# The Dozenal System

Should we count by dozens instead of by tens? In many situations, we already do. Eggs, donuts, and months in a year are all counted by twelves, and many goods are ordered by the gross (a dozen dozen). Some advocate that we use a base-12 numbering system for all counting, switching from the decimal system to the duodecimal or dozenal system.

Some other base number systems are already in regular use for different purposes. Binary or base-2 is used by computers, and hexadecimal or base-16 is used to encode colors. The dozenal system is less well-known, but it has distinct advantages over counting by tens.

The decimal system may seem pre-ordained, perhaps because we have ten fingers, or because metric measurements have become ubiquitous. But counting by tens is arbitrary. Other cultures have used base-12, base-20, even base-60 number systems. The Babylonian number system still influences us today, as we continue to divide hours into 60 minutes and minutes into 60 seconds.

The dozenal system makes fractions easier. One-third is a common fraction, but when it is converted to decimal it becomes 0.333 … and the threes repeat forever. In dozenal, one-third is simply 0.4. (Note that that is four-dozenths, not four-tenths.) One-quarter is 0.3 in the dozenal system, neater than 0.25 in decimal. Most fractions are simpler in dozenal, because more numbers divide evenly into 12 than 10. (Some use a semicolon instead of a point when writing dozenal fractions.)

We have ten fingers, so you may think counting by tens is natural. Here’s a better way of counting on your fingers. With only one hand, use your thumb to count off the three segments on each of your four fingers. That’s a dozen (10 in dozenal). Now use your other hand the same way, to keep track of how many dozens you have counted. Using this method, you can count to one gross (100 in dozenal).

But wait, how is 10 a dozen? This is the fundamental change in switching to a different base. In any positional notation, the first position to the left of the radix point (decimal or dozenal point) counts the units, and when you have counted to the base, you put a 1 in the next position to the left (the tens or dozens place). This means that whenever we use a base larger than 10, we need extra symbols. Hexadecimal uses letters for its 6 extra symbols. For dozenal, we need two more. Some use X and E for the former ten and eleven, now called “dek” and “el.” Others use an upside-down 2 and an upside-down 3. Here is a table to compare base-10, base-12, base-16 and base-2.

## Dozenal and Other Number Systems

00zero00
11one11
22two210
33three311
44four4100
55five5101
66six6110
77seven7111
88eight81000
99nine91001
10dekA1010
11ƐelB1011
1210dozen (or “doh”)C1100
1311dozen-oneD1101
1412dozen-twoE1110
1513dozen-threeF1111
1614dozen-four1010000
1715dozen-five1110001
1816dozen-six1210010
1917dozen-seven1310011
2018dozen-eight1410100
2119dozen-nine1510101
221ᘔdozen-dek1610110
23dozen-el1710111
2420two dozen1811000
3630three dozen24100100
4840four dozen30110000
6050five dozen3C111100
7260six dozen481001000
8470seven dozen541010100
9680eight dozen601100000
10890nine dozen6C1101100
120ᘔ0dek dozen781111000
132Ɛ0el dozen8410000100
144100gross (or “groh”)9010010000
17281000great gross (or “moh”)6C011011000000

I’ll leave you with this 1973 episode of Schoolhouse Rock, which does a fine job of explaining dozenal counting.