Theoretically, which base would you prefer our numeration to be in?

For 'alternative' bases, I'm not sure if six or twelve is my favorite.

I actually kind of like decimal, too. Ten may not be terribly rich in factors, but it's not exactly burdensome.

Twelve.
Sixty is attractive - but much too big.

I guess it depends on how you do your arithmetic. The Babylonians didn't memorize their multiplication table.

Sixty is a second to twelve in my book. There is a way to multiply and divide in sixty without memorizing the entire table. When I get back to St. Louis and have a little more time I can explain; it isn't too hard for those who want to learn. (I am at Cornell University on business.) And yes, ten is surprisingly not too bad a base. But twelve is the optimum base for humankind...
Every day I grow more fond of base 60, now that I can wield it, but the grouping size is far above the apparent natural size a human being can easily recognize (7 to 12, I once read). Grouping things in sixties seems to lack simplicity.

When I say eight, I am referring to octomatics, a clever way of representing base 8 digits in a way that they can instantly be read in binary. Each character contain three bits of binary which form to make the numerals for 0-7. It allows for easy addition and subtraction, and since it is binary it is remarkably simple to multiply without memorizing the multiplication tables. Here is an example:

Alternatively, you could also multiply the ancient Egyptian method. In the left column you write the digits of the first factor in reverse order, and in the second column you continuously double the number. By adding the numbers on the right that correspond to ones on the left, you get the product.

There are many ways you could think of these principles to shorten the process. So since addition and subtraction is nearly entirely visual, and multiplication requires no memorized times tables and can be thought of in many different ways that are still very simple, it has a clear advantage over other bases. It also has the advantage of being close to ten, so large numbers can still be written fairly compact. On the flip side, it is radically different from Arabic numerals, and will take some getting used to. On another note, the primary reason why larger bases such as sixty are considered impractical is that it requires memorizes tons of numerals and has an impossible times table. There have been several creative methods of building the numerals out of other parts, much the way octomatics is built out of binary bits. Times tables are not required and the Egyptian method of multiplying would handle base sixty just as readily as it would binary, simply write the left-hand column in binary lie you would in octal, but write the right hand column in sexagesimal digits.

For a better explanation of the Egyptian multiplication and to see their method of division (which is simply the reverse of the method for multiplying), go to http://www.youtube.com/watch?v=fCy-HmCDB_w For a good numeral set for sexagesimal go to http://autonomyseries.com/sexagesimal-numerals/

Neogenesis, I like your octomatics as a logical system of building up numbers.
At first sight, I thought I was looking at a remarkable property of the number eight then I realised that instead the property is more rooted in the number 4. If instead of taking three digits, we take two, this will give the binary representations from 1 to 3.

If underneath the line, we also record in binary how many times we have repeated, then the digits from zero to eleven can be built up as follows:



This could be called dozomatics.

And as a clock:


Zero is left "as is" for stacking in threes.

Thus:

I was a little worried that the symbol for three looks too like a four, ten and eleven look like atches. This can be overcome by angling the up and down strokes thus:



The other thing that comes to mind is that I am not even sure that it is best to build up numbers from the greater powers on the left to the lesser powers on the right. I can literally remember right back to when I learned addition, subtraction multiplication and division and could not understand why I had to work backwards. Everything else is done from left to right. Why not mathematics?

This would mean that one would have to use mirror images of these symbols.

At the risk of making this a monologue, the beauty of this system is that it is essentially a base twelve system which manages to slip in base two and four in the formation of its symbols.

How come?
You use the BODMAS rule (Brackets, order, division, multiplication, addition, subtraction) and work from left to right, don't you?
For example 2 + 3 x 5 is worked out as 3 x 5 + 2 since the multiplication is done before the addition - is this what you were thinking of?

