I must say from the outset that I beg forgiveness for not introducing myself sooner  if I had of known that there was such a section to post details of myself ( so that you all could get to know me ) I would have posted here first before posting on “Calendar Reform “ but I only just now scrolled down far enough to see that there was indeed this section available to inform all you members a little about myself.
My name is Gregory Neil Collie. I am a Leo and my birthdate is a perfectly balanced 5 in numerology. I live in the Tropics in Cairns, Queensland, Australia but I rent. I hold a degree in Economics from La Trobe University and it was there that the Resident Professor of my college invited me to his apartment for afternoon tea one day and first introduced me to the concept of a 12 base number system  damn I wish I could remember his name but it continues to allude me! Anyway having come from a typical working class family (my father was a fitter and turner with Kenworth Trucks and my mother was a bookkeeper) and me studying “the dismal science” I grew up quite hard and ended up so depressed that I was diagnosed with Bipolar Disorder  a condition that, after multiple hospital admissions later, I continue to take medication for up until this day. But don’t worry  I haven’t had a relapse or been back in hospital for nearly 10 years  I am quite stable now. And that’s about it except to say that I have never married nor had any children and that I continue to be happy living a bachelors lifestyle (probably because of the drugs I take ha ha ha) although if the right girl came into my life I would enjoy having her company in my later years.

hotdog8Casual Member
 Joined: Feb 21 2018, 01:52 PM
Thank you Icarus  I must say from the outset that, without knowing too much about you yet, I concur with your description as being a Dozenal Demigod seeing that you joined the DSA at the very young age of 35 and have posted over 2000 times! I myself joined at the old age of 52. Can I take a guess and say that your field of expertise is either Mathematics or Computer Science??icarus wrote: Hello hotdog8 and welcome to the forum!

KodegaduloObsessive poster
 Joined: Sep 10 2011, 11:27 PM
Welcome to the forum, hotdog8!
If you're looking for someone here who actually gets paid to develop software, that would be me. And the first time I stumbled upon this forum (and indeed the whole topic of dozenal as an actual movement) I too was already at a ripe age (50_{d} in my case, although I prefer to think of it as 42_{z} )
No, professionally icarus is in the architecture business (as in, designing actual physical buildings). His interest in number theory (particularly cooking up interesting number sequences for the OEIS) and mathematical programming to support those investigations, is an amateur side interest (although it's fair to say he's gotten quite adept at that).hotdog8 wrote:Can I take a guess and say that your field of expertise is either Mathematics or Computer Science??
If you're looking for someone here who actually gets paid to develop software, that would be me. And the first time I stumbled upon this forum (and indeed the whole topic of dozenal as an actual movement) I too was already at a ripe age (50_{d} in my case, although I prefer to think of it as 42_{z} )
Last edited by Kodegadulo on Mar 25 2018, 04:04 PM, edited 1 time in total.
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Baseneutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Baseneutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)

ShaunDozens Disciple
 Joined: Aug 2 2005, 04:09 PM
And for many years ... I joined the DSGB when it was founded in the late fifties. Which also makes me a wee bit older than the rest of the mob here  *67 this year.hotdog8 wrote:So I take it that you reside in England at present ?Shaun wrote: Admin, and retired Secretary of the DSGB. I look in now and again to see if there are any new posts.

