, very few of them only contain factors of 2 and 3; 60 = 2^2 * 3 * 5, 840 = 2^3 * 3 * 5 * 7 are highly composite numbers, for instance. How about the partial sum of a harmonic series? 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + ....

Or the arithmetic mean, what most people call the "average"?

5+11+7+2+6 = 31 Mean = 31/5 = 6 + 1/5

alysdexia May 17, 2015 +Cooper Gates need statistics like binomial distribution of group sizes

https://www.youtube.com/watch?v=U6xJfP7-HCc&amp;lc=z130ihgjwkzhgpjmh22wjv5z5s2yttx2n Stephen Anthony 5 months ago Forget base 12, just learn how to deal with base 16 from the start. Children are going to learn how to program computers eventually anyway, so why not teach them to count in hexadecimal?

Cooper Gates 4 months ago (edited) I believe that sixteen symbols is not too many, I just don't prefer A-F when dealing with hex because I get stuck on thinking that C is the third letter in the alphabet so it stands for thirteen.... no, it's twelve in that convention.. same with the other letters, overshooting by one.

Dozenal is only more useful for small fractions, where a noticeable portion of denominators don't have factors other than two and three... otherwise, most still recur in doz. This is where the secret of hex comes in - since each hex digit is four bits, any recurring fraction in binary with a pattern that has an even length will be only half as long in hex... if the length is divisible by four, it will be four times shorter!

In bases like decimal and dozenal, which are not square numbers, reciprocals of some primes repeat as many digits as one below the prime (1/7 can repeat six, 1/17 (decimal) can repeat sixteen digits). Except for 1/2 in odd bases, because all those periods have EVEN lengths, it is strictly impossible in hex!

1/7(Dec.) = 0.249249249249249249 (Hex.) One thirteenth is a good example, repeating the full twelve digits in binary, but only three in hexadecimal...

1/D = 0.13B13B13B13B13B13B13B13B13...

The only base that might be able to have even shorter periods for any recurring patterns is sixty-four, since it can cut periods with lengths that are divisible by three OR by two... but sixty-four is much too high (too many symbols). Hex is the way to go for reciprocals of ugly primes or semiprimes such as five times thirteen (65 in decimal).

alysdexia 2 months ago (edited) Until the trinary computers come. Quantum?

alysdexia 1 month ago (edited) Forget about my endorsement of base-12 in another thread; I changed my mind. A base must be a product of 8, the first nontrivial cube for 3D packing and the vertical cubic coordination, and a product of 6, the first nontrivial semiprime and the facial cubic coordination, thus base-24. Notify NIST.

Z=0 A=1 B=2 C=3 D=4 E=5 F=6 G=7 H=8 I=9 J=10 K=11 L=12 M=13 N=14 O=15 P=16 Q=17 R=18 S=19 T=20 U=21 V=22 X=23 AZ=24

incidentally, https://www.youtube.com/watch?v=U6xJfP7-HCc&amp;lc=z12bc3ri4naodzv2h231w5obxlf4i3ine Unhidden Polymath? 2 months ago The whole thing about base 12 having more factors seems kinda inconsequential when you take into account that 3 and 7 were the only division issues in decimal. When you switch to 12, you're just switching the problem from 3 to five, leaving it just as problematic.

Cooper Gates 3 weeks ago +Robin Powell I like base thirty but it's too high for the seven-segment display (I only managed to fit twenty-four distinguishable symbols at most into it, and the highest number for symbols which never have the same shape I discovered was twenty).

https://www.youtube.com/watch?v=U6xJfP7-HCc&amp;lc=z13lxhygoqvsxrcix04cjddpwtzdefwpl3w ... Improbabilities 1 month ago +atsay714 Decimalizing time and the calendar would have been stupid, but I quite like the consistency of unit relations when using metric. ... Cooper Gates 3 weeks ago +Improbabilities Or even a consistent sixty like degrees, arc minutes, arc seconds, maybe dividing days into 60 fractions instead of 24 lmfao

Not much of a choice with the number of days in a year, 365.2422 is pretty ugly using any divisions

alysdexia 3 weeks ago +Cooper Gates 365.2422/24 = 15,5,5,'19,12,4,3,12,18,21,10,6,1,14,16,23,8,7,23,16,15,1,6,9,21,18,13,3,4,11,19,20,11,5,2,13,17,22,9,7,0,15,16,0,7,8,22,17,14,0,2,5,10,0,20'. 365.2422/60 = 6,5,14,31,55,12. 365.2422/30 = 12,5,7,7,29,12. 365.2422/15 = 24,5,3,9,7,6,'5,9'. 365.2422/90 = 4,5,21,71.73,72. 365.2422/120 = 3,5,29,7,81,72. 365.2422/150 = 2,65,36,49,75. 365.2422/180 = 2,5,43,107,50,72. 365.2422/210 = 1,155,50,181,4,42. 365.2422/240 = 1,125,58,30,172,192. 365.2422/270 = 1,95,65,106,102,162. 365.2422/300 = 1,65,72,198. 365.2422/330 = 1,35,79,305,191,132. 365.2422/360 = 1,5,87,69,43,72.

Looks like the year likes base 300.

Cooper Gates 3 weeks ago +alysdexia As you wrote it, that base 300 approximation has eight digits (using your decimal sub-basing), more than the seven I started with. You would also not make a version of a month that fits 300 times into a year. Dividing by fifteen is even more sensible since each fifteenth would have 24 or 25 days, and a system of twenty months would have eighteen or nineteen in each. Now that I look at it, February having only 28 or 29 is dumb.

alysdexia 3 weeks ago (edited) +Cooper Gates Wrong, the base-300 year has four digits (nondecimal of course).

