Another Senary Advocate

This forum is dedicated to discussions of bases {6, 18, 24, 36, 48}, cousins to twelve.
Kodegadulo
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Kodegadulo
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Apr 18 2018, 11:30 PM #25

SenaryThe12th wrote: *chuckle*  I confess, one of my favorite aspects of dozenalism is that we need two new cool symbols!!   Which opens up all kinds of neat things to think about:  which two symbols should we pick?   Or should we just chuck them all and come up with 12 new ones!!  Endless opportunities for fun.
What I found annoying about "ian misali's" video is the implied criticisms and subtle lampooning of dozenal and its usage by all the past generations of dozenalists. Lots of strawman and reductio ad absurdum arguments, nested in satirical snark. I hear echoes of that from you here. I'd prefer it if you and he would lay off the indirectness and just lay out the critique forthrightly. Then see what parts those of us present  here, and now actually defend or disavow. I personally don't think every past idea of the DSA necessarily has merit, but somehow the critics seem to think we now have to answer for all of them.

For instance, dozenalists in the past have advocated separate identity numerals. Some dozenalists. I am a dozenalist. I'm vehemently in the least change camp. I think most are today. Should I be painted with the separate identity brush?  Guilt by association? In fact it's my opinion that the whole question has pretty much been settled long ago. So why should any of us have to answer to this so-called "controversy", simply because some wags can find historical writings about it from the WWII era, on the DSA website?

Besides, the question is by no means unique to dozenal. I could just as easily snark about how any twinkling now some senarist will be lobbying us all to use the dot-shapes from dice or dominos to express base 6 numerals, just to assert the separate identity of senary.

As for what to use for transdecimals, well there have been various answers to that, all juggling the relevant trade offs of practicality and availability, and the tastes of different people on either side of the Atlantic. The DSA has modestly tried not to dictate what people should use. But it seems we aren't to be taken seriously unless there's just one universal solution established by unanimous fiat. Well we've come pretty far in getting Unicode to recognize the Pitmans, which is remarkable given how little clout any of us actually have. The Pitmans WILL be supported in all fonts, eventually.  What more do you guys want from us?

And we get similar lampooning about the various schemes people have come up with for dozenal pronunciation. Yeah, some of us advocate using what dozenal words English already has. But of course it needs more. Is it our fault it doesn't? Is it our fault that the only way to enhance it is by conlanging? How is that different from the fact that there are _no_ English words for any senary power at all, above six itself?  There's no choice but to conlang that too, and indeed that's exactly what "ian misali" engages in. Not very artfully or originally, imho. He's certainly swiped a lot of ideas from this forum, without acknowledging them.

Now, don't get me wrong. I think senary is a fun thought experiment. Indeed I've tried to help others flesh out their own thought experiments with senary, and even dreamed up Xohox numbers on a whim. I just think dozenal is a better fit for non-fictional people.
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
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SenaryThe12th
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Apr 19 2018, 01:03 AM #26

Kodegadulo wrote: I hear echoes of that from you here. I'd prefer it if you and he would lay off the indirectness and just lay out the critique forthrightly.

I'm afraid you are reading far too much into my post Kodo.   I've lived in NYC long enough to have the residual passive-aggressiveness beaten out of me.  If I don't like something, I'll say it.  You can take my words at face value.

When I say that thinking about extra symbols is one of my favorite things about dozenal, I meant it.  Come on, who doesn't like playing and thinking about cool new symbols?   My lighthearted tone isn't snark; its an attempt to enjoy a shared delight with you. 

Far from criticizing dozenal, I was actually *AGREEING* with you.  Dozenal numbers are less boring, more interesting, etc to look at.  They even have cool new symbols.
I'd prefer it if you and he would lay off the indirectness and just lay out the critique forthrightly.
Well that's just the thing; I haven't critiqued anything, *except* senary.  My request was explicitly for a list of critiques of senary. 

At no time did I try to dump poop on any other base.   Really, what should have tipped you off to that is your trying to lump me together with some other guy on the internet.  "You know those snarky-ass senarists, they are all the same."   Some forthright criticism: nobody likes to be stereotyped.  Read what I actually wrote; if you have a beef with somebody else, take it up with them, don't bring that baggage into conversations with me.  Please.