icarusDozens Demigod
 Joined: Apr 11 2006, 12:29 PM
I joined at two dozen eleven (currently will celebrate four dozen years this July), but "discovered" dozenal when I was a dozen years old (coincidence). I liked how the fractions "came out", meaning 1/2 = .6, 1/3 = .4, 1/4 = .3, 1/6 = .2, 1/8 = .16, 1/9 = .14, 1/14 = .09. Erroneously I thought 1/5 = .24 (it is .24972497...). This was by influence of my aunt who had been in computer science; till then my "first love" was hexadecimal, since age eleven. But really got deep into dozenal during sophomore geometry class. I am a registered architect. Over the last duodecade (dozenade? unquade?) I have become increasingly overcome with numbertheoretic questions that did arise from trying to multiply in sexagesimal without using the full table of over a great gross unique values. This lead to the "discovery" of two species of necessarily composite nondivisors that are not coprime to the base, the "semidivisor" and the "semitotative", the former's relationship as "regular" to the base (meaning it divides some power of the base), the divisors also a special case of "regular" numbers in a given base. This was used to examine number bases for their performance regarding a set of three principal applications: arithmetic, divisibility tests, and expansion of fractions, that became the "Tour des Bases" (which is now in ruins on this forum, but we shall rebuild!) The aim of the Tour was to allow people to "test drive" each base to arrive at their own conclusions about them. Personally it has helped steep my own conviction that dozenal is the optimum base for general arithmetic. In the past 4 years I've become involved with the OEIS. So in that capacity my later writing might have the feel of mathematics, as most of what we discuss when we discuss base twelve must frame with maths. (There are some nonmathematical considerations that are perhaps more related to human cognition, like the speed of learning multiplication facts, or how many digits can someone hold in mind over a given time, etc. There are also "design" considerations, like the form of numerals, and numeral names. Systems of measure are also design problems.) Most of the maths I've done came about through trying to knock down my favorite number. Was it just a favorite number, or is it deeper. This attempt has led to a greater appreciation for decimal and other bases. After a dozen years of trying to smash the notion through logic and maths, I remain convinced base twelve is best.
Look forward to your input!
Look forward to your input!

hotdog8Casual Member
 Joined: Feb 21 2018, 01:52 PM
LOL Yes Kodegadulo both yours and my age sounds a lot better expressed dozenally hey, and sorry to everyone who has read my posts for me primarily using decimal numbers in them  it's like I am Pavlov's Dog  conditioned over time to think decimally and not dozenally! But thanks to your quip Kodegadulo I will remember to express myself dozenally in the future.Kodegadulo wrote: Welcome to the forum, hotdog8!
No, professionally icarus is in the architecture business (as in, designing actual physical buildings). His interest in number theory (particularly cooking up interesting number sequences for the OEIS) and mathematical programming to support those investigations, is an amateur side interest (although it's fair to say he's gotten quite adept at that).hotdog8 wrote:Can I take a guess and say that your field of expertise is either Mathematics or Computer Science??
If you're looking for someone here who actually gets paid to develop software, that would be me. And the first time I stumbled upon this forum (and indeed the whole topic of dozenal as an actual movement) I too was already at a ripe age (50_{d} in my case, although I prefer to think of it as 42_{z} )

hotdog8Casual Member
 Joined: Feb 21 2018, 01:52 PM
Wow! You must have read a lot of interesting articles and posts in your time! So Shaun are we getting anywhere? Do we have the answers and solutions needed to introduce for example base 12 to the Western World or are we still debating and arguing over whether senary, duodecimal, hexadecimal or sexagesimal is the better number base? Do we know any U.S. congressmen, or any delegates to the United Nations (I can totally understand if you don't want to tell me that!) etc. I wonder to myself how can the DSA influence the world (and in my opinion make it a better and fairer place) when it only has 125z members  I would like to see change within my lifetime Shaun. I agree with Icarus that base 12 is the way to go (and he should know seeing that  according to what Kodegadulo told me  he has had multiple "entries" into Sloane's) simply because I think it would be the easiest to learn and teach our children and hence be easily accepted by the worldwide community  children can count in dozens simply by using their fingers on their hands (3 sections on each finger by 4 fingers to a hand gives 12 sections). And I believe that we could persuade the Jewish Faith and hence then the Christians to accept a dozenal calendar if we allude them to the Star of David that they display so proudly on their flag and synagogues. Just as the pentagram philosophically represents both the human body (like da Vinci represented with his Vitruvian Man) and also our hand (the pentagon in the middle of the pentagram being the palm) that gave our ancestors and us the reason to count in decimal tens, so too could we possibly persuade the Jewish Faith that their symbol is like a hand with 6 fingers giving rise to a duodecimal counting system. Of course nature determined that we do only have five fingers on each hand, which in hindsight today may have been a mistake in evolutionary terms compared with theoretical number systems. But maybe their God gave them the vision to adopt the Star of David because he wanted them to count in sixes and twelves!Shaun wrote:And for many years ... I joined the DSGB when it was founded in the late fifties. Which also makes me a wee bit older than the rest of the mob here  *67 this year.hotdog8 wrote:So I take it that you reside in England at present ?Shaun wrote: Admin, and retired Secretary of the DSGB. I look in now and again to see if there are any new posts.

