Base 960: Faithful Companion

TreisaranDozens Disciple
 Joined: Feb 14 2012, 01:00 PM
[b]Base 960:[/b]
SDN hexoctanunqual or hexoctnilial; [url=http://z13.invisionfree.com/DozensOnline/index.php?showtopic=887]encoding[/url]: plain decimal (numbers from the range 0 to 959)
The upper limit of generalpurpose bases is most probably 120, the Long Hundred, and this is so because its returns are worth its size  fortunate neighbourrelationships that its halfsized, quartersized and doublesized relatives all lack  and because of the ability to encode it as two humanscaled bases: twelve and ten in alternation. The price of handling two bases in one is probably already too high for most people, so it's safe to say there's no point in prospecting for arithmetic gold beyond the height of tendozen. Why consider base 960, then? Because, as a fractionpart auxiliary rather than a generalpurpose base, we can go higher, and 960 has quite a few desirable properties for its use together with integerpart decimal.
Free of the considerations of multiplication size and coprime digits and others that beset the choice of a generalpurpose base, a fractionpart auxiliary base is evaluated according to its precision, the recognisability of its fractional values, and the possible extra feature of its omega relationship. Base 960 is a clear winner in all three respects: its closeness to the third decimal power means you get nearly three decimal digits' worth of precision with just one figure, and nearly six with two figures; this closeness also contributes to the recognisability of base960 fractions, which look familiar to one used to decimal fractions; and as a bonus, 960 like the Long Hundred has 7 as its omega inheritor (959 = 7Â·137), meaning you're close enough to one or more sevenths if you've got a multiple of 137, even a single instance of it, let alone repeating.
Here's a table of the fractions covered by base 960, also comparing them to their decimal representations:
The divergence between the base960 and the decimal fractions increases as the fractional value approaches the maximum, but even the highest base960 fractionals don't stray too far from the decimal appearance. Of all the fractions given, the unreduced ninths may be the hardest to use, because they need two places; given the lesser need for ninths, we may as with sevenths be content with gauging proximity to them rather than accuracy. For the most important fractions, as well as binary subdivisions, base 960 delivers the goods: the basic dozenal fractions are represented (though their appearance is like the one we see in decimal fractions, not a clockface correspondence as in the Long Hundred), and only after the sixth binary power is an additional place required.
Arithmetic with base 960, as with any fractionpart auxiliary, is plain decimal arithmetic with a different carry rule, in this case a carry that comes into force once you're past 959. The strength of a fractionpart auxiliary, however, lies in its notational advantage rather than arithmetic; for those who value dozenal for its terminating thirds ([i]contra[/i] decimal) but think a change of integerpart base, the adoption of new numerals ('X' and 'E' in dozenal or the Long Hundred) and alternatingradix arithmetic are all too high prices to pay for this feature, base 960 as a fractionpart auxiliary to be used together with integerpart decimal may be just what they need.
In closing, here are the most important irrational numbers using base960 fractionals, rounded to two places:

gingerbillRegular
 Joined: May 20 2012, 10:15 PM
960 is very useful. In web design, instead of designing for 800px, 1024px, or 1280px 960px is very common (or 1000px with 20px margins either side).
X = Ten, E = Elv
or
É¤ = Ten, Æ = Elv
or
á˜”= Ten, á˜ = Elv
Dozenal number will always use radix point let it be a ; (semicolon) e.g. 103; 3;4
Decimal number may not use radix point but only when needed e.g 14 or 9.2
Preferred system of units: IDUS or UUS (or if needed SI)
Preferred prefixing system: SDN (excluding multipliers)
Preferred language: English, FranÃ§ais
or
É¤ = Ten, Æ = Elv
or
á˜”= Ten, á˜ = Elv
Dozenal number will always use radix point let it be a ; (semicolon) e.g. 103; 3;4
Decimal number may not use radix point but only when needed e.g 14 or 9.2
Preferred system of units: IDUS or UUS (or if needed SI)
Preferred prefixing system: SDN (excluding multipliers)
Preferred language: English, FranÃ§ais

TreisaranDozens Disciple
 Joined: Feb 14 2012, 01:00 PM
Precisely, this trick of playing it both ways, decimally with a choice of 1000 units and dozenally with 960 units, leaving 40 for padding, is a very good one. I've often found myself setting the canvas size for a new image to be 960 in at least one dimension, if not both.gingerbill wrote: 960 is very useful. In web design, instead of designing for 800px, 1024px, or 1280px 960px is very common (or 1000px with 20px margins either side).
Having a useful auxiliary with a nearness to a power of the base gives some more possibilities. For decimal, there's 1080 (septhexunqual), a fractionpart auxiliary going a long way back: 1/1080 is a helek, used to divide the hour in Hebrew calendar reckoning (it's given in the new moon announcement on the Sabbath before the beginning of a month). In comparison to 960, it gains slightly better fractional recognition because it's over 1000, not under, and it also has singleplace ninths; the price is a loss of binary resolution (1/16 already needs two places) and an omega that no longer divides by 7, but by 13. Yet closer to 1000, but even less appealing, is 1020 (septununuqal), which has just two binary powers and one ternary power and is flanked by primes.
In a dozenal context, searching for a fractionpart auxiliary that would take care of the fifths, but without sacrificing recognition (ie the familiar appearance) of the basic dozenal fractions like base *500 does, and without introducing decimal arithmetic like twelveonten Long Hundred, base *1010 (unnilununqual, decimal 1740) preserves the ataglance recognition of the dozenal fractions while rendering fifths as multiples of *25 with a trailing zero:
 Half: *606â€²
 Thirds: *404â€², *808â€²
 Unreduced quarters: *303â€², *909â€²
 Unreduced sixths: *202â€², *X0Xâ€²
 Unreduced dozenths: *101â€², *505â€², *707â€², *E0Eâ€²
 Fifths: *250â€², *4X0â€², *730â€², *980â€²
 Unreduced eighths: *161â€² *606â€³, *464â€² *606â€³, *767â€² *606â€³, *X6Xâ€² *606â€³
 One ninth: *141â€² *404â€³
This could be explored further on, of course. Since all I want is clean thirds, base 960 as a fractionpart auxiliary for use with decimal brings the most bang for the buck as far I'm concerned.

