Folks

Here is a challenge:

Make a strong case for using dozenal over decimal and any other radix. Why dozenal?

I've been challenged by plenty of people about WHY twelve is better than ten or hexadecimal. We are all familiar with eggs and donuts, feet and inches. This doesn't cut it. Dozenalists need a stronger, more basic rationale with which we can bring our message to the public and to the professionals. I realize information lies on the websites regarding the advantages of dozenal. Some of the reasons seem to rely on properties that are defined by man, like 24 hour days and things we box up (which, in the case of boxes, is indeed a good rationale). Suppose we are to spread this "gospel" to the public. We ought to have the very best case for such a fundamental and radical change in order to convince the public that twelve is worth it. I am a dozenal believer, but want to build a strong case.

The case for hexadecimal appears on the following website: http://www.intuitor.com/hex/index.html
This is a well-documented and finely illustrated set of cases, and I would venture to say the cases, though they illustrate hex has advantages over decimal, they do not illustrate hexadecimal is ideal.

The best mathematical rationale I can assemble (given humble non mathematical background) is that twelve is an optimized colossally abundant number. It is third in the series A004490 (http://www.research.att.com/~njas/sequences/A004490). These numbers seem to best present the user with a means to represent a quantity because they can better "dissolve" the quantity. What does that mean for the common guy? Because I am not a mathematician, it's difficult for me to say. I arrived at colossally abundant from believing the best bases would be composite numbers, then numbers that were "highly composite" (Wiki says: A highly composite number is a positive integer which has more divisors than any positive integer below it). Later the best radix seemed to be a "superior highly composite number, which I arrived at using the sigma values associated with Euler's totient function. The higher numbers in these series are tempered by our human ability to employ the simpler bases. So instead of "competing" with decimal and hexadecimal, maybe we ought to consider why bases 2, 6, or 60 aren't appropriate. All this was spurred by reading Bill Lauritzen's "Versatile Numbers: Self-Organization, Emergence and Economics" (http://www.earth360.com/math-versatile.html)

I think Twelve is "optimized" because, within the span of its digit range, it has few relatively prime numbers. When we get to 5040, it can accommodate plenty of small prime numbers, but it spans 5040 digits, many of which are not accommodated by the radix. Additionally, it is too large to function as a base for humanity. Sexagesimal is inferior to dozenal for the same reason, although less so. It has to wrestle with 13, 17, 23, 29, 31, 37, etc within its digit range.

Anyone have any thoughts?

Icarus wrote:

How about this:
when accuracy and precision are paramount such as in the pharmaceutical industry, rounding errors can cause death particularly where thirds are involved. OK, base twelve won't eliminate rounding errors but it will considerably reduce them.

It is my experience in presenting the case for Dozenal that no amount of logical argument will convince the existing system to be replaced. Any case for dozens will need to identify all the logical and technical advantages over the current decimal system and will also need to present a complete set of replacement measures, but still this will never be enough.

There is one and only one argument that will result in the introduction of dozenals, MONEY!

The only argument with sufficient weight will be one that shows a very significant economic advantage for adopting dozenals!

So while I agree that we must present all the technical advantages, we must find overwhelming economic advantages for dozenals. All of those who see the use of dozenals as a goal must examine our past experience in all areas of industry and identify any such advantages. For instance, although it is now too late, the New York stock exchange changed from quoting stock prices down to 1/8 ths but has now changed to decimal points. Would there have been an economic advantage if they had used dozenals where eighths would have been specified with only 2 digits and all other fractions that were previously used by just single digits?

Another example that I am currently investigating is something I was alterted to in a local pub. My daughter who worked there as an assistant manager told me shortly before moving to another job, that the manager reported the number of bottles of various drinks in their inventory in dozens and units and likewise reported the number of empties the same way. I have also learned that the cabinets used for storing the bottles hold bottles in dozens or sixes and of course the cases that bottles are delivered in hold 24 bottles. The manager said he had little difficulty in adding all the dozens and units then converting the total units to dozens and units and then adding the number of dozens. These numbers are reported to the brewery daily by in-pub computer where the numbers are entered in dozens and units. I have yet to receive further information on how these are used by the brewery.

Do any of our members and guests have any similar examples that could be investigated and can any of these be shown to have a cost benefit if dozenals are adopted? Perhaps you have some comments criticisms or suggestions to my emphasis on economic advantages and/or that this would be the critical argument for adopting dozenals.

Nevertheless Icarus, I am impressed with your enthusiasm for dozenals and particularly your active use of dozenals in your business. Perhaps you could also investigate if there is an economic benefit in your use of dozenals. Does the use of dozenals reduce your costs in any way such as reducing time or eliminating errors and if so can you quantify the benefits?

PS Could you provide some concrete examples of your use of dozenals in your business if they could serve as examples for others to identify further examples?

Ruthe,

I am on the run and will add detail soon, but here is a brief synopsis on how I use dozenals.

I am an architect, and when taking measurements in the field, in the US customary system, I can note feet and inches in dozenal and save a lot of time and confusion. Dozenal notation eliminates the quotes and vulgar fractions. The four or five digit numbers I jot down allow me to make more measurements, because it is easier to write a legible measurement in dozenal e.g. *5730 than 5'-7 1/4". Once an employee is trained to read dozenal notation, the placing of the measurement onto a drawing is very easy. No more unreadable (ambiguous) fractions. (ps I don't write an asterisk, it is assumed on the field if I write a dimension, its dozenal.)

