Tuesday, June 22, 2010

Duodecimal

My paternal grandfather enjoyed doing arithmetic using base-12. That's perhaps not surprising, he was an engineer, and he lived at a time when Britain used £sd. The British currency was pounds, which consisted of 20 shillings each containing 12 pence.

And the number 12 pops up all over the place: between noon and midnight there are 12 hours, 12 months in a year, 12 signs of the zodiac, a dozen is used as a common measure of eggs, there are 12 inches in a foot, ...

He referred to base-12 as duodecimal. At school I had to learn the times table up to 12 x 12. And the English language even has special words for 11 and 12.

Part of the reason that 12 is such as nice number is that it has a lot of factors: 2, 3, 4 and 6. Compare that to just 2 and 5 for 10 (as in base-10). With lots of factors numbers that are common expressed as multiples of 12 have easy to calculate 1/2s, 1/3s, 1/4s and 1/6s.

To use duodecimal you 'simply' add two symbols for 10 and 11: for example, you could use A and B and so you'd count like this: 1, 2, 3, 4, 5, 6, 7, 8, 9, A B, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B, 20, ... There 10 is the number we usually call 12.

It's possible that my grandfather was influenced by the 1935 book New Numbers: How Acceptance of a Duodecimal Base Would Simplify Mathematics, part of that book appeared in the Atlantic Monthly under the title An Excursion into Numbers.

Although it's unlikely that duodecimal will replace decimal in everyday use, especially since metric is used in place of imperial weights and measures across the world, and since the British pound was decimalized in 1971, other non-decimal base systems are in common use.

Computers use base-2 (binary), and programmers often use base-16 (hexadecimal). Vestiges of another computer base, base-8 (octal), still remain: aircraft transponder codes are four digit octal codes.

8 comments:

pob said...

Why not go the whole hog and use the Babylonian sexagesimal system?

The factors are 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30.

pob

miles said...

Do you know how to count in duodecimal on your fingers? Count on your finger joints, and use your thumbs as pointers. To count in hex, count your fingertips as well. This technique is apparently widespread in the Indian subcontinent. Discussion here.

Peter Risdon said...

Pedantry: you add two characters (A and B), not three.

Pádraig Brady said...

We'll be using base 12 at some stage
12 fingers

Ryan said...

Every base is base 10. http://cowbirdsinlove.com/43

Alen said...

"Dozenal FTW," check it out:

http://flud.org/blog/2008/01/12/dozenal-ftw/

jbaltz said...

For those of you who know or remember "Schoolhouse Rock", this was done very well in "Little Twelvetoes".

Still not very convincing, though. When teaching number bases to elementary schoolers, I had enough trouble connecting base ten to their fingers...

Osvaldo Doederlein said...

The decimal base is a foundation of the International System (S.I.) of units; only a very few S.I. units (like time) are easier to handle under base 12 or any other base. I know that the UK and US use barbarian non-SI units, but even the massive economic/military/cultural dominance of these countries didn't make most of the civilized world to prefer SI... ;-)