Skip to main content

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.

Comments

pobicus 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
Unknown 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.
We'll be using base 12 at some stage
12 fingers
Ryan said…
Every base is base 10. http://cowbirdsinlove.com/43
Unknown said…
"Dozenal FTW," check it out:

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

Popular posts from this blog

Your last name contains invalid characters

My last name is "Graham-Cumming". But here's a typical form response when I enter it:


Does the web site have any idea how rude it is to claim that my last name contains invalid characters? Clearly not. What they actually meant is: our web site will not accept that hyphen in your last name. But do they say that? No, of course not. They decide to shove in my face the claim that there's something wrong with my name.

There's nothing wrong with my name, just as there's nothing wrong with someone whose first name is Jean-Marie, or someone whose last name is O'Reilly.

What is wrong is that way this is being handled. If the system can't cope with non-letters and spaces it needs to say that. How about the following error message:

Our system is unable to process last names that contain non-letters, please replace them with spaces.

Don't blame me for having a last name that your system doesn't like, whose fault is that? Saying "Your last name …

All the symmetrical watch faces (and code to generate them)

If you ever look at pictures of clocks and watches in advertising they are set to roughly 10:10 which is meant to be the most attractive (smiling!) position for the hands. They are actually set to 10:09.14 if the hands are truly symmetrical. CC BY 2.0image by Shinji
I wanted to know what all the possible symmetrical watch faces are and so I wrote some code using Processing. Here's the output (there's one watch face missing, 00:00 or 12:00, because it's very boring):



The key to writing this is to figure out the relationship between the hour and minute hands when the watch face is symmetrical. In an hour the minute hand moves through 360° and the hour hand moves through 30° (12 hours are shown on the watch face and 360/12 = 30).
The core loop inside the program is this:   for (int h = 0; h <= 12; h++) {
    float m = (360-30*float(h))*2/13;
    int s = round(60*(m-floor(m)));
    int col = h%6;
    int row = floor(h/6);
    draw_clock((r+f)*(2*col+1), (r+f)*(row*2+1), r, h, floor(m…

Importing an existing SSL key/certificate pair into a Java keystore

I'm writing this blog post in case anyone else has to Google that. In Java 6 keytool has been improved so that it now becomes possible to import an existing key and certificate (say one you generated outside of the Java world) into a keystore.

You need: Java 6 and openssl.

1. Suppose you have a certificate and key in PEM format. The key is named host.key and the certificate host.crt.

2. The first step is to convert them into a single PKCS12 file using the command: openssl pkcs12 -export -in host.crt -inkey host.key > host.p12. You will be asked for various passwords (the password to access the key (if set) and then the password for the PKCS12 file being created).

3. Then import the PKCS12 file into a keystore using the command: keytool -importkeystore -srckeystore host.p12 -destkeystore host.jks -srcstoretype pkcs12. You now have a keystore named host.jks containing the certificate/key you need.

For the sake of completeness here's the output of a full session I performe…