In schematics, if I use dozenals in the layout of a building, I can add, multiply, or divide US coustomary dimensions rapidly and efficaciously. As I've stated in previous entries, I believe common people are already dozenal thinkers.construction crews are already familiar with twelve and its powers, and natuarlly use it, even when it isn't necessarily part of the nature of the US customary system. (they use 12 and 24 feet rather often, when they could use 120 or 160 inches.)

more later, gotta go...

When I was in high school (and lost $4000 in imaginary money in my economics class' stock-picking contest), the prices were quoted in 16ths, which require four decimal digits. Those were a PITA to type into my calculator. But dozenal would still only need 2 digits.

Hi folks,

The financial aspect is difficult for another reason. I do not keep books in dozenal, my CPA would refuse; the IRS does not respect anything but normalcy when accounting is concerned. That is a barrier, but it is only part of these times.

Yeah, Dan a cad guy like me finds ourselves typing in things like 2'8.1875 and 1'2.5625 so many times its sick. The good thing is if you do it often enough it gets through - but it shouldnt have to be that hard. How about 28.23 or 12.69? oh thats so much better, and faster, especially if you're drafting a window extrusion, which goes to the 1/32 of an inch or so each and every dimension. (Now it is often metric, but a kind of metric that is a "transliteration" of the US customary. Those guys didn't change their dies.)

I think, as Dan and Ruthe have suggested, there would be advantages to finance that would be brought about through dozenal use. (pondering that...) I think the greater impact would be to productivity. It's hard to make a case for improved productivity through dozenal use, though. We have the cost of conversion to dozenal, so many legacy documents and processes to convert, some literally written in stone. But ultimately, I believe conversion to dozenal would benefit society far more than hinder it. I think we are missing a chance with China and India, because their infrastructure is now being created around decimal.

It would be great to use dozenal when possible, and think that this is one option if the entire system cannot be influenced to make the switch.

This is why I would like to create a case for dozenal rather like intuitor has, to put dozenal thought out there, so that it might be debated by a wider public.

Here's a stab...
What's really great about hex is:
1. Because it's a power of 2, it has the ability to represent powers of 2 fluidly, whether through logarithms or in multiplication. It "abbreviates" binary "words".
2. It has gained great visibility since the 1970s because of its use in the computer fields. People are somewhat used to hexadecimal. It has an application.
3. Its multiplication table, which arguably may be as difficult as the decimal, is still manageable for a twentyfirst century human mind to encompass. Therefore, it can function as a working radix.

What's unfortunate about hex is:
1. It is "inbred", that is, the factor 2 is far too represented for the size of the base. It would be nice to be able to resolve powers of other primes, like a three instead of all those twos. You'll probably come across 1/3 or a power of three of something before you come across an eighth of something.

What's wild about 60 is:
1. It resolves powers and reciprocals of 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60 fluidly. Due to resonances with the factors of 60, plenty of other common numbers are accommodated within two sexagesimal digits. This is its power. It is a "colossally abundant number".
2. Its indomitable application regarding time and angle have withstood metrication; it is rather indelible due to the power of its versatility.
3. It can massively compact extreme quantities, very large or very small numbers, due to its breadth.

What is a bummer about 60:
1. Try to memorize 1800 multiplication combinations. I guess if society had to, it would. This necessitates using a "sub-base", something rather necessary. It surely isn't optimized.

What is great about dozenal:
1. Dozenal resolves powers and reciprocals of 2, 3, 4, 6 and 12 fluidly. Due to resonances with the factors of 12, the numbers 8, 9, 16, 18, 24, 36, 48, 72, and 144 are handled with two dozenal digits. It handles the commonest fractions well.
2. Dozenal resolves the greatest proportion of powers and reciprocals of numbers within its span for its size. (Is this true?)
3. It is larger than decimal so presents a more compact representation of quantities.
4. The present civilization already memorizes a multiplication table that includes 12.
5. The multiplication table is simpler and not much larger than the decimal.
6. Society already uses the dozen as a unit for many of the commonest applications.

What are the drawbacks of dozenal:
1. It handles the third prime poorly.

What is wrong with decimal:
1. It resolves 2 and 5, really the same total factors as base 6 but spanning 4 extra positions. These primes are split, so decimal shuns the third, which crops up more often than the fifth. (This is something Endi mentioned) So the third appears to the users of decimal as a "strange" fraction. Bases 6 and 8 would be finer choices. Base 14 is almost as rational as decimal (and behaves rather like it), except four additional digits are added without any benefit.
2. It's multiplication table is not as simple as it could be if the base was dozenal, senal, or octal.

What is wonderful about base ten:
1. If you stick out your fingers and count them, you can use decimal. (its only helpful for so long...maybe first grade...)
2. Society has crystallized around decimal. The inertia of its use is massive and maybe insurmountable.

Just stuff to think about. Boy I'd love to make that case so it can be presented in an absolutely awesome way. I just think my descriptions are too "squishy" for advanced thinkers, and thus do not provide a strong enough framework to simplify for children and common folk. (BTW, yes, I will teach my daughter dozenal - but she's only *3 and needs to get through her early years of school).

