P(l)aying with Euro's
Paying a small amount of money in Euro's can always be done with the following
set of coins and (bank)notes:
0.01 , 0.02 , 0.02 , 0.05 , 0.10 , 0.20 , 0.20
, 0.50 , 1.00 , 2.00 , 2.00 , 5.00 , 10.00 , 20.00 , 20.00 , 50.00 .
Some multiple of this is what people commonly have in their wallet when they go
to the local supermarket, here in the Netherlands.
Paying an amount of money now goes like here. € 99.99 = 50.00 + 20.00
+ 20.00 + 5.00 + 2.00 + 2.00 + 0.50 + 0.20 + 0.20 + 0.05 + 0.02 + 0.02 (though
50.00 + 50.00 and receiving back 0.001 would have been much simpler).
It
is remarked that paying the exact money involves a series of binary decisions,
twelve (12) in our case. This is due to the above standarized subdivision in
coins and notes. Herewith, counting all the way up to 99.99 (like paying with
single cents 0.01 ) is replaced by the much more efficient divide and conquer
approach which is common in our cash money economy.
Some redundancy is
involved, too, with most divide and conquer approaches. If we restrict attention
to cents only, 3 yes or no - 1 or 0 - decisions about 0.01 , 0.02 , 0.02 ,
0.05 - in that order, then we find:
0.00 = free of charge (0000) , 0.01 =
0.01 (0001) , 0.02 = 0.02 (0010 or 0100) , 0.03 = 0.02 + 0.01 (0011 or 0101) ,
0.04 = 0.02 + 0.02 (0110) , 0.05 = 0.05 (1000) or 0.02 + 0.02 + 0.01 (0111) .
It is seen that all
of the 23 = 8 possible combinations of yes (1) or no (0) decisions
are present here. But some of these denote the same transaction of money and the
maximum amount involves only 5 cents (while 7 cents could have been possible
with a binary decision tree instead of one matching base 10). There is a
redundancy with the cashing of Euro's.
Some of the redundancy could be
removed by allowing multiple
decisions. With ( 0.01 , 0.02 , 0.05 ) it becomes:
0.02 = 0.02 (010) , 0.03 = 0.02 + 0.01 (011) , 0.04 = 2 x 0.02 = (020) , 0.05 =
2 x 0.02 + 0.01 (021) or 0.05 (100) .
Thus 0.02 and 0.03 now become unique, but such is still not the case with 0.05
and, moreover, the whole Boolean Objects character of our decision support
system is destroyed, with the advent of more than one decision at a time.
In retrospect, the above is only a very primitive 'Ansatz' to a far more
sophisticated (and far less trivial) mathematical theory, which is known as
"The Coin-Exchange Problem of Frobenius". I've discovered this only
recently on the
Internet.