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 Is The Decimal 12x Table Still Relevant?, Let's double down on decimal!
Oschkar
Posted: Oct 13 2017, 05:59 AM


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QUOTE (Double sharp @ Oct 12 2017, 11:11 AM)
To be fair, we don't really use the multiples of eleven that much in decimal, and in tetradecimal the use of the multiples of zeff-one is more to facilitate the lines of three and five, because 3 ◊ 5 = 11. The individual "off-grid" numbers like decimal 77 are quite unimportant, and I think tetradecimal 99 is about the same thing: it is an awkward mess of threes and fives, and the only thing going for it is that it is obvious that it is an awkward mess of threes and fives.

Iíd rather equate these numbers to something like decimal 189 and 147. Theyíre fairly obviously made up of small primes, but they donít seem round in any useful way.
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Oschkar
Posted: Oct 13 2017, 07:45 AM


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QUOTE (Double sharp @ Oct 12 2017, 11:11 AM)
Which makes me wonder: how much benefit would senary and octal gain from memorising the tables to the dozen?

Let me see.

{6}

Here are the senary products with factors up to six.

1 = 1◊1
2 = 1◊2
3 = 1◊3
4 = 1◊4 = 2◊2
5 = 1◊5
10 = 1◊10 = 2◊3
12 = 2◊4
13 = 3◊3
14 = 2◊5
20 = 2◊10 = 3◊4
23 = 3◊5
24 = 4◊4
30 = 3◊10
32 = 4◊5
40 = 4◊10
41 = 5◊5
50 = 5◊10
100 = 10◊10

For one, this list seems severely underwhelming, especially when coming from tetradecimal. It just seems to stop so early, and miss many useful numbers. However, I donít think the list needs any explanatory prose, that is, it seems crucial to be able to recognize any product on the senary multiplication table.

The only 5-smooth numbers under 100 that are missing from this list are 43 and 52. The first is a sort of visual pun: it represents both three-quarters of 100, and a cube with side length 3. The second is the fifth power of two, small enough that itís not even necessary to have computers for it to be recognized; itís just twice four times four.

Iíll add four more rows now, stopping just before the first off-kilter prime.

11 = 1◊11
22 = 2◊11
33 = 3◊11
43 = 3◊13
44 = 4◊11
52 = 4◊12
55 = 5◊11

These are just the multiples of seven, with their repeated digits, and the two prime powers I already talked about.

104 = 5◊12 = 4◊14. This is the decimal 40, whose convenience is only slightly shadowed by the base.
110 = 10◊11. Obvious, but it still needs to be mentioned.
113 = 5◊13. One and a quarter exents. Surprisingly divisible for an odd number.
120 = 10◊12. A very convenient 3-smooth number leading a run of three consecutive products.
121 = 11◊11. The middle of the triad, a slightly awkward square of a prime.
122 = 5◊14. The one on top, another weird number that is twice the square of a prime.
130 = 10◊13. One and a half exents. 3-smooth, but with too many threes.
132 = 11◊12. And weíre back to a number with a lot of binary powers.
140 = 10◊14. A superior highly composite. Absolutely important.
143 = 11◊13. One and three quarters. Another divisible odd number.
144 = 12◊12. The eighth power of two. Another really important number.
154 = 11◊14. This is the product of three small primes, but it doesnít seem too useful.
200 = 12◊13. Twexent is pretty much a given. Two threes, three twos.
212 = 12◊14. Another number with a lot of binary powers.
213 = 13◊13. The fourth power of three, recognizable because itís a surrogate for a twex-fourth.
230 = 13◊14. Another convenient number with three prime factors.
244 = 14◊14. Beside the fact that itís the sum of the easy squares 100 and 144, I donít really think is one is too interesting.

Now Iíll add the other two rows.

We start off with a lot of opaque multiples of eleven.

15 = 1◊15
34 = 2◊15
53 = 3◊15
112 = 4◊15
131 = 5◊15
150 = 10◊15
205 = 11◊15

220 = 11◊20. Largely composite, and about as round as the 140 we already met.
224 = 12◊15. Another opaque multiple of eleven. I wonít call out the rest of them.
240 = 12◊20. A larger, but still very convenient 3-smooth number.
243 = 13◊15
300 = 13◊20. Threxent. Extremely convenient, and impossible to miss.
302 = 14◊15
320 = 14◊20. This is the long hundred, another SHCN.
321 = 15◊15. At least this one is a square. The descending pattern in the digits make it stick
340 = 15◊20.
400 = 20◊20. And at we come to another 3-smooth square: forexent, or a gross.

In general, I think itís more useful to memorize the senary products up to ten, ironically. The multiples of 20 are very useful, but theyíre just the even numbers with a zero at the end. The multiples of 15, however, are pretty much useless, and actually memorizing them seems more appropriate for the advanced mental calculator who wants to factor numbers quickly rather than for a primary school student. Surely, people will recognize the smaller multiples {15, 34, 53, 112}, just as we do in decimal for the lower multiples of 13{a} and 17{a}, but they arenít necessary by any means.
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Double sharp
Posted: Oct 13 2017, 04:08 PM


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I started writing a reply to this, but more and more the topic drifted to small bases, so I'm going to move this to the lower natural scale thread instead. (EDIT: Okay, done.)