I'm a base pluralist.  Just like anybody else on here, I've got my favorite bases.  But professionally, I use 4 bases on a daily basis, and I use another one every time I look at a clock.  I take delight in every cool thing I see somebody doing in base 12 or any other base.  
Lots of strawman and reductio ad absurdum arguments
Well, if anybody has call to complain about being strawmaned, I think that would be me.  And I didn't do any reductio's, ad absurdum or ad anywhere else :-)  BTW, what *is* your beef with reductio ad absurdum?  Its a valid form of argument. 

P.S.  you ok bro?  Sometimes I wonder whether you come down a bit too hard because you are in a bad place.  Too many internet trolls got you down?
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Kodegadulo
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Apr 19 2018, 01:38 AM #27

Okay fair enough, Sen. Yeah, my beef is with "ian masali". Sorry to read unintended connotations between your lines. I guess I have felt a bit embattled over the years here. It's gotten to the point where I'm half-expecting it.

Re: reductio ad absurdum. I think if it's a matter of taking the opponent's points and showing how they logically lead to absurd conclusions, then it's a legitimate argument.  But if it amounts to painting an absurd caricature of one's opponent and then demonstrating that the caricature is absurd, well that's dirty pool.  But then that's just a straw man fallacy, isn't it?  Okay, my bad on the terminology. Anyway, I didn't intend to strawman you.  Like I said, I'm generally supportive of thought experiments like senary.

I guess I'm tired of the whole idea of everyone having to assert that their favorite base is "superior" and having to conquer the world for  its sake.  I've been kind of preaching coexistence for a while now.
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
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SenaryThe12th
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Apr 19 2018, 02:13 AM #28

Kodegadulo wrote:   Like I said, I'm generally supportive of thought experiments like senary.
Indeed, you helped me out with nomenclature in this very thread.  And I religiously read all your stuff on the Hexican thread.  You're a good egg man.  Can't wait to see what you and the rest of the folks on this forum come up with next.
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Kodegadulo
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Apr 19 2018, 10:32 AM #29

Funny, some months back I gave another member here what I intended to be a compliment. On face value, I was just saying something nice. But he (and apparently his brother) interpreted it as sarcasm (possibly because of the tenor of the exchange with yet another member going on at the same time). I was completely taken aback, and thought, for goodness sake, you can take anything nice and completely turn it around if you repeat the exact same words, but emote them with dripping scorn. I had to reassure him that I meant what I said plainly.  And now here ironically I've made the same misinterpretation of something Sen meant merely as a compliment. My bad.

Is there an emoticon that means "plain-speaking"?😉
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
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SenaryThe12th
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Apr 19 2018, 01:19 PM #30

Is there an emoticon that means "plain-speaking"?
THAT is what the world needs now!!!
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Double sharp
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Apr 19 2018, 02:18 PM #31

SenaryThe12th wrote:Well of course, if 5's are particularly important to you, a base which has a factor of 5 will beat anything.   Kind of like how my tribe (which finds powers of 2 important) uses binary, octal, and hexidecimal.  

But really, this isn't really an objection against senary per se, as a generally used base, is it?     I mean, senary *is* better at 5's than dozenal.   And your original point, that 2 is waaaaay more important than 3 is, and 3 is waaaay more important than 5 is--that was a good point, IMHO.
I didn't mean that 5 ought to be a top priority; it's just the treatment of 1/5 in senary is not that much better than its treatment in dozenal. It still doesn't terminate. Yes, it certainly looks nicer, and you get a nicer divisibility test for it, but you're only getting half of the possible improvement - and not the half that lets you use fifths in contexts with a limited number of significant figures to play with, such as measurement.

