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Walabio


A Skeptic & So Also Therefore Now A godless Agnostic Atheist

More Blog Posts77

Oct
22nd
2015

Radix-Economy & Monetary Denominations. · 4:21am Oct 22nd, 2015

The most apes use decimal currency. In most currencies, the largest currencies are 100 or 1,000. The smallest denominations is generally .01. One could have these denominations:

1,000.00
0,100.00
0,010.00
0,001.00
0,000.10
0,000.01

The denominations are far apart, thus making change difficult. Most countries use pseudonbinary:

1,000.00
0,500.00
0,250.00
0,100.00
0,050.00
0,025.00
0,010.00
0,005.00
0,002.00
0,001.00
0,000.50
0,000.25
0,000.10
0,000.05
0,000.02
0,000.01

That is the sequence one gets from starting at the largest denomination at works toward smaller denominations. If one starts at the smallest denomination and works toward larger denominations, one gets this different sequence:

0,000.01
0,000.02
0,000.05
0,000.10
0,000.20
0,000,50
0,001,00
0,002.00
0,005.00
0,010.00
0,020.00
0,050.00
0,100.00
0,200.00
0,500.00
1,000.00

¿Does a better way to create denominations which is unambiguous as to which denominations to use? ¡Yes!:

1,000.00
0,300.00
0,100.00
0,030.00
0,010.00
0,003.00
0,001.00
0,000.30
0,000.10
0,000.03
0,000.01

No ambiguity exists about which is the next logical denomination and although sometimes one must hand out 2 of a denomination, the reduction in denominations more than compensates for this. This is obviously derived from powers of 3. ¿Why powers of 3?:

3 has the lowest Radix-Economy. Let us look at 1,000 in 2 bases:

In unary (base1), one only needs to know one symbol, but one needs 1,000 symbols for writing 1,000.

In Base 1,000,000, one only needs only 1 symbol for representing 1,000, but one must learn 1,000,000 symbols.

The complexity of a base-system is equal to the number of symbols one must learn times the length in symbols of a particular number. This is Radix-Economy. The number with the number with the lowest radix-economy is e. The natural number with the lowest radix-economy is 3. ¿What does this have to do with denominations and making change?:

The number of denominations a cashier has to handle times the number of coins and bills the cashier has to hand out in change is lowest using powers of 3.

Comments ( 9 )

3488700

I do not have time to check your work now because I have to go to my work, but I appreciate the participation. I also wrote about the 1 true circle-constant Tau, Unicorn-Teleportation, an half-finished blogpost about the teleportation of Miss Pinkamena Diane Pie, Why ponies might use balanced ternary, the Martian, and about the Antiscience of an Admin who turned out to be Meeester who banned OnlyANorthernSong for for debunking creationism because it hurt the feelings of creationists (Meeester went on to ban me for complaining about the wrongful ban of OnlyANorthernSong).

I am off to work.

Post Scriptum:

You might want to join my group:

The Skeptics' Guide to Equestria

Interesting idea...

One constraint on any real system is its compatibility with people..

3490105

One can use Radix-Economy for fictional currencies too:

I once created a campaign for Dungeons and Dragons with ponies using sexagesimal (Base60). The ponies 1st used electrum for money, but discovered that electrum is a natural alloy of copper, silver, gold, and some minor components. They switched to a system where 1 Gold-Coin is worth 60 Silver-Coins and 1 Silver-Coins are worth 60 Copper-Coins.

3 has the best Radix-Economy, but does not work well for sexagesimal; so now, it uses every factor of 60:

01
02
03
04
05
06
10
12
15
20
30

I chose sexagesimal because it is both highly composite and 3 does not work well as I can show:

Small to large:

01
03
10
30

Large to small:

20
06
02

We can actually do better than this. Let's consider for the moment only the problem of making change. How can we minimize the number of coins needed to make change? Assuming any amount is equally likely, the current US system (1, 5, 10, 25) has an average cost of 4.7 coins. We can cut this down to 3.89 by replacing the dime with an 18-cent coin, and optionally replacing the quarter with a 29-cent coin (this doesn't affect the average cost). The roughly base 3 system you propose has an average cost of 4.2 coins, for comparison. Euro-style change (1, 5, 10, 20) has an average cost of 5 coins. If we include the 50-cent piece for US and Euro, those systems also have an average cost of 4.2 coins.

