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 Compatibility, dozenalist views on bridges to decimal
Treisaran
Posted: Sep 22 2012, 09:08 PM


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Ever wondered why so many bases are considered by dozenalists? Shouldn't dozen as a base be a closed case? The answer that pops up right away is that people who have broken outside the box of decimal will inevitably take interests in number bases in general, and while that's true, I don't think it explains why some non-dozenal bases like the Long Hundred aren't just a subject of consideration, but proposed for one's main use.

It's just my guess, but I think compatibility has to do with it. The alternatives to decimal that aren't dozenal tend to be multiples of the dozen, which says a lot about the matter. Obviously dozenalists think an alternative to decimal needs to have all or most of the advantages of the dozen. Where the other bases differ from dozenal, it's in the degrees of compatibility they offer with the decimal world out there. As in the computing world, when a new standard is proposed, there are revolutionaries who want a clean break with all past legacies, there are stick-in-the-muds who think the new standard will never catch on as is but must be set as part of the old-standard world, and there's every shade of grey in between. Let us explore those ranges, if only briefly.

Revolutionaries

The purest in dozenalism is to look upon decimal in contempt as a technological dinosaur awaiting extinction. With the decimal ancien regime there should be no connection maintained, no bridges allowed to stand. Of necessity for dozenal purism is an anti-quinary stance: the prime 5 is believed to be objectively as useless as 7, inflated in its importance only due to decimalism. Therefore, any accommodation for 5 in dozenal divisibility testing and representation of fractions is a waste of time at best.

Of the dozenal-multiple bases that burn the bridges with decimal totally, those that fulfil 5n±2 are to be chosen: 48 (quadranunqual), 72 (hexanunqual), 108 (ennunqual) or 132 (levanunqual). The fundozmentalist believer, the member of the Ultra-Orthodoz community waiting for 10th Imam (that's dozenth, not tenth, ye of little faith!) to shine the light on the benighted decimal world, would likely champion base four-dozen as a pure manifestation of dozenal fervour, the kind that leaves no bridge to decimal and its dastardly divisor 5. Of course, dozenal itself might be suitable, but the SPD test for 5 devised by one of the compromisers would have to be avoided; in base *40 such a test is out of reach, so it would be the ideal choice for the one who wishes to avoid temptation.

Satire aside, a dozenal purist would not actually reject something like the ability to test for divisibility by 5 in dozenal, yet he would consider it an extra, something nice to have but not that important. He would not stress himself over the fact that fifths will never be clean in dozenal. Compatibility with decimal is, as far as he's concerned, nothing to be striven for. If achieved, well and good, but dozenal is so much superior to decimal that the one who truly appreciates it no longer cares about keeping in touch with the decimal world. Once he's settled on names and symbols and a system of metrology, the pure dozenalist never looks back.

Minimal Compatibilists

Less stringent than the revolutionaries are those who think there needs to be some handy provisions for decimal compatibility. They hold that the factor 5, though not so important that it needs to be a divisor of the base, should not be treated in such hostility as 7 is in dozenal or decimal. It should be a neighbour-totative of the base, in order for it to have an easy divisibility test and to avoid maximal recurrence for its fractions.

The choices here depend on which neighbour relationship one would prefer and how many powers of 5 one is to be able to test for. The α-totative relationship gives dozenal multiples like 24, 84 and 624, and the ω-totative relationship, 36, 96, 276 and 876. Since the base chosen as one's main base tends not to be too large, the choice will be between 24 (double dozen) and 36 (triple dozen), the former winning because of its smaller size, the latter because of the benefits of the omega relationship (the digit-sum test and fractions with only one recurring digit). Both require a lot of thinking about symbols and names for them, which is why dozenal purism might be the easier route to take after all; but they offer much more in the way of compatibility with decimal than dozenal does, especially in the case of fractions.

Decimal Auxiliarists

The next level of compatibility is bases that give decimal (fivefoldness) and dozenal (threefoldness) equal stature. With those multiples of the dozen that also have the prime factor 5 in them, we get to have that gorgeous trailing zero in both representations: *50 = 60, X0 = 120. We have no need to compromise on either side, because prime factors 3 and 5 are both first-class. Another advantage is that we can maintain compatibility with decimal names and symbols up to a much further point than with other dozenal-multiple bases. For example, in the usual representation of base 120 (dozenal-on-decimal encoding), all numbers up to 99 inclusive have the same names and meanings as in decimal, and only after that do we have to break out of the mainstream with 'teenty', 'elfty' and so on.

