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Post by Buffoonery on Oct 11, 2015 15:46:16 GMT -8
Ahhh, I see. That's interesting.
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Post by fcg710647 on Oct 16, 2015 11:36:28 GMT -8
I have a text file where I've written a lot of documentation involving high bases, and I'm using the font Candara; it makes Gs look a bit less like 6s, but Os look exactly like 0s. I prefer to keep O and S out and most likely I as well. An X has too much confusion with a multiplication sign or an unknown quantity (hence your Dec symbol looks more like a d even though the silly Dozenal Society of America still uses a script x lol).
Anyway, when we divide by 5s, it's pretty much because we use currencies or approximations derived from decimal. On the other hand, we would get into total chaos if we tried to redefine a week (which days we work, go to school, etc. etc. and which days we don't, not to mention the names of the seven days). This should be an advantage of 42 relative to 30. It sounds like we have 4 choices to keep looking into:
12 - The ease of adding only 2 more symbols and still having more finite fractions.
18 - All the fractions that are finite in bases 6 and 12 are finite here, and 18 has some special recurring properties, although it does not shorten recurring periods much on average.
36 - As in dozenal, common fractions terminate in only one significant digit, and the same ones already mentioned (dividing by 2s and 3s) terminate. The real advantage of 36 is its shorter recurring periods because it's a square base. The same remainder tends to turn up in long division sooner, indicating that a cycle of the recurring digits has been completed and can be written out without continuing the tedious division.
42 - After 2 and 3, the only other prime we're stuck with using is 7. 42 has a lot of divisibility power and even more finite fractions. It is fast at identifying a lot of numbers as being composite (last digit tells if the number is even or divisible by 3 or 7).
Beyond 30 and 42, there are only two strategies to have still more finite fractions, which I don't really recommend as they should not be worth the complications.
Sub-basing: Use a base with four or more primes in it, such as 210, 330, 420, etc. but split each digit into smaller bases so you don't have hopelessly many symbols. We do this currently in order to handle sexagesimal. Each base 60 digit is a digit of decimal and another of senary, so one sexagesimal digit is 00 to 59. For base 210, one could alternate bases 14 and 15, and alternating 15 and 22 would give 330. Sub-basing causes the same operations to require manipulating more digits because of the splitting. If you pull up Wolfram|Alpha and type in, without the quotation marks, "1/16 in base 2310", you'll see how lengthy that gets when compared with an expression such as 0.0625 for 1/16 in decimal.
The factorial system: The first digit to the right of the radix point is a bit, the next one a ternary digit, then quaternary, and so on. This means that 0.1 is a half, 0.01 is a sixth, 0.001 is 1/24(decimal), etc. etc. reciprocals of factorials. This strategy actually terminates any fraction, as any integer is present as a base digit somewhere along the line. The biggest disadvantage is the confusion that comes from different base digits in different places. It also requires a lot of extra translation when using scientific notation for a similar reason, taking digits and converting them to smaller or larger base digits.
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Post by alysdexia on Apr 20, 2019 0:30:23 GMT -8
You were right, we should just write here due to G+ deletions. I can't read your entire message each time. Anyway, quadrivigesimal needs a 12*12 to fit packings of 6*4 or 2*12 into a square, and trigesimal needs a 30*30 square LOL 12 and 18 prevail big time if you want that 6*6 packing. OOOOOO @@ OOOOOO @@ OOOOOO 24<36 @@ OOOOOO 12/12= OOOOOO OO 1 OOOOOO OOOO 36/20= OOOOOO @@ 1·8 OOOOOO @@@@ vs. OOOO @@@@ OO OO @@ OOOO 24/12= 24/16= OOOO 2 1·5 OO 12/8= 24 wins 2D. 24 wins 3D. 1·5
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Post by alysdexia on Apr 20, 2019 2:04:55 GMT -8
I'm just wondering how you'd get a computer to not run on binary. I can't imagine how else the transistors would work. en.wikipedia.org/wiki/IEEE_1164Two transistors at different thresholds could make 0, L, Z, H, 1, where 0<L<H<1 and Z={0, L, H, 1}.
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Post by fcg710647 on Jul 5, 2020 20:06:49 GMT -8
You were right, we should just write here due to G+ deletions. I can't read your entire message each time. Anyway, quadrivigesimal needs a 12*12 to fit packings of 6*4 or 2*12 into a square, and trigesimal needs a 30*30 square LOL 12 and 18 prevail big time if you want that 6*6 packing. OOOOOO @@ OOOOOO @@ OOOOOO 24<36 @@ OOOOOO 12/12= OOOOOO OO 1 OOOOOO OOOO 36/20= OOOOOO @@ 1·8 OOOOOO @@@@ vs. OOOO @@@@ OO OO @@ OOOO 24/12= 24/16= OOOO 2 1·5 OO 12/8= 24 wins 2D. 24 wins 3D. 1·5
Sure, you could probably tile a plane or 3D space with different arrangements of a given number of identical units, but you're not going to see egg cartons and racks shaped in octagons... maybe hexagons. Thirty-six is also unusual in being both a square and triangular number (and triangular arrangements can pack more efficiently).
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