Five of one, half a dozen of the other

Take a book, any book. It might be the Chambers Dictionary, for example, and turn to that page, usually a left-hand one near the beginning, that you never even glance at, containing all the guff about not reproducing or transmitting it by an means, electronic or mechanical including photocopying.

Anyway, near the bottom of the guff, you'll find the ISBN - in this case 0-304-34452-4. The first digit, if the book is in English, will always be zero. Now what I want you to do with the remaining digits is this: multiply the first by 9, the second by 8, the third by 7, continuing until the final digit is multiplied by 1, and add all the products together. In the above case, we have:

(9x3)+(8x0)+(7x4)+(6x3)...+(4x1)

giving a grand total of 132. And the remarkable thing is that the number you arrive at will always be divisible by 11. (For foreign books you add the first digit multiplied by 10 to the sum). The final ISBN digit is included only as a check, with the simple divisibility by 11 test to guard against incorrect entry.

That little trick is explained in an old issue of The Dozenal Journal, the annual bulletin of the Dozenal Society of Great Britain (incorporating the Duodecimal Association).

The original aim of the Dozenals was to dispense with the silly finger- based counting system anchored to the number 10 by a more versatile numerology based on 12.

Look at the fractions: 1/2, 1/3, 1/4, 1/5, 1/6. In decimal notation, they emerge as 0.5, 0.333..., 0.25, 0.2, 0.1666... - two infinitely recurring decimals, one of two digits after the decimal points, and two with one such digits.

In duodecimal, however, we would have 1/2, 1/3,1/4, and 1/6 all coming out cleanly as 0.6, 0.4, 0.3 and 0.2, with only 1/5 causing problems. But how often do you want to divide something exactly among five people anyway?

Practicality and tradition motivate the Dozenals, and the first practical question is how to write the two new digits required for a duodecimal system and what to call them.

The British Dozenals opt for an inverted 2 and 3 filling the gaps we now call 10 and 11. The Americans, however, write them as X and pounds . Both sides of the Atlantic call them "dek" and "el", though the British call the round dozen "zen" while the Americans call it "do".

When an ISBN needs a final rounding digit of 10, you'll see an X as the last figure. So the ISBN men are half-way to dozenal conversion already.

In recent years, the pure Dozenal aim, of a counting system that allowed easy divisibility into halves, thirds and quarters, has become concentrated on the fight against metrication.

When we had 12 pence in a shilling and 20 shillings in a pound, dozenality made even more sense. We are now losing the 1,760-yard mile in favour of the decimal kilometre. With the latest Statutory Instruments threatening to ban all Imperial measures from 1 Jan 2000, the Dozenals see themselves as a bastion of common sense, retaining the pint, acre and troy ounce that have served us so well.

"Measurement of the people, by the people, for the people" is how Arthur Whillock of the Dozenal Society expresses their current aims. The "decimal transmogrifiers" are, they say, even having a go at the 12-hour clock and 60-second minute with plans to devalue the second to 0.864 of its present length, just to ensure a 100,000-second day.

The Dozenal Society produces a once-yearly journal, and the subscription is £4. The present membership is "about a gross". For further information, contact: Arthur Whillock, Walnut Bank, Underhill, Moulsford, Oxford OX10 9JH.