The Higher Natural Scale

This forum examines bases other than twelve and less than sixty.
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Jan 15 2018, 10:42 AM #433

In fact, I am increasingly quite convinced that the bulk of this "higher natural scale" is just the mid scale. I might even provocatively claim that the higher natural scale does not actually exist, because it lacks defining properties. Instead, there is simply a "human-scale approach" and a "mid-scale approach" to both addition and multiplication: in the former you don't break things up, while in the latter you do to avoid manipulating some numbers directly. The bits of the "higher natural scale" that could most justifiably be claimed to be distinct - {16, 18, 20} - could more reasonably simply be considered a reflection of the break not happening at the same place for each operation, not to mention for each person, which creates a transition where addition is single-step and multiplication isn't.

Consider bases 28 and 84 as {2:14 or 4:7, 6:14}. Now it seems to me that in both bases there simply isn't any way to memorise a full addition table: it's just a lot more ridiculous to try in 84 than in 28. Rather, we use breakpoints at every seven or fourteen, and split the problem into chunks, breaking the base down into factors. (I would've used 20 and 60 for examples, but you can actually almost finish the addition table in vigesimal with Sauter's strategies.)

Now consider multiplication. In both of these bases, every factor has a line that is simply too long to be memorised, so we end up splitting problems multiplicatively. To do 2*m in octovigesimal, we'd simply see that m is 8 over e, so the answer is 1g. The same goes for 2*23 in tetraoctogesimal: you'd surely see first that 2*20=40, then add 2*3=6 to give 46. (In fact, not even the whole of icarus' abbreviated table seems to be memorised.)

Similarly, whenever complementary divisors arise, they are the obvious and pretty much the only way to see where you are in a line. 7*j is pretty obviously dividing j0 in four to give 4l, since l is 3/4 of twenty-eight. And in base eighty-four, 17*42 = 42'00 / 4 = 10'30.

Lastly, when we do numbers that are "off the grid", we simply choose from a number of strategies, like multiplicative breaking (like hexadecimal regrouping to bunch up powers of two), additive breaking (into factors), and the quarter-square identity if the numbers happen to be close. We just see the last a lot more often in a small base, because then the digits can't be too far apart, and the method works in a higher proportion of all cases. Even DD-shifting human-scale products from the smaller base finds a parallel in how we just tweak the upper end of the multiplication table slightly to do things in {6:10}, writing 8*9=1'12 instead of 72; similarly, in vigesimal, we might think "seventy-two" and write "3c". (Surely if the French can think "soixante-douze" and write "72" it should be no different the other way around.)

It therefore seems to me that the difference is purely cosmetic, in the sense that a small mid-scale base like {4:6, 4:7} may be felt as though each place had one digit, rather than two subdigits. So, for example, the Babylonian use of {6:10} has equally spaced senary and decimal figures, while the Mayan use of {4:5} has each vigesimal digit as a separate block containing both the fives and the units. In fact, the latter rather remind me of the Chinese rod numerals: despite the fact that this and the abacus are constructed as {2:5}, not only is there no evidence of any biquinary alternation in arithmetic at all, there is instead copious evidence of the use of pure decimal and a multiplication table.

This would then make {(20), 24, 28, 30, 32, 36} look like human-scale bases even if the tables needed to make them run as such are enormous. However, the clear omnipresence of their factors should facilitate decay once fast multiplication stops being the preserve of a few learned scribes, and the algorithms these scribes would have been using beforehand would surely have been thoroughly mid-scale.

(I am considering 20 a special case because it is small enough that you can largely proceed with addition much as you would in a smaller base, so that you have enough of a foundation to start before abacus-like fiddling with fives and ones fills in the leftovers, which are mostly adding {3, h} and {4, g}. This being said, its multiplication table for me is definitely quite mid-scale, or at least it would be if I wasn't just DD shifting from decimal for pretty much all of it. I think that might be a significant barrier for us to find out how vigesimal feels like intuitively as a base in itself, assuming of course that it can be thought of that way: I somehow don't do this to get hexadecimal from octal, so we cannot rule out the hypothesis that this isn't only because we're all immersed in decimal.)
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Jan 15 2018, 03:26 PM #434

Come to think of it, this suggests that you might be able to save vigesimal by a sort of numeric diglossia with decimal. We could consider adding with the "teens" digits (that go down to the feet) to be an exercise that is covered only after adding with single digits (on the hands) is completely understood, so that the gradation of addition facts would be:

1. The trivial facts (adding 0 and 1);
2. The facts whose answer doesn't surpass a;
3. The facts whose addends don't surpass a;
4. The facts where one addend surpasses a;
5. The facts where both addends surpass a.