These bases are large. The smallest of them, sexagesimal, already needs to be encoded as two sub-bases for each digit. The alternating-radix encoding means we have a different carry rule for each slot, so that for example in the Long Hundred there is a carry after 9 in one slot but after E (eleven) in the other. That is the price to be paid for the good measure of decimal compatibility gained.

Timid Accommodationists

And then are those who think decimal is here to stay, differing from those who write out dozenalism completely only by their agreement that the advantages of dozenalism are real. They do not think any change of number base in the main, meaning any change of names and symbols, will ever catch on; what they propose in order to gain the benefits of dozenal in an incurably decimal world is to integrate twelve-valued scales into niches.

They point out that we already have thirds represented as 4 inches of a foot ruler or 8 hours of a 24-hour day, yet we do not change our decimal numerals and names one tiny bit for those gains; this, they say, is the way forward for dozenalism. Theirs is a proposal to use base-12 modular arithmetic in a decimal world: count up to 11, and write it thus (not something like E as the other dozenalists do), and just carry to a new unit above when adding to it. So 11 plus 4 inches equal 1 foot 3 inches, and that's the spirit. In numeration, their scheme would have the dozenal 6E4 (decimal 1000) represented as 6,11,4. It is the type of dozenalism that demands the least change of its adherents, at the price of the lack of the benefits of base-n calculation. Also, in a decimal world that is itself not much predisposed towards compromise (look at SI and its roughshod attitude to anything outside the confines of straight decimal multiples), attempts to integrate dozenal scales do not produce a more palatable result than real dozenalism, except for established practices like the foot/inch ratio.

Summary

The outline above shows that the gradation of decimal compatibility advocated by various dozenalists is not primarily a technological, arithmetic issue; the biggest factor here is one's attitude, the measure of one's optimism about the ability of dozenal to become mainstream. The first two groups are of those who believe that the vision of Douze notre dix futur ('Twelve, our future ten', the title of Jean Essig's 1955 book) can be achieved, and decimal, if accommodated at all, needs just a little help from the base that is going to replace it. Members of the other two groups think decimal must stay one's main base as it will always be so in the world, and the benefits of dozenal are to be gained through optional-use auxiliary superbases or special-purpose dozenal scales. Yours truly has had times of belonging to all four groups, and as of this writing is vacillating between the first and the third, occasionally flirting with the second group by pondering the triple dozen as a base. Hopefully harmless...
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icarus
Posted: Sep 23 2012, 12:00 AM


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A masterpiece, once again, Treisaran. Printworthy!

[no swearing, please]
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Treisaran
Posted: Sep 23 2012, 01:55 AM


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QUOTE (icarus)
A &$€#% masterpiece


Symbols for a new separate-identity set, Icarus? wink.gif

Thanks. smile.gif You know, I'm thinking maybe I should open a dozenalism website of my own, just like Don Goodman has done. That way a print form of articles that are too long for the DSA Journal could be published. For example, on my image gallery's LiveJournal blog I've posted a primer on dozenal and another primer on dozenal divisibility tests, both of which are better published in PDF form for print. Luckily, today's free hosting services are less limiting than they used to be in the past.
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dgoodmaniii
Posted: Sep 23 2012, 12:49 PM


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QUOTE (Treisaran @ Sep 23 2012, 01:55 AM)
QUOTE (icarus)
A &$€#% masterpiece


Symbols for a new separate-identity set, Icarus? wink.gif

Thanks. smile.gif You know, I'm thinking maybe I should open a dozenalism website of my own, just like Don Goodman has done. That way a print form of articles that are too long for the DSA Journal could be published. For example, on my image gallery's LiveJournal blog I've posted a primer on dozenal and another primer on dozenal divisibility tests, both of which are better published in PDF form for print. Luckily, today's free hosting services are less limiting than they used to be in the past.

Go for it! The more dozenal sites there are lurking around the Web, the better.

I'd encourage you not to neglect the Bulletin, though; I can think of many of your posts that would probably be excellent Bulletin material.
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dgoodmaniii
Posted: Sep 23 2012, 12:51 PM


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I guess this makes me a "pure dozenalist," then, as the lack of five as a divisor really never gave me any trouble at all, and while SPD is a neat addition, I'd still be a confident dozenalist if it didn't exist.
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Treisaran
Posted: Sep 23 2012, 03:03 PM


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QUOTE (dgoodmaniii)
Go for it! The more dozenal sites there are lurking around the Web, the better.

I'd encourage you not to neglect the Bulletin, though; I can think of many of your posts that would probably be excellent Bulletin material.


I'm all for having an article appear in the Bulletin if it's suitable. The idea of having a website of my own is for being able to publish material right away even when the Bulletin editor's workload is too high to attend to it.

