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Post by fcg710647 on Aug 4, 2015 13:18:06 GMT -8
Oh 36, 20(Od.) and 30(doz.) is a triangular number anyway he he Pi is 3.184809493B9.... in doz. and 3.29FCEH0G771... in Od. (since you use A-H for it)
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Post by Buffoonery on Aug 4, 2015 15:24:36 GMT -8
yeah, those three are pretty good. However, I think base 12 is the most reasonable for everyday use. -Senary and Octadecimal are great, but not the best on quarters. And I see this may not be the most important factor of bases in mathematics, but it allows for those who say, "I can't do arithmetic, I'm terrible at it," to have an easier time. Adding 9+3 or 8+4 to get 10, is nicer to work with than senary's quarters, 1.3 and 4.3, or octadecimal's quarters, 4.9 and D.9. Having advantage over both 3 and 4 is very nice and it's something I think the average worker would appreciate. Dozenal also has better placement in 1/(2^3), 1/(2^4), 1/(2^5), etc., which is very helpful for the average Joe. I feel strongly that dozenal is the sweet spot of all bases, especially for people using simple mathematics. We can use base 10 to get some of the advantages octadecimal has and base 12 for the rest. Dozenal seems to excel in all the areas of math that I'd use on a daily basis.
I did a test to see if arithmetic was faster in dozenal. I got a deck of cards out and set these rules: A: 1 2-9: 2-9 Queen/King: ᕍ and Ɛ 10: 10 Jack and Joker: would double my sum. So by putting cards down and adding them onto my sum, I found that when playing both decimal and dozenal, dozenal was quicker than decimal for me. In decimal, we have certain patterns we have to remember, like: 8+7 = 15 4+9 = 13 6+4 = 10 7+3 = 10 With those examples, there's no reason behind them. What is 6 or 8 represent? The closest number to 6 that has relevance is 6.666...(2/3), the number 8 is .5 more than 7.5(3/4). We tend to just have to memorize combinations rather than reason why they exist. The only numbers between 1-10 that have significance are 2 and 5 (1/5 and 1/2). Other than that, you'll have to memorize the combinations, or compare them to closer numbers Now if we compare that to dozenal, what do 3,4,6,8 and 9 represent? 3 is 1/4, which links to 9 (3/4) 4 is 1/3, which links to 8 (2/3) 6 is 1/2 So, already we've improved the speed of calculating enormously.
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Post by fcg710647 on Aug 4, 2015 15:38:37 GMT -8
Why would decimal carry any of the advantages of octodecimal? What about shorter expressions for 1/9 = 0.2 and 1/3^4 = 0.04? My usual reason for Od. was because it does more interesting things when you divide by multiple 5s or 7s, or by thirteen (thirteen is adjacent to twelve so it doesn't count for dozenal's case, Od. would do a similar thing when dividing by nineteen for the same sort of reason). Both doz. and Od. have shorter expressions when dividing by either one of their primes multiple times (2s for doz. and 3s for Od., looks like you see the 2s as a lot more important). Od. makes up for it with the repeating digits upon dividing by a lot of 2s, which doz. doesn't show. One of the tools posted on wikipedia's dozenal page is a conversion of exponents table, ranging from something like b^-2 to b^6 or maybe higher, and both 5 and 7 are included in the base b of the exponents. When you do 5^-2 or 7^-2, you get super long patterns and even writing the bar over them to denote the recurring digits is a nuisance. Maybe that doesn't matter to you hahaha
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Post by Buffoonery on Aug 4, 2015 16:19:04 GMT -8
Yeah, I put more importance on 2's rather than 3's. I can think of so many uses for 2's, but only a few for 3's. So in my mind, 2's are more important. The easiest example is cutting something with a knife equally, which uses powers of 2. - I can't think of any situation where 5's and 7's would be useful, or at least would make it worth my while, so I wouldn't miss them very much. - Another thing is having to learn 8 new numerals and the multiplication table that goes with it. Whereas in dozenal, it's 2 just more columns and rows. Learning 2 extra numbers is a small price to pay in exchange for easier arithmetic and better fractions (fifths and sevenths excluded).
