Re: OT: coins and currency (was: [Theory] Types of numerals)
| From: | Nik Taylor <yonjuuni@...> |
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| Date: | Monday, January 9, 2006, 2:41 |
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tomhchappell wrote:
> As far as minimizing the [number of coins] (not the number of types
> of coins) you have to get and/or give in change, and keep carrying
> around in your pocket-or-whatever (not the number in existence),
>
> the optimum ratio is either 3 or 4. That is, each coin (or
> denomination of note) is either 3 or 4 of the next lower, and either
> one-third or one-fourth of the next higher.
Wouldn't two be better at that?
>
> So, 1, 3, 9, 27, 81, 243, 729 ... is a good series;
> 1, 4, 16, 64, 256, 1024, 4096 ... is a good series;
> 1, 3, 12, 36, 144, 432, 1728 ... is a good series;
> 1, 4, 12, 48, 144, 576, 1728 ... is a good series.
>
> ----
>
> A base for a number system has extra conveniences if it has many
> factors. Defining, for the moment, a "good base" to mean "a number
> that has at least as many factors as any smaller number", we get the
> following "good bases";
> 2 factors: 2, 3
> 3 factors: 4
> 4 factors: 6, 8, 10
> 6 factors: 12, 18, 20
> 8 factors: 24, 30
> 9 factors: 36
> 10 factors: 48
> 12 factors: 60, 72, 84, 90, 96, 108
> 16 factors: 120, 168
>
> and so on.
>
> Natlangs, and ordinary uneducated people, aren't likely to use bases
> greater than about 40 (in spite of the Mesopotamian/Egyptian/Greek
> scholars' fondness for base 60).
>
> Besides, the base 12 -- a "good base" -- fits neatly with the third
> and fourth example series I wrote towards the beginning of this reply.
>
> So, I plan to use base 12.
> I might use something like:
> 1 "knuckle" or "joint"
> 3 knuckles = 1 "finger" or "digit"
> 12 knuckles = 4 fingers = 1 "hand"
> 36
> 144
> 432
> 1728
> ... etc.
>
> Or, since I'd rather have the higher factor used first, I could use
> the series
> 1
> 4
> 12
> 48
> 144
> 576
> 1728
> ... etc.
> but then I don't know what I'd name them.
1, 3, 6, 12, 36, 72, 144, etc. (or 1, 2, 4, 12, 24, 48, 144) would work
better at minimizing the number of coins needed. The number of coins
needed for values of 0-11* in the four different systems (adding binary
- 1, 2, 4, 8, etc.) for further comparison):
Value 1/3 1/4 1/3/6 1/2/4 Binary
0 0 0 0 0 0
1 1 1 1 1 1
2 2 2 2 1 1
3 1 3 1 2 2
4 2 1 2 1 1
5 3 2 3 2 2
6 2 3 1 2 2
7 3 4 2 3 3
8 4 2 3 2 1
9 3 3 2 3 2
10 4 4 3 3 2
11 5 5 4 4 3
Avg 2.5 2.5 2 2 1.67
*I'm going with 0-11 on the basis of calculating coins used in change,
which is why 0 is a valid entry. Multiples of twelve would follow the
same pattern, which means that for 0-143, you'd double those averages
(since you'd need coins for both twelves and ones)
Binary-based coins are the best system for minimizing coins per
transaction. 1/3/6 or 1/2/4 comes close.
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