Le Tour Des Bases

icarus
Dozens Demigod
icarus
Dozens Demigod
Joined: 12:29 PM - Apr 11, 2006

3:01 PM - May 01, 2012 #1

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icarus
Dozens Demigod
icarus
Dozens Demigod
Joined: 12:29 PM - Apr 11, 2006

3:04 PM - May 01, 2012 #2

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icarus
Dozens Demigod
icarus
Dozens Demigod
Joined: 12:29 PM - Apr 11, 2006

3:14 PM - May 01, 2012 #3

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Dan
Dozens Disciple
Dan
Dozens Disciple
Joined: 2:45 PM - Aug 08, 2005

12:39 AM - May 02, 2012 #4

[quote="icarus @ May 1 2012, 09:04 AM"] [/quote] I'm curious as to how you decided that 10 is male but 8, 9, and 18 are female.
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icarus
Dozens Demigod
icarus
Dozens Demigod
Joined: 12:29 PM - Apr 11, 2006

2:06 AM - May 02, 2012 #5

Dan,

Indeed there is a pattern! Even if it is accidental… (recalling from memory…) I speak Russian, that should be a hint, and the pattern is number theoretical at least with part of the set of possible categories.
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Dan
Dozens Disciple
Dan
Dozens Disciple
Joined: 2:45 PM - Aug 08, 2005

3:45 AM - May 02, 2012 #6

I don't know Russian, so the hint is lost on me.
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dgoodmaniii
Dozens Demigod
dgoodmaniii
Dozens Demigod
Joined: 1:45 PM - May 21, 2009

6:08 PM - May 02, 2012 #7

Dan @ May 2 2012, 03:45 AM wrote: I don't know Russian, so the hint is lost on me.
Grammatical gender. I don't speak (much) Russian, so I can't so for sure, but I think he's ascribing sex to these numbers based on their grammatical gender in Russian.
All numbers in my posts are dozenal unless stated otherwise.
For ten, I use :A or X; for elv, I use :B or E. For the digital/fractional/radix point, I use the Humphrey point, ";".
TGM for the win!
Dozenal Adventures
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icarus
Dozens Demigod
icarus
Dozens Demigod
Joined: 12:29 PM - Apr 11, 2006

9:11 PM - May 02, 2012 #8

Gentlemen,

Actually all I meant was I think the assignment of gender is completely random but there are three of them; feminine {(6), 8, 9, 12, 15, (16), 18} masculine {(6), 10, (20)}, and neuter {7, 11, 13, 14, 17}. So it appears that powers of two, multiples of three are feminine, multiples of ten are masculine, and primes and multiples of 7 are neuter.

I say 6 can be masculine or feminine, maybe stretching it, because my wife is into MMA. So women can be fighters, tho I am not sure the whole "welterweight" deal applies to them.

Anyway, the entire masculine/feminine/neuter thing is entirely random, just there to make the stories a little more lively. The 8 "because it has more beauty" had me thinking female, the "beauty" remark comes from the poster of the thread and not me. Sixteen may not be female from the title, as the title refers to a Stones song I happened to be listening to at the time: "Please allow me to introduce myself / I'm a man of wealth and taste...". No, the masculine/feminine/neuter gender trends won't necessarily be carried up or down scale; 21 may be male or neuter or something else. I think I am trying to impose order given dan's question. No this isn't an identity politics statement either. It's simply for levity.

I do hope you enjoy the posts. This is simply a proof of concept for a particular audience that isn't necessarily on the forum, or at least active on it. It also happens to serve to illustrate the properties of other number bases for folks who'd like to see things for themselves (skeptics/sceptics).
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icarus
Dozens Demigod
icarus
Dozens Demigod
Joined: 12:29 PM - Apr 11, 2006

2:15 PM - Aug 29, 2012 #9

Folks,

I've been able to take my multiplication tables from a print document and fold them into HTML so now I can produce tours of mid-scale bases pretty easily. I am working on a method of transferring Mathematica tables to Excel. When this happens I can produce any size table, though the notation won't be the same, instead it would be decimal coded. I think we can continue to explore the very high bases if desired.