The beginning, in fact, would look exactly like decimal, except that once we get to group 3, we're writing "8 + 6 = e" instead of "8 + 6 = 14". Of course we'd still say "fourteen" or something like that, but we'd write it with its own transdecimal digit. Starting group 4 would presumably be facts like "plus a, without carries", "plus 5, without carries", and "plus f, without carries", as Oschkar's exposé has them. (I may write a revised version going through this route.) Group 5 can either round off from 10 using subtractive strategies or simply group the "teens" together into a carry and then use group 3 strategies for the remainders.

In this sort of vigesimal, you'd learn the tables up to ten times ten, but in base twenty, and you could have a lot of code-switching involved with a decimal nomenclature. So you could say "eight nines make seventy-two", but you'd always write it as "8*9=3c". Once again we have:

1. The trivial facts (multiplying by 0 and 1);
2. The facts whose multiplicands don't surpass a;
3. The facts where one multiplicand surpasses a;
4. The facts where both multiplicands surpass a.

I imagine that the effect of multiplying by a (halving a number) has to be taught early, somewhat apart from these groups. Then we can learn the tables up to a*a=50 using something close to the decimal strategies (although I confess I haven't really thought much about the alterations needed because the base is double the size). To do the facts in group 3, you break up the larger multiplicand into a and some leftovers; so g*7 = a*7 + 6*7 = 3a + 22 = 5c. (I think the facts with a need to be thoroughly drilled in, so that they feel almost as round as the true scores. Perhaps there needs to be finger-only counting as well as finger-and-toe counting?) To do the facts in group 4, you use the quarter-squares identity, just as you can for the high facts in group 2 if the decimal nine-times strategies aren't obvious enough. For example, d*e = (a+3)(a+4) = (h*a) + (3*4) = 92. Of course you can in some cases use 10 instead, like in g*h = (10-4)(10-3) = d0 + c = dc. And if you do them enough, it might well be possible to memorise them all.

I'm not sure if this can really be taken past twenty, because of the need to learn lots of different numerals. But if it can, it seems to be one of the best hopes available for such a double-sized base.

P.S. I think I've forgotten the even bigger problem that this doesn't really reduce the number of facts you have to learn, but only it makes it possible to learn the ones that are already there, so that I think it's massively unlikely that this would actually work to get people to memorise the facts for {24, 28}. This being said, these bases are larger, so it takes less efficiency to get them to work as long as the bottom of the range is well-understood. I think I'll have to try them all out again. This is honestly still using mid-scale strategies with the exception that removing the alternation perhaps removes the biggest hurdle, but they might not feel that way.
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Piotr
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Jun 13 2018, 06:59 PM #435

Oschkar wrote: It has been pretty much established that the bases within the human scale can be set at {(6), 8, 10, 12, 14}.
Nope, it's not estabilished... And we can't as we don't have an Earth simulation.
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Jun 19 2018, 03:52 PM #436

Indeed not, but since the 16*16 decimal table has been taught in the past, it seems reasonable to err on the high side rather than the low side. I think a reasonable inspection of bases larger than 12 reveals that the benefits they have over the dozen are outweighed by the increased amount of totative junk they have to carry around anyway; the reason we bring in the "human scale" is mostly for the case for dozenal, and ending it at anywhere from 14 to 20 instead of 12 doesn't really change the final outcome that much.

It's worth noting that if you demand only "passive competence" in a base, rather than "active competence", you can probably push the limit further. By that I mean that it seems possible to be able to read and confirm results in a slightly oversized base even if you can't quite remember what they are. Even when you cannot remember every multiplication table result, at least for a while it should be possible to estimate them; when that peters out, it should still be possible to estimate addition results; and when that peters out, it should still be possible to meet the low bar of being able to count and compare numbers.

Now, to be fair, this ends up bordering on simple innumeracy in the end, but after all; a big base around 36 is kind of like two digits of a human-scale base already. We don't complain that people can't memorise the result of 27*83 right off the bat if they know how to do it slowly and can quickly guess that it must be somewhere near 30*80=2400. And in a just-beyond-manageable base - I am thinking mostly of 18 and 20, so that only the skill of remembering every multiplication table result is missing - it seems possible that while not everyone will achieve full fluency, people will be able to estimate what they can't quite remember, and perhaps then recognise what the right result should approximately be. (In situations when the exact answer is very important, there should be time to work it out properly.)