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Post by fcg710647 on Aug 4, 2015 17:10:39 GMT -8
6^12 = 509000000 in dozenal and 3A000000 in octodecimal... seems subtle but I often enjoy shortening representations by using higher bases. Doz. has a slight advantage over dec. here; URLs and other encoding schemes (including security codes) tend to use bases at least as high as 36. I remember playing with calculators as a little kid and getting bored with the ones that only displayed 8 decimal digits. Perhaps a much more noticeable case is significant figures. 5.A56 in octodecimal differs from 5.A55 by only 1/5832, where dozenal numbers which differ by 1 in the third place right of the dozenal point differ by 1/1728. The dozenal approximation to Pi 3.185 has a ~0.00604% error, while the octodecimal one 3.29G has a ~0.00126% error. Numerous middle school exams with areas and volumes of cones, cylinders, and spheres say "use 3.14 for Pi" and Ugh it's kind of inaccurate. This is also a funny thing because sometimes I think twelve is too close to ten. The biggest mistake people have when attempting to use dozenal is saying that 100(doz.) is 120 in decimal instead of 144 (120 in dec. is A0 in doz.). They're so used to multiplying by tens that they forget that adding zeros in doz. is multiplying by twelves.
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Post by fcg710647 on Aug 4, 2015 17:42:28 GMT -8
You are right about cutting. If I have a giant pizza and I want it in eighteen slices, I can't do the order of 3, 2, 3 that I mentioned for ruler segments, I would need 2, 3, 3. In other words, cut it cleanly in half first, then divide the halves by three (which does permit cutting across the whole thing because it's already symmetrical by two) and then cut the resulting sixths into threes. My tactic for estimating a third of something is that it's half the size of the remainder (the 2/3 left over).
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Post by fcg710647 on Aug 5, 2015 16:27:14 GMT -8
Which is a bigger priority to you, then? Divisibility by 3s or by 4? For instance, 20 is not as big as 24, and its factors are 1, 2, 4, 5, 10, and 20. 1/2 = 0.A, 1/4 = 0.5, and 1/5 = 0.4. Maybe vigesimal is no good if you care very little about fifths and need the exact thirds. Six divisions for a vigesimal chart though would be 1/5 (40), 1/4 (50), 1/2 (A0), 3/5 (C0), 3/4 (F0), and 4/5 (G0).
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Post by Buffoonery on Aug 6, 2015 14:57:37 GMT -8
Haha yes, "120". Most people I've shown the dozenal system to have made the same mistake. - Woah, I never knew that they encoded with base 36+, that's cool. I'm curious on what they use for their symbols, since (10 + Alphabet) = 36 - For Pi, since it's an irrational, I can understand how a higher base would be more accurate since each there's more information provided in less space. Having more numerals allows for more information to be stored in any given area. The problem I see is having that many symbols, having to remember the multiplication tables, it would be a chore to say the least. I heard engineers commonly use 3.1416, which is close enough to 3.14159. It gets the job done, but still. - What about this method for getting thirds (no protractor): prntscr.com/81p30f- Ooo tough choice 3s or 4s, they're both useful, but I guess 4's are more common. One of the reasons I think this is due to the thirds in decimal recurring. So transferring to another system with thirds may be less appetizing than quarters. It's also easier to draw quarters so long as you have a straight edge, whereas thirds you need to measure half the radius. - I think fifths are really uncommon compared to the other numbers from 1-10. Even though we use the decimal system, we don't seem to use fifths that much. This is how I would order them: 2>4>3>8>16... That's a just rough idea, but I'm not 100% certain about them. - Vigesimal would be pretty cool, but I'd rather have both 3's and 4's if I can. I think the deal breaker though is how high it is. 10 extra symbols would be a bit much.
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Post by fcg710647 on Aug 6, 2015 15:33:23 GMT -8
I wonder if I would have time while holding that pizza cutter to estimate a quarter of the way across the first cut and imagine a perpendicular to divide it into threes. Once I have it sliced into sixths, though, I have to use my own method to divide those by three. Well, then, rank the following bases in preference: 16 (2^4), 18 (2*3^2), and 20(2^2*5)
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Post by fcg710647 on Aug 6, 2015 15:51:17 GMT -8
One does not have to memorize the multiplication table, as long as it is possible to make it manually (as you have done for bases up to 24), to be able to derive each product either directly or from neighboring ones in the table until it's all finished. I have memorized the decimal one but I expect most people haven't.