hotdog8Casual Member
 Joined: Feb 21 2018, 01:52 PM
If you take a moment to read my reply to Shaun before reading this post Icarus you will see that I totally support your view that base 12 is the best. I am however accepting your opinion on trust based on your experience and qualifications  I must admit that I could not understand the number theory you were explaining to me about the faults of the sexagesimal base system. My only real understanding is that all positive integers are either composite or prime and by definition what is not composite must be prime. What I couldn't understand was the terms "necessarily composite nondivisors that are not coprime with the base"  Ohhhhhh I think I understand you now!!!! They are composites that do not divide into the base BUT are not prime either!!! Please post me back and tell me if i am right Icarus.icarus wrote: I joined at two dozen eleven (currently will celebrate four dozen years this July), but "discovered" dozenal when I was a dozen years old (coincidence). I liked how the fractions "came out", meaning 1/2 = .6, 1/3 = .4, 1/4 = .3, 1/6 = .2, 1/8 = .16, 1/9 = .14, 1/14 = .09. Erroneously I thought 1/5 = .24 (it is .24972497...). This was by influence of my aunt who had been in computer science; till then my "first love" was hexadecimal, since age eleven. But really got deep into dozenal during sophomore geometry class. I am a registered architect. Over the last duodecade (dozenade? unquade?) I have become increasingly overcome with numbertheoretic questions that did arise from trying to multiply in sexagesimal without using the full table of over a great gross unique values. This lead to the "discovery" of two species of necessarily composite nondivisors that are not coprime to the base, the "semidivisor" and the "semitotative", the former's relationship as "regular" to the base (meaning it divides some power of the base), the divisors also a special case of "regular" numbers in a given base. This was used to examine number bases for their performance regarding a set of three principal applications: arithmetic, divisibility tests, and expansion of fractions, that became the "Tour des Bases" (which is now in ruins on this forum, but we shall rebuild!) The aim of the Tour was to allow people to "test drive" each base to arrive at their own conclusions about them. Personally it has helped steep my own conviction that dozenal is the optimum base for general arithmetic. In the past 4 years I've become involved with the OEIS. So in that capacity my later writing might have the feel of mathematics, as most of what we discuss when we discuss base twelve must frame with maths. (There are some nonmathematical considerations that are perhaps more related to human cognition, like the speed of learning multiplication facts, or how many digits can someone hold in mind over a given time, etc. There are also "design" considerations, like the form of numerals, and numeral names. Systems of measure are also design problems.) Most of the maths I've done came about through trying to knock down my favorite number. Was it just a favorite number, or is it deeper. This attempt has led to a greater appreciation for decimal and other bases. After a dozen years of trying to smash the notion through logic and maths, I remain convinced base twelve is best.
Look forward to your input!

ShaunDozens Disciple
 Joined: Aug 2 2005, 04:09 PM
I don't think anyone is seriously suggesting a base other than dozenal to replace decimal, but what we are trying to do  and since the introduction of the Internet, doing quite well  is spread the word to introduce people to the concept of dozenal application. I don't know about the DSA statistics, but our DSGB site has regularly been attracting some two gross or so of visitors from all over the world, and the forum here attracts people who are interested in finding out more.hotdog8 wrote:Wow! You must have read a lot of interesting articles and posts in your time!
Do we have the answers and solutions needed to introduce for example base 12 to the Western World or are we still debating and arguing over whether senary, duodecimal, hexadecimal or sexagesimal is the better number base? Do we know any U.S. congressmen, or any delegates to the United Nations (I can totally understand if you don't want to tell me that!) etc. I wonder to myself how can the DSA influence the world

hotdog8Casual Member
 Joined: Feb 21 2018, 01:52 PM
Yes Shaun the internet IS a powerful tool to spread ideas and information to people all over the world and hopefully those newcomers to the DSGB and also the DSA will spread the word about Dozenal to everyone they know  word of mouth is another way to raise awareness especially when combined with tools like Facebook and Twitter ! Also I’m glad that you too think that base 12 is the only serious option  I think it is what we should exclusively be promoting.

icarusDozens Demigod
 Joined: Apr 11 2006, 12:29 PM
The DSA site reaches great grosses of people monthly (from the stats I used to see when I managed the site).
