### How to Do Duodecimal, Dozenal, Base 12 Number System Conversions—Includes Examples

Latest update: September 17, 2020
Article originally published 11/17/2013. Copyright, updating, and historical notes at end of article.

If you understand the everyday decimal (base 10) number system, then you already understand the duodecimal, base 12, dozenal counting and numbering system. You just don’t know you know yet. The complete lesson immediately follows the short introduction.

And at the end of the page is a video, What Is the Sound of Pi in Base 12. Kind of induces a strange feeling after awhile...

### An Interesting "Political" Side Note.

The base 12 numerical system, also known as the duodecimal or dozenal system, is just like all the other base numbering and counting systems. However, this is the only base numbering system which has a "political" aspect to it. This has to do with the number 12 being a very useful number and as to which symbols to use for the base 10 numbers "10" and "11".

If one wishes to remain within the standardized structure of hexadecimal and the other base numbering and counting systems up to and including base 36, then the use of sequential numbers and letters should be used. Thus, as in hexadecimal, the base 10 number "10" is equal to the base 12 number "A", and the base 10 number "11" is equal to the base 12 number "B".

Others advocate the use of different symbols, some examples being:
• 10 = T
• 10 = X
• 11 = E
This how-to tutorial will stick with the base 2 through base 36 mathematical "standard" of 10 being designated by "A" and 11 being designated by "B".

### Complete Lesson and Examples

A quick review of the decimal, base 10 structure...

Base 10, Decimal Orders of Magnitude
1 · 10 · 100 · 1,000 · 10,000 · 100,000
Positional
100,000 · 10,000 · 1,000 · 100 · 10 · 1

We use the base 10 numbering/counting system in our day-to-day living. Base 10 has ten numbers (0-9) and orders of magnitude that are times ten.
• The lowest order number represents itself times one.
• The next order number represents itself times 10.
• The next order number represents itself times 10 x 10, or itself times 100.
• The next order of magnitude would be 10 x 10 x 10, or 1000.
And so on.

A base 10, decimal example would be the number 7824. This number means there are:
• Four 1’s,
• Two 10’s,
• Eight 100’s,
• And seven 1000's.
Which represents 4 + 20 + 800 + 7000 for a total of 7824.

### The duodecimal, base 12, dozenal numbering system...

...uses the same structure, the only difference being the orders of magnitude. Base 12 aka duodecimal has twelve numbers (0 thru B). The numbers are:
• 0 = 0, 1 = 1, 2 = 2, 3 = 3, 4 = 4, 5 = 5, 6 = 6, 7 = 7, 8 = 8, 9 = 9
• A = 10
• B = 11
The orders of magnitude are times twelve.
• The lowest order number represents itself times one.
• The next order number represents itself times 12. The next order number represents itself times 12 x 12, or itself times 144.
• The next order number represents itself times 12 x 12 x 12, or itself times 1728.
• The next order number represents itself times 12 x 12 x 12 x 12, or itself times 20736.
And so on.

#### Duodecimal, Base 12 Orders of Magnitude

1 · 12 · 144 · 1728 · 20736 · 248832

#### Positional

248832 · 20736 · 1728 · 144 · 12 · 1

A basic, first example of a duodecimal number would be the base 12 number 11111. This would mean there is:
• one 1,
• one 12,
• one 144,
• one 1728,
• and one 20736.
Which represents 1 + 12 + 144 + 1728 + 20736 for a total of 22621 in base 10 decimal.

Another base 12 example would be the number 2B9A. This number means there are:
• Ten 1’s,
• Nine 12’s,
• Eleven 144’s,
• And two 1728’s.
Which represents 10+108+1584+3456 for a total of 5158 in base 10 decimal.

Another base 12 example would be the number A51B. This number means there are:
• Eleven 1’s,
• One 12,
• Five 144’s,
• And ten 1728’s.
Which represents 11+12+720+17280 for a total of 18023 in base 10 decimal.

#### Convenience relist...

Base 12, Duodecimal Orders of Magnitude
1 · 12 · 144 · 1728 · 20736 · 248832
Positional
248832 · 20736 · 1728 · 144 · 12 · 1

### Side notes...

• Latitude and longitude are heavily reliant on the number 12 and its multiples and composites.
• Dice probability theory loves the number 12 composites.
• Astrology, the zodiac, and ancient cultures recognized the uniqueness of the number 12.

### More Duodecimal, Dozenal, Base 12 to Base 10 Conversion Examples

Column headings in the following table are simply a convenience relist of the relevant positional orders of magnitude as applies to each column. There is no significance attached as to where one column ends and the next one begins.

12 · 1
144 · 12 · 1
1728 · 144 · 12 · 1
0=0
92=110
B00=1584
1=1
100=144
BBB=1727
5=5
101=145
1000=1728
9=9
110=156
1001=1729
A=10
200=288
1010=1740
B=11
202=290
1100=1872
10=12
20A=298
1111=1885
11=13
20B=299
2000=3456
18=20
210=300
42BB=7343
20=24
7B6=1146
AB2B=18899
5A=70
A00=1440
B460=19656
5B=71
A2B=1475
BBBB=20735

#### Convenience relist...

Base 12, Duodecimal Orders of Magnitude
1 · 12 · 144 · 1728 · 20736 · 248832
Positional
248832 · 20736 · 1728 · 144 · 12 · 1

The Dozenal Society of America has all sorts of information regarding the mathematical and societal aspects of the base twelve number, counting system.

### What Is the Sound of Pi in Duodecimal

And last, but not least. Here is a video, The Sound of Pi in Base 12.

## Some of the Comments From the Previous Platforms

• Commenter, Will Apse, said: I'm a bit weird about the number 12. As a kid I used pounds, shillings and pence for money with 12 pennies in the shilling and twenty shillings in the pound (decimalized when I was 12, lol). This might be why I often think about the oddities of 12's. Money and time are rather important.
• Commenter, DreamerMeg, said: I was brought up in the UK pre decimal money, same as Will Apse, but it's still difficult for me to get my head round the idea of base 12.
• Commenter, BradMastersOCcal, said: Binary to base 12 is not as clean as base 8 (3 bits) or 16 (4 bits); this allows binary to just overflow into the next number. Base 12 has to use 4 bits but stop at 1011 (B). It is more like BCD. I had no idea this was a popular common base. Thanks, it is a very interesting article.
• Commenter, DreamerMeg , said: As a child, we had to learn to count in 12's for the UK's monetary system. Counting in 10's is a lot easier. We had a class of 7-year-olds chanting 12 pence is 1 shilling, 18 pence is 1 and sixpence, 24 pence are two shillings, 30 pence is 2 and sixpence, 36 pence is three shillings! The UK went decimal in 1971 but I can still calculate between new money and old money and between decimal measurement systems (SI units) and the old pounds and ounces. Keeps the brain active but I don't know that it's useful. Good article.

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