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:: FAQ > Editorial > Denomination, culture, and math (by webmaster) ::
 
The denomination design is not for convenience only, but it also show some characteristics about a country/region*, just like a business card of a person.

The entire world has been ruled by Europeans some time or another, except China, Japan, Korea, Thailand, Nepal, and Ethiopia. Let's take a look at East Asian currencies first.

 coinbanknote
Thailand0.25, 0.5, 1, 510, 20, 50, 100, 500, 1000
Japan1, 5, 10, 50, 100, 5001000, 2000 (comm.), 5000, 10000
Korea, South1, 5, 10, 50, 100, 5001000, 5000, 10000
Korea, North0.01, 0.05, 0.1, 0.5, 11, 5, 10, 50, 100, 500
Taiwan1, 5, 10, 20 (new), 50100, 200 (new), 500, 1000
Mainland China0.1, 0.5, 11, 2 (withdrawing), 5, 10, 20 (new), 100

As we can see, currency with prefix 2 is not common, I call this system 1-5.

Let's look at UK and its former colonies

 coinbanknote
UK0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1, 25, 10, 20, 50
US0.01, 0.05, 0.1, 0.25, 0.5 (rare), 1 (rare)1, 2 (rare), 5, 10, 20, 50, 100
Canada0.01, 0.05, 0.1, 0.25, 1, 25, 10, 20, 50, 100
Australia0.01, 0.02, 0.05, 0.1, 0.2, 1, 25, 10, 20, 50, 100
New Zealand0.01, 0.02, 0.05, 0.1, 0.2, 1, 25, 10, 20, 50, 100
Singapore0.01, 0.05, 0.1, 0.2, 0.5, 12, 5, 10, 20, 100, 1000, 10000
South Africa0.01, 0.05, 0.1, 0.2, 0.5, 1, 2, 510, 20, 100, 200

Prefix 2 is common, I call it 1-2-5. And their exchange rate are similar, with 100 as maximum.

The Dutch part is also interesting

 coinbanknote
Netherlands0.05, 0.1, 0.25, 1, 2.5, 510, 25, 50, 100, 250, 1000
Netherlands Antilles0.01, 0.025, 0.1, 0.25, 15, 10, 25, 50, 100, 250
Suriname0.01, 0.05, 0.1, 0.25, 15, 10, 25, 100, 500, 1000, 5000, 10000, 25000
Aruba0.05, 0.1, 0.25, 0.5, 15, 10, 25, 50, 100, 500

I call it 1-25-5 or 1-25.

And communism countries often have prefix 3, like
People's Republic of China 1953 3 Yuan
Vietnam 1985 30 Dong
Mongolia 1966/1983 3 Turik
USSR 1947/1961/1991 3 Rubles
Cuba 1989/1995 3 Peso

You might think this is a product of soviet communist, but Russia had 3 Rubles in 1840 already.

Ok, after so much crap with culture, what else does it show? To collectors, a very important factor is rarity. In order words, the quantities produced are different.

Consider cash transaction, 1 Dollar, 2 Dollars, 3 ..., and assume each amount appears with equal probability. Although in real life, the probabilities are not equal. Large amounts have less chance, high value banknotes market values are mostly determined by face value, rather than rarity.

In 1-5 system
The last digit of cash exchange could be 0, 1, ... 9, each with 10% probability. So the expected number of 1 Dollar notes used in one cash exchange is

0.1 (0 + 1 + 2 + 3 + 4 + 0 + 1 + 2+ 3 + 4) = 2

And the expected number of 5 Dollars is

0.1 (0 + 0 + 0 + 0 + 0 + 1 + 1 + 1+ 1 + 1) = 0.5

Therefore the quantity of 1 Dollar needed is 4 times the number of 5 Dollars needed. Similarly, it can be applied to 10 Dollars/50 Dollars.

What about 1-2-5 system

First we have to further assume that we use large denomination as many as possible, so 6 = 5+1, not 2+2+2. After computation, the expected values are 0.4, 0.8, 0.5. Besides knowing the demand ration, we know that 0.4 + 0.8 + 0.5 = 1.7 < 2.5 = 2 + 0.5, which means in average, each transaction only needs 1.7 notes from the pool of 1, 2, and 5 Dollars, which meets our expectation. The theoretical minimum is 1, with each number having its own note. The theoretical maximum is 4.5, with 1 Dollar only.

In addition, if a number is further away from its higher neighbor (in terms of ratio), the expected quantity needed is higher. But how come the expected quantities of 1 and 5 are different under 1-2-5? Because 1-2-5 is not a perfectly divisible sequence, meaning that some number is not divisible by numbers that are lower than itself. In this case 5 is not divisible by 2. So when we attempt to derive the formula, we must distinguish the two cases. (actually one is a special case of the other).

Let's consider the simpler case, perfectly divisible sequence
In base N world, define sequence a0, a1, a2, ..., an, ai < N. Let quotient sequence qi = ai+1 / ai, qn = N / an. Both ai and qi are positive integers.

For simplicity, the resulting expecting quantity of ai is (qi - 1)/2, which also meets our earlier intuition that the higher the qi is, the more needed.

In the case of non perfectly divisible sequence, let ai+1 = aiqi + ri. ai, qi are both positive integers. ri is non-negative integer.
The expected quantity for ai is (1/N)((qi - 1)aiqiqi+1...qn/2 + f(ri)qi+1 qi+2...qn + f(ri+1)qi+2 qi+3...qn + f(rn-1)qn + f(rn)), where f is f(x) = the number of ai needed to assemble all of 0, 1, ..., x-1, with the help from a0 to ai-1. When ri is small or qi is big, the formula approached to the one of perfectly divisible sequence.

No matter it is 1-5 or 1-2-5, we can see imperfection and imbalance of distribution. And regardless how you modify it, there is no way to create a perfect system. This is because the base 10 we are using, 10 = 2*5. If we use base 16, 16 = 2*2*2*2, we can use either 1-4 (need 1.5 each) or 1-2-4-8 (need 0.5 each). Both systems gives a symmetrical harmony. If we use the latter one, then we need at most 1 note for each denomination for any total amount, which is a greater perfection.

There is a downside of using base 16. We will have to memorize larger multiplication table that is 2.78 times the base 10 table.

*Sigh, every time I have mention "country/region" to be political correct. For example, Hong Kong is never a country, but it has its own currency. There are more than 10 countries in the Euro zone, but they all share the same currency.

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