A regular number k is one whose prime divisors p are a subset of those of the base n. Rather we could say that there is no prime factor q that does not divide base n. Another way to look at it is that k divides some power n^e with integer e ≥ 0. Divisors k = d divide n^e with 0 ≤ e ≤ 1, while semidivisors k divide n^e with e > 1. The divisor d is a special case of regular number k in base n. In decimal we have the divisors {1, 2, 5, 10} and semidivisors {4, 8} thus the regulars are {1, 2, 4, 5, 8, 10, 16, 20, 25, ...}. The unit fractions of all these numbers terminate when expanded decimally: {1, .5, .25, .2, .125, .1, .0625, .05, .04, ...}. The regular numbers have the "regular" divisibility test: look at the end digit, if it is one in a list, then it is divisible by that number. In base ten, we know a number is divisible by 5 if it ends in {0, 5}. But we also can see if it is divisible by 8 if it is divisible by one of 125 threedigit endings (not very practical). Fractions are keener in dozenal: we have divisors {1, 2, 3, 4, 6, 10} and semidivisors {8, 9} to make regular {1, 2, 3, 4, 6, 8, 9, 10, 14, 16, 20, ...} and these terminate {1, .6, .4, .3, .2, .16, .14, .1, .09, .08, .06, ...} and are as can be seen generally briefer terminations for smaller, more commonly encountered numbers, as the prime divisors of the dozen are the two smallest primes.
A "neutral" number m is one that neither divides nor is coprime to n. The numbers 3 and 10 are coprime since the latter is 2 × 5 and thus their greatest common divisor (highest common factor) is the empty product 1. Looking at ten, we have {1, 3, 7, 9} coprime to ten (these are called "totatives"), thus, the cototient, those numbers less than ten that are not coprime to it, consists of {2, 4, 5, 6, 8, 10}. Among these, {2, 5, 10} divide ten, and we are left with the nondivisors {4, 6, 8} in the cototient. These are neutral to ten. Neutrals are necessarily composite. There are two species of neutrals: the first are the semidivisors {4, 8} that divide some power of ten, and the semitotative {6} that does not divide any power of ten. The nature of a semitotative is twofold: it has a regular factor and a coprime factor. Decimal 6 is the product of the divisor 2 and the "omega" totative 3; the "richness" of the regular and the "relatedness" of the totative make semitotatives "kind" or "mean" (we say "transparent" or "opaque" here on the forum). The unit fraction of a semitotative has a mixed recurrent expansion in its base. 1/6 = .1666666... in decimal. 1/14 = 0.0714285714285... There is a nonrepeating "preamble" then the decimal never ends. Semitotatives have a compound divisibility test. In decimal, an arbitrary number x is divisible by 6 if it is even (divisible by divisor 2) and if the digits of 6 total to a number divisible by 3 (the "omega" totative rule, commonly known as the Rule of 3 and 9 in base ten). In base twelve we have the totatives {1, 5, 7, b}, thus the cototient is large: {2, 3, 4, 6, 8, 9, a, 10}. Of these we have many divisors {2, 3, 4, 6, 10} leaving us with {8, 9, a}; the first two divide the gross, only ten does not divide any power of twelve. Dozenally, 1/a = .124972497... and 1/12 = .0a35186a35186... We can tell if a number is divisible by ten in base twelve if it is even, and if it is divisible by 5. Now 5 is an "opaque totative" in base twelve, so it does not enjoy the "omega divisibility rule" ("wasted" on eleven!) But we can use a technique that a Member here called Treisaran promoted. Since 1 more than the gross (decimal 145) is divisible by 5, we can group the digits of a number in pairs and do the following. We know that dozenal 5954 is divisible by 5 since 59 − 54 = 5 (that was an easy one). Since it is even we also know it is divisible by ten. (dozenal 5954 = decimal 10000). Thus decimal 6 is "nice", a transparent omegarelated semitotative, while dozenal ten is "not nice": it is an opaque semitotative, but we can mollify it by learning our fives well in base twelve.
