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icarus 
Posted: May 1 2012, 03:01 PM

Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
Le Tour des BasesGet your road gear on, we’re pedaling into the hills visiting various number bases along the way! This forum and the Dozenal Societies often discuss number bases, especially base twelve. Here, we’ll cruise through the main features of each number base. Each tour consists of a brief “travelogue” from the standpoint of general human computation — what would it be like to actually use the number base on a daily basis? The travelogue refers you to “Icarus’s Standard Nomenclature for Number Bases”, a sort of “road map” explaining some of the sights along the way. Like a phrasebook for foreign travel, or a guidebook describing the kind of attractions you’ll likely see in every city on your trip, the map is important because it defines some of the terms used in the narrative. We need to know a few things common to all the number bases so that we can describe their properties “apples to apples”. We’ll dive into the digits of the number base, as all number bases represent the world through their digits. A short synopsis of the kinds of digits and their relationships to the base follows; digits are like the main attractions in a town. Here, some key terms link to expanded descriptions and references to maths texts so you can make sure we’re not taking you for a ride. Next, we’ll look at the addition and multiplication tables, basic tools of computation. Here you can “test drive” maths in the number base — try your hand at memorizing the facts. Running some arithmetic given these tables, you can get a little sense of what it’s like to “live” in the base we’re visiting. Fractions are a key consideration when thinking about a good number base. Will the base help us with fractions, or will it just make things more difficult? We’ll look at fractions in four different ways. First, we take a look at reciprocals of the smallest integers (example, the reciprocal of 2 is ½). Everyone loves fractions that have short terminating digital representations, e.g., ½ = 0.5 decimal. The second exploration of fractions examines the primes less than 30 to see whether their reciprocals have terminating or recurrent digital representations. If the prime reciprocal has a recurrent digital representation, the table lists the length compared to the maximum length, and the reptend (group of numbers repeated), e.g., one seventh decimally has a reptend length of 6 out of a maximum length of 6; it repeats “142857”. The third table looks at reciprocals of coprimes with the shortest 8 reptends. (Coprimes are integers “out of phase” with the base, such as 3 or 7 in base 10). The final fractionrelated table looks at the reciprocals of regular numbers smaller than the cube of the base. (Regular numbers are integers whose reciprocals terminate when expressend in the base in question, such as 4 or 25 in base 10). Our tour takes us through intuitive divisibility rules for the number base, then a quick shot through logarithms for each digit and the values of some irrational numbers. Oftentimes you may read claims on this forum and in the writings at the Dozenal Societies of America and Great Britain. Dozenalists tend to examine number bases more closely than many accomplished mathematicians, as number bases are an elementary form of mathematics. Their writings tend to incorporate notions with which the writers are familiar with in several bases, but with which many readers are not at all familiar. This tour is devised to let you examine the main properties of each number base so you can form your own opinion. The travelogues are indeed written from the perspective of someone looking for a base to facilitate computation and are a sort of introduction, but the data that follow it derive from elementary number theory, which anyone can test. It is my endeavour to maintain a fair tour, especially in the data portion of the visit, but the introductions indeed work like a travel guide celebrating the noble features of a base as well as some potential pitfalls you might encounter during your trip. I hope you enjoy the Tour des Bases! If you get a “flat tire” or spot a flaw in the route, post something and we’ll be happy to chat about it or amend the route. If you get winded on the hills, take a break and join us another day. Responses are welcome, as well as additional information you’d like to post, after all, this is a forum! Get your helmets, fill those water bottles, and pack a spare tube, let’s roll! (Start the tour with base 6, 8, 10, 11, 12, 14, 15, 16, or 20; new tours will be posted now and then). 
icarus 
Posted: May 1 2012, 03:04 PM

Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
Le Tour des Bases · List of ToursLinks to tours of number bases follow. Note that tours to new bases will be posted from time to time. I’ve written notes for bases between 2 and 30, all of which will eventually find their way on the tour.
Base 2 • Busy Binary · Keep It Simple, Smily! “Human Scale” Bases Lower MidScale Bases Upper MidScale Bases Grand Bases “Randomly Considered” Bases 
icarus 
Posted: May 1 2012, 03:14 PM


Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
Map of the TourThis map arranges each number base between 2 and 30 inclusive in rows, the base listed in the leftmost column. Each digit of each base appears on the same row as the listed base. The decimal values of the base and the digits are given for your convenience. Read more about each type of digit here.