365.2422/100 = 3,65,24,22. 365.2422/200 = 1,165,48,88. 365.2422/300 = 1,65,72,198. 365.2422/320 = 1,45,77,161,89,192. 365.2422/330 = 1,35,79,305,191,132. 365.2422/400 = 0,365,96,352. 365.2422/600 = 0,365,145,192. 365.2422/900 = 0,365,217,882.

Besides the original base-10 significant bias, base-300 represents the year most compactly. (I don't know; maybe there are in-between odd bases even more compact; I'd hav to check hundreds of them.)

Submitted by anonymous - 2015-11-17 01:38

Base 6 does have some advantages for counting on fingers. Since we have five fingers, we can use one hand to represent 6 digits (0..5) and so with two hands we can count to 35.

I agree however, that 6 is not a good base for all situations. Neither is 12, 10, 8, 16, 2 or any other base useful in all situations. But what is the best overall?

Most people can divide a stick in half fairly accurately by eye. Not many people can divide into thirds accurately. If we are cutting a cake we almost always start by cutting it in half. If we really care about equal sized pieces we then cut in half again. This leads me to think that binary or pseudo-binary like 8 or 16 is most natural. And 8 has some obvious advantages when you move to 3D. But that only assumes you are using an xyz coordinate system. What about tetrahedral coordinates? That actually makes 12 look like an interesting choice again.

So, while I agree that there are some bad base choices (10 being one of them), there is no single base that is better than all of the others. We must learn to see the beauty of all bases and try to use the one that makes the most sense for our particular application.

Submitted by anonymous - 2015-12-09 01:26

I think it will be better to change the whole digital system rather than retrofitting the one we are using today with two new characters if we want to use the dozenal system. That will be twelve new digits to match with the dozenal system.

Submitted by anonymous - 2015-12-09 01:28

I agree a lot. We should be conversant with number radixes.

Submitted by paul_rapoport (dozenal.ae-web.ca) - 2016-01-07 18:45

Good, helpful article, thanks. For me there are too many problems with bases 6, 16, and 24. Meanwhile, almost all the discussion of bases is theoretical. I've actually been living with a thoroughly dozenal clock/watch and calendar for some time. The clock (actually three dozenal clocks) and calendar are on the website http://dozenal.ae-web.ca. The clock is also available for the iPhone, soon with additional features. The digital output of the clock will also be available in various configurations for Pebble smartwatches.

Comments on any of this are welcome.

### Re: Fourbang 2

Submitted by graham - 2017-05-29 21:47

autymn_d_c/alysdexia:

There's a lot to unpack in your various comments and honestly I don't really follow much of it, but regarding base twenty-four, etc., again I just feel it's too far quantitatively from decimal to really be intuitively useful. Twelve is close to ten; we have an idea of twelve things being somehow "like" ten things, which is convenient in terms of everyday math and quantitative conceptualization if nothing else.

I agree, I think, that the whole three-order "superbase" convention is not in itself critical to a new terminology, and that alternative conceptions exist, but I do think such a system provides some useful intuitive gloss over, say, straight up scientific notation. There definitely seem to be alternative approaches in this area, involving different groupings and scalings that might make sense, etc., but again I'm not sure I entirely follow everything you are trying to say here.

Regardless, one should be mindful, I think, that the whole issue here is somewhat psychological—whether or not some radix or approach is "theoretically" ideal in some sense is beside the point if it isn't something humans can use and acclimate themselves to. And I'm just not convinced replacing, say, positional notation itself with some radically (so to speak) different representation of number would be all that useful, particularly in the short term. Dozenal is ultimately a cheap and easy swap-out for decimal—we keep all our familiar conventions and customs of thought around arithmetic, etc., while vastly improving our handling of common fractions, simple multiplication, and so forth.

### C.O.F.

Submitted by wendy_krieger - 2018-07-09 09:08

One of the interesting developments of a dozenal measurement system is the C.O.F. system. This is essentially similar to Kodegadulo's Primel, but greatly extended into uses not generally considered.

The sorts of systems such as Primel and other systems where gravity is set to one, are unlikely to take off, because to keep in an acceptable range, you need a unit much smaller than a second. The only suggested metric system of this nature is the dm-kg-ds, proposed by Professor W.Moon in the 1890's, and glossed by Steven Dresner in 1971 (Encyclopedia of scientific units and quantities).

The most likely system to take off on this, is a system based on the DD 'decimetre', the cube of water, and the day divided to 12 hours of 144 minutes of 144 seconds. The magic that C.O.F. gives (and no one really discusses) can be hidden in 'unit values'.

What unit values do, is that they implement g=j=1 etc, and allow you to directly convert from units like 'pounds per square inch' into coherent units. A pound is '600', an inch is '3' dozenally, so you get a psi is 600/3^2 = 80. The unit here is pieze. Atmospheric pressure is 35 feet of water, or 2E (doz), a foot is 30, so the atmosphere is 890 'pz'. Divide this by 80 (doz), and you get 11.16 (ie 13.125 dec).

This was being developed at DozensOnline, but the 'moderator' engaged in a spot of 'abuse of power'. So I have been forced to create my own board at Tapatalk TwelftyOnline. Here we are free to devise both bases and measurement systems to any base, free from the hassle of conforming to the ever-demanding DSK (duodecimal society of Kodegadulo).

Wendy

Kode borrows the electrical and magnetic systems directly from C.O.F., also developed at the same place, but not relying heavily on the baroque prefixes.

But C.O.F. has now had to move to TwelftyOnline at tapatalk, because it's too classy for the dozenalists.

In any case, I converted to base 120 in the 1970s and have overcome most of the serious problems. You will see the advantages when you write the reciprocals of the factorials and 2^n n! in the various bases. Base 120 does not pull a two-place period until n=11, and shorter 'leaders' all the way through.