QUOTE
I guess this makes me a "pure dozenalist," then, as the lack of five as a divisor really never gave me any trouble at all, and while SPD is a neat addition, I'd still be a confident dozenalist if it didn't exist.


Heh. smile.gif When I was writing the last paragraph of the 'Revolutionaries' section, I thought to myself, 'Don Goodman must belong here'.

My category changes as my optimism fluctuates. Like Icarus, I cherish sexagesimal and think that a civilisation starting over would best do as the Babylonians did and use decimal for everyday life, sexagesimal for serious mathematical stuff. But in our reality, the use of sexagesimal beyond its established niches is problematic because of the way SI steamrolls any non-decimal usages. So, while revolutionism usually has a bad rep, for dozenalism it may be the most rational course: in order to succeed, dozenalism must be as pervasive in one's use as decimalism is now, leaving all purposeful (as opposed to fortuitous, like the Calg/Kelvin correspondence) compatibility routes behind.

On the other hand, two things holding me back from pure dozenalism are the difficulty of thinking up acceptable-sounding dozenal names in my native language and the fact that no symbols except the ASCII 'X' and 'E', not even the script-capitals (available in Unicode but not guaranteed to display correctly, especially not the script-capital X), are accessible in plain text. The former issue draws me to the Long Hundred, where I already have a set of nice names in my language, while the latter means having to rely on PDF publications. However, the Long Hundred like sexagesimal needs its own TGM equivalent in order to be viable, and the mixed-radix approach takes getting used to, and PDF files are less versatile than plain text. As you can see, there are lots of dilemmas to contend with.
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icarus
Posted: Sep 23 2012, 07:01 PM


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I am an aging revolutionary, having internalized my own use of duodecimal early on. I do think fivefoldness is part of a continuum of "importance", roughly like the function 1 − φ(p). Seven is not quite so important to merit accommodation, and I don't think higher primes contribute anything significant to a system, diminishing returns.

I don't know if I am on the spectrum here. I use a variety of bases, but mostly {12, 60}. my usage of sexagesimal is increasing, but I have deep seated vestigial and highly integrated uses for dozenal. I think the prospect of returning to a sexagesimal civilization are dimmer than conversion to dozenal, which itself is not likely. Thus I think people should consider using bases as tools in their own practices. This way there might arise a niche-need for a number base (cf. hexadecimal in the coding of bits). I feel pretty strongly that duodecimal is optimum for human arithmetic using today's arithmetic algorithms. Sexagesimal is superior if one can wield it, not everyone can or wants to work with it. I have a feeling base 120 is about as good as sexagesimal. I think folks who use 60 vs. twelfty find themselves attempting to confine themselves to regular numbers. I have not done enough with 120 to say that it is any different but surmise, given the favorable relationships with {7, 11, 17} I'd expect some relief in rendering these somewhat common primes (if that is the word, like the summer on Pluto can be considered "warm").

When I use a number base I am a purist. I dislike compound digits because they blunt the natural pure divisibility of a base by a group of its divisors, which is the very reason we are going there in the first place. I suppose given the fact I am a native decimalist, threefoldness is easily recognized by rote of nearly 4 decades of exercise. This makes the situation with decimal coded sexagesimal tolerable. When I try to use base 2520 as 42-on-60, I have an extreme difficulty in recognizing sevenfoldness, resorting to attempting to memorize a table of exempt sexagesimal totatives (I.e., those that are multiples of seven) that number something like 200. Hence my development of argam.

Because I am a purist I do not truck decade and unit figures in sixty and twelfty as well as others and it probably shows. I'd write single-piece and think in similar ways especially in addition-subtraction. But if I think of base sixty as base-root-sixty, a bit less than 8, then the alternating system makes sense. Go figure. Probably a nerd thing lol.
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wendy.krieger
Posted: Sep 24 2012, 07:18 AM


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My main problem with twelve as a base, is that all of the less-used primes tend to rise to the top, and were hard to get rid of. Twelve itself is quite useful if you stick to twos and threes. But when 5's and 7's turn up in the mix, they tend to migrate to the main places.

For example, the classical division of the circle in geometry is triangles, pentagons, and bisection of the angles. This gives 2^x.15. Gauss found 17 some time later.

The general divisions of the circle, is into 6 if you're standing inside it, or 3 if you are outside it (the Ancient sumerian division of a circle on the ground was into 3.00 ells of 24 digits, supposing the diameter was 1.00. The division into 6.00 and finer sixtieths, applied only to right ascession (ie stars in the sky).

What happens is with the significant integer. It can never be a multiple of the base, for example. So, for example, in decimal, one can not have a significant integer a multiple of 10, but 12 works (like 360 = 12 × 30). You always have this sense of breaking a unit (like tens or hundereds) up. In twelve, you can't have a multiple of 3 and 4 there, and so to get both, you have to break up a place.