Dan 
Posted: May 2 2012, 12:39 AM


Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
I'm curious as to how you decided that 10 is male but 8, 9, and 18 are female. 

icarus 
Posted: May 2 2012, 02:06 AM

Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
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. 
Dan 
Posted: May 2 2012, 03:45 AM

Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
I don't know Russian, so the hint is lost on me.

dgoodmaniii 
Posted: May 2 2012, 06:08 PM


Dozens Demigod Group: Admin Posts: 1,927 Member No.: 554 Joined: 21May 09 
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. 

icarus 
Posted: May 2 2012, 09:11 PM

Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
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). 
icarus 
Posted: Aug 29 2012, 02:15 PM

Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
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 midscale 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, base24, base22, 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. 
icarus 
Posted: Aug 31 2012, 01:58 PM

Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
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. 
icarus 
Posted: Sep 6 2012, 09:27 PM

Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
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. 
Dan 
Posted: Sep 7 2012, 12:34 AM

Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
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 09 and AZ for digits), and coincidentally a highlycomposite number.

Oschkar 
Posted: Sep 7 2012, 01:08 AM


Dozens Disciple Group: Members Posts: 575 Member No.: 623 Joined: 19November 11 
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. 

icarus 
Posted: Sep 7 2012, 02:41 AM

Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
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 beneficialflank 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 nonnumberbase 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. 
Dan 
Posted: Sep 7 2012, 02:50 AM


Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
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. 

wendy.krieger 
Posted: Sep 7 2012, 07:50 AM

Dozens Demigod Group: Members Posts: 2,432 Member No.: 655 Joined: 11July 12 
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.

Oschkar 
Posted: Sep 7 2012, 08:37 PM


Dozens Disciple Group: Members Posts: 575 Member No.: 623 Joined: 19November 11 
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. 

icarus 
Posted: Sep 7 2012, 10:04 PM

Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
Oschkar: surely we will separate bases 60 and 120: that was a comparison and a standin 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! 
m1n1f1g 
Posted: Sep 8 2012, 11:16 PM


Dozens Disciple Group: Members Posts: 826 Member No.: 591 Joined: 20February 11 
What about complementary divisor method ("truncated") tables? I'd like to see what they're like in terms of size and memorability. 

Treisaran 
Posted: Sep 9 2012, 12:25 AM

Dozens Disciple Group: Members Posts: 1,221 Member No.: 630 Joined: 14February 12 
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 (halfdozen; already here), 18 (dozen halfdozen, 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 240lovers 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 neardozenal 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:
and yet, they don't have the same utility. The diprime 34 is at a considerable disadvantage in comparison to the divisorrich 120. That's another case for the argument that handy neighbour relationships are no substitute for richness in divisors. 
icarus 
Posted: Sep 9 2012, 02:05 AM

Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
M1n1f1g,
I'll add abbreviated tables soon, till then here are AMTs for bases 1056 and 60120 at my web portfolio.
treisaran thanks for the feedback! Indeed I haven't posted base 18 yet, and have skipped the mid scale bases thus far. I have started base 2520, a leviathan, will take a couple weeks, jobs rolling in soon.
Indeed I am trying to integrate the tour with your thoughts as well as those of others. I'll also direct folks to the SPD test in the 144 and 12 entries.

Oschkar 
Posted: Sep 9 2012, 03:49 AM

Dozens Disciple Group: Members Posts: 575 Member No.: 623 Joined: 19November 11 
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 alpha2, 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 â†’ 47+9=56 16447242 56 164472 42 1644 722=70 â†’ 446=38 16 383=35 â†’ 169=7 
Treisaran 
Posted: Sep 9 2012, 01:50 PM


Dozens Disciple Group: Members Posts: 1,221 Member No.: 630 Joined: 14February 12 
Good luck! But how are you going to squeeze the huge tables required for this monstrosity of a base into forum posts?
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 cubeα).
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'. 

Oschkar 
Posted: Sep 9 2012, 05:32 PM


Dozens Disciple Group: Members Posts: 575 Member No.: 623 Joined: 19November 11 
Thanks. Fixed. 

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