Sorry for overlong posting but you people are so awesome that I am excited and can't stop thinking dozenal...good evening, time for supper. maybe *10 toasted raviolis, if I'm lucky :-)

I suppose this is a case of 'Render unto Caesar...'.
This doesn't stop you from keeping internal accounts whichever way best suits you as long as its not a big problem to output the final results in decimal.

icarus, do you ever need to calculate areas and volumes and if so do you use dozenal multiplication tables? For instance, if you had a room of 25.730 by 16.546 for 29' 7 1/4" by 18' 5 3/8" {ps I am using a point to delineate feet and inches here} how would you calculate the area?

Note that the value after the point in the final total is the dozenal value of square inches and the value before the point the square feet (I'm sure you know that but I mention it for any guests who may look in).

That's not the only reason. You will find in Europe timber and laminate board sizes in multiples of 300cm. I wonder why?

Why would documents need to be converted? Were existing Customary/Imperial documents converted or were they just replaced when the new measures were introduced? I think this is just another horror story by the entrenched S.I. pundits and the true cost of conversion would be magnitudes lower than they suggest. If their projections were correct, then the same cost would have been needed to go from Imperial to metric in the UK and it would not have been attempted. Their argument is false!

That form of evidence does not have any effect with the general populcae for a very simple reason. For the majority of the population, mathematics has been presented as "hard". As a result, start talking to most people in the street and they just turn off saying "I was no good at maths." No matter what you try, they don't want to know. Until all forms of negative depictions of mathematics (and science) are banned from public media this attitude will continue. How many times have you seen adverts on TV that include something like "Ugh, I've got maths next" or "I've made my dentist appointment the same time as my maths class"? As a consequence, arguments can only be made to the mathematically cognizant.

PS Many people THINK they are no good at maths, but in fact they are. I talked to man many years ago when I used to take my grandad's bets to the betting shop. This man said he was useless at maths, but could he work out the odds on the gee gees, could he!!!

No need to worry about long posts, it's nice to hear from somebody who is actually using dozenal maths. Keep it up!

True, but how would dozenal accounting books deal with cents?

I have, however, found a practical use for dozenalized accounting. The sales tax rate here (including state, county, and city taxes) happens to be 8¼%. In dozenal, this is :B.:A7 pergross, which conveniently rounds to *10 pergross. So converting a price to dozenal gives me a good estimate of the tax.

Because 300 is divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20, etc.?

About as often I've had conversations like:

Them: So you're a grad student?
Me: Yes.
Them: What's your major.
Me: Mathematics.
Them: Wow! That sounds hard.

Attitudes depend on the type of school. I started teaching Maths many years ago -in an all girls school. I wouldn't say they particuarly cared for Maths; but the thing I noticed was that the girls who were good at Maths were very good at Maths.

For a while I alsotaught in an all boys school; again the better pupils were very good at Maths.

For the rest of my teaching "career" I taught in a mixed school. The boys tended to shine at Maths and most girls thought of it as a boys' subject.

I don't know if the attitudes are any different nowadays; but I have noticed that few children seem to be able to do any calculations without their calculators, and even simple estimating seems beyond them!

(I wish I could do that box thing but am afraid I don't know how...)

When I learned to oil-paint, my colleagues advised me to "take it to the top", that is, master the construction of the image and its good rendering before focusing on your expression. I didn't want to paint lifelike pictures of oranges, but they advised that that was essential, because, once one can manage the finest rendering techniques, their simplification in the mission to communicate a message becomes an easy task. This is analogous to my desire to build a strong case for 12. If a dozenist can speak to the power of 12 in the strongest way, with the force of mathematical logic and science behind it, we can then bring that to language and reason for the average bloke (like me). We can select what parts of that case to use when we talk to the public. The dozenal message is akin to any other complex message. We advocates should find out all the reasons why we believe what we believe, take it to the top, then excerpt what we need when we are out there in the world. I'd so love to have a strong case (maybe we already have it, but I'd like to have it in one place.) With that, we can help the world save time and money. Imagine all the manhours spent typing .1875, avoiding the option of dividing something into three, pondering how many total cans are in that optimally packed case of beer (6, *10, or *20 to the case, completely transparent to the dozenist.)

Now this doesn't mean to render everything else unimportant, but without a case, dozenal becomes a hobby (and that's okay, but I believe it is a best practice that can save people time and money). We can discuss digits, names, and units of measure. But we should know why we believe what we believe first so that we're sure we aren't in a folly (and I wholeheartedly believe we are NOT in folly). I am an ex-hexadecimal prophet (too much teenage computing, back in the 80s). Hex is a beautiful thing, but clumsy in its work. Dozenal may not have that great symmetry, compounded four times over, the four itself a square of the progenitor power of Two. But overworship of Two is its weakness. So I left hex, even though I reserve the right to use 16 in conjunction with twelve, and use sixties and 360s for their power. Absent an ideal working tool that will help a human resolve any quantity instantly (apparently impossible), dozenal is the optimized alternative; everything else, even 60, 120, 360 (because they are unwieldy) falls somewhat short.

