The Dozenal System

With yesterday being 12.12.12, I thought I would discuss a numerical base system–the dozenal (or duodecimal) system.

The dozenal system is a base-12 system.  When I first heard about it, I dismissed it as just another base system for digits, which was popular because of the date.  However, there is a rather large contingent of people–scientists, accountants, engineers–that all actually propose that we teach both the decimal (base 10), and the dozenal system.  Again, I dismissed them as crackpots at first, but I was curious enough to read up on the matter, and I have to say that they have a point.

Dozenal Basics

Before we go any further, let’s explore what I mean when I say base 10 or base 12.  Any numerical system must provide some kind of symbolic representation of numbers–it is inconvenient to use tally marks or the like to represent all numbers.  Preferably, there would be a small set of symbols that can be ordered to represent all numbers.  In our familiar decimal system, those are:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

In binary, those are simply

0, 1.

For the computer scientists, hexidecimal would be

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.

The symbols are arbitrary, but how you group the symbols makes up the number system. There are 10 symbols in base 10.  Once you use those, you start to repeat, combining those symbols, e.g. 13 is one ten, and three ones.  In binary, 11 is one two, and one one, or 3 in the decimal system.  To put it another way, in decimal 13 is

1*10^1 + 3*10^0

and in binary, 11 is

1*2^1 + 1*2^0.

Dozenal is no different.  Instead of 10 symbols, we have 12

0,1,2,3,4,5,6,7,8,9,χ,ε (the last two pronounced dek and el, respectively).

Here, χ represents 10 in the decimal system, and ε represents 11. Please note that I have used an epsilon to represent el, which isn’t quite right but I don’t have the proper character set on WordPress…

Now that we have the symbols, let’s make some numbers.  Think back to second grade and those hauntingly boring words “we carry the three to the tens column…”  Well it’s the same here, except they are dozens columns.  This makes sense when thinking back to our exponential (the exponents below are in base 10.  Damn this is getting confusing).  In dozenal, the number 10 (pronounced do as in dough) is

1*12^1+0*10^0

or 12 in base 10.  The dozenal number 3χ7 in base 10 exponents is

3*12^2 + 10*12^1 + 7*12^0 = 432 + 120 + 7 = 559,

and going the other direction (exponents now in dozenal)

5*χ^2 + 5*χ^1 + 9*χ^0 = 3*10^2 + χ*10^1 + 7*10^0 = 3χ7.

If you are more word-oriented, try

3 gross and 10 dozens and 7 ones in dozenal is equivalent to 5 hundreds and 5 tens and 9 ones in decimal.  See?  It’s just that easy (yeah right; it hurt my head too).

OK, so those are the absolute basics.  There is more, like how you name the equivalents of tens and hundreds (dozens and grosses and such), but my brain hurts.

Why the heck would we do this?

So now the question is why the devil would you propose such a system?  Proponents argue that it’s simpler to do arithmetic.  Basically, 12 has more factors, so there are more even fractions.  For example, 1/4 of 10 is 2.5.  But 1/4 of 12 is 3 (a whole number).  Same story with thirds and so on.  They have somewhat of a point–bakers operate in dozens, the foot was divided into 12, shillings were 12 pence, etc.  12 is a more natural base, though by virtue of us having 10 fingers, we chose 10.  There are also some useful patterns that pop up in arithmetic that can be used, but I won’t go into those here.

Is this going to catch on?

Well in short, no.  We can’t even go metric in the United States.  A system being useful doesn’t mean it will be adopted–look at Plank Units if you’re not convinced of that.  It is an entertaining exercise though.  Honestly, for my own use, I file this under ‘recreational maths.’

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About faradaysheadache

Research Geophysicist with the US Geological Survey.
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