It would then not be a big deal if most kids can't remember the whole multiplication table for octodecimal and vigesimal. The important thing would be that they should be able to add very quickly (which I suppose is easier, given the ability to visualise either an abacus or counting fingers and toes). Then multiplication and addition can kind of be taught together to save time (which has already started for the doubles); you drill in some results, and it's okay if you use addition for what you don't remember because of the coverage of the divisors. After those strategies are taught, then we could teach tricks like complementary divisors or "complement" rows to provide another way of looking at the medium-hard rows. I suppose rows like vigesimal h will always be dealt with via addition or just quick estimation (which can through tricks like (z+a)(z+b) = z(z+a+b) + ab be turned exact), but at least the majority of rows should be reachable. Estimation is then rendered multi-levelled, since you can round off to quarters of the score in about the same way we in decimal round off to half a division.

Okay, I think I've managed to convince myself that this can be made to work pretty well! I will try it out for bases 18, 20, and 24 and see how it ends up working.
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Jun 19 2018, 05:14 PM #437

Yeah, it's pretty much estabilished that we can't estabilish completely with certainty whether base 6 and 14 are usable or not. Bases 7, 9, 11 and 13 are odd, which has the side effect of their multiplication tables being harder to remember, especially 11 and 13. I don't think base 13 would be usable due to its difficulty.
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Jun 19 2018, 10:23 PM #438

I think the notion of "usability" (utility) is not a binary question, meaning base x is usable, but base y is not usable. It is clear that some bases, like 45!, are not usable, nor is base 19#, these are gigantic numbers and no person could fathom them. I would be careful about base 6, because it is usable. I would also suggest that 7 would be usable, that 11 could be usable, just not efficient, and that base 14 might be usable, but unwieldy. Senary presents a tricky situation. It doesn't "fail" or prove "inferior" purely on the grounds of its arithmetic, which at once resembles that of decimal (being a semiprime) and uncial (for its factors 2 and 3; indeed for being the squarefree root of any {2,3} base). It has a very simple multiplication table. The question we would raise about it is different from the one we'd raise about tetradecimal or septenary. Is it at a point where there are too many digits needed to represent everyday numbers, and are these digits too similar to one another? There are non-mathematical grounds to wonder about base 6; do we have sufficient lexicographic space in senary? Meaning, do we have enough numbers with which to index items, like room numbers in buildings, before we "run out" and have to turn to ever larger groups of monotonous digits? Those are questions we would need to explore with cognitive science, and can't quite answer fully given number theory. The answer given by experiment would be "fuzzy" and not as crisply distinct as those given by maths proof.

Base 7 is surprisingly not too bad, given that it is a prime number. Its multiplication table is relatively easily memorized compared to its double, base 14 since it is so small. It has nice neighbors. It certainly isn't a clement choice. Merely because a base is prime does not mean it cannot be used.

Base 14 is similar to decimal in many ways, but larger. It would require more time to teach children, and the things we find ourselves not liking in decimal are made worse oftentimes in tetradecimal.

I presume by "use" we mean "general human computation", like measurement and kitchen use, simple mental sums and products, divisibility testing, and expansion of commonly needed fractions. There are other applications of bases that constitute a type of "usage" but perhaps not a question of "everyday use by people".

I would say that senary is questionable but certainly usable. Septenary might be usable. Octal, decimal, uncial are usable, the middle one for certain. Nonary and undecimal are likely usable, just not very efficient; they are "harsh mistresses" like Heinlein's moon. Base 13 has interesting properties; at what point are these no saving grace? At 11? At 13? In many ways base 9 behaves "prime", meaning it proves as difficult as a prime base. I think we could use base 14; it would be a poorer choice than 10. This might be the limit, unless we use different arithmetic algorithms and special numerals in hexadecimal. The societal usage of hexadecimal would prolong acquisition of higher mathematical concepts, probably doubling the acquisition time for numeral recognition and basic arithmetic. If we didn't "need" to learn algebra in middle school, then sure, hexadecimal is in there, perhaps. So all of these questions have a lot of moving parts. It would be interesting to identify the parts. Then we might be able to measure and come up with meaningful ways of determining which bases are useful and which are not. Maybe we say some are "useful" {(6), 8, 10, 12, 14}, which are quasi-useful {7, 9, (11), 16}, and some are not useful (anything else). Again, I would say that {20, 60, (120)} are useful (with an asterisk they use in baseball statistics, perhaps); the former two because they were useful. We just can't use them without thinking differently, and given the burden that modern life puts on everyday arithmetic.

Maybe this thread already went through definitions of these, I just haven't read them in a long while.

I need to sign off, I have a tremendous headache.
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