Because 12 is so good at providing 3's and 4's, you get some rather odd looking numbers if you want to go past 2 and 3. The auxhiliary bases suggested here for duodecimal are 500, 50, and 2E00. The general sense is that you are dividing the circle into 35's and then into regular fractions.

In base 18, for example, you can have a multiple of 12, and '360' and '6Y0' both work quite fine, these are respectively, 1080 and 2160. The numbers are just large, though.

The difference with 120, is that there really does seem to be a use for a pair of 5's, and thus, one can use two places of 120 to get 576 and a pair of 5's. 7 and 11 living next door are a bonus, especially 11². Since ultimately one is effectively getting an '11-smoth' system over 2-places of period, 120 is well worth the price.

On the other hand, one feature that really outstands of 120 is that it is not just a multiple of 12 and of 10, but it is the [i]product[/] of said numbers. 60 is a multiple, but as one might get from looking at a clock, one is not looking at a division of 12's but of sixes and halves of sixes. The minute hand is on the 7, but you have to relate it to 5 minutes past what it's saying on the six (ie 35). In twelfty, the minute hand on the seven means seventy.
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Treisaran
Posted: Jan 8 2013, 02:59 PM


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My thinking about compatibility on this thread has so far been as a one-dimensional line, where the only variable is the closeness of the approach chosen to the numeric appearance of decimal. That is, I've evaluated the approaches according to how, for example, a number like '37' is related to: is it 'thirty-seven', filling out three decimal multiples and being less than the fourth by just three units, or is it 'threezen-seven', filling three dozenal multiples and being less than the fourth by five units? A system like six-on-ten encoded sexagsimal, as on our digital clocks, keeps the decimal appearance for the last twistaff digit (eg, in 15:37 the last '37' means thirty-seven of the smallest units).

However, following a lengthy, winding and fruitful discussion on a thread on the Long Hundred, it's now clear to me compatibility is not a one-dimensional line but at least a two-dimensional axis of lines like Cartesian coordinates. On the one axis we have numeric compatibility, which I've explained above and has been the focus of this thread; on the other axis, the line denotes arithmetic compatibility, which is compatibility with the mechanics of working with the numbers.

The world today has not only settled on a number base; it has also settled on a particular notation to be used with it, and the attendant mechanisms of arithmetic. It is not only the choice of decimal that marks our current age, but also the choice of having a single radix throughout the number, with no alternation of radices within the number (contrast Babylonian six-on-ten alternation) and no utilisation of a different base for the integer part and the fractional part of a number (contrast the Romans' use of decimal for integers but dozenal for fractions, or the European practice of using int-decimal with frac-sexagesimal up to Simon Stevin's time). The arithmetic practices that have trickled down as a result are optimised towards uniform treatment of all parts of a number, such that we can prove the infinite recurrence of the digit 3 in decimal thirds by doing long division on đ10/3.

In my opening post, the rubrics of 'Decimal Auxiliarists' and 'Timid Accomodationists' rank as compatibilist options only on the axis of numeric compatibility; on the other axis, of arithmetic compatibility, they are actually on the incompatible side of the line, while it is none other than pure dozenal ('Revolutionaries' or 'Minimal Compatibilists') that are the most compatible with the current mainstream way of working with numbers! By choosing pure dozenal, one needs to make no change to the arithmetic methods one is already accustomed to from the decimal world; only a change in evaluating numeric appearance (eg '25' and '29' now trade places as to their possibility of being prime) is needed.

I hear people say, 'Only a change in evaluating numeric appearance?! Only?!' I'm not going to deny this is a big change, an alteration of one's ingrained thinking, which demands effort. In ignoring the other axis, however, one is liable to rush into solutions where the gain of numeric compatibility is offset by the loss of arithmetic compatibility. If we liken the change of number base to learning a new language, then the change of arithmetic method is analogous to adopting a different culture. While the two axes are intertwined, it is far more difficult to change one's cultural customs than it is to acquire a new language. It's roughly the difference between learning static facts and mastering dynamic know-how.

Switching from pure decimal to pure dozenal is far less revolutionary than opting for a decimal compatibility auxiliary, for it is more difficult for most people to internalise new ways of doing arithmetic than to think in a different number base. That's my latest conclusion and a necessary addition to any discussion of compatibility with regard to the mainstream usage.
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Treisaran
Posted: Mar 1 2013, 11:55 AM


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Ladies, gents,

For the past few weeks I've been flailing wildly among the options listed in the opening post: full dozenalism, a partially dozenal alternating-radix base (the Long Hundred) and decimal with a special fractional notation that covers the dozenal fractions. If anybody's found occasion to raise a few eyebrows at all these changes, that's nothing to what I've felt: like going crazy. But it's worth the insights gained, and since I believe those insights about number base usage are of more interest generally than 'I, me and mine', let's have at it.

