## 21251Re: [PrimeNumbers] Re: Set of prime numbers

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• Jan 3, 2010
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--- On Wed, 12/23/09, Paul Leyland <paul@...> wrote:
> On Tue, 2009-12-01 at 23:54 +0000, Phil Carmody wrote:
> > > > and was certainly known some 1000
> > > > (maybe 25 000) years earlier.
> > >
> > > I'm intrigued by the "certainly";
> > > I would have said "probably" for 1k BCE.
> >
> > I'd have said "definitely" for >3k BCE. Base 60 just screams
> > knowledge of divisibility properties.
>
> Sorry for the late response to this thread but I've been
> rather tied up
> with Real Life(tm) recently.
>
> There is a persuasive suggestion that the divisibility
> properties of
> radix-60 arithmetic is a consequence of its choice, not a
> reason for its
> choice.  The argument goes as follows.
>
> A number of cultures have independently invented quinary
> arithmetic, for
> reasons which should be obvious.  There are still
> relics of this in
> modern culture --- the five-bar-gate tallying method, for
> instance.
> Bi-quinary has also been widely used throughout
> history.  This uses four
> different symbols for the digits 1-4 (the symbols are
> frequently 1 to 4
> identical lines or dots) and another symbol for 5.
> Digits 6 through 9
> are then represented by the juxtaposition of the 5-symbol
> and the
> appropriate symbol for 1 through 4.
>
> A number of cultures have independently invented duodecimal
> arithmetic.
> Many relics of this exist: 12 ounces to the Troy pound; 12
> inches to the
> foot; 12 pennies to the shilling and so on.  The most
> convincing
> survivors to my mind are the survival of the English words
> "dozen" and
> "gross".
>
> Some time around 4000 to 3500 BCE the Sumerians moved into
> Mesopotamia
> and merged with a pre-existing culture.  One culture
> used quinary or
> bi-quinary and the other
> duodecimal.   Neither culture supplanted the
> other, rather their notations
> merged.   Indeed, the symbols of early
> Mesopotamian arithmetic and accounting documents show
> strong evidence
> for a bi-quinary (later decimal) sub-structure in the
> sexagesimal
> notation.

That looks like the choice of 60 precisely because of its divisibility properties. They didn't take the LCM and later make a shock discovery that it had all the factors of the two original numbers, shall we say.

Phil
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