Re: More on number bases

Raymond Brown writes:
 > At 3:04 pm -0500 17/5/02, Danny Wier wrote:
 > [snip]
 > >
 > >The ancient Egyptians used base-8 fractions and base-10 whole numbers, if
 > >I'm not mistaken.
 > >
 >
 > Dunno about that - but the ancient Romans used base-12 fractions and
 > base-10 whole numbers.  The Latin for 1/12 is _uncia_ [u:Nkia] which, by
 > way of Old English _ynce_, gives us "inch" (1/12 of a foot) and, by way of
 > Old French _unce_, gives us "ounce" (1/12 of a pound troy* - still used by
 > gold- and silversmiths when I was young, but now legally obsolete in the
 > UK).
 >
 > *named from Troyes in France, not the place inhabited by Trojans of old!
 > Why the pound avoirdupois assumed 16 ounces, I know not - but 'twas nothing
 > to do with the Romans  :)
 >
 > Indeed, Latin had special words - all nouns - for all twelve divisions of a
 > unit.  In case anyone might like to imitate them or, indeed, base conlang
 > forms on them, I give them below:
 >             NOMINATIVE    GENITIVE
 > 1/12        u:ncia,       u:nciae
 > 2/12 = 1/6  sexta:ns,     sextantis
 > 3/13 = 1/4  quadra:ns,    quadrantis
 > 4/13 = 1/3  trie:ns,      trientis
 > 5/12        qui:ncu:nx,   qui:ncu:ncis
 > 6/12 = 1/2  se:mis,       se:missis
 > 7/12        septu:nx,     septu:ncis
 > 8/12 = 2/3  bes,          bessis
 > 9/12 = 3/4  do:dra:ns,    do:drantis
 > 10/12 = 5/6 dexta:ns,     dextantis
 > 11/12       deu:nx,       deu:ncis
 >
 > All are masculine, except _uncia_ which is feminine.
 >
 > Just to complete the picture, the word for 'a unit' was _as_ (gen.
 > _assis_), also masculine.
 >
 > Ray.

Come to think of it, if you confine yourself to proper fractions, and
have no fractions with a denominator greater than than the base you're
using, there's no distinction unless your fractional base is larger
than your integer base.  By which I mean, 1/2 is the same in base 8 as
in base 10.  Maybe the Egyptians only expressed fractions which were
multiples of 1/8, but that's not quite the same as saying they were
using base 8 for fractions.


Similarly for the Romans - we're talking about special words they
used, here, but if they expressed them numerically they'd have given
deu:nx as XI/XII, assuming they had that fractional notation.  Of
course, Roman numerals are a tally system rather than a place-value
system, so it's not quite correct to speak of them as having a
particular base anyway.


I can't quite remember what the Egyptians did, anyway.  I must get a
copy of Ifrah's _Universal History of Numbers_.  Anyhow, I _think_
they didn't really understand the idea of a fraction as a ratio of two
numbers - they knew about halves, and could apply them iteratively,
but they lacked the concept (or at least the notation) to express
something like 5/18.  I could be wrong about this, though - it was
something I read a while ago.  Well, I might as well look it up before
posting this...



Well, it seems it's a little more complicated than I remembered.  They
had fractions for all numbers, but they only ever expressed them as
unit fractions, that is 1/x.  Non-unit fractions were always expressed
as a sum of unit fractions, and the same denominator couldn't be used
twice.  So you got things like 6/7 = 1/2 + 1/3 + 1/42.  This does have
certain advantages over expressing it as a decimal, or as a simple
fraction.  Say you had 3 identical loaves of bread you wanted to
divide between 4 people.  If you use the Egyptian fraction 3/4 = 1/2 +
1/4, and divide the loaves in this way, everyone gets not only the
same amount of bread but also the same shaped pieces.  Or another
example (from the website linked to below) - if you have a number of
sacks of grain to divide up amongst a number of people, and they don't
divide neatly, Egyptian fractions reduce the problem to one of
dividing up each sack, rather than of dividing them all up in
aggregate.

 http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fractions/egyptian.html

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