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Post by fcg710647 on Aug 6, 2015 16:08:39 GMT -8
When reviewing my own multiplication table creations, I noticed an issue with multiple of 4 bases: For any multiple of 4, cut it in half (b/2). The prime factors are still the same because half of the base is still even. Now, take this even number and add or subtract 1 and you get a totative. In other words, if b mod 4 = 0, then (b/2) +- 1 is a totative. In hexadecimal, 16/2 - 1 = 7 and 16/2 + 1 = 9, both totatives. This is how you get the only two totatives present in dozenal (12/2 + 1 = 7 and 12/2 - 1 = 5). Even in decimal this doesn't happen since 10 mod 4 = 2. 10/2 = 5 so the two numbers are 4 and 6, the first of which is a semidivisor and the second a semitotative.
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Post by Buffoonery on Aug 6, 2015 16:53:26 GMT -8
Ha, I was thinking about that too, cutting it up into threes, pausing while others stare at me in confusion, then finally cutting up my 3s. - I would rank those: 18>16>20 I like 18 for its factors, 16 for its symmetry and it's not too big of a base. But 18 and 20 have quite a few symbols. I could get used to 6 more, maybe 8, but 10 more might be a bit much.
What about you? What's your list? - Yeah, I remember in elementary school, we did speed tests for our 1-12 time tables. I used to think it was stupid to memorize 11s and 12s, but I found out the importance soon enough. I guess I can see your point to a certain degree. Say for example in decimal, if you know that 12x5 is 60, then you know 12x6 must be 72. The thing I love about my favourite base, is that there are more of these easy neighbouring multipliers.
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Post by Buffoonery on Aug 6, 2015 17:00:07 GMT -8
What are the advantages of having more totatives in a base? I heard it's good for cryptography. It's generally bad to have them eh? Say for example base 60, it has 23 totatives and semitotatives, that'd be extremely difficult to remember.
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Post by Buffoonery on Aug 6, 2015 18:07:28 GMT -8
End-digit sequences for: octodecimal: 3: {3, 6, 9, C, F; 0}. 4: {4, 8, C, G; 2, 6, A, E; 0}. 5: {5, A, F; 2, 7, C, H; 4, 9, E; 1, 6, B, G; 3, 8, D; 0}. 6: {6, C; 0}. 7: {7, E; 3, A, H; 6, D; 2, 9, G; 5, C; 1, 8, F; 4, B; 0}. 8: {8, G; 6, E; 4, C; 2, A; 0}. 9: {9; 0}. Hexadecimal: 3: {3, 6, 9, C, F; 2, 5, 8, B, E; 1, 4, 7, A, D; 0} 4: {4, 8, C; 0} 5: {5, A, F; 4, 9, E; 3, 8, D; 2, 7, C; 1, 6, B; 0} 6: {6, C; 2, 8, E; 4, A; 0} 7: {7, E; 5, C; 3, A; 1, 8, F; 6, D; 4, B; 2, 9; 0} 8: {8; 0} Dozenal: 3: {3, 6, 9; 0}. 4: {4, 8; 0}. 5: {5, A; 3, 8; 1, 6, B; 4, 9; 2, 7; 0}. 6: {6; 0}.
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Post by fcg710647 on Aug 6, 2015 20:41:37 GMT -8
Yea sexagesimal is kind of gargantuin. Well, if you want a simple multiplication table, you would go with a tiny base like quaternary or senary. For instance, the base 4 table is just this: 1 2 3 10 2 10 12 20 3 12 21 30 10 20 30 100 Too bad, those are too small. We want the factors 2 and 3 and larger than 6 so we're stuck with those 5 and 7 totatives (vigesimal's totatives are 3, 7, 9, 11, 13, and 17 so that isn't great). Od. is the highest with 4 or fewer non-trivial totatives (ignoring 1 and b-1). More Od. patterns... A: {A, 2, C, 4, E, 6, G, 8, 0} (ten, semitotative for it and doz.) C: {C, 6, 0} // It's happy with twelve, vigesimal would have a period of 5 here E: {E, A, 6, 2, G, C, 8, 4, 0} (fourteen) F: {F, C, 9, 6, 3, 0} // fifteen, semitotative G: {G, E, C, A, 8, 6, 4, 2, 0} // Two below the base, always b/2 pattern length for an even base Oh, if you have a compass set to the radius of the circle, just draw a diameter and then use the compass to mark off a straight distance of the radius in both directions from each end of the diameter and you have six equal sections Lovely head start toward both doz. and Od.
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