Re: [HM] Irrationals in Mesopotamia (was: Music and Incommensurability)
Luigi Borzacchini (gibi@pascal.dm.uniba.it)
Wed, 28 Jul 1999 09:09:39 +0200
>
> It would be easy for the Babylonian mathematicians to decide that "7
> does not divide" etc. 1/7 has period 3 in the sexagesimal system (6 in
> the decimal one), 1/11 has period 5, 1/13 period 4. Thus only few easy
> divisions were necessary, and then they realized the computation enters
> into an infinite loop. So they gave approximations instead. One wonders
> how long they worked on the square root of 2 before giving up. Did the
> distinction between repeating and non-repeating positional fractions
> even occur to them? If it did not, one can hardly speak about
> incommensurability in that connection. Another intriguing question is:
> since base 60 was used for scientific calculations, but base 10 for
> everyday writing of numbers, e.g. in dates, did they realize that some
> fractions, say 1/3, are finite sexagesimal fractions, but that "3 does
> not divide" in decimal notation?
>
> Avinoam Mann
>
Right. But it is difficult to find an answer to the second question, almost
impossible to the first. Anyway my locution "a numerical fact" didn't go to
this level of analysis: it simply stated the difference between the ancient
numerical approximation experience and the proof by absurd.
Luigi Borzacchini