We want more regular numbers, particularly divisors in our number base, and we want fewer totatives, except that a highly divisible number "omega" n − 1 (or slightly less desirable, "alpha" n + 1) is useful in fractions and divisibility tests (the decimal 9 is a "saving grace" for that base). We want semitotatives that have divisor and "omega" or "alpha" as factors; and we want few of those in the smallest range of numbers; especially as digits of the base. Dozenal beats decimal in these regards, with the exception that decimal has that keen relationship with 9 that ameliorates the "skipped" prime 3 (1/3 = .33333..., to see if divisible by 3, add 'em up) and mollifies 6 and 9. In dozenal, we "skip" no prime, but the next prime 5 is not friendly (1/5 = .24972497..., already described the test for 5). In octal half the digits are totatives; 1/3 = .25252525..., 1/5 = .14631463.... In hexadecimal we have half the digits totative, but TWO omega totatives, so 1/3 = .555555555... and 1/5 = .33333333... (but its multiplication table has 136 unique values to memorize: dozenal only 78 and decimal 55, but we commonly memorize 78 anyway, so learning arithmetic in the lower grades of school might take twice as long). Only senary has a keener set (i.e., the same set of regulars, totatives, and semitotatives; it is the "core" of twelve, meaning that it is the squarefree root; 6 = 2 × 3 but 12 = 2² × 3.
The problem we have in senary is the loss of that additional power of 2, lengthening the expansions of fractions of the powers of 2: 1/4 = .13, 1/12 (one eighth) = .043, 1/24 (one sixteenth) = .0213, while in dozenal 1/4 = .3, 1/8 = .16, 1/14 = .09. This also serves to illustrate another, nonnumber theoretic problem with base 6. It becomes "verbose". What takes dozenal one digit to convey requires 1.079 decimal digits, but 1.387 senary. Now for small numbers, fine. But human memory only can ably deal with so many digits. In base 6 the numerals are fewer and so the numbers begin to look monotonous. Finally, there are more uses for number than mathematics. One of the problems we have with senary is that we run out of numbers of a certain number of digits, say twodigit senary. There are only three dozen such (counting 00, 01, etc.); then we need three digits. But with dozenal we have a gross, and even decimal has a hundred, which is 2.777... times as many as senary has. To sum, dozenal is superior to senary, despite the latter's simplicity, because it is more flexible. Senary life is like rowing a boat with teaspoons when you could use oars.
The numbertheoretic and humancognitive qualities* of the dozen seem to make it the best base. (* this latter set of qualities would require scientific testing to be a solid leg upon which to stand. This is the reason why I try to stick to maths. It would stand to reason that a multiplication table nearly 3/4 larger than the one we memorize in second and third grades would take kids longer to memorize, especially if it is loaded with facts like the hexadecimal thirteens table: {d,1a,27,34,41,4e,5b,68,75,82,8f,9c,a9,b6,c3,d0} ).
A regular number k is one whose prime divisors p are a subset of those of the base n. Rather we could say that there is no prime factor q that does not divide base n. Another way to look at it is that k divides some power n^e with integer e ≥ 0. Divisors k = d divide n^e with 0 ≤ e ≤ 1, while semidivisors k divide n^e with e > 1. The divisor d is a special case of regular number k in base n. In decimal we have the divisors {1, 2, 5, 10} and semidivisors {4, 8} thus the regulars are {1, 2, 4, 5, 8, 10, 16, 20, 25, ...}. The unit fractions of all these numbers terminate when expanded decimally: {1, .5, .25, .2, .125, .1, .0625, .05, .04, ...}. The regular numbers have the "regular" divisibility test: look at the end digit, if it is one in a list, then it is divisible by that number. In base ten, we know a number is divisible by 5 if it ends in {0, 5}. But we also can see if it is divisible by 8 if it is divisible by one of 125 threedigit endings (not very practical). Fractions are keener in dozenal: we have divisors {1, 2, 3, 4, 6, 10} and semidivisors {8, 9} to make regular {1, 2, 3, 4, 6, 8, 9, 10, 14, 16, 20, ...} and these terminate {1, .6, .4, .3, .2, .16, .14, .1, .09, .08, .06, ...} and are as can be seen generally briefer terminations for smaller, more commonly encountered numbers, as the prime divisors of the dozen are the two smallest primes.