The Long Hundred is a good base, but alternating-radix arithmetic, I now realise, isn't easy even when you know the ropes. I'm glad I've managed to grasp the operations involved, as well as the importance of a notation that marks the staves of each twistaff (Long Hundred double-digit place), but I don't see myself using a base that requires me to use the rules of two bases, decimal and dozenal both, according to context; although the notation guards against confusing the sub-bases quite well, the switching between rules is tiresome even if you've mastered both multiplication tables (which I have).

Alternating-radix arithmetic is hard beyond necessity, and apart from the better compatibility of nomenclature of the Long Hundred (up to ninety-nine inclusive), the gains aren't compelling. There are ways of dealing with fifths in dozenal, while sevenths and elevenths may be interesting for number theory studies but not practically useful. A solution to the problem of dozenal nomenclature was all I needed in order for the Long Hundred to cease to appeal to me, and I suspect other non-English native speakers looking at dozenalism would agree on that.

The third direction, of keeping decimal for integers while using a fraction-part auxiliary such as hexoctunqual (đ960), is unsatisfactory also, and now I understand just why this is so. The catering to various coprime factors through a fraction-part auxiliary is a one-time device; by this I mean it enables you to divide by the coprime factor once, but trying more than that will send the arithmetic complexity spinning beyond control. If, in employing TGM for length measurements for example, you have a quantity and divide it by 3, and after that you perform a couple of operations on that new length and divide yet another new length by 3, then you can be sure the result isn't going to be messy beyond the level that the operations have made it so. But with a fraction-part auxiliary I can only divide an original quantity, a nicely-chosen integer, whereas after a slew of operations I'm going to get a mess, and even more so if I try to divide that mess according to the rules of the fraction-part auxiliary.

The fact is a fraction-part auxiliary is useful, for its notational capacity, only in contexts where intensive transformations aren't called for, such as marking angles or statistic distributions. So the fraction-part auxiliary base 1010 (unnilununqual), which gives dozenal fifths without sacrificing the basic dozenal fractions, is ideal for those relatively few contexts where a dozenalist would need fifths. But the contexts where a decimalist needs thirds are not so few, and many of them involve multiple operations, thus the fraction-part auxiliary solution is inadequate. In fact, this brings us back to decimal's square one: the prime factor 3 is too important for compromises. The main point regarding decimal versus dozenal is made starkly clear by such solutions as a fraction-part auxiliary: the importance decimal accords the primes 3 and 5 is the reverse of their actual importance, while in dozenal the order is just right (or at least once you've accommodated 5 with the SPD test and some creative fractional solution).

But the greatest drawback of all in my looking aside to the Long Hundred and a fraction-part base for decimal has been, without doubt, the loss of TGM. It is a testimony to the relative maturity of full dozenalism that it already has a compelling application that can keep a user locked in, so to speak. A metrology could be devised for dozen-on-ten Long Hundred, but so far there is no equivalent to the systematic TGM. For decimal with a fraction-part auxiliary base, devising a metrology is impossible by nature, for the reason I've given above. TGM is, at least for me, the one great enticement of dozenal. Other attractions are the comfort of a pure-radix base, free of any manner of kludges, hacks, inconsistencies, scalability failures and so on that beset the other solutions. I had looked at those other solutions because I'd said to myself that my only requirement was solving the problem of thirds in decimal; I now see that full, pure-radix dozenal is the best solution to that problem.

I can't guarantee and daren't announce anything, but I do hope this'll be the end of those flip-flops.
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Zenarchist
Posted: Mar 1 2013, 06:19 PM


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Well there's definitely nothing preventing someone from creating a Long Hundred metrology that's for sure. But I'd say the biggest problem with it is still the mixed-radix rule remembering. Pure dozenalism is for sure the most elegant, but I sometimes feel sorry for poor 5 in it tongue.gif.
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Zenarchist
Posted: Mar 1 2013, 06:25 PM


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I've been very flip-floppy myself as well ever since I started posting here. I've been strangely gravitated toward sixty lately tho. But school's been really busy so haven't had much time for recreational mathematics. But only 36; more days to go!
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dgoodmaniii
Posted: Mar 1 2013, 07:29 PM


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QUOTE (Zenarchist @ Mar 1 2013, 06:25 PM)
I've been very flip-floppy myself as well ever since I started posting here. I've been strangely gravitated toward sixty lately tho. But school's been really busy so haven't had much time for recreational mathematics. But only 36; more days to go!