Now eagerly awaiting your exposť of octal!
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Double sharp
Posted: Nov 3 2017, 03:46 PM


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On reflection, it seems to me that the same thing largely applies in octal: {9, 10} are fairly useful to know, and so is {12}, but the great divisibility of the dozen means that you already know most of the products. The dozen already is doubly even, so you just need to multiply it by an even number and you get a trailing zero immediately.

{8}

The odd multiples of the dozen are split into two groups: the three that are already known:

14 = 1 ◊ 14 = 2 ◊ 6 = 3 ◊ 4. The dozen itself.
44 = 3 ◊ 14 = 4 ◊ 11 = 6 ◊ 6. Six squared, and a clear alpha product.
74 = 5 ◊ 14 = 6 ◊ 12. Decimal 60, a superior highly composite, of absolute importance.

and the three that are not:

124 = 7 ◊ 14. Largely composite and fairly round.
154 = 11 ◊ 14. Also largely composite, but a bit too heavy on the threes.
204 = 13 ◊ 14. Eleven dozen, but this really does seem fairly useless.

I would think that octalists would not really explicitly memorise most of the dozens, although they might pick up {124, 154} through exposure, in about the same way they would know squares and cubes. This is one way in which ten seems to pattern as a "small human-scale base", although it usually does not. It seems borderline: as 2*5 it is getting to the point where half the base is a usable stretch in itself that is getting out of the subitising zone, but it also stops just before the base starts incorporating more and more weird and not very useful primes. The next threshold seems to be crossed around vigesimal, after which there seems to be practically no way to imagine the stream of consecutives {25, 26, 27}.

Decimal seems to get away with leaping through eleven (a silly prime) to get to the dozen because eleven is the alpha, and so incurs zero mnemonic cost. But in senary and octal, it's not so nice.

{e}

Even in tetradecimal, when 12 beckons, its extra products aren't quite as useful as the dozens, because 12 is rather unbalanced onto twos at the expense of anything else (I am rather understating this). The tetradecimal products of sixteen that I can see people remembering are 92 and 144, but surely people would remember these instead as part of the "powers of two" sequence? The other multiples of sixteen are either in the table already or don't seem worth the effort.

{a}

One thing I find is that when I am immersed in the "small human-scale bases" like {(6), 8}, it requires a bit of adjustment to get back to working in the "large human-scale bases" {16, 18, (20)}, and vice versa. The main trio of {10, 12, 14} seem to stand firm and unambiguous in the middle, though, and you've said much the same thing before at the higher natural-scale thread:

QUOTE (Oschkar @ Jan 14 2017, 08:32 AM)
QUOTE (Double sharp @ Jan 14 2017, 04:57 AM)
P.S. After having tried hexadecimal, I am beginning to find octal small with its lack of opaque digits aided by SPD, so that {6, 12} and {8, 16} as comparisons seem to favour {12, 16} instead. {10, 20} seems more balanced and probably 10 wins that one. It's interesting to note that octal has been overtaken by hexadecimal, I think partially because the range of numbers wanted in a byte needs an extra digit in octal. I think we prefer to expand the range instead of limit it. I would continue thinking of the range as {6, 8, 10, 12, 14, 16, 18, 20}, but I think the "sweet spot" in it varies depending on what you try a lot, so that it would make sense to argue for either 6-12, 8-16, or 10-20 as a range. Some are so similar that I would perhaps think of them in pairs of {6, 12}; {8, 16}; {10, 20}; {14}; {18} (which makes the gender-assignment for the characters make a great deal of sense). Including the rest of the understandable scale gives {6, 12, (24)}; {(4), 8, 16}; {10, 20}; {14, (28)}; {18}; {(30)}.

I think this effect is partly due to you becoming accustomed to working in large bases like {16, 18, 20}. The lack of concision and low digit diversity of octal compared to these higher natural-scale bases is becoming evident, and therefore youíre starting to feel it to be unsuitable.

My "sweet spot" is {10, 12, 14}; {6, 8} are a little small for me, and {16, 18, 20} are somewhat big, but I can use all of them without too much effort. Some minor base conversion might be necessary because I donít know the tables of any bases other than {10, 12, 14}, but Iíll always choose the closest in prime factorization to try to immerse myself in that base.

But then again, choosing between human-scale bases really depends on what youíre looking for in a base: octal represents binary purity and confidence in the innate ability to divide things in half, decimal showcases pseudo-5-smoothness and friendly auxiliaries, duodecimal stands for terminating expansions of the most common fractions at the expense of 5 and 7, and tetradecimal is large and hefty, but is overall concise and handles the first four primes rather well. In any case, you lose about as much as you gain, and once a civilization already using one of {8, 10, 12, 14, 16}, the cost of conversion to another base is large enough that it may never be paid off entirely.
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Double sharp
Posted: Dec 11 2017, 03:47 PM


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QUOTE (Oschkar @ Oct 12 2017, 08:07 AM)
{e}

cc0 and 1ba0 seem very round, more so than any number in decimal or dozenal in fact. However, Iím not sure if they would ever be used as divisions. Itís practically impossible to discern an angle of 1/cc0 of a turn, and, to avoid confusion, concentrations of a solute would be best expressed in parts per some tetradecimal power. Perhaps their niche use would be in packaging; it doesnít seem unlikely that at least some small items would be sold in packs of 30 or 60, and stored in boxes with 44 packs in them.