When I said "not the top priority", I didn't mean that fifths ought to be given the top priority, but the opposite; I meant that improving fifths isn't the top priority, and I'm sorry for not being clearer. While 5 is quite important in its niches, it tends to be not important at all elsewhere. Senary then sits at an awkward compromise which fits neither situation very well: in the first type of situation, the fifths are not good enough to actually use, and in the second type, the switch from dozenal to senary has allocated valuable resources that could have been spent on greater concision and better powers of two onto the much less important 5.
Double sharp wrote:Compounded on that, there is still no good way to use an auxiliary base containing 5 in senary, because the fifths always get absorbed into the mantissa instead of the trailing zeroes and end up outshining the more important fractions. 
SenaryThe12th wrote:I don't understand this; can you unpack it a bit for me?
We use numbers like 60 or 360 in decimal as common groupings instead of 10, 100, and 1000: on the forum, we've been calling such things auxiliary bases. The mantissa of a number is the significant part, before the trailing zeroes start.

The point of using a number like 60 is that it is highly divisible, adding the missing factor 3 to decimal: in fact, 60 and 360 are both highly composite (they have more divisors than any smaller number). More importantly, the mantissa of 60 is 6, itself highly composite, and crucially, the lower factors are in the mantissa. This means that to divide it by 2 or 3, you don't need to add significant digits:

1/2 = 30'
1/3 = 20'
1/4 = 15' (which still feels pretty round as a half-division)
1/6 = 10'

and fifths, while still catered for, are given the least round position, because 5 doesn't divide the mantissa:

1/5 = 12'

Last of all, the auxiliary 60 is a round number in decimal. An auxiliary like 1152 in decimal isn't terribly useful because even though it's highly divisible, it's not very compatible with the base. Using such an auxiliary, as in 60 seconds making a minute and 60 minutes making an hour, we can stay in the decimal framework and still get the benefits of high divisibility and giving 3 at least preferential treatment over 5 and 2 at least preferential treatment over 3. (In fact, this is a necessity for being a highly composite number. If you have a number 2^p * 3^q * 5^r, and q < r, then 2^p * 3^r * 5^q is smaller and yet has the same number of factors. The same conditions hold for the other primes.)

The problem with senary and dozenal with auxiliary bases is that they already have the first three primes catered for in the right order, and you have to emphasise the largest and hence least important one for the auxiliary. Suppose you want a senary auxiliary that is divisible by 5. You can't use 506 (decimal 30), because then the mantissa is 5, and fifths are emphasised over everything else. So you double it to 1406 (decimal 60), which is better; but then the mantissa is ten, which is not very divisible, and means that 3 is valued below 5. So you triple it - except that, since 6 = 2 * 3, doubling and then tripling it (as you would need to get the factors in the right order of prominence) just adds another zero. You end up with 5006 (decimal 180), which has a mantissa of 5, and then the cycle just repeats itself. You simply can't have a senary mantissa that is divisible by 2, 3, and 5, because then it'd be divisible by 6 and hence wouldn't be the mantissa. In fact, dozenal has a slightly better situation because you can at least give up completely clean quarters: 26012 (decimal 360) is about the best you can do for a dozenal auxiliary base, using 2*3*5 as the mantissa.

Another way of looking at it is that auxiliary bases have to give up a factor of the base in order to get one that isn't there already. With decimal, this is a no-brainer; you give up 5 to buy a better 3 and maybe 4. With dozenal, this is harder; the first thing you can buy is 5, and the only thing you can give up for it to keep 2 better than or equal to 3 better than or equal to 5 is 4. With senary, there's only 2 and 3 to give up, and neither can be given up for 5 if you want to keep {2, 3, 5} with the right priority.
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Kodegadulo
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Apr 22 2018, 01:03 AM #32

jan Misali makes the point that each of our hands can naturally represent a single senary digit, and using them together we can actually count to 556 = 35d = 2Ɛz. This is essentially the same point I demonstrated in the Xohox Numbers thread, with this image of how my fictional "Xanadunni" supposedly finger-count:



But I should point out that it's perfectly possible for one human hand to represent a single dozenal digit as well, if one allows a difference of orientation to indicate "add 6":


In other words, with the hand held upright so its ventral surface is facing toward the viewer, a fist means digit zero, and one to five fingers exposed means the digits one to five. But turning the hand across one's chest so its dorsal surface is towards the viewer, a fist now means a digit six, and one to five fingers exposed means the digits seven through eleven.  You could think of zero through five as the "front" half of the digits, and six through eleven as the "back" half.  Doing this with both hands we could therefore count to ƐƐz.
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
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Double sharp
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Apr 22 2018, 05:25 AM #33

I must confess that I'm a bit uncomfortable with finger-counting schemes that aren't just unary, because it seems almost inevitable that different gestures will be used with different meanings in different cultures - or possibly in one culture, judging from the Chinese number gestures, in which extending the thumb and the index finger together may be interpreted as "7" or "8" depending on which region of China you are in.