Friends, we have the optimal tool for the human understanding of numerical information. It won't solve everything, but it is the best tool we can make short of changing the laws of math. So I want to build a dozenal TANK. Oh it's going to be a mighty instrument, that strong dozenal case. I'll bring it personally to NYC in October, credit where that is due, along with a nice contribution so that it gets somewhere, in sha' Allah, and share it with the DSA at the annual meeting. It is time for the dozenal 1728-ennium.

Professor William Lauritzen!

Maybe this is the answer to the case for dozenal. I read it in 2000 and again in 2003, and have just re-read it. Professor Lauritzen gives a very detailed rationale why dozenal is best suited, above all other possible bases. This is perhaps a response as well to shoelace's case for eight.

its at http://www.earth360.com/math-versatile.html. Printing it out is probably the best option. The guy is an absolute genius. A full bibliograph nonetheless.

Whaddaya think!

I read Bill Lauritzen's document on Versatile Numbers and was somewhat, no very disappointed given your enthusiasm. Why?

Well, undoubtably he has provide several examples of why versatile numbers and 12 in particular would form more usefull sets of values for use in daily commerce and usage. But, and it's a big but, he left me deflated when he gave no solid proofs or even strong arguments for using 12, and even didn't attempt to prove his thesis stated in paragraph seven which I quote here.

Instead, he finishes with five conclusions, only the first three of which have been illustrated.

Conclusions 1, 2 and 3 have been demonstrated and I believe he has made the case for these statements. These three are quoted below.

Conclusions 4 and 5 however are just possibilities and although I agree with his conjectures, they are only that. This can be seen by the wording of both these statements starting with "Homo sapiens might benefit from". In order for these two statement to be accepted as proven, it would be necessary to conduct extensive real world studies to provide evidence to support his claims, and given such paradigms, I feel strongly he would be proven right.

So in the end, the major point of his article gives us, as proponents of a Dozenal system of numeration and corresponding system of weights and measure, little to advance our case. His thesis remains just that and this paper only an esoteric mathematical musing. This will not convince the CEOs or bean counters of the world, and it will be they we need to convince.

What we need instead are clear and demonstrable real world cases where dozenal can be shown to have a significant advantage over decimal. This may also mean that such examples must be advanced with a dozenal number base alone without the desired dozenal weights and measures, but these must be ready in the wings as this is likely be the focus of the first objection to be fielded.

I am always reminded of the example described by F.E.Andrews in his article MY LOVE AFFAIR WITH DOZENS of the plea by the (US) Army Transport Service in 1943 for his aid in using Duodecimal arithmetic to aid them in calculating cubic capacity of goods packaged in boxes specified in feet and inches to speed their task and improve accuracy for their unit "engaged in measuring cargo loaded aboard ships sailing to our armed forces".

These are the types of example we should be looking for. Surely, with the numbers of people In the DSGB, the DSA, members of this site and guests, there must be some situations in their current or past working or recreational lives that would benefit from Dozenals, and if so could they respond, perhaps by starting a new thread such as "Applications of Dozenals". We already have one example, that quoted by icarus. If he could find the time to extract the salient points from previous postings and post into the proposed thread, we would have a start that may prompt other inputs. We might then have something concrete to start with to convince yet one more disciple. Which Chinese individual said, "The journey of a thousand (miles) begins with a single step."?

PHEW!!! :mellow:

But that example is actually worthless in order to make a case for dozenal, because it only means that, if one is already using twelve-based measurement units such as feet (which, remember, most of the world isn't using anymore), it is the most convenient to handle those measures using dozenal arithmetic. Which should be as obvious as saying that it's best to use decimal arithmetic if you're using decimal units (which happens to be what most of the world is doing, so they'll tell you the world at large doesn't need dozenal at all because their metric units are simply not dozenal and it would be as disadvantageous for them to use dozenal arithmetic with meters as it is for you to use decimal arithmetic with feet). In the end, no case for dozenal here at all, just a case to be coherent and use the same base for measuring and for doing arithmetic with those measures.

The population of the USA is over 200 million and a very large proportion of these still use feet and inches. In fact, there is a move to reverse the use of metric measures due to the pressure of the citizens. One only need to look at Caltranswho have completely reversed their use of metric measures in all new projects simply based on the overwhelming pressure from their suppliers and from the majority of similar organizations in other states of America. Perhaps Caltrans could benefit from Dozenal arithmetic.

Ruthe,

I will try to gather instances of where dozenal would present advantages over decimal, trying to avoid the issue I think uaxactum (am recalling issue from memory) presented regarding the use of dozenal-based units. It's my contention that 12 has natural benefits over decimal (see "case for base 6" thread). Now the public may indeed regard the benefits as only an improvement, but not worth the "hassle" of conversion (if we think metrication was a folly, imagine changing to dozenal!) I believe that, regardless of current public belief, we as dozenalists ought to have the ability to present a great case for why we believe 12 is superior to 10. I think Prof. Lauritzen has generated the most solid kernel for that belief; we may need to contribute to that case, or consider the "versatile economics" case as part of a "canon" of our beliefs. I think, though the proofs are not necessarily present in that document, that "versatile economics" presents the best case thus far for dozenal; it gives us a great framework, though there may exist other considerations. BTW I have chatted with Prof. Lauritzen last week, motivated by this conversation we are having. Maybe the professor could be convinced to expand his thoughts. I do think that every one of us on this website, plus those that dance around its margins, can contribute to a case for dozenal.