A "neutral" number m is one that neither divides nor is coprime to n. The numbers 3 and 10 are coprime since the latter is 2 × 5 and thus their greatest common divisor (highest common factor) is the empty product 1. Looking at ten, we have {1, 3, 7, 9} coprime to ten (these are called "totatives"), thus, the cototient, those numbers less than ten that are not coprime to it, consists of {2, 4, 5, 6, 8, 10}. Among these, {2, 5, 10} divide ten, and we are left with the nondivisors {4, 6, 8} in the cototient. These are neutral to ten. Neutrals are necessarily composite. There are two species of neutrals: the first are the semidivisors {4, 8} that divide some power of ten, and the semitotative {6} that does not divide any power of ten. The nature of a semitotative is twofold: it has a regular factor and a coprime factor. Decimal 6 is the product of the divisor 2 and the "omega" totative 3; the "richness" of the regular and the "relatedness" of the totative make semitotatives "kind" or "mean" (we say "transparent" or "opaque" here on the forum). The unit fraction of a semitotative has a mixed recurrent expansion in its base. 1/6 = .1666666... in decimal. 1/14 = 0.0714285714285... There is a nonrepeating "preamble" then the decimal never ends. Semitotatives have a compound divisibility test. In decimal, an arbitrary number x is divisible by 6 if it is even (divisible by divisor 2) and if the digits of 6 total to a number divisible by 3 (the "omega" totative rule, commonly known as the Rule of 3 and 9 in base ten). In base twelve we have the totatives {1, 5, 7, b}, thus the cototient is large: {2, 3, 4, 6, 8, 9, a, 10}. Of these we have many divisors {2, 3, 4, 6, 10} leaving us with {8, 9, a}; the first two divide the gross, only ten does not divide any power of twelve. Dozenally, 1/a = .124972497... and 1/12 = .0a35186a35186... We can tell if a number is divisible by ten in base twelve if it is even, and if it is divisible by 5. Now 5 is an "opaque totative" in base twelve, so it does not enjoy the "omega divisibility rule" ("wasted" on eleven!) But we can use a technique that a Member here called Treisaran promoted. Since 1 more than the gross (decimal 145) is divisible by 5, we can group the digits of a number in pairs and do the following. We know that dozenal 5954 is divisible by 5 since 59 − 54 = 5 (that was an easy one). Since it is even we also know it is divisible by ten. (dozenal 5954 = decimal 10000). Thus decimal 6 is "nice", a transparent omegarelated semitotative, while dozenal ten is "not nice": it is an opaque semitotative, but we can mollify it by learning our fives well in base twelve.
We want more regular numbers, particularly divisors in our number base, and we want fewer totatives, except that a highly divisible number "omega" n − 1 (or slightly less desirable, "alpha" n + 1) is useful in fractions and divisibility tests (the decimal 9 is a "saving grace" for that base). We want semitotatives that have divisor and "omega" or "alpha" as factors; and we want few of those in the smallest range of numbers; especially as digits of the base. Dozenal beats decimal in these regards, with the exception that decimal has that keen relationship with 9 that ameliorates the "skipped" prime 3 (1/3 = .33333..., to see if divisible by 3, add 'em up) and mollifies 6 and 9. In dozenal, we "skip" no prime, but the next prime 5 is not friendly (1/5 = .24972497..., already described the test for 5). In octal half the digits are totatives; 1/3 = .25252525..., 1/5 = .14631463.... In hexadecimal we have half the digits totative, but TWO omega totatives, so 1/3 = .555555555... and 1/5 = .33333333... (but its multiplication table has 136 unique values to memorize: dozenal only 78 and decimal 55, but we commonly memorize 78 anyway, so learning arithmetic in the lower grades of school might take twice as long). Only senary has a keener set (i.e., the same set of regulars, totatives, and semitotatives; it is the "core" of twelve, meaning that it is the squarefree root; 6 = 2 × 3 but 12 = 2² × 3.
The problem we have in senary is the loss of that additional power of 2, lengthening the expansions of fractions of the powers of 2: 1/4 = .13, 1/12 (one eighth) = .043, 1/24 (one sixteenth) = .0213, while in dozenal 1/4 = .3, 1/8 = .16, 1/14 = .09. This also serves to illustrate another, nonnumber theoretic problem with base 6. It becomes "verbose". What takes dozenal one digit to convey requires 1.079 decimal digits, but 1.387 senary. Now for small numbers, fine. But human memory only can ably deal with so many digits. In base 6 the numerals are fewer and so the numbers begin to look monotonous. Finally, there are more uses for number than mathematics. One of the problems we have with senary is that we run out of numbers of a certain number of digits, say twodigit senary. There are only three dozen such (counting 00, 01, etc.); then we need three digits. But with dozenal we have a gross, and even decimal has a hundred, which is 2.777... times as many as senary has. To sum, dozenal is superior to senary, despite the latter's simplicity, because it is more flexible. Senary life is like rowing a boat with teaspoons when you could use oars.