On the other hand (sorry), this multitude of possible conventions easily suggests to us that it would be quite easy to make reasonable schemes for any reasonable base, as you have just done for dozenal.
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Kodegadulo
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Apr 22 2018, 10:31 AM #34

Double sharp wrote:On the other hand (sorry), this multitude of possible conventions easily suggests to us that it would be quite easy to make reasonable schemes for any reasonable base, as you have just done for dozenal.
Indeed, one need look no further than the fact that we can co-opt the standard Latin 1 alphabet for transdecimal digits supporting up to base 62. American Sign Language has long mastered the use of one hand to express all the decimal digits plus all the letters of the alphabet (and more):


Although I do note the ambiguity between 0 and O. And the subtlety of the differences between 2 and V, and 6 and W.
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
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SenaryThe12th
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Apr 22 2018, 04:59 PM #35

Double sharp wrote:  Senary then sits at an awkward compromise which fits neither situation very well: the switch from dozenal to senary has allocated valuable resources that could have been spent on greater concision and better powers of two onto the much less important 5.
 I agree with this:  if powers of 2 are important to you, switch to binary, octal, or hexidecimal.  If powers of 5 are important to you, switch to decimal.  Use the right tool for the right job.  Switching from dozenal to senary would absolutely make division by 4 and higher powers of two more prolix, even though it would make powers of 5 marginally easier.
the fifths are not good enough to actually use,
This I still don't quite understand.  5ths are unusable in senary?   No disrespect to dozenal, but 5ths are even worse in that base, but I suspect that dozenalists wouldn't have any more trouble with 5ths than the whole metric-using world has with 3rds.  If you want 1/3rd of a metre, there's not even a line on the meter stick for that.  Yet they use thirds all the time.  Honestly, the success of the metric system is proof to me that there's no problem that any base has w.r.t. measurement which can't be overcome *chuckle*.
Another way of looking at it is that auxiliary bases have to give up a factor of the base in order to get one that isn't there already. With decimal, this is a no-brainer; you give up 5 to buy a better 3 and maybe 4. With dozenal, this is harder; the first thing you can buy is 5, and the only thing you can give up for it to keep 2 better than or equal to 3 better than or equal to 5 is 4. With senary, there's only 2 and 3 to give up, and neither can be given up for 5 if you want to keep {2, 3, 5} with the right priority.
Is there *any* human scale base which both makes 2,3, and 5 easy to work with, and still keeps them in the right priority?  I guess base 2*2*2*3*3*5 would, but that's hardly a human-scale base.

Wouldn't the fact that the sumerians used a 6-on-10 base be an existence proof that senary works just fine with auxilliary bases?  I mean, that actually was a civilizational base.   One of the few non-decimal civilizational bases, for that matter.

P.S.  sorry it took so long to get back to you; I've been feeling under the weather.
Last edited by SenaryThe12th on Apr 23 2018, 12:21 AM, edited 1 time in total.
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SenaryThe12th
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Apr 22 2018, 05:11 PM #36

Kodegadulo wrote: jan Misali makes the point that each of our hands can naturally represent a single senary digit, and using them together we can actually count to 556 = 35d = 2Ɛz. This is essentially the same point I demonstrated in the Xohox Numbers thread, with this image of how my fictional "Xanadunni" supposedly finger-count:
delightful!  Kodo, how did you make the drawings of the hands?
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Kodegadulo
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Apr 22 2018, 08:02 PM #37

SenaryThe12th wrote:delightful!  Kodo, how did you make the drawings of the hands?
Theft. Ahem, strategic borrowing. 😉 I found the ventral drawings on some site about American Sign Language somewhere, and then generated the dorsal views by copying, rotating, and erasing interior lines.
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
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Double sharp
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Apr 23 2018, 02:32 PM #38