I really am interested for constructing a case because I believe 12 is optimal; it is a best practice, so to speak. That, maybe the current civilization will not adopt dozenal, but perhaps future people might. Dozenal may have to wait until the strong standards that have evolved around decimal weaken so that something as basic as optimized radix can be introduced. (That would be a good time for spelling reform, endi). We individuals can't wait for that, so perhaps we can live with using dozenal in daily life when practical on our own or in small groups. The greatest thing a dozenalist can do is propagate dozenal, but the second greatest thing is EMPLOY dozenal as much as is practical, given that we operate in a decimal society. That is what I try to do, to try to personally use dozenal whenever it is possible. The Roman Empire did not (to my knowledge) employ the current "hinduarabic" numbers; the idea maybe could have never practically arisen while its hegemony was operating.

It's essential to me that all dozenalists, especially the great contributors to this site, find a way to agree on a standard so that, when we are using dozenal, we are using the same dozenal, making our efforts to convince who we can a more powerful endeavor. After we construct a case (and that might be a project that, though it might attain a practical solidity initially, can be amended as needed when new instances arise that reinforce the case) we ought to adopt standard digits. (I prefer DSGB, though I like my backward-seven eleven: it looks more prime through resonance with seven and isn't married to my language (midwestern american english)).

Okay I will take the next week and try to gather instances of dozenal improvement over decimal, avoiding the case when the system involves preexisting dozenal units of measure. Hmm. Unless a rush project requires all of my time like last week's project...maybe will be posting from Ithaca NY.

Onward!

Icarus,

Why avoid the case that use pre-existing dozenal units of measure? The USA is the one place where feet and inches are still in common use. Applying a dozenal number base system to calculations of feet and inches is a valid example of the benefits of dozenal. So why exclude it? You use it and have your staff use it also. Surely that constitutes an concrete example of an advantage.

PS Have you seen the film "Contact". I always thought it should have had the initial contact sequence of primes ending at 149 decimal or *105. If it ever does happen, I bet that's the radix ET uses.

The combined populations of the USA and the few other countries where dozenal units are still in use amount to about 6 pergross of the world's population; a very small minority by any standard. Sure it's easier to find compelling practical reasons to switch to dozenal in the context of those countries, because people there are in fact already using dozenal to some extent in some way, but such reasons don't work anywhere else where people are metric since long ago. For example, where I live, where no one would see any point in your argument if you tell them that dozenal is nice because it makes it easier to find the area or the volume of something measured in inches. They'll tell you who needs those "complicated" and "outdated" inches in the first place, when everybody is now using the "easy" and "scientific" centimetres. I'm not saying you couldn't or shouldn't use the imperial units argument to promote the use of dozenal within the USA. What I'm saying is that, if we are to make a real case for dozenal, we need to find practical advantages that work for the world at large; ones that do not presuppose dozenal measures are already in use for a complete switch to dozenal to be shown as beneficial. Otherwise, our dozenal case risks being largely perceived and ignored as a pointless anglocentric parochialism.

I'd say it could as well have had the list of versatile numbers (1, 2, 4, 6, *10, *20, *30, *40, *50, *A0, *130, *180, *260, *500, *5A0, *890, *B80, *1560, *2B00, *4460, *5A00...), or the subset of those that Lauritzen has labelled dominant versatile numbers (those that have more factors than any other number up to at least double themselves: 2, 6, *10, *50, *A0, *260, *1560, which is the initial sequence of the superior highly composites). We shouldn't assume ETs place a comparable disproportionate interest in prime numbers and disproportionate disregard of versatiles as we humans have mostly done so far. ;) It could be they have chosen to call the prime numbers something meaning "rigid numbers" or "minimally composite numbers" (labels Lauritzen has suggested somewhere), which would make those numbers look much less appealing. Compare how differently the situation is pictured when using the terminology "prime number vs. highly composite number" and when using the terminology "rigid number vs. versatile number".

You're completely right and I agree. If anyone is listening then they should be expecting nothing specific but open to as many possibilities as they can. Likewise, I think it possible that any such contact would be using some completely different media possibly beyond our current ken.

Ruthe

Right said! I am finding it is a little difficult to completely avoid units.

I think and am suspicious that metric is part of a group of ideas that arose during a rebellious teenage phase of "civilization", along with modernism (I am a Modernist, but a reformed one if that makes you feel better), communism, and other ideas that tend to say "what came before was uncivilized and irrational, so we're going to use our science as it stands to remake the world." The sense was that human beings did not need to heed traditional behavior; we could be up all the time with no respect to time or season, be made to live at any climate. With modernism, we could build using basic principles the same glass box everywhere, and it will be totally appropriate. With communism, yes, everyone will gather together and shine on their brother and we'll all live wonderfully, sharing everything. But what is happening now (I guess in the USA, from most recent trip to Tuscany I think my friends in the Val di Chiana are going to experience what American inner cities experienced, unfortunately) is that we are beginning to discover that our grandparents, though they could not put it in scientific terms, actually built excellent communities, and that pre-communist societies weren't perfect, but at least were sustainable (as sustainable as a human system could be.) So we are discovering that the traditions were not as bad as we thought. Metric may be part of that. It is a symptom of decimal that it is a bad idea; if it were based on the dozen, it would be a pretty acceptable idea (see takashi Suga's work) Metric has validity merely because it has been ascribed validity, so we will need to work with it. I think Suga did a fine job with it. (The time thing though, wow we earthlings will need to resort to a dozenal division of the day until the other worlds he mentions come on line as places to live.)