As a "true believer," I don't personally flip-flop about dozenalism; I was convinced long ago and remain so.

But it's not as if changing one's opinions, or getting interested in other bases, is something that needs to be apologized for. Other bases are interesting. We're really a pack of nerds here, and that's fine; if we weren't, we'd have other interests, like NASCAR or competitive eating.
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Treisaran
Posted: Mar 2 2013, 08:22 PM


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QUOTE (Zenarchist)
But I'd say the biggest problem with it is still the mixed-radix rule remembering.


It's advanced. It requires a firm grasp of the addition and multiplication rules of two bases. There's simply no way kids could be raised on it; the Long Hundred would be serviceable, but it would be in dichotomy to the everyday decimal just as sexagesimal was to decimal for the Babylonians. We would be amiss, however, to pretend the Long Hundred lacks substantial advantages:

QUOTE (Zenarchist)
Pure dozenalism is for sure the most elegant, but I sometimes feel sorry for poor 5 in it tongue.gif.


That's one. For fractions with the prime factor 5 in the denominator, dozenal has no solution that smoothly integrates with the base. Approximating using the multiples of twozen-five {25, 4X, 73, 98} to the desired precision (1/5 as 0;25 or 0;24X or 0;2498 or 0;24973 or 0;249725 and so on) introduces the difficulty of handling we know all too well from decimal thirds, only harder, while fraction-part auxiliaries such as unnilununqual (base dozenal 1010) have the same problem as fraction-part auxiliaries anywhere else: they don't scale well for multiple arithmetic operations.

Twelve-on-ten Long Hundred may be used a fraction-part auxiliary for dozenal, but that again doesn't scale well. It's only when you use it as the main base that the problems of scaling, integration and handling are solved (at the cost of having to deal with alternating-radix arithmetic, of course. TANSTAAFL).

The real big advantage twelve-on-ten Long Hundred has is it doesn't clash with the decimal legacy so forcefully as dozenal does. The compatibility is absolute in the range 0 to 99, and after that, while the names ('hundred' etc.) have different meanings, the written forms are still disambiguated from the decimal ones because of the necessity of twistaff notation: '100' is the short hundred (dozenal 84), '1:00' is the long hundred. In dozenal I have still yet to become accustomed to thinking of '14' as 'dozen-four' rather than 'fourteen', and even if I manage it, the clash with mainstream usage (decimal...) is inevitable. The only solution for that is to use a separate-identity dozenal numeral set, which presents great difficulties for computer use.

You can see I'm still very much wavering. I think I'd better put my signature out of commission, seeing as the daily changes make it useless.

QUOTE (dgoodmaniii)
But it's not as if changing one's opinions, or getting interested in other bases, is something that needs to be apologized for.


If only it were about apologising to others and nothing more than that; the worse problem is that these frequent changes are driving me up the wall. I think I need to take a break.

QUOTE (dgoodmaniii)
We're really a pack of nerds here, and that's fine


*sigh*

Me being a nerd is not fine in my eyes. I've got to be off for a while, I'll be back when my mood improves hopefully soon. No hard feelings towards anyone here of course, but I shouldn't post when under the influence of a deteriorating mood, so goodbye for now until this wave is past.
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Zenarchist
Posted: Mar 2 2013, 09:54 PM


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Nerds are the new Cool Guys, Treisaran. But I hope you feel better soon smile.gif
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Treisaran
Posted: Mar 10 2013, 04:04 PM


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QUOTE (Zenarchist)
Nerds are the new Cool Guys, Treisaran.


I think the status has seen fluctuations over the years, from 'desirable' to 'definitely uncool' in periodic waves. But it may be a red herring; when I'm in my downs, any issue will masquerade as the cause. All right, now that I'm in better shape I ought to address the matter of compatibility again. (That is the topic, after all.)

Le sigh... I've been changing between 'decimal default, dozenal marked' and 'dozenal default, decimal marked' so many times now that my older posts would be so confusing. But even without those changes of mine, it's become pretty clear to me that the possible confusion of number bases - in those numbers that don't have transdecimals in them - is something I've found frustrating from the start (and even before being a dozenalist, back when hexadecimal was my thing).

Whether or not I manage to straighten native-language dozenal nomenclature out, I wish to dispense with the precarious method of prefix-marked numbers and use a system where one always knows if I'm using decimal or dozenal. In other words: a separate-identity set. The font support issues for a sep-id dozenal numeral set are magnified over the mere two numerals required for least-change dozenal, but fortunately I've made the one sep-id dozenal numeral set that can be posted without special font support. It takes getting used to parsing it, but I far prefer this hurdle to the chaos resulting from changes of marking when using a least-change dozenal numeral set.