SenaryThe12th wrote:This I still don't quite understand.  5ths are unusable in senary?   No disrespect to dozenal, but 5ths are even worse in that base, but I suspect that dozenalists wouldn't have any more trouble with 5ths than the whole metric-using world has with 3rds.  If you want 1/3rd of a metre, there's not even a line on the meter stick for that.  Yet they use thirds all the time.  Honestly, the success of the metric system is proof to me that there's no problem that any base has w.r.t. measurement which can't be overcome *chuckle*.
They're "unusable" in the sense that you can't use them directly. If you want to use exact 5ths in senary, then you have to start with something that is already divisible by 5, so that you can divide it into 5ths and still get an integer - which is the same situation if you want exact 3rds in decimal, or exact 5ths in dozenal. So it doesn't matter that "0.111..." is a nicer recurrence than "0.24972497..."; neither terminates, and so in measurement you have to use an auxiliary if you want to deal with 1/5 in a senary or a dozenal context, just like how we deal with 1/3 in a decimal context.

I agree that this can be overcome indeed: since we can deal with a non-terminating 1/3, we should be able to deal with a non-terminating 1/5, which would come up less often. Nevertheless, it does mean that the slightly improved handling of fifths in senary over dozenal is not really a direct advantage, because both need to be worked around in pretty much the same way. It only really comes into handy when testing for divisibility by 5.
SenaryThe12th wrote:Is there *any* human scale base which both makes 2,3, and 5 easy to work with, and still keeps them in the right priority?  I guess base 2*2*2*3*3*5 would, but that's hardly a human-scale base.

Wouldn't the fact that the sumerians used a 6-on-10 base be an existence proof that senary works just fine with auxilliary bases?  I mean, that actually was a civilizational base.   One of the few non-decimal civilizational bases, for that matter.
You can't possibly have 2, 3, and 5 all as factors, because then your base is at least 2*3*5=30, and that's certainly not human-scale; so you have to give up one of them, and it may as well be 5 as the least important among them. This is similar to Jean Essig's argument for dozenal: you'd like to have all of {1, 2, 3, 4, 5, 6} as factors, but that way lies pure sexagesimal, which is too big, so you have to jettison at least one of those factors. If you want to do so, but retreat as little as possible from sexagesimal, then you ought to be giving up the least important factor among these, 5 (since having 2 and 3 guarantees 6 already), and that leavs us with lcm(1,2,3,4,6) = 12. Choosing decimal instead would be giving up {3,4,6} for {5} instead of the other way round; although the way decimal gives up {3,4,6} is significantly better than the way dozenal gives up 5, you're still giving up three factors for one. Choosing senary is essentially giving up 4 as well as 5 when you don't actually need to; as stated above, the shift from dozenal to senary means that 5ths, while better, still aren't fully usable without provisions being made for them, and this doesn't feel to me like sufficient justification for worsening quarters and concision.

6-on-10 is not really a senary auxiliary base but a decimal one. You can see this from the fact that it looks like decimal if you start counting with it: it goes 1, 2, 3, 4, 5, 6, 7, 8, 9, and then overflows to 10. This easily harmonises with decimal as you don't have to relearn how to count. If you wanted to use sexagesimal as a senary auxiliary, it would have to be 10-on-6 instead, and that is not as good as 6-on-10 because the top subdigit is decimal and hence 1/5 is represented more nicely ("20") than 1/3 ("32"). (It is also not as good as 6-on-10 because the top subdigit uses a larger subbase than the bottom, so senary users seeking to use 10-on-6 have to agree on and learn to use extra figures beyond the standard range.)
SenaryThe12th wrote:P.S.  sorry it took so long to get back to you; I've been feeling under the weather.
Sorry to hear that, and I hope you're feeling better now!
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Apr 23 2018, 07:17 PM #39

They're "unusable" in the sense that you can't use them directly.
Well, its true that you can't use them *directly*, but:

So it doesn't matter that "0.111..." is a nicer recurrence than "0.24972497..."; 
actually, that makes all the difference.   Just *because* 1/5 =  0.1 repeating, dividing by 5 is is as close as you can get to being directly usable.    Take a look at how easy this is in senary:   To find 1/5th of a number, all you have to do is multiply it by 0.1 repeating.  

e.g. to take 1/5th of 12 (as I'm Senary the 12th I am..)  you just multiply 12 * 0.1 repeating.  This is actually just a very simple addition:

Code: Select all

1.2
+ 1.2
+   1.2
+     1.......
----------
1.3 3 3 ......
There will only ever be 1 digit in the repeating part, so as soon as you see which one that is, you are done.  
Note: since we've taken into account the repeating digit, this is an *exact* answer.  