Okay I owe you some examples, forthcoming next week, barring a big push for that Atlanta work.

Dozenal | The Optimized Base | Example 1.

The following is a true story that took place in Saint Louis, Missouri on 5 May 2006 at 9 pm local daylight time.

My wife is a graphic designer; she needs to reduce an image so that someone reading the magazine can resize that reduced image back to original size using a copier. (The image serves as a template for sewing aficionados). Now we are familiar with the "enlarge/reduce" function on a common copier. You need to use percentages to resize a copy, you are limited to three digits (two if under 100%), and usually limited either by capability or paper availability to 200%. We should assume the "lowest common denominator", as a member of the general public will likely use the most common copier available, and these aren't the latest models. She asks me, "What is a good clean number for a reduction so that I can print a clean number for the readers to enlarge the image." A figure like 75%, a clean reduction, for example, yields an imprecise 133%, which for some (picky! not on my wife's account, though) reason is not acceptable. But 80% gives us 125%, perfectly usable, but because she needs a smaller size, required her to blank an entire page or use the other available clean reduction-enlargement that qualifies, of 50%. This is DIRECTLY attributable to DECIMAL!

Now I haven't done a thorough assessment, but I think the only sensible numbers she had available (results that yield a necessity to enlarge more than 200%) were:

80% reduction for a 125% enlargement. (4/5 * x * 5/4 = x).
50% reduction for a 200% enlargement. (1/2 * x * 2/1 = x).

We were confined by decimal! Man I nearly jumped for joy! Because dozenal under the same constraints would enable:

*a8% reduction for a 116% enlargement. (8/9 * x * 9/8 = x).
*90% reduction for a 140% enlargement. (3/4 * x * 4/3 = x).
*80% reduction for a 160% enlargement. (2/3 * x * 3/2 = x).
*69% reduction for a 194% enlargement. (9/14 * x *14/9 = x).
*60% reduction for a 200% enlargement. (1/2 * x * 2/1 = x).

This is obviously a finer range of choices than being forced to use decimal 80%. This is the magic of using a base that tests for 2, 3, 4, and 6. There are so many fine divisions that yield concise terminating digital fractions. So copier manufacturers wouldn't be shutting out thirds when they limit a person to a three digit percentage.

Now I'm not being thorough (am on the run) but let's examine octal (figures in octal):

40% reduction for 200% enlargement. (1/2 * x * 2/1 = x).

Now Hex (in hex):

80% reduction for 200% enlargement. (1/2 * x * 2/1 = x).

This paucity of choices results in the depth of representation of the factor of the same prime. Though they have plenty of factors, they are all based on the same prime, and there is no opportunity for a different prime to mate up with 2 to give a clean choice except halving - There is no occasion for multiples of different primes to "interfere" and yield a nonterminating result. (These bases are too inbred.) And at the risk of a misstatement because of my inability to be thorough, these fare worse than decimal.

Now base 6 (in base 6):
40% reduction for 130% enlargement (2/3 * x * 3/2 = x).
30% reduction for 200% enlargement (1/2 * x * 3/2 = x).
Which is just as many results as we had using decimal. Senal has two factors and yields fewer results than dozenal because of it.

I would imagine (but don't have time to conduct) that base 60, another versatile number, would yield a basket of choices, because it includes 5, and three primes can "interfere" and yield vastly more choices. BUT, the base is too unwieldy for twentyfirst century people (in too much of a rush to memorize 1800 permutations of the multiplication table and 1800 of the addition table) to practically deploy. Imagine a copier with *50 digits on the keypad! That would be as big as a regular keyboard! (Actually it wouldn't necessarily have to be that way, but the image is kinda cool.)

(Correct me if I am wrong on any of the above). No units used!!!

Assuming that Base-60 would be implemented as a mixed 6/10 base, there's:

• 50% reduction for a 1:12% enlargement.
• 48% reduction for a 1:15% enlargement.
• 45% reduction for a 1:20% enlargement.
• 40% reduction for a 1:30% enlargement.
• 36% reduction for a 1:40% enlargement.
• 30% reduction for a 2:00% enlargement.

Yeahhhh, but, couldn't copier manufacturers just provide a set of fractional reductions/enlargements which may be approximated by decimals to three or four places internally and display these choices as fractions alongside the percentage facility?

Thus they could have :
[doHTML]
<PRE>
R E
1/2 - 2/1
2/3 - 3/2
3/4 - 4/3
4/5 - 5/4
8/9 - 9/8
9/14 - 14/9
</PRE>
[/doHTML]

This way they could provide what your wife requires without the need to even mention dozenals.

Oh damn, I hate to dampen anybody's enthusiasm and particularly yours since you are the only person I know that actually uses dozenal arithmetic for real, but I'm only playing Devil's advocate to ensure we have a solid response when somebody else raises the same questions.