I think he refers to our custom of writing numbers from left to right but with the powers of the base in descending order rather than ascending. To add numbers written that way, you have to start from "the end" of the number to take into account the carries, that is, you have to do the operation "backwards" with respect to the direction of writing. Writing the larger powers the first and the smaller powers the last is an arbitrary convention, and the convention could very well be viceversa (and, in fact, for example the units are spoken before the tens in languages such as German). Tolkien's languages, for example, are written consistently from left to right, both the words and the numbers (which, OTOH, can be decimal or dozenal), so our 123 is written with the digits ordered like 321 in Tengwar script.

That is what I was referring to Uaxuctum.

Perhaps I did not put it clearly enough. You put it very clearly with the exception of 123 as 321.

In Tengwar, one hundred and twenty three is written 321. If you were to write out the numbers from one to a given base you would still do it from left to right.

I see - hadn't thought of it that way.

At this point, I think I need to check understanding. This is just a dozenal adaptation of the octomatics system that neogenisis posted.

Do you all understand how I have built up each symbol?

For what it's worth, here's a multiplication table.
I decided to give the zero a downstroke afterall:


One problem with these symbols seems to be that unless they are very tall, narrow and well spaced, they become difficult to read.

I'm sorry, but I don't understand the process by which you built up the progressive digits.

Neogenesis wrote:

OK, I'll explain step by step.

Instead of using three digit binary numbers as you used in octomatics, my system uses two digit binary numbers on the top and two digit binary numbers on the bottom.

The top strokes count in binary from zero to four and the bottom numbers count how many times this process has been done.

Thus the symbol for zero is a horizontal line with a downward stroke on the right (I've changed my mind on this one - it should have a downward stroke.) This counts the fact that it is the first iteration 01 in binary.

One is a downward stroke on the right and an upward stroke on the right as this is the first time we have a non-zero number to count 01 in binary.

Two is 10 in binary and this is duly recorded on the top as a stroke on the left and no stroke on the right while the fact that it is still the first iteration is recorded below.

Three is 11 in binary so there are two strokes on top still the first iteration below.

Four is where things now change since we have run out of digits on top (remember we are only using two) we get back to zero and record the fact that it is the second iteration in binary below two is 10 in binary so we have a left down-stroke but no right down-stroke.

Then we repeat the top digits as before for five, six and seven but recording underneath each time that we are in the second iteration.

Eight brings us to the third iteration remember three is 11 in binary so we have two downward strokes.

Finally the top digits are repeated for nine ten and eleven at which point we run out of two digit binary numbers and we use the symbols as place-holders in increasing powers of twelve in exactly the way we are used to doing already gross, dozens units, etc.

I have a leaflet from Gene Zirkel about Binary Coded Digits which look the same as yours Endi. :huh:

The lower right stroke stands for 1, lower left stroke stands for 2, upper left stroke stands for 4, upper right stroke stands for 8 and it is recomended to use 0 rather than just the bar for zero.

Doing addition this way does help. As 1+1 means cancel the one stroke and add a two stroke etc.

What do you think Endi?

Sorry I can't draw them, I don't know how to do that. :unsure: It would be great if we could just type them. :)

Ged wrote:

So basically, you are saying that Zirkel's digits use downstrokes to count from one to three and the iterations are counted on top. The opposite to what I came up with.

There is no obvious reason for using one direction or the other provided one is consistent.

My computer is a bit slow at the moment and I am loath to do any graphics work at the moment (which is bound to further slow it) so I won't post what you described for now. However, it is very easy to design your own fonts.

Design and save a truetype font using this program:
Pentacom

And copy and paste it into "fonts" in the control panel.

You can code any letter or number to be whatever you like provided it fits into the grid. You can be typing normally in whatever font you like then just change to your own designed font to type your newly designed numbers. You won't be able to post them on a message board but you will be able to display and print them out in a word-processing or spreadsheet program.

If you want to display them here, draw them in a program called "Paint" you've probably got it. Save as a .gif upload to a website and post the URL between IMG tags here.

PS could you write the text of the leaflet, Ged.

Ah! The penny has droped. :huh: What confused me was that your iteration started at zero.