I put it to you that:
  • This isn't *that* much worse than dividing by 5 in decimal, and
  • Its a whole heck of a lot nicer than dividing by 5 in dozenal.
So, by your own criteria, if you want a base which is easy to divide by 2,3,5, in that order, and still get an exact answer, may I humbly suggest senary for your kind consideration.
6-on-10 is not really a senary auxiliary base but a decimal one. You can see this from the fact that it looks like decimal if you start counting with it: it goes 1, 2, 3, 4, 5, 6, 7, 8, 9, and then overflows to 10. This easily harmonises with decimal as you don't have to relearn how to count. If you wanted to use sexagesimal as a senary auxiliary, it would have to be 10-on-6 instead, and that is not as good as 6-on-10 because the top subdigit is decimal and hence 1/5 is represented more nicely ("20") than 1/3 ("32")
I guess I see where you are coming from: would it be correct to say that (in twistaff notation) the upper  staff is the auxiliary base and the lower staff is the ....uh, non-auxilliary base?

.
(It is also not as good as 6-on-10 because the top subdigit uses a larger subbase than the bottom, so senary users seeking to use 10-on-6 have to agree on and learn to use extra figures beyond the standard range.)
How about a 5-on-6 twistaff notation then?  But perhaps you are right.  I mean, given how easy 5's really are in senary, using an auxilliary base to make it even easier does seem like gilding the lily.
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pcyrus
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Apr 24 2018, 07:20 AM #40

Just a thought on the senary vs. dozenal debate: a reverse (balanced) dozenal system (such as Shwa, discussed elsewhere) has some characteristics of both.  
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SenaryThe12th
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Joined: Mar 1 2018, 02:03 PM

Apr 24 2018, 09:02 AM #41

pcyrus wrote: Just a thought on the senary vs. dozenal debate: a reverse (balanced) dozenal system (such as Shwa, discussed elsewhere) has some characteristics of both.  
We should see if we can persuade Icarus to expand his tour-des-bases to the ballenced bases as well :-)
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Double sharp
Dozens Demigod
Double sharp
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Joined: Sep 19 2015, 11:02 AM

Apr 24 2018, 11:44 AM #42

{c} default dozenal

Actually, dividing by 5 in dozenal is also pretty easy, precisely because the period length is the maximum it could be. You just need to remember 1/5 = 0.2497..., and then all the other fifths are just a frameshift: 2/5 = 0.4972..., 3/5 = 0.7249..., 4/5 = 0.9724.... Again, there is no need to do any real multiplication.

As a result, it is quite easy to work out other fractions containing 5 (not squared) in the denominator. Take 7/a for example. Well, just reduce that to 3.6/5 and do mental short division. You start with 0.8, because 5*8=34. Then the next digit gets us to 0.2/5 and we can instantly get 0.849724972... immediately if we remember the fifths, or if we don't we just take 5*4=18 and now we know we need the version starting with 4. This isn't really different from dividing by 5 in any other base where you can remember the tables; just run through them to do short division, and when you're done you immediately know what the repeating part must be.