The numbertheoretic and humancognitive qualities* of the dozen seem to make it the best base. (* this latter set of qualities would require scientific testing to be a solid leg upon which to stand. This is the reason why I try to stick to maths. It would stand to reason that a multiplication table nearly 3/4 larger than the one we memorize in second and third grades would take kids longer to memorize, especially if it is loaded with facts like the hexadecimal thirteens table: {d,1a,27,34,41,4e,5b,68,75,82,8f,9c,a9,b6,c3,d0} ).

hotdog8Casual Member
 Joined: Feb 21 2018, 01:52 PM
I'm sorry Icarus  I realise that you have tried to be as logical and simple for me to understand a little about number theory but I will not pretend to you that I understand what you are saying. But that doesn't matter  you and (I am sure) many others in the Forum understand how you can basically prove that duodecimal is the best system overall and again I am sure that if we ever get conflict from advocates of decimal or other nonefficient bases by certain mathematicians, computer programmers, architects etc then Dr. Goodman III will have you on the frontline of defense!icarus wrote: The DSA site reaches great grosses of people monthly (from the stats I used to see when I managed the site).
A regular number k is one whose prime divisors p are a subset of those of the base n. Rather we could say that there is no prime factor q that does not divide base n. Another way to look at it is that k divides some power n^e with integer e ≥ 0. Divisors k = d divide n^e with 0 ≤ e ≤ 1, while semidivisors k divide n^e with e > 1. The divisor d is a special case of regular number k in base n. In decimal we have the divisors {1, 2, 5, 10} and semidivisors {4, 8} thus the regulars are {1, 2, 4, 5, 8, 10, 16, 20, 25, ...}. The unit fractions of all these numbers terminate when expanded decimally: {1, .5, .25, .2, .125, .1, .0625, .05, .04, ...}. The regular numbers have the "regular" divisibility test: look at the end digit, if it is one in a list, then it is divisible by that number. In base ten, we know a number is divisible by 5 if it ends in {0, 5}. But we also can see if it is divisible by 8 if it is divisible by one of 125 threedigit endings (not very practical). Fractions are keener in dozenal: we have divisors {1, 2, 3, 4, 6, 10} and semidivisors {8, 9} to make regular {1, 2, 3, 4, 6, 8, 9, 10, 14, 16, 20, ...} and these terminate {1, .6, .4, .3, .2, .16, .14, .1, .09, .08, .06, ...} and are as can be seen generally briefer terminations for smaller, more commonly encountered numbers, as the prime divisors of the dozen are the two smallest primes.
A "neutral" number m is one that neither divides nor is coprime to n. The numbers 3 and 10 are coprime since the latter is 2 × 5 and thus their greatest common divisor (highest common factor) is the empty product 1. Looking at ten, we have {1, 3, 7, 9} coprime to ten (these are called "totatives"), thus, the cototient, those numbers less than ten that are not coprime to it, consists of {2, 4, 5, 6, 8, 10}. Among these, {2, 5, 10} divide ten, and we are left with the nondivisors {4, 6, 8} in the cototient. These are neutral to ten. Neutrals are necessarily composite. There are two species of neutrals: the first are the semidivisors {4, 8} that divide some power of ten, and the semitotative {6} that does not divide any power of ten. The nature of a semitotative is twofold: it has a regular factor and a coprime factor. Decimal 6 is the product of the divisor 2 and the "omega" totative 3; the "richness" of the regular and the "relatedness" of the totative make semitotatives "kind" or "mean" (we say "transparent" or "opaque" here on the forum). The unit fraction of a semitotative has a mixed recurrent expansion in its base. 1/6 = .1666666... in decimal. 1/14 = 0.0714285714285... There is a nonrepeating "preamble" then the decimal never ends. Semitotatives have a compound divisibility test. In decimal, an arbitrary number x is divisible by 6 if it is even (divisible by divisor 2) and if the digits of 6 total to a number divisible by 3 (the "omega" totative rule, commonly known as the Rule of 3 and 9 in base ten). In base twelve we have the totatives {1, 5, 7, b}, thus the cototient is large: {2, 3, 4, 6, 8, 9, a, 10}. Of these we have many divisors {2, 3, 4, 6, 10} leaving us with {8, 9, a}; the first two divide the gross, only ten does not divide any power of twelve. Dozenally, 1/a = .124972497... and 1/12 = .0a35186a35186... We can tell if a number is divisible by ten in base twelve if it is even, and if it is divisible by 5. Now 5 is an "opaque totative" in base twelve, so it does not enjoy the "omega divisibility rule" ("wasted" on eleven!) But we can use a technique that a Member here called Treisaran promoted. Since 1 more than the gross (decimal 145) is divisible by 5, we can group the digits of a number in pairs and do the following. We know that dozenal 5954 is divisible by 5 since 59 − 54 = 5 (that was an easy one). Since it is even we also know it is divisible by ten. (dozenal 5954 = decimal 10000). Thus decimal 6 is "nice", a transparent omegarelated semitotative, while dozenal ten is "not nice": it is an opaque semitotative, but we can mollify it by learning our fives well in base twelve.
