The Dozenal system
In our society we are used to using base 10 for our calculations. However there are good reasons not to. It would be very sensible to use the dozenal system (base 12). Dozenal notation will be in Bold on this page.
The symbols for the numbers 0 to 11 on this site are:
0 1 2 3 4 5 6 7 8 9 X E
Why base 12?
There are several good reasons for choosing base 12. The main reason
is its divisibility. Whereas 10 can only be divided by 2 and 5, 12 can
be divided by 2,3,4 and 6. This has good consequences:
Reciprocals
A comparative table:
| Base 10 | Base 12 |
| 1/2 | 0.5 | 0;6 |
| 1/3 | 0.(3) | 0;4 |
| 1/4 | 0.25 | 0;3 |
| 1/5 | 0.2 | 0;(2497) |
| 1/6 | 0.1(6) | 0;2 |
| 1/7 | 0.(142857) | 0;(186X35) |
| 1/8 | 0.125 | 0;16 |
| 1/9 | 0.(1) | 0;14 |
| 1/10 | 0.1 | 0;1(2497) |
| 1/11 | 0.(09) | 0;(1) |
| 1/12 | 0.08(3) | 0;1 |
As you can see the table for base 12 looks much easier, especially for
the often used fractions.
Divisibility Criteria
Divisibility criteria are "how do you check if a number is divisble by x" rules.
They are listed here.
| Base 10 | Base 12 |
| 2 | Ends in 0,2,4,6,8 | Ends in 0,2,4,6,8,X |
| 3 | Digit sum divisible by 3 | Ends in 0,3,6,9 |
| 4 | Last two digits divisible by 4 | Ends in 0,4,8 |
| 5 | Ends in 0,5 | Subtract first or last two digits from rest |
| 6 | Divisible by 2 and 3 | Ends in 0,6 |
| 7 | Subtract last digit twice from rest | Add last digit three times to rest |
| 8 | Last three digits divisible by 8 | Last two digits divisible by 8 |
| 9 | Digit sum divisible by 9 | Last two digits divisible by 9 |
| 10 | Ends in 0 | Divisible by 2 and 5 |
| 11 | (Odd digits - even digits) divisible by 11 | Digit sum divisible by E |
| 12 | Divisible by 3 and 4 | Ends in 0 |
One of the methods might be unfamiliar. The method in base 10
to check divisibility by 7 works like this. Take the last digit,
double it, and subtract it from the rest of the number. If this is
divisible by 7 then the original number is. An example:
Is 178444 divisible by 7?
17844 - 2*4 = 17836
1783 - 2*6 = 1771
177 - 2*1 = 175
17 - 2*5 = 7 => yes
Similar in dozenal. 178444 = 87324
Is 87324 divisible by 7?
8732 + 3*4 = 8742
874 + 3*2 = 87X
87 + 3*X = E1
E + 3*1 = 12 = 14 => yes
The same test for division by 5 in dozenal:
Is 87324 divisible by 5?
873 - 24 = 84E
4E - 8 = 43 = 51 => no
Number theory
All prime numbers (except 2 and 3) in base 12 end in 1,5,7,E
Perfect squares in base 12 end in 0,1,4,9
Time in base 12
Clocks
Instead of our 24-hour clock one can achieve the same accuracy with
less digits in base 12. A day will be cut up in 12 hours, each of 144
minutes (each of which takes 50 seconds). Each minute is divided into
144 seconds, which take 25/72 of a second). A digital clock would therefore need only 5 digits:
H:MM:SS
Your current time is calculated here:
Calendar
The year is divided into 12 months of 30 days. Each month is
divided into five 6-day weeks. We then have 5 (or 6) extra days. For a
suggestion where to put those have a look at Newhall's
Dozenal Calendar.
Length measurement
Starting from the basic time unit, the dozenal second (1 s =
25/72 s) and the speed of light we define a dozenal "foot" as basic
length unit. It is defined as the distance light travels in
12-8 dozenal seconds:
1f ~ 0.24209 m
Other length units are defined by using standard metric prefixes. A
dozenal inch or decifoot is 1/12 f: 1df ~ 2.0174 cm. A dozenal
mile is 1kf ~ 418.33 m.
Weight measurement
For this I use the mass of one cubic
dozenal foot of water and define this as 1 kg. This gives us a
basic unit of: 1g ~ 8.2108 g. This is more useful than the
current unit because it gives rise to smaller numbers in every day
life.
Other units
One mol is now of course measured with respect to the new unit of
mass, i.e. the amount of substance of a systems that contains as many
elementary entities as there are atoms in 0;01 kg of
10C.
The unit of velocity is 1f/s = 0.697 m/s = 2.51 km/h
Wind speeds
Instead of using the Beaufort scale we can have a continuous scale: Take the base 12 logarithm of the wind velocity in f/s, then:
Wind scale = 12 * 12log(wind velocity) - 6
| Scale | Velocity | Beaufort | Hurricane |
| 0;0 | 8.7 km/h | 2 |
| 2;0 | 13.2 km/h | 3 |
| 4;0 | 19.9 km/h | 4 |
| 6;0 | 30.1 km/h | 5 |
| 8;0 | 45.6 km/h | 6 |
| 9;0 | 56.1 km/h | 7 |
| X;0 | 69.0 km/h | 8 |
| E;0 | 84.8 km/h | 9 |
| 10;0 | 104.3 km/h | 11 |
| 11;0 | 128.2 km/h | 12 | 1 |
| 12;0 | 157.9 km/h | 12 | 2 |
| 13;0 | 194.2 km/h | 12 | 3 |
| 14;0 | 238.9 km/h | 12 | 4 |
| 15;0 | 293.8 km/h | 12 | 5 |
Money
There are several ways of making base 12 money. With 3 different coins or notes for one order of magnitude there are 3 possibilities:
| 1 | 1 | 1 |  |
| 1/2 | | 1/2 |
| 4 | |
| 1/4 | | |
| 2 | 2 |
| 1 | 1 | 1 |
The first of these methods looks most like the American Dollar
system, the last of these methods looks more of the European system
for the Euro. The only difference is that the "big" step is 3 units
and not 2 1/2 units. As is shown by the Dutch 3 guilder coin
from 1830 a base 12 currency system has been in use in the past on the continent as well as in Britain with its old pence / shillings system.
Constants
c = 100 000 000 f s-1
π = 3;18480 9493E 91866 4573X 6211E E1515 51X05 72929 0X780 9X492 ...
Links
The American Dozenal Society
Great Britain Dozenal Society
Dozens Online Forum