Let me usher a new kind of experiment that to the best of my knowledge hasn't been tried on the boards before: the exclusive use of a separate-identity set for dozenal numerals. I hope I manage to cling to it, because I don't want to go back to all that maddening switching-around. Wavering between decimal and dozenal as defaults is bad enough, but having one's meaning unclear is intolerable. Here's to the first full separate-identity try!
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Double sharp
Posted: Sep 27 2017, 06:20 AM


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I have to wonder how much of the advocacy of bases like 60 or 120 here comes from the fact that they have decimal as the smaller subdigit, and thus retain a sense of apparent compatibility, as Treisaran mentions. It really does seem that the drilling in of decimal from a very young age makes the association of "13" = "thirteen", with all its properties (prime, etc.), pretty much indelible, and making the numerical appearance "13" mean "one dozen three" requires almost constant fighting against this deeply ingrained association, at least at first. It is deeply jarring at first to train oneself to "9+6=13" when all one's instincts are screaming that "15" should be written instead. It's not that we don't know that nine items plus six items makes one dozen three items; we know that. The problem is that we have been conditioned to well into calling that "fifteen" and writing it as "15", so much so that on the Wikipedia page for hexadecimal it seems that I always need to correct some helpful passers-by who think that the largest digit in hexadecimal must represent sixteen: evidently they think of "10" as being vaguely indivisible, despite it being two digits!

Hence the temptingness of calling 13{c} "thirteen", to surrender to the visual-verbal associations, and just relearn the language the same way you would relearn the numerals. Except that if you look more deeply at this natural urge, it stops making any sense, because language isn't a place value system, instead giving a separate name for some integer powers of ten. Kode and I have criticised it in more detail elsewhere very often, but in spite of that I understand why it is so tempting.

Much of this must be because the "teens" are almost as common as the single digits thanks to their appearance in the addition table, and get similar associations. Just as "6+3" inevitably brings up the answer "9", so "7+4" similarly brings up "11", because we are all native decimalists. This may even have a place in why vigesimal doesn't seem all that intractable, even though some additions (e.g. d+d) don't seem to have any good way to be done without splitting the base into double decimal, because the latter is just so obvious to a native decimalist. This may be why I find 20 tractable and 24 and 28 not, although to be fair the latter two have more sums like that than vigesimal does.

This is jarring when trying dozenal, but bases 60 and 120 sidestep this completely. Especially for sexagesimal, I often end up thinking of each place as a decimal two-digit number that just carries at 60, so I don't really notice the senary part at all. To me, 38 + 47 sexagesimally automatically carries to "85 = 1'25". And if you use complementary divisors, you really do avoid everything senary, and because the multiplication table row of 6 is well-known, even the decimal-to-sexagesimal conversions are quite straightforward for places like "17*19=323=5'23" because we know that 5*6=30 instantly.

So sexagesimal keeps on appearing as being really easy, which it is provided you already know decimal backwards and forwards, but the problem is that making it the first base in use completely removes this advantage. If we were somehow thrust into the position of unethical overlords training kids to use some other base from the start, forgetting that this is going to be immensely unhelpful for them to function in the real world, I have no doubt that a sensibly-sized pure base like 12 would show up as much easier than base 60 as 6-on-10. I'm not quite prepared to say that base 18 or 20 would be easier than sexagesimal in a vacuum, but because of the way the latter would have to be taught (with decimal first as a stepping stone), I strongly suspect that they would show up as easier.

To that end, perhaps a better way to test the true complexity of a mixed base is to try using base 84 as 6-on-14. It has the same prime shape as sexagesimal, and keeps 5 in the alpha, but substitutes the alien tetradecimal for decimal. That should nullify the advantage of apparent compatibility, while keeping the presence of 3 prime factors.
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Double sharp
Posted: Dec 29 2017, 02:11 PM


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QUOTE (Double sharp @ Sep 27 2017, 06:20 AM)
Much of this must be because the "teens" are almost as common as the single digits thanks to their appearance in the addition table, and get similar associations. Just as "6+3" inevitably brings up the answer "9", so "7+4" similarly brings up "11", because we are all native decimalists. This may even have a place in why vigesimal doesn't seem all that intractable, even though some additions (e.g. d+d) don't seem to have any good way to be done without splitting the base into double decimal, because the latter is just so obvious to a native decimalist. This may be why I find 20 tractable and 24 and 28 not, although to be fair the latter two have more sums like that than vigesimal does.