Others have written great pieces on other bases. This series will continue to expand in the same character as the previous essays, with links to the other conversations (i.e., tridecimal, base-24, base-22, etc.). This way this forum will eventually have the broadest such number base tour on the internet, if it isn't already so.

I may revisit a few earlier threads and inject some new data so that they are on par with the latest essays. If there is anything (practical!) you'd like to add in the tour of each base, please suggest it.

Again, the frequent posters on this site continue to influence me in seeing the "sights" to be seen in these number bases. Please don't wait till I cover a given base to chat about it. I can mention a "sight" and link to whatever conversation has been had about it in other threads, so that we have an integrated experience.

I've written most of bases 9, 24, and 21; these will come next. I may try to write 60 or 120, but want to do my "experiment" first. As we push into September, I may not have much time to write, so the progress will be discontinuous.
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icarus
Dozens Demigod
icarus
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Joined: 12:29 PM - Apr 11, 2006

1:58 PM - Aug 31, 2012 #10

I've updated decimal, dozenal, and hexadecimal threads to show "regular figures", which are equivalent to "units" defined by Wendy Krieger at her "Number Theory 102" thread in the number theory forum. Also, I've added diagrams of patterns in the decimal, dozenal, and hexadecimal multiplication tables, and diagrams showing the relationship of the positive primes less than the base in the same threads.

There is a limit to the size of posts and this has affected the hexadecimal thread the most. The usual format is interrupted, but the same data is linked from the introductory post.
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icarus
Dozens Demigod
icarus
Dozens Demigod
Joined: 12:29 PM - Apr 11, 2006

9:27 PM - Sep 06, 2012 #11

I've added some limited tours of "grand bases" 210, 240, and 360. These were laying around this summer in partial form when I was mulling over other matters and away from the forum. If you've got a "grand base" to inspect, I'll put it on the queue. I am producing base 2520 but that's experimental and we'll see what needs to be done to do it. Hopefully some of these might spark some debate. I think they are curiosities, mountains that are fun to climb but impractical places to live.

Look at the tour menu, second post in this thread.
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Dan
Dozens Disciple
Dan
Dozens Disciple
Joined: 2:45 PM - Aug 08, 2005

12:34 AM - Sep 07, 2012 #12

While it may be a little too small to count as a "grand base", I'd like to see an analysis of Base 36, the highest base supported by the C strtol function (due to the convention of using 0-9 and A-Z for digits), and coincidentally a highly-composite number.
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Oschkar
Dozens Disciple
Oschkar
Dozens Disciple
Joined: 1:07 AM - Nov 19, 2011

1:08 AM - Sep 07, 2012 #13

Dan @ Sep 7 2012, 12:34 AM wrote: While it may be a little too small to count as a "grand base", I'd like to see an analysis of Base 36, the highest base supported by the C strtol function (due to the convention of using 0-9 and A-Z for digits), and coincidentally a highly-composite number.
I would like to see 36, 40, 42, 48, 56, 60, 72 and 84 as possible midscale bases on the way to grand 120 and higher.
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icarus
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icarus
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Joined: 12:29 PM - Apr 11, 2006

2:41 AM - Sep 07, 2012 #14

I wrote about base 144 tonight, and it ties to many posts on this forum (Shaun, m1n1f1g et al.) in a cool way. Will do 100 too. These things tend to illustrate a tidbit, but aren't terribly useful. Then 36 will be next.

I have tables for 36 and 60. I am not sure the board will handle the tables, but maybe I could split them. I've dispensed with the tables for "grand bases", but may circle back and do other things done for the small bases. A link to a PDF for bases larger than 40 or so might relieve the board of so many big tables. This stuff can fold over into indesign to be a big book of bases one day, that's exciting (to me).

