Re: OT: coins and currency (was: [Theory] Types of numerals)

The best ratio between adjacent denominations for minimizing the
number of tokens exchanged in a transaction, whether your require
exact amounts or allow overpayment with change, is 2 - this is
mathematically provable.  There are papers by economists claiming that
it's actually 3, but the claim is based on a flawed analogy with the
Bechet problem of weights, the flaw being that you're not allowed to
use multiples of a given weight in the Bechet problem.   (Imagine
being able to use at most one penny per transaction.  Ridiculous!)
This set of criteria (fewest tokens exchanged) is based on the
economic principle of least effort.

As I said, however, there is counter pressure for keeping the total
number of denominations down, given that people have to be able to do
the math, recognize each token quickly, etc etc.  The Bechet problem
does have relevance here: you want the fewest denominations overall
that handle arbitrary amounts reasonably efficiently, and 3 turns out
to be a pretty good ratio (though not necessarily ideal).

Van Hove makes a compelling argument that the principle of least
effort should be given greater weight.  But he also points out that
the powers-of-two system yields computational difficulties for human
brains used to thinking in decimal terms.  Successive 1-2-5 triplets
(1,2,5,10,20,50,100,200,500,...) make a reasonable compromise; the
mean ratio is 2.25, not far from the least-principle ideal of 2.0, but
the math is easier and the number of denominations slightly lower.

Of course, in a conculture, any number of variables may differ from
what's true here and now, causing different choices to be superior.