P.S. Given that I find it increasingly tough to visualise digits as indivisible units past 15 or 16 (for hexadecimal), and pretty much impossible past 21 or 22, I have to wonder if users of octodecimal or vigesimal would find this easier to wrap their head around. For us a number like "12" or "16" feels pretty basic, but to them it would likely not be so. Even for hexadecimal, it is already starting to feel like there is a difference to me, where the "high digits" start to feel pretty different from the "low digits", so much so that what in decimal is a digit-teen divide has become a low digit-high digit divide; it is admittedly weak in hexadecimal for me but really quite strong for octodecimal and vigesimal.

I'm really not sure where the divide is, or even if it really makes sense to talk about it as if it happens in a constant place for everybody, but it seems a reasonable conjecture that there is one, and that users of a large base might therefore have fewer problems using alternative bases with their arithmetic compatibility but lack of numeric compatibility. Admittedly the drawback is that that makes decay rather easy, because one important mental block to it (the ingrained meanings of numbers past the base) is removed - and that is assuming that the society has figured out place value in the first place. But if vigesimal can be propped up somehow via making use of decimal, it seems almost as though it could remain, though on life support, though I hesitate to say if tetravigesimal and octovigesimal might be able to use such a method.
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The Mighty Dozen
Posted: Dec 31 2017, 02:10 PM


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QUOTE (icarus @ Sep 23 2012, 07:01 PM)
I am an aging revolutionary, having internalized my own use of duodecimal early on. I do think fivefoldness is part of a continuum of "importance", roughly like the function 1 − φ(p). Seven is not quite so important to merit accommodation, and I don't think higher primes contribute anything significant to a system, diminishing returns.

I don't know if I am on the spectrum here. I use a variety of bases, but mostly {12, 60}. my usage of sexagesimal is increasing, but I have deep seated vestigial and highly integrated uses for dozenal. I think the prospect of returning to a sexagesimal civilization are dimmer than conversion to dozenal, which itself is not likely. Thus I think people should consider using bases as tools in their own practices. This way there might arise a niche-need for a number base (cf. hexadecimal in the coding of bits). I feel pretty strongly that duodecimal is optimum for human arithmetic using today's arithmetic algorithms. Sexagesimal is superior if one can wield it, not everyone can or wants to work with it. I have a feeling base 120 is about as good as sexagesimal. I think folks who use 60 vs. twelfty find themselves attempting to confine themselves to regular numbers. I have not done enough with 120 to say that it is any different but surmise, given the favorable relationships with {7, 11, 17} I'd expect some relief in rendering these somewhat common primes (if that is the word, like the summer on Pluto can be considered "warm").

When I use a number base I am a purist. I dislike compound digits because they blunt the natural pure divisibility of a base by a group of its divisors, which is the very reason we are going there in the first place. I suppose given the fact I am a native decimalist, threefoldness is easily recognized by rote of nearly 4 decades of exercise. This makes the situation with decimal coded sexagesimal tolerable. When I try to use base 2520 as 42-on-60, I have an extreme difficulty in recognizing sevenfoldness, resorting to attempting to memorize a table of exempt sexagesimal totatives (I.e., those that are multiples of seven) that number something like 200. Hence my development of argam.

Because I am a purist I do not truck decade and unit figures in sixty and twelfty as well as others and it probably shows. I'd write single-piece and think in similar ways especially in addition-subtraction. But if I think of base sixty as base-root-sixty, a bit less than 8, then the alternating system makes sense. Go figure. Probably a nerd thing lol.

Indeed, icarus. If you've heard of IOTA, the cryptocurrency, their programming is based on ternary.
~*~

If anyone wants to publish stuff, I'm sure Shaun or myself could arrange to put decent articles on the DSGB website(s).
~*~

And please, guys, no insulting each other or swearing. It violates the rules of the board and is just generally unacceptable behaviour.
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Ruthe
Posted: Feb 6 2018, 12:44 AM


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QUOTE (icarus @ Sep 23 2012, 07:01 PM)
I don't know if I am on the spectrum here. I use a variety of bases, but mostly {12, 60}. my usage of sexagesimal is increasing, but I have deep seated vestigial and highly integrated uses for dozenal. I think the prospect of returning to a sexagesimal civilization are dimmer than conversion to dozenal, which itself is not likely.

I also think the chances of changing the world to an uncial system are very slim, but that the only real stimulus would come from a distinct financial advantage of uncial usage.

Having said that, so far the only thing I have come across as a possible drive is that to be gained by sales. As we all know, prices of goods espaceially in the higher cost goods are traditional stated as xx.99, xxx.99, $xx.99 or $xxx.99 etc.

But if we switched to an uncial system, the prices would be stated with the glyph 9 replaced by the glyph for uncial eleven. Would that increase across all their goods be a strong enough incentive for companies to switch?
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