Ruthe
I am glad to see devil's advocate material.

This is true, that "vulgar fractions" could be used on a copy machine ( I haven't checked my terminology but I mean fractions like 1/4, 1/3, 2/5, etc). The fact is, and I've read this somewhere, that in the USA, people perceive percentages to be more exact than vulgar fractions. Many Americans have trouble with fractions. Yes, it would be great if the manufacturers allowed vulgar fractions.

I use cad software; there are several ways to enter fractions of a unit of measure. One can type 5.5625 for 5-5/16 inches. Or one can write 5-5/16, or 85/16. This ability to process fractions comes handy when offsetting lines for brick courses: 8/3 inches (3 courses in 8 inches of height). That's because 2.66666666666666666 is, well, unwieldy. There is software that won't allow vulgar fractions, so one must resort to the same digital fraction abbreviations. For a while Adobe Illustrator wouldn't let you put in "1/3" for 0.3333... inches (picas, cm, etc.) You needed to use digital fractions ("decimals").

If one is limited to digital fractions, dozenal will yield more clean options.

My wife is intelligent and perceptive, but when I tried to explain to her that she could figure the problem herself using vulgar fractions (not using that term, though) and their reciprocals, she was pretty confused. "How can there be more than three thirds?" Many people try to "convert" vulgar fractions to "decimals", so when you say "one third", they think "oh, that's that fussy 0.333333... how weird" but thirds come up more than fifths, despite their decimal "cleanliness". (Practically every architect knows the sixteenths 0.0625, 0.1875, etc. because of customary fractions of an inch, so that when you say 7/16, the dude automatically thinks 0.4375) People trust digital here. I think the problem here is the education climate in the US, that it's cool to be dumb; you don't want to stick out and be a nerd.

So in summary, true, the machines could be built to handle input like "1/3", but the limitation of the solution to digital fractions as stated in the example is best solved if one were to use dozenal digital fractions.

I will be accused here of generalization, but you should not think that attitude is reseved for US students. Although there are many students in the UK who belie this categorization, the publicly perceived feeling is similar, perhaps not so much "cool" to be dumb but nerdy to be capable.

Which is going to be easier, place more emphasis on teaching fractions or dozenal airithmetic? Even more frustrating is the realization that if the teaching of fractions in a decimal environment is difficult, what chance is there for teaching dozenals? Is there some way the problem could be turned around by using dozenals to teach fractions?

Comments anyone! <_<

Now that you mention the sixteenths:

Not precisely easy to learn and work with. And, as a result, practically no-one (except architects in the US and maybe a few others) knows them by heart or would figure them out quickly or easily.

In dozenal, the sixteenths only require two significant digits, and follow the (not difficult) multiplication table of 9. That makes them much easier to learn and use than in decimal.

This is part of the case for dozenal, although admittedly many of its more obvious practical applications are due to the use of sixteenths in imperial measures. It has use in converting hexadecimal/binary non-integers to "common numbers", too, so the argument of easier sixteenths (of easier binary fractions in general) might appeal to some computer geeks also (although hexadecimal advocacy is rather popular among them, so some might answer that we should be switching to hex instead).

Back from vacation!

Folks here's a summer problem:
Devise a way to determine the optimum base - in a rational manner. Best done under a palm tree, before the tropical storm swings in.

I've tried to do this, responding to the gripes by hexadecafanatics that claim measures that indicate dozenal is optimum are slanted to favor 12.

Here are some key ingredients for a good base, and examples (in my small vacation groggy mind) of how to arrive at a measurable.

1. Factors.
1a. Number of divisors of n (known by some as "sigma-sub-zero") divided by n. This is the ratio of a base's number of divisors to the base itself. A higher ratio is better. A book called the "CRC Standard Mathematical Tables", 28th edition, page 104, yields an excellent survey of sigma sub zero for the first 1000 integers.
1b. Euler's Totient Function, (designated by phi(n)) "number of integers not exceeding and relatively prime to n". This measures which digits within a base's "span" harbor repeating digitals (decimals).
1c. An assessment of the individual prime factors of a number, where each prime factor x is assessed as ln(x)/ln(2) so that the simplest primes are accentuated. The importance of smaller factors is illustrated in the sequence of primes (google Sloane's A073751) that, when multiplied in sequence, generates the Colossally abundant number series (Sloane's A004490).

2. Compression power. This can be measured by the natural logarithm of n, the base in question. When compared to other values of n, the logarithm is a gauge of a base's ability to compress a quantity.

3. Ability of the human mind to wield computations in the given base. This is where we have to be careful, because this measure can be pretty subjective. We can assume that decimal multiplication tables give a good gauge of what a human mind can live with. It seems that the users of base-60 may never really have computed in that base, but used the decimal sub base to arrive at answers. Their use of base-60 would be more like our use of "base-100", composed of two ranks of base 10 rather than 60's ranks of 6 and 10. So I assume anything smaller than decimal is easily mastered. I could also presume that a 72-value mult table is just as easily mastered, because many memorize the 11 facts and 12 facts. To satisfy the hex fans, I can stretch the assumption to 16, but after that, we would have to gradually penalize larger bases because the mult tables get too unwieldy. A better way would identify individual cells in a base's mult table that doesn't follow an easily-identifiable pattern and flag these. Here's a dude that DQs dozenal for one early instance of a "number that looks like a prime but isn't": http://forum.darwinawards.com/lofiversion/....php/t6246.html, see Neosisani's passage at 23 Mar 05 14:10, very close to the bottom of the page. (Ooh, you should read what this guy/gal says...)