Indeed many of the bases Oshkar suggested are on the list. I had HCNs, and the "runner up" bases to the HCNs, I think they are {24, 30, 36, 48, 60, 72, 84, 90, 96, 120, 144, 168, 180} plus Wendy's/Triesaran's beneficial-flank numbers {21, 34, 55, 99, 120} (doing from memory). Will add 42, 56. I'd written the full list on the menu post but commented them out. (it looked daunting! And I didn't want to promise and leave it stand) I also felt averse to starting a bunch of threads, only to have a "partial" set. Now I think we start threads and folks can comment and see patterns etc., with other info coming later. I think I'll start the bases off like the grand bases then fill in the details later. If you've written something about the bases please post a link to your discussions in the tour thread so people reach your thoughts, if I hadn't done that already. I am not as familiar with what happened between May and mid August this year, due to all the mulling on another non-number-base issue. (my midlife crisis, lol. Good thing: it's mostly over!)

Any idea on a really big prime (between say 60 and 360), just to illustrate what many of these look like? I think 55 will serve as a really big semi prime.
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Dan
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Dan
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Joined: 2:45 PM - Aug 08, 2005

2:50 AM - Sep 07, 2012 #15

icarus @ Sep 6 2012, 09:41 PM wrote:Any idea on a really big prime (between say 60 and 360), just to illustrate what many of these look like?
61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359

Pick one.
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wendy.krieger
wendy.krieger

7:50 AM - Sep 07, 2012 #16

70 has a place in there somewhere. This is the first abundant number that can not be represented as the sum of its divisors. Also, i am rather fond of it. It has, for example, among the squares, 2.00.01.
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Oschkar
Dozens Disciple
Oschkar
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Joined: 1:07 AM - Nov 19, 2011

8:37 PM - Sep 07, 2012 #17

Dan @ Sep 7 2012, 02:50 AM wrote:
icarus @ Sep 6 2012, 09:41 PM wrote:Any idea on a really big prime (between say 60 and 360), just to illustrate what many of these look like?
61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359

Pick one.
In this case, I'd choose 109, probably, for its intutive relationships with 108=2^2x3^3 and 110=2x5x11. It is not exactly the typical large prime base, but it is rather versatile for such a base. On the other hand, 173 is an example of the worst possible type of prime base, with 172=2^2x43 and 174=2x3x29 as neighbors. You could do both as contrasting bases.

And I suggest that you separate 60 from 120.
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icarus
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icarus
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Joined: 12:29 PM - Apr 11, 2006

10:04 PM - Sep 07, 2012 #18

Oschkar: surely we will separate bases 60 and 120: that was a comparison and a stand-in till I get a proper page set up.

I like the suggestion 109 and 173 for the very reasons you've described.

The weird thing that occurred to me is with any composite, I can "shorten" the table by using two divisors. For the primes, the "shortening" will look odd, because the numbers don't have nontrivial integer divisors. I am trying to respect 30 cells as the maximum width, so might use some convenient number below that to "fold" the cells.

Wendy: 70 is now on the list!
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m1n1f1g
Dozens Disciple
m1n1f1g
Dozens Disciple
Joined: 10:15 AM - Feb 20, 2011

11:16 PM - Sep 08, 2012 #19

icarus @ Sep 7 2012, 03:41 AM wrote: I have tables for 36 and 60.
What about complementary divisor method ("truncated") tables? I'd like to see what they're like in terms of size and memorability.
A few little conventions:
- Dozenal integers suffixed with prime (′). This is the uncial point.
- Decimal integers suffixed with middle dot (·). This is the decimal point.

You may see me use * prefix for messages before 11Ɛ7-03-1X, and a whole range of similar radix points. I will often use X and Ɛ for :A and :B.

Sometimes, I will imply that an integer is in dozenal, so I won't add any marks to it. You should be able to tell that "10 = 22 * 3" is in dozenal.
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Treisaran
Dozens Disciple
Treisaran
Dozens Disciple
Joined: 1:00 PM - Feb 14, 2012

12:25 AM - Sep 09, 2012 #20

Very impressive, Icarus! (Yeah, mood's improved a bit the past few days :) ) I'm especially interested in the rundowns of those bases that have a relationship to the dozen: 6 (half-dozen; already here), 18 (dozen half-dozen, or as I call it, dozen ha'zen ;) ), 24 (double dozen), 36 (triple dozen), 96 (octodoz?) etc. The last three I find intriguing because of the slight improvements (at an undeniable price, of course) they offer over dozenal, through their neighbour relations (96 is for the 240-lovers who want something better than a primeflank but don't want to abandon the sixteenfold division). All in contrast to 18, that 'total loss' of near-dozenal bases, which gains nothing and loses quite a lot.

