-: Twelfty for Decimal-Users :-


Intro:Home twelfty thorn revived Links Twelfty is an alternating base, using staves of 10 and 12 in order.

History

The Early indo-europeans were decimal counters, using a number form of the type one hund six ty seven. They had no word for thousand. Thousands came later: the Germanic/Slavic form is different to the Latin or Greek forms.

When people absorb other cultures, bits of the absorbed cultures are preserved. So we see the celtic use of base 20 (where welsh [uigan] [mean twenty] is cognate to twenty, superimposed onto a non-IE number system.

The oldest records of the germanic languages all contain words corresponding to the count by tens. These are cognate to either ten-count, or teenty-wise. This points to the existance of a non-decimal counting system. When this is revealed, it is always the hundred of 120 in number and a thousand of 1200 in number.

In Gothic there is a marginal in Wilfus's Bible, in I Cor 15:6, which includes
      ... þau fimf hundam [taihuntejam] broþre sums ...
It reads ... five hundred [teentywise] brothers ... Adding a kind of thing like this is necessary only if the normal counting is not teentywise.

In practice, that the hundred is 120 everywhere, suggests that this was the practice in the proto-language. and that the different terms for decimal, comes only after the split. There are papers on the subject by
      Stevenson: The Long hundred and its use in England'
      Goodare: The Long Hundred and its use in Scotland'
      Menninger: Number words and Number Symbols
      esp pp 154-158
      Gordon: An Introduction to Old Norse, esp 107 Numbers.

The Practical Base

The decision to use base 120 was based on practical needs. What i needed was some base that allows all sorts of vulgar fractions to be added together and easily recognised.

I considered many different approaches to the problem of how to express numbers in a readily identifiable way. In the end, twelfty won, not just because of its divisors, but also because it has a preferred interval.

The two-place period of twelfty is a product of small primes: 7, 11 and 17. Even though decimal, base 21, base 99 also have preferred intervals, the twelfty-system is better equiped for the problems.


              decimal        twelfty       base 60
  17/63   0.269 841 269    0:3245 8585    0:16 11 25 42 11
  23/77   0.298 701 298    0:35V1 35V1    0:17 55 19 28 49 .. (15 places)
   2/ 7   0.285 714 285    0:3434 3434    0:17 08 34 17 08

What makes the periods reckonisable is that the periods of say 17/63 is in twelfty and sixty, the same as a seventh. So we see that the number is something of the form 2^x 3^y 5^z 7.

Differences to the Historical Forms

The abacus is read differently, which means that we name the columns, rather than the digits. So we have eg

            million  cention  thousand  hundred  units  :  primes  secs, third  fourth

     tens

     units


© 2003-2009 Wendy Krieger