Of course, 3490105 has a point. There's a big problem with (1, 5, 18, <25 or 29>) that is not found with (1, 5, 10, 25), (1, 5, 10, 20), or (1, 3, 10, 30) don't have, and this problem makes it much less user-friendly. How do you make change with US, Euro, or pseudo-base-3 coins? Take the biggest coin that is no larger than the remaining amount, then repeat that until you've made change. This is called a "greedy algorithm", and it's nice, simple, and in these cases gets you the optimal result. With the 18-cent coin in play, how do you make change for 36 cents? Obviously, two 18-cent coins would be the best option...but if you use a greedy algorithm, you'll need to use four coins (25+5+5+1 or 29+5+1+1). Oops! In this case, there is much less of an advantage, with average costs of 4.45 or 4.58, depending on if you replace the quarter or not, respectively, which is probably not a net gain when you factor in the additional effort for cashiers/people getting upset about inefficient change. We can still beat your system while restricting ourselves to denominations that work with greedy algorithms, but then you're talking about replacing everything but the penny, and these numbers are harder to reason with (example sets: (1,3,11,37) or (1,3,7,18,44)). You don't have to reason with them, since the greedy algorithm works, but people probably wouldn't like it.

3492988

I like your style, but you forgot something and I was not clear:

The European System has a .02 too:

.01
.02
.05
.10
.20
.50

My mistake is that I mean to simplify the complexity of the whole system by minimizing both coins handed out to customers and denominations of coins. Number of coins handed out to customers times number of denominations is what I optimize. This is Radix-Economy.

As an example, the 4.2 coins for roughly Ternary times 4 (.01, .03, .10, and .30) is 16.8

You are right, but for the wrong question. You answered the wrong question because I was not clear. Sorry.

As far as what My Little Economy wrote, that leaves us with a pseudopower-system shoehorned into decimal. Let us look at the options minus powers of powers because we might as well just use the lower power:

1
2
3
5
6
7
10

Those are all of the powers up to 10 meeting the criteria. These shoehorned into decimal:

2:

.01
.02
.10
.20
.50

Or

.01
.02
.10
.25
.50

3:

.01
.03
.10
.30

5

.01
.05
.25

6

.01
.05
.30

7:

.01
.05
.35

10:

.01
.10

They are human compatible, and they use the greedy algorithm (as power-sequences of natural numbers raised to natural numbers, they must fit the greedy algorithm).

3493753
>My mistake is that I mean to simplify the complexity of the whole system by minimizing both coins handed out to customers and denominations of coins. Number of coins handed out to customers times number of denominations is what I optimize.

Ah, but in fact you do not! Your system of (1,3,10,30) also has four denominations but requires an average of 4.2 coins (∏=16.8). (1,5,18,25) has four denominations and requires an average of 3.89 coins to make change (∏=15.56), although the greedy algorithm is not optimal here. If we restrict ourselves to systems where the greedy algorithm produces the optimal result, we can still do better than that, with (1,3,11,37), which requires an average of 4.1 coins (∏=16.4).

3493839

But my system is the best if we limit ourselves to denominations with only 1 nonzero figure. Indeed, if we go from apes using decimal to ponies using balanced ternary, then powers of 3 become optimal:

Let us suppose that the largest unit is 243 bits and that ponies divide the bit into 243 subunits and that ponies use balanced ternary. The best possible radix economy is thus:

1,000.000,0
0,100.000,0
0,010.000,0
0,001.000,0
0,000.100,0
0,000.010,0
0,000.001,0
0,000.000,1

I have a BlogPost about why ponies might use balanced ternary:

Ponies might decide to use balanced ternary.

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