On another note, I think a comparison of 34 and 120 is warranted: together with their neighbour relationships, they offer the same factors:
  • 34 = 2·17, ω = 3·11, α = 5·7
  • 120 = 2·2·2·3·5, ω = 7·17, α = 11·11
and yet, they don't have the same utility. The diprime 34 is at a considerable disadvantage in comparison to the divisor-rich 120. That's another case for the argument that handy neighbour relationships are no substitute for richness in divisors.
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icarus
Dozens Demigod
icarus
Dozens Demigod
Joined: 12:29 PM - Apr 11, 2006

2:05 AM - Sep 09, 2012 #21

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Oschkar
Dozens Disciple
Oschkar
Dozens Disciple
Joined: 1:07 AM - Nov 19, 2011

3:49 AM - Sep 09, 2012 #22

To me, the relationship between 21 and 22 looks a lot like the one between 14 and 15, only backwards, since 10, 14 and 22 are all manageable semiprimes. For example, like 10 uses the omega for 3, 14 the alpha, and 22 the omega again. Bases 10 and 14 both are compatible with 5, and 22 could use SPD to reduce the numbers with the alpha-2, since duovigesimal 101=decimal 445 is divisible, and there are still only 88 multiples to memorize (easily reduced to 22 by subtracting multiples of duovigesimal 50). For 7, 14 has it as a divisor, and in 22 it is an omega inheritor, but in decimal, 7 is out of SPD reach (143 multiples of 7 less than the decimal thousand, and subtracting multiples of 70, or even of 98, is rather impractical for a 3 digit number).
Although now that I think about it, the fact that 301 is so close to 300 could be used as an extension of SPD in decimal (not a neighbor of the square of the base, but of a multiple of it). To test for 7, instead of adjusting the remaining digits by what was added to the final 2, adjust them by triple that amount:
164472424704
1644724247 04+3=7 → 47+9=56
16447242 56
164472 42
1644 72-2=70 → 44-6=38
16 38-3=35 → 16-9=7
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Treisaran
Dozens Disciple
Treisaran
Dozens Disciple
Joined: 1:00 PM - Feb 14, 2012

1:50 PM - Sep 09, 2012 #23

icarus wrote:I have started base 2520, a leviathan, will take a couple weeks, jobs rolling in soon.
Good luck! But how are you going to squeeze the huge tables required for this monstrosity of a base into forum posts? <_<
Oschkar wrote:and there are still only 88 multiples to memorize (easily reduced to 22 by subtracting multiples of duovigesimal 50).
Compacting the number of SPD sequences, eh? Sounds good. I just wonder how far it could be carried. For base 22 it looks feasible, but I wonder about base 32, or the dozenal SPD test for 7 (which is based on *1001, the dozenal cube-&#945;).
and 22 could use SDN [...] 7 is out of SDN reach [...] could be used as an extension of SDN in decimal
SDN stands for 'Systematic Dozenal Nomenclature', a comprehensive dozenal number naming scheme devised by Kodegadulo on this board; SPD stands for 'Split, Promote, Discard', a divisibility testing shortcut method stumbled upon by yours truly. :)

Reminds me of that passage from the 1989 film Hunt for Red October: 'Pavarotti is a singer, Paganini was a composer'. :)
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Oschkar
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Oschkar
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Joined: 1:07 AM - Nov 19, 2011

5:32 PM - Sep 09, 2012 #24

Treisaran @ Sep 9 2012, 01:50 PM wrote:
and 22 could use SDN [...] 7 is out of SDN reach [...] could be used as an extension of SDN in decimal
SDN stands for 'Systematic Dozenal Nomenclature', a comprehensive dozenal number naming scheme devised by Kodegadulo on this board; SPD stands for 'Split, Promote, Discard', a divisibility testing shortcut method stumbled upon by yours truly. :)

Reminds me of that passage from the 1989 film Hunt for Red October: 'Pavarotti is a singer, Paganini was a composer'. :)
Thanks. Fixed.
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