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

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...

Timekeeping is heavily reliant on the number 12 and its composites: 2, 3, 4, 6.
Timekeeping is heavily reliant on the number 12 and its composites: 2, 3, 4, 6.

How to Learn the Duodecimal, Dozenal, Base 12 Numbering System

And a "political" note.

The emergence of a 12-vortex structure in a rotating Bose-Einstein condensate. About the base 12 aka duodecimal system.
The emergence of a 12-vortex structure in a rotating Bose-Einstein condensate. About the base 12 aka duodecimal system. | Source

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 all the other base X numbering / counting systems up to and including base 36, then the use of sequential numbers and letters should be used. Thus, as in hexadecimal, base 10 "10" is equal to base 12 "A", and base 10 "11" is equal to base 12 "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

Another Twelve-vortex array in a rotating Bose-Einstein condensate. About the base 12 aka duodecimal system.
Another Twelve-vortex array in a rotating Bose-Einstein condensate. About the base 12 aka duodecimal system. | Source
Astrology, the zodiac, and ancient cultures recognized the uniqueness of the number 12. About the base 12 aka duodecimal system.
Astrology, the zodiac, and ancient cultures recognized the uniqueness of the number 12. About the base 12 aka duodecimal system. | Source
Latitude and longitude are heavily reliant on the number 12 and its multiples and composites. About the base 12 aka duodecimal system.
Latitude and longitude are heavily reliant on the number 12 and its multiples and composites. About the base 12 aka duodecimal system. | Source

Duodecimal Base 12 Numbering System

Dice probability theory loves the number 12 composites... About the base 12 aka duodecimal system.
Dice probability theory loves the number 12 composites... About the base 12 aka duodecimal system. | Source

Duodecimal Orders of Magnitude

1 · 12 · 144 · 1728 · 20736 · 248832

Positional

248832 · 20736 · 1728 · 144 · 12 · 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.

An 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 (base12 ) or dozenal numbering system...

...uses the same structure, the only difference being the orders of magnitude. Base 12 aka duodecimal has twelve numbers (0 through 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 12x 12, or itself times 1728 And so on.

An 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.

Another 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.

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.

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

Many people advocate the teaching of base 12 as a societal standard.

Do you think this is a good idea?

  • Yes
  • No
See results without voting

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.

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Comments 2 comments

Will Apse profile image

Will Apse 2 years ago

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.


DreamerMeg profile image

DreamerMeg 2 years ago from Northern Ireland

I was brought up in the UK pre decimal money, same as will apse but its still difficult for me to get my head round the idea of base 12. Useful hub.

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