I wonder if this would continue to odd bases: how round does 2520 feel to you as b30{f}? Or non-7-smooth bases with more primes to fill in: how about 55440 as 38720{b} or 54c0{m}?

Actually, what does a round number even feel like in a prime base? I can see that in an odd base you could still have surrogates for the thirds, but the ninths may not be useful enough. Certainly any multiple of a power of the base must feel round, with "1, 10, 100, ..." especially so, but there must be some "divisible roundness" as well. But is it enough for something with as many significant figures as 360 = 1023{7}? I'd note that 1ba{e} still feels somewhat round, because of the transparent 3 and 5 content.
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Oschkar
Posted: Dec 16 2017, 01:08 AM


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Anyway, back to the topic of the thread.

I ran a few tests estimating the products of pairs of random 6-digit numbers in bases {8, 10, 12, 14} and calculating the mean relative error. Iím sorry, icarus, but the spikes and bumps you noticed in your study were sampling artifacts; when taking 720720 samples, the graphs for all four bases look uninterestingly similar.

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Oschkar
Posted: Dec 16 2017, 01:08 AM


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[continued]

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Oschkar
Posted: Dec 16 2017, 01:09 AM


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[continued]

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Double sharp
Posted: Dec 16 2017, 03:58 AM


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{8}

It seems like the only patterns of note are the progressively smaller "spikes" that occur just after every multiple of the base, and the way in which smaller bases result in the mean relative error decreasing faster (presumably because the number you're rounding off to is always closer). Large bases also get you to memorise more and more high rows with less and less improvement along the way, which is rather irksome as the high rows are also precisely the less patterned ones because none of them can be easy divisors.

Surprisingly this seems to be a fairly good argument for octal! While it is true that many of the useful products live in the dozens line, it doesn't feel like senary in which too many useful numbers are shut out, especially since binary powers are always easy to see. I don't think it's unlikely that an octal civilisation would simply stop at 100 and recognise a few more useful numbers via immersion (things like 74 and 170, for instance), the way we do for things like 216{a}, 360{a}, 864{a}, and so on.
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Oschkar
Posted: Dec 16 2017, 05:17 AM


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QUOTE (Double sharp @ Dec 16 2017, 03:58 AM)
Surprisingly this seems to be a fairly good argument for octal! While it is true that many of the useful products live in the dozens line, it doesn't feel like senary in which too many useful numbers are shut out, especially since binary powers are always easy to see. I don't think it's unlikely that an octal civilisation would simply stop at 100 and recognise a few more useful numbers via immersion (things like 74 and 170, for instance), the way we do for things like 216{a}, 360{a}, 864{a}, and so on.

{6}

It might also be an even better argument for senary, provided we learn the tables up to 14 or 24, both for infill and for extension to larger numbers. A few of the products will go up to four digits (24*24 = 1104, 23*24 = 1040, 23*23 = 1013, 22*24 = 1012), but it seems like a relatively small price to pay for better number sense in general, both for estimation and for number recognition. After all, the senary products are easier to learn than those in the human scale proper, even up to 24.

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Double sharp
Posted: Dec 16 2017, 06:02 AM


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{6}

Well, I'm not sure if other considerations aren't also valid for senary. After all, binary and quaternary are even easier, but they would run into the other problem: they're tiny. (Though I'd be interested in seeing what happens when you plot them.) And we also need to take into account the increased difficulty of three-digit as opposed to two-digit products.

In other words, I think that the argument works for senary only if its magnitude is comfortable for you. Of course, the other bases also have good arguments, so I suspect that {6, 8, 10, 12, 14, 16} already all stand successfully on their own merits: senary for sheer ease of use, octal for easy binary purity, decimal for pseudo-5-smoothness, dozenal for terminating fractions, tetradecimal for pseudo-7-smoothness coupled with concision, and hexadecimal for concise binary purity. And once you're there, you can deal with it, though 6 and 16 at the limits may feel a bit "off" to some people. In the end, all these factors (ease of use, concision, presence of auxiliaries, nice fractions, pseudo-smoothness, etc.) are present, and we're only analysing one; the choice of which to prioritise is presumably highly personal.
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icarus
Posted: Dec 23 2017, 02:29 PM


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Iím sorry, icarus, but the spikes and bumps you noticed in your study were sampling artifacts; when taking 720720 samples, the graphs for all four bases look uninterestingly similar.

Oschkar, thanks for the deeper study. We're after "truth" or a semblance of it that we can quantify, after all.
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