Hmm, anything else? I have an old Excel spreadsheet that figures a lot of this, trying to improve it. It's important that it be as impartial as possible, so that I can use it to evangelize democratically.

The properties described in the recent "Base 8" thread are part of the case for dozenal. Note that there are several categories of properties, each category acting like options on a new car. Some cars feature four wheel drive, and maybe that's a plus, some get better mileage, some can seat 4. No car possesses all the best features, but there certainly are cars that have features that are optimum for each consumer, and for all consumers in general. In this way, dozenal can be seen as having that proper balance of features, beneficial (compact prime composition, for example) and detrimental (hmm, maybe larger size, for example, or maybe that it is a change from what everyone is used to), that make it optimum for general everyday use by humanity.

The dozenal prime composition accounts for the most common two primes which together account for *80% of all numbers. The dozen accounts for the 2nd power of two, so that it has additional power to screen for that prime. The dozen is a compromise among the possibilities of bases having three prime factors. Eight puts all of its power in one prime (2*2*2), while thirty hedges its bet, making for a large integer (2*3*5). Twelve accomplishes some diversity and some concentrated power with (2*2*3). Eighteen (2*3*3) and fifty (2*5*5) would do the same, but their power is concentrated at less-common primes. They also represent relatively large integers. Twelve's prime factors are compact; no prime is skipped in favor of a larger prime like decimal (2*5). So the dozen can screen successfully the prime composition of more numbers than any other integer in its size range (less than 20). Because of this, the user of dozenal can tell at a glance whether or not a figure is divisible by 2, 3, 4, 6, or 12, and if that user extends the base to its powers, 8, 9, 16, 18, 24, 36, 48, 72, and 144. No other base in its size range offers so much power. ("no other car it its class..." :-) )

The dozenal totatives, the digits 1, 5, 7, 11, make dozenal a great base for analyzing any random quantity of items. This means that dozenal "round numbers", those divisible by 12, aren't as necessary as when one uses decimal or most any other base. This is because so many of dozenal's digits represent some directly evident divisibility by one of 12's factors. In decimal, we tend to use multiples of 5 or 10, or 25, so that we can get a "round" figure that might stand in as an estimate, for example. With dozenal, a figure is more apt to stand on its own, so that we can consider *73 or *84 as somewhat "round" despite the fact they aren't multiples of the dozen. The only base I think possesses a superior or equal similar quality is base 6. The principal drawback of base 6 is that it fails to fully engage the human capacity to mentally calculate: we can handle a larger multiplication table and should, in order to capitalize on a second factor of 2.

Dozenal "percents" and "per-mils", that is, digital fractions expressed in terms of the second or third powers of the dozen, are more flexible than any other integer base's below 20. This means that the human propensity to turn to digital fractions ("Percents", etc.) will more often yield exact representations of a rational fraction. Of course, like options on your next car, we can't get EVERYTHING. Rational 1/5 will throw dozenal digital fractions into a tailspin at *24.972497%. While decimal percents are strong, dozenal "percents" or per-gross provides exact support for more rational fractions, and the rational fractions the per-gross supports are arguably more common.

Most everyone has learned the multiplication facts associated with 12. Most people know intuitively that 12 x 7 is 84, almost without thinking. So 12 has a nicely manageable multiplication table that allows one to compute with it readily. Something else dozenal has going for it is that many of the lines of facts in the multiplication table are easier to remember than their decimal or other analogs. The 3, 4, 6, 8, and 9 facts are beautifully easy to remember. I use multiples of both twelve and sixteen all day long and facts like 7 x 16 and 13 x 16 are still operations I have to double-check.

These three features are some of what makes the dozen the optimum base for everyday use by humanity. Individual applications like counting bits will indicate usage of the "binary" bases but because of its versatility, the dozen beats everything else.

Today, someone gave me a book as a birthday gift, it was a book that mentioned our friend Fibonacci. Speaking of Fibonacci, maybe I am spelling his name wrong, I think it is keen to see that dozenal is a wonderful base for the sequence. Decimal, sexagesimal do not illustrate the cycle as cleanly as dozenal. I think it's right to mention that octal also features the kind of keenness that dozenal sponsors. (All figures are dozenal.)

1
1
2
3
5
8
11
19
2a
47
75
100 (cool!)

First dozen terminates in the gross.

175
275
42a
6a3
b11
15b4
2505
3ab9
6402
a2bb
14701
22a00

Second dozen, the last two digits are reset.

37501
5a301
95802
133b03
309705
441608
74b111
b90719
171b82a
27b0347
430bb75
6b00300 (holy cow!)

Well I don't know how significant Fibonacci's sequence is to real life matters, but this is one exciting thing to pen out. (Simply add the last two numbers in the sequence to get the next number in the sequence.) Okay this might be a little funky because I added in my head kinda quickly.