TreisaranDozens Disciple
 Joined: Feb 14 2012, 01:00 PM
Ooh, this thought actually occurred to me when I was writing the opening post! But I decided against mentioning it, because the farthing is even more of a curio than Â£sd  inflation had already made it obsolete before British currency was decimalised.Dan wrote:Coincidentally, the Brits used to have 960 farthings to a pound.
This does make me wonder about a nomenclature for base960 fractionals. It wouldn't do to use the term 'farthing', because 'farthing' is the native English counterpart to the Latinate word 'quarter'. The farthing of currency was a quarter of the penny, hence 240Â·4 farthings to the pound of the Â£sd system. Furthermore, Iceland was historically divided into farthings, four divisions, and by inspiration, so was Tolkien's Shire. I don't wish to coopt a traditional name like this.
I've thought of a more suitable name: on analogy with the permille, one part in a thousand, though unlike the percent it's not much used, one can devise a term for base960 as a fractionpart auxiliary when noting the nearness of 960 to 1000. From the Latin paene mille meaning 'almost a thousand', I've coined the term 'penemille' /pÉ›nÉ™ËˆmiËl/ for these units. Taking the twoplace approximation of âˆš2, 1Â° 397â€² 619â€³, one can read it as 'one [unit], three hundred ninetyseven penemille and six hundred nineteen'.
It looks like all the bases duplated from 30 have been given a survey (mostly by Icarus), except one: 30, 60, 120, 240 and now 960 are accounted for, leaving 480 as a gap. However, as 480 is an unremarkable base even as a fractionpart auxiliary, unless an application to which it is well suited is found (as I have for base 960), only a brief look is warranted:
480, SDN triquadranunqual, isn't a primeflank but it might as well be one, as its omega is prime while its alpha is 13Â·37, composed of high primes (and the alpha doesn't help much for fractionpart representation anyway). As for the recognisability of its fractions, 480 or *340 or 0x1E0 doesn't give familiarlooking fractions in any of decimal, dozenal or hexadecimal. An octalist might value it as a fractionpart auxiliary, however: octal 740 is close to the third octal power, the way 960 is near the thousand in decimal, and in octal it provides for the representation of thirds and fifths, both problematic in that base. 1/3 becomes octal 240â€² to correspond to 0.252525..., and 1/5 becomes octal 140â€², matching the maximally recurrent 0.14631463..., with the binary fractions exhibiting divergence downwards.
Generally, the strategy of choosing a nearpower base as a fractionpart auxiliary for a particular base depends on the relationship of the base to the coprime factors that need to be provided for. Thus, in decimal we can leverage the fact that 3 divides the omega (9); in dozenal, 5 inheriting the squarealpha (*101); in hexadecimal, both 3 and 5 dividing the omega (0xF); and in unbinal, both 3 and 5 inheriting the alpha (basefourteen 11). Thus we have the fractionpart auxiliary bases 960 and 1080 for decimal, *1010 for dozenal, 0xFF0 and 0xFFF0 for unquadral, and unbinal 1100, to name a few examples near the third power. Octal is unfortunate in this regard, because the two coprimes we wish to fill in have different relationships to the base: 3 inherits the alpha (octal 11), 5 the squarealpha (octal 101). The choice between octal 1010 and 1100 brings one coprime factor in while leaving the other out. Instead, one needs an octal multiple of either 74 or 113 (decimal 60 and 75 respectively). Because the former is closer to an octal power, we can see why base 480 would be valued as a fractionpart auxiliary for octal.
All this is an interesting venture into number theory; for application by us here, who are, if not dozenalists then at the very least wish to bring the benefits of dozenal into decimal usage, bases 960 and *1010 as fractionpart auxiliaries for decimal and dozenal respectively are a boon for covering important missing prime factors. One must also note, however, that in our civilisation, because of the familiarity with dozenal divisions (from the clockface and other items in everyday life), a fractionpart auxiliary for any base has recognisability if it presents dozenal fractions. So bases like 120 or 1200 or 120120 in decimal, 0xC0 or 0xC0C0 in hexadecimal and CC0 in unbinal are all suitable as auxiliaries, because we get the basic dozenal fractions in them just like we see on the clockface. Whether or not we ever make use of those auxiliaries, they are enlightening from another point of view: showing how our civilisation has a substantial dozenal component in it, despite being decimal at its core, and despite attempts (coughSIcough) to level nondecimal 'anomalies'.
That'll be all for now...