Re: More on number bases

At 9:51 pm +0100 18/5/02, Tim May wrote:
[snip]
>
>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.
Yes, but 1/3 can be expressed exactly in base-12, but may be only
approximated in base-10 or base-8 where they are recurring fractions

[snip]
>
>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.
They didn't.  Deunx was expressed as:  S followed by the sign for quincunx,
i.e.
. .
 .
. .

>Of
>course, Roman numerals are a tally system rather than a place-value
>system,
Exactly, see above.

>so it's not quite correct to speak of them as having a
>particular base anyway.
But they did.  In the spoken language, integer numbers were clearly
base-10, and expressed in numerals (and calculated on an abacus) in
bi-quinary form.

[snip]
>
>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.
Yes, that's right.  Somewhere I've got notes on this - it's a standard
problem for those learning to program: write a program that will express a
given numerator and denominator as an "Egyptian Fraction".  But IIRC the
Egyptians themselves actually complicated the matter by including a symbol
for 2/3   :)

Ray.

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Speech is _poiesis_ and human linguistic articulation
is centrally creative.
                                 GEORGE STEINER.
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