Dis, Dat, Dix & Douze

How many fingers has Mickey Mouse? A glance at the figure shows that he has three fingers and a thumb on each hand, so eight in all. Thus, we may expect Mickey to reckon in octal numbers, with base eight. We use decimals, with ten symbols from 0 to 9 for the smallest numbers and larger numbers denoted by several digits, whose position is significant. Thus, 47 means four tens plus seven units.

Mickey has three fingers and a thumb on each hand. How many toes has he got?

Mickey has three fingers and a thumb on each hand. How many toes has he got? [Image from The Walt Disney Company logo].

But the base ten is divisible only by 2 and by 5. There are advantages to having a highly composite base – one with many divisors. The Sumerians and Babylonians used base 60, divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30. We still use remnants of this sexagesimal system in reckoning time and measuring angles, but base 60 is uncomfortably large for general use.

Question: How many toes has Mickey got? In the notorious phrase beloved of maths-book writers, this is left as an exercise for the student.

The Dozenal System

The duodecimal, or dozenal, system with base twelve has been proposed, as 12 is divisible by 2, 3, 4 and 6. We need two extra symbols for the numbers ten and eleven, which are less than the base. Let’s write them as X and E and call them dek and el. Then twelve is written 10 and called do (pronounced doh and short for a dozen). The system continues with 11, 12, … 1X, 1E and 20 or do one, do two, … do dek, do el and twodo.

Then, jumping in twelves, threedo, fourdo, up to eldo and gro. This gro is short for gross or twelve twelves, written 100. Twelve gro is one mo (twelve cubed, or 1728 in decimal). So, the decimal number 47 becomes 3E, threedo el or three twelves and el units. And we are currently in the year 11E9, or mo gro eldo nine.

Prospects

What advantages has the dozenal system? For one thing, multiplication tables are substantially simpler in dozenal. And many small fractions (one quarter, one third, three quarters, etc.) have a simpler form in this system.

So, why don’t we move from dix to douze? The Dozenal Societies of America and of Great Britain would favour this. We already have twelve months in a year and twice twelve hours in a day.

Computers convert numbers to binary form, using only zeros and ones, an convert the answer back to decimal before presenting it. We are generally oblivious to what goes on under the bonnet, and unconcerned about it. But computers calculate without grief. For us, a number base change would be massively disruptive, causing unimaginable confusion.

I believe that the chance of the Dozenal Societies persuading us to change to base twelve is about the same as the likelihood of Mickey Mouse converting us to octal.

 


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