What's more difficult is a case like hexadecimal 7 or 9 where the period length is not maximal: then you have neither the divisibility tests nor the frameshifts substituting for multiplication. In any case, though, this is only really useful for situations where a number with infinitely many significant digits is meaningful, and not all situations are like that.
Last edited by Double sharp on Apr 24 2018, 03:38 PM, edited 2 times in total.
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SenaryThe12th
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SenaryThe12th
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Joined: Mar 1 2018, 02:03 PM

Apr 24 2018, 02:41 PM #43

Double sharp wrote: Actually, dividing by 5 in dozenal is also pretty easy, precisely because the period length is the maximum it could be. You just need to remember 1/5 = 0.2497..., and then all the other fifths are just a frameshift: 2/5 = 0.4972..., 3/5 = 0.7249..., 4/5 = 0.9724.... Again, there is no need to do any real multiplication.
Wow!  That is a very cool trick.  I totally didn't know about that.  Thanks!  
{c} default dozenal
Follow your bliss, man.   I mean, the name of this board isn't "senaryonline"; It would be pretty reasonable to expect that the default position here would be dozenal :)   I'm just glad ya'll tolerate a few of us wingnuts.
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Double sharp
Dozens Demigod
Double sharp
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Joined: Sep 19 2015, 11:02 AM

Apr 24 2018, 03:42 PM #44

No problem; I originally liked senary better myself, because of the better arithmetic relationships. Actually trying it out was what convinced me that this probably isn't as important as it looks, and that the smaller size is probably more important than it looks; nevertheless, this may vary between individuals, and I'm convinced that senary is at least worth consideration even if I think dozenal is better.

I'd prefer to say that in 6-on-10 sexagesimal, senary is simply the upper subbase and decimal is simply the lower subbase. Only if you are using this in specific niches in a decimal context, instead of as the default base for everything, is it an auxiliary base. (I originally added this as an edit to the previous post, but since you've already replied to it, I've deleted it there and written it here.)

I think 5-on-6 trigesimal should be worse than 6-on-10 or even 10-on-6 sexagesimal, because now one of the subbases is odd and 1/5 ends up being emphasised beyond 1/3 or even 1/2. Furthermore, it is even worse at handling binary fractions than senary itself (you need two subdigits just to represent 1/2), and it is even smaller (you only have 30 possible numbers before hitting the third subdigit, instead of 36 before hitting the third digit in senary).
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Kodegadulo
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Kodegadulo
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Joined: Sep 10 2011, 11:27 PM

Apr 24 2018, 04:32 PM #45

Sen, there's no presumed default of dozenal here. Nothing like the presumed default of decimal out in the mainstream. Many of us in this forum subscribe to the ethic that we should not act like "base-chauvinists", even if we do have a personal preference. One consequence of that ethic is that, because we use and discuss a lot of bases, we think every number quoted here needs to be qualified with its base. One way or the other.

I tend to take that to the logical extreme, putting a base subscript on every number. That's just like the way mainstream mathematicians would do it, except for them the subscript is itself a number, presumed to be base ten. Well that in itself  is base chauvinism, so we've developed conventions here where the subscripts aren't numbers as such, but rather single symbols indicating the base. So we might use a subscript 6 for senary or a subscript 8 for octal, but we choose subscript lowercase d, z, and x for bases ten, dozen, and fourzeen (sixteen), respectively. Or a different convention where subscript uppercase A, C, G represent those. DS has yet another convention where he starts with lowercase letters for transdecimals and reserves uppercase, and even some foreign alphabets, for even higher digits; but you know when we're using his convention because we wrap it in squiggly braces: e.g. subscript {a}=decimal, {c}=dozenal, {g}=hexadecimal.

One compromise we often make on this extreme position is to declare a "local temporary default". When you see something like a [z] or a [C] or a {c} on a line by itself, it means "until further notice, but not past the end of this post, assume any unmarked numbers I write are all dozenal." Likewise, [d] or [A] or {a} signals a local temporary default to decimal. This can be done for any base, e.g. [6] or {6} for senary. It makes things easier when you're trying to write tables or other masses of numbers all in the same base.

Other folks use other conventions (some dating back dozens of years) to specially mark dozenal numbers, with or without any special marking on decimal numbers. Some members here use their signature to declare their own personal blanket default; in effect, they're saying "You can presume my numbers are always dozenal unless otherwise marked." That's everyone's prerogative here, but I think that borders on base chauvinism. It's a nice courtesy to insert some explicit prompts close to your numbers so readers can know how to interpret them, without having to be puzzled until they finally get down to your signature, and without having to memorize your personal idiosyncrasies (and everyone else's).
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
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