How about each stroke being 1, 2, 3 and 6 instead of the binary count? <_<

Ged wrote:

Now I'm afraid it's me that doesn't understand. Could you explain more fully? :)

Yes, you've got it. The iteration starts at zero to make things more balanced. If I started recording iterations at 4, I would end up with a dozen and four symbols.

Useing 1, 2, 3 and 6 I mean:

Right down stroke equals 1, left down stroke equals 2, left upper stroke equals 3 and right upper stroke equals 6.

So all strokes 'H' equals *10 (twelve).

Of course down and upper can be swaped likewise left and right.

„Ÿ„¢„Ÿ„§„¤„¢„¤„§„¡„Ÿ„¡„£„¥„Ÿ„¥„£„¡„¢„¡„§„¥„¢„¥„§
Boom?

Anyway, I absolutely love your system. It doesn't have the ease of adding that octomatics has, but it solves the problem of creating digits for dozenal. Instead of trying to come up with letter-like glyphs for each numeral based on their uses in other cultures, this allows a digit system with serious reason behind it. You won me over. Just get rid of the name dozomatics >_> And also keep in mind that straight edges aren't necessary, you could easily curve the lines togive it a handwriting feel.

*gasp internet explorer doesn't support the symbols*

Do you mean how it seems to make sense how the smallest numbers should be at the beginning of where you write, that is, the left, because we write from left to right, and yet the smallest numbers are on the right? e.g. 321 = 3 hundreds 2 tens and 1 unit.

Funnily enuff, that never occurred to me, before.

Yes, Bryan. However, IMHO although it might be nice to think about how such things are in fact arbitary, I don't think it is a good idea to mess about with the order we write powers.

Ged, I've found this:
Binary Coded Digits
Which may be a reference that includes Gene Zirkel's symbols.

This is a pretty obvious idea and it is not surprising that someone else came up with it before me. However, from what you have said, it sounds like Zirkel uses the upstrokes and down-strokes in the opposite way to me. This being arbetary, I had an evens chance of getting the same figures. I don't want to start a competing system if this is in use already. Could you post the leaflet here, Ged.

OK, I've been corresponding with Ged to get the Zirkel numerals I am posting them here:

Zirkel also talks about the need to write quickly and says that the position of the horizontal is unimportant.

Hi this is Gene Zirkel. I am happy to see my Binary Coded Digits here. The simple device is basically a letter H. The lower right leg is 1, the lower left is 2. Moving clockwise the upper left is 4 and the upper right is 8.

Endi states that Ged stated that he has a leaflet containing my Binary Coded Digits.

I don't think I wrote a leaflet, but they did appear in The Duodecimal Bulletin some time ago. If anyone has a copy could you let me know the date or the volume and issue number so that I could locate my copy. I have all 96; (9 dozen and 6) issues.

You can visit Dozenal Society of America at www.Dozenal.org or contact me at Contact@Dozenal.org

The Annual meeting of our Society will be held on Saturday 6 October 11#3;(2007.) at Nassau Community College on Long Island in NY. If interested contact me.

Gene

You're welcome.

So if I understand you correctly, in order to preserve the binary nature of the notation, you view the top line as if the whole thing had been rotated 13 (dozl) degrees.

Think of the 4 legs of the 'H' symbol as 4 binary digits 'dcba' as indicated below.

c|_|d
b| |a

A 1 indicates the presence of the leg and a 0 indicates its absence. The 4 legs of the 'H' symbol proceed clockwise from the lower right.

Thus 0101 or 5 is

|_
|

and
_|
| most be 1001 or 8+1 and so it represents 9.

Genito

Yes, that would amount to the same thing.

Can you just confirm then that the symbols are as I posted them.

One thing I will say is that your system avoids producing any symbol that looks like a 4 that isn't a 4 but it isn't as intuitive as the system I came up with after looking at optomatics.

How widespread is its use, btw.

I have visited the American duodecimal society website but unfortunately most of the links are broken.

I suggest leaving out the grande, as having "mon-" denote 2 groups of -s may be confusing. And don't skip by 3s. That's always confusing. My system is in my sig.
Think of is as such:

a=8s place
b=4s place
c=2s place
d=1s place