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An easy way to explain how this works is to start by counting to ten. You would say, "0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10". But when we reach 10, you notice there are two digits, 1 and 0. This means there is 1 set of ten and 0 more. Now, if you continue, you say, "11, 12, 13...", like saying, 1 set and 1 more, 1 set and 2 more, etc.
Okay, now imagine we start again, but we make twelve the set. Let's count, "0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ". Hold it now, there's a problem. We cannot say, "10" since we don't have a full set. We have two more digits before a set is complete. Let's just mark the two digits, "ᕍ" and "Ƹ", then let's count, "0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ᕍ, Ƹ, 10".
In the decimal system, we count like this:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, the last three numerals being "eight, nine, ten".
In the dozenal system, we count like this:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ᕍ, Ƹ, 10, the last three numerals being "dec, lev, do".
What happens when you count to 100? "...98, 99, 100," right? No, we only do that for the decimal system, not in the dozenal system. If we counted from 98 in the dozenal system, it would be, "98, 99, 9ᕍ", since we have two extra symbols per set. Now that we have two extra symbols, ᕍ and Ƹ, continuing on we would say, "9ᕍ, 9Ƹ, ᕍ0, ᕍ1." Here's an illustration:
Okay, now imagine we start again, but we make twelve the set. Let's count, "0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ". Hold it now, there's a problem. We cannot say, "10" since we don't have a full set. We have two more digits before a set is complete. Let's just mark the two digits, "ᕍ" and "Ƹ", then let's count, "0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ᕍ, Ƹ, 10".
In the decimal system, we count like this:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, the last three numerals being "eight, nine, ten".
In the dozenal system, we count like this:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ᕍ, Ƹ, 10, the last three numerals being "dec, lev, do".
What happens when you count to 100? "...98, 99, 100," right? No, we only do that for the decimal system, not in the dozenal system. If we counted from 98 in the dozenal system, it would be, "98, 99, 9ᕍ", since we have two extra symbols per set. Now that we have two extra symbols, ᕍ and Ƹ, continuing on we would say, "9ᕍ, 9Ƹ, ᕍ0, ᕍ1." Here's an illustration:
We find that in order to reach 100, we need to first reach the ᕍ0's, then Ƹ0's, and finally after reaching ƸƸ, the next number is 100. That's how you would count to 100 in the dozenal system.
Basic Building Blocks
Unfortunately, since all numbers after 9 in the dozenal system are exclusively in dozenal territory, we have to rename them. We can no longer use the hundreds, thousands, and millions to describe them, those are base ten constructs and must not be confused with dozenal.
There is no agreed upon standard for the pronunciation of dozenal digits. Some dozenalists say that dec and lev should be represented with an X for dec and an E for lev. Some say that dec should be called ten and that twelve should be called "do". However, what I have found is that "do" can cause problems, since it literally means "2," this causes a problem of overlapping nomenclature. An example is the "dodecagon," a 12 sided polygon. Should this be read as, "twelve and ten sides"?
Since the system is still at a premature stage, I propose a simple, yet in my opinion, elegant solution.
There is no agreed upon standard for the pronunciation of dozenal digits. Some dozenalists say that dec and lev should be represented with an X for dec and an E for lev. Some say that dec should be called ten and that twelve should be called "do". However, what I have found is that "do" can cause problems, since it literally means "2," this causes a problem of overlapping nomenclature. An example is the "dodecagon," a 12 sided polygon. Should this be read as, "twelve and ten sides"?
Since the system is still at a premature stage, I propose a simple, yet in my opinion, elegant solution.
My version will be written in Blue:
Decimal

Dozenal

Zero
One Two Three Four Five Six Seven Eight Nine   Ten 
0
1 2 3 4 5 6 7 8 9 ᕍ Ƹ 10 
Nil
One Two Three Four Five Six Sem Eight Nine Dec Lev Do 
If you have any suggestions, or would like to view other nomenclatures, please create a thread at: http://dozenal.boards.net/board/2/nomenclature
("ty" as suffix) Ten
Hundred Thousand Ten Thousand Million Billion  Tenth Hundredth Thousandth Ten thousandth Hundred thousandth 
10
100 1,000 10,000 1,000,000 1,000,000,000 ... 0.1 0.01 0.001 0.0001 0.00001 
Do ("zy" as suffix)
Gro Mo Do mo Bimo Trimo (latin numeral prefixes) Edo Egro Emo Edomo Ebimo 
Example Numbers
Decimal

Dozenal

54
fiftyfour 278
two hundred seventyeight 100,000
hundred thousand 6,982
six thousand nine hundred eightytwo 4,568,269
four million, five hundred sixtyeight thousand, two hundred sixtynine 84,000,000,293
eightyfour billion, two hundred ninetythree 2300 twentythree hundred 2,300 two thousand, three hundred 
ᕍ4
dec'zy four 27Ƹ
two'gro sem'zy lev 100,000
gro'mo 6,Ƹᕍ2
six'mo lev'gro dec'zy two 4,568,269
four'bimo, five'gro six'zy eight mo, two'gro six'zy nine 84,000,000,2ᕍ3
eight'zy four trimo, two'gro dec'zy three 2300 two'zy three gro 2,300 two mo, three gro 
Fractionals
Decimal

Dozenal

Period denotes a decimal point

Semicolon denotes a dozenal point

83.46
eighty three, point four six 2,007.24
two thousand seven, point two four 5.00548
five, and fivethousand, fourten thousand eighthundred thousandths 
83;4ᕍ
eight'zy three, dit four dec 2,007;24
two'mo seven, dit two'zy four 5;005,48
five, and five'mo four'zy'mo eight egro'mo 
Communicating Numbers With Your Fingers
For counting with your fingers, it's much better to use the 3 sections per finger we each have. However, if you were to show someone across a reasonable distance, you're going to need a more distinct form of communication.
Decimal

Dozenal

This system is universally known with the only differences some have are:
using the thumb to represent one. versus using the index finger to represent one. Other than that, most people around the world will understand what you mean when you try to convey numbers up to ten. 
Representing 6 with a “perfect” hand signal is pretty memorable since there are 3 fingers shown, and the 2 others to me atleast could be seen as x2. Then using 6 as a “full hand” is also useful in that people will notice the difference between 10decimal and ᕍ.
If anyone can come up with a less intrusive way of presenting Dozenal numbers, I’d love to hear. The system is not set in stone as of now, so it's best to give ideas while the concrete is solidifying. 