We want more regular numbers, particularly divisors in our number base, and we want fewer totatives, except that a highly divisible number "omega" n − 1 (or slightly less desirable, "alpha" n + 1) is useful in fractions and divisibility tests (the decimal 9 is a "saving grace" for that base). We want semitotatives that have divisor and "omega" or "alpha" as factors; and we want few of those in the smallest range of numbers; especially as digits of the base. Dozenal beats decimal in these regards, with the exception that decimal has that keen relationship with 9 that ameliorates the "skipped" prime 3 (1/3 = .33333..., to see if divisible by 3, add 'em up) and mollifies 6 and 9. In dozenal, we "skip" no prime, but the next prime 5 is not friendly (1/5 = .24972497..., already described the test for 5). In octal half the digits are totatives; 1/3 = .25252525..., 1/5 = .14631463.... In hexadecimal we have half the digits totative, but TWO omega totatives, so 1/3 = .555555555... and 1/5 = .33333333... (but its multiplication table has 136 unique values to memorize: dozenal only 78 and decimal 55, but we commonly memorize 78 anyway, so learning arithmetic in the lower grades of school might take twice as long). Only senary has a keener set (i.e., the same set of regulars, totatives, and semitotatives; it is the "core" of twelve, meaning that it is the squarefree root; 6 = 2 × 3 but 12 = 2² × 3.
The problem we have in senary is the loss of that additional power of 2, lengthening the expansions of fractions of the powers of 2: 1/4 = .13, 1/12 (one eighth) = .043, 1/24 (one sixteenth) = .0213, while in dozenal 1/4 = .3, 1/8 = .16, 1/14 = .09. This also serves to illustrate another, nonnumber theoretic problem with base 6. It becomes "verbose". What takes dozenal one digit to convey requires 1.079 decimal digits, but 1.387 senary. Now for small numbers, fine. But human memory only can ably deal with so many digits. In base 6 the numerals are fewer and so the numbers begin to look monotonous. Finally, there are more uses for number than mathematics. One of the problems we have with senary is that we run out of numbers of a certain number of digits, say twodigit senary. There are only three dozen such (counting 00, 01, etc.); then we need three digits. But with dozenal we have a gross, and even decimal has a hundred, which is 2.777... times as many as senary has. To sum, dozenal is superior to senary, despite the latter's simplicity, because it is more flexible. Senary life is like rowing a boat with teaspoons when you could use oars.
The numbertheoretic and humancognitive qualities* of the dozen seem to make it the best base. (* this latter set of qualities would require scientific testing to be a solid leg upon which to stand. This is the reason why I try to stick to maths. It would stand to reason that a multiplication table nearly 3/4 larger than the one we memorize in second and third grades would take kids longer to memorize, especially if it is loaded with facts like the hexadecimal thirteens table: {d,1a,27,34,41,4e,5b,68,75,82,8f,9c,a9,b6,c3,d0} ).

Paul RapoportDozens Disciple
 Joined: Dec 26 2012, 01:59 AM
Thanks for the summary, icarus. I wonder whether there's another way in dozenal to make use of the fact that 6 is next to 5 (and 7).