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I'm not sure which SE network this question belongs to, but I'll try to formulate it for this one.

What determines the choice of banknote denominations?

In the real world we usually use powers of ten - 1, 10, 100, etc. (because it's the simplest thing to do in base-10). It seems like most of the world also issues 2 and 5 multiples of of the above denominations (2, 5, 20, 50, etc.). There are some exceptions like the banknotes of 3 and 25 rubles used in USSR. Most currencies also form subunits and they do it by dividing into 100. The romans divided by 12.

Do these numbers arise naturally as a consequence of using base-10? What would be the choice for e.g. base-12?

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closed as off-topic by Brythan, Pavel Janicek, Hohmannfan, Cort Ammon, Thucydides Jun 10 '16 at 19:01

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not appear to be about worldbuilding, within the scope defined in the help center." – Brythan, Pavel Janicek, Hohmannfan, Cort Ammon, Thucydides
If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ I think this is most-certainly off-topic here. I recall that there is money.SE. I am not sure whether it is on-topic there, it might be a better fit. Have a look at the linked help center page. The same for economics.SE $\endgroup$ – T3 H40 supports Monica Jun 10 '16 at 8:16
  • $\begingroup$ Calculate 10^(1/3) and 10^(2/3) and round to integer. What do you get? $\endgroup$ – celtschk Jun 10 '16 at 8:30
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    $\begingroup$ As this is currently written, I really don't think it's on topic for us. If you can rewrite it in terms of how to choose a set of banknote denominations for your fictional world, then it might be on topic, but the answers might not be what you want. I would suspect that Economics is the place to ask this; I think it would be closed as not about personal finance on Personal Finance & Money. You could always drop into their respective site chat rooms and ask. $\endgroup$ – a CVn Jun 10 '16 at 8:55
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    $\begingroup$ The short answer is coin systems, sets of denominations for currency, are designed to be amenable to the "greedy algorithm" of change making, which is the most natural and intuitive way for humans to operate. If a particular set of denominations prodded the minimum number of coins/notes for any value returned using the greedy algorithm, then that system is called a "canonical change system". You can read more about the math behind it in this CS.se question. $\endgroup$ – Dan Bron Jun 10 '16 at 10:39
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    $\begingroup$ @MichaelKjörling It's on-topic at Math.se too. I'd suggest if OP just wants the nuts-and-bolts of the theory, he ask on Math.se; if he's using it towards some computational end, then CS. But if he wants to apply these theories in some novel / interesting / fantastical way to build a fictional world, to drive some story, then this is the best place. Of course I'm mostly talking about out of my rear-end here, 'cause for the most part I'm just a lurker on most sites, except English. $\endgroup$ – Dan Bron Jun 10 '16 at 11:36
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There's no universal law for what denominations of bills must be made, but there are some general guiding principles. I come up with a slightly different list than @InstantMuffin. Here's mine:

  1. It must be possible to come up with a set of bills totaling any desired amount. But that's easy: You have to have a 1.

  2. You should be able to make any likely amount with a manageable number of bills. For example, if all you had were 1's, and you had to pay 653, that would require counting out 653 bills. Or if you had just 1 and 100, 653 would still require 6 hundreds and 53 ones, still a pain to count.

  3. It should be easy to do the arithmetic to add up the total value of a stack of bills. If your denominations were 1, 6, 13, 28, and 87, it would be much easier to make arithmetic mistakes if trying to count in your head than if they were 1, 10, and 100.

  4. The number of different denominations shouldn't be too large, or it gets hard to manage them. Like in the US, cash registers generally have 5 bins, for \$1, \$5, \$10, \$20, and "bigger". If you had 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 20, 25, 30, 35, 40, 45, 50, and 100, this would be a pain for anybody who has to sort money: the number of piles gets too large.

In the U.S., few people use cash for very large transactions any more. So despite inflation, our largest bill today is \$100, while in the past we had \$1,000, \$10,000, and for a time in the 1930s even \$100,000 bills. I believe nothing bigger than \$100 has been printed since 1969 (wouldn't swear to the detail). (I can't imagine wanting to walk around with \$10,000 bills in my pocket. What if I lost it? I remember once when I worked for a non-profit taking a deposit to the bank, I had \$30,000 in my briefcase. And I was thinking, I would really hate to get mugged right now. But anyway.)

So it makes sense to have 1, 10, 100, etc, as high up as makes sense for your currency. It's easy to count, keeps the number of bills you have to carry around manageable, etc. Adding 5, 50, etc also cuts down on the number of bills you need to carry and how many you have to count out for a transaction, without making the arithmetic difficult. Other denominations can be problematic. So you see 2's and 20's. I think any other denomination is rare.

I presume a society that used base 12 would use denominations that are round numbers in base 12, i.e. 1, 10 base 12 (i.e. 12 base 10), 100 base 12 (144 base 10), etc. In the middle they'd likely have 6, 60 base 12 (72 base 10), etc. As 12 neatly divides by 4, I wouldn't be surprised if they also had a 3 and a 9. Then you'd have 1, 3, 6, 9, 10, 60, 100, 600, 1000, as high as makes sense to go. I'd guess they'd be more likely to have 30's and 90's than 20's, but I'm just speculating wildly.

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1) One should be able to pay any size of debt.

2) It should require the least amount of units as possible.

3) One should be able to easily calculate the number of coins/notes one needs for a specific debt.

4) Make frequently used notes into coins, or into smaller notes, due to wear and tear, and use the size also so easily distinguish individual coin/note value. To achieve individual value distinguishment they are also made up of several materials.

5) Higher denominations are also usually notes because there are more measurements available to battle counterfeiting.

So ideally initially you'd use a binary system (points 1) and 2)), however due to 3) you tend to have easy setups like 1 2 5 10 20 50,... 4) and 5) depend on inflation. Also if you run out of distinguishable coins (they'd have to be too big or small, or they run out of differently colored metals), notes are preferred. There are frequent discussions regarding the borders:

1) Should the last note be removed due to it being less needed or to make counterfeiting less profitable?

2) Do we need a higher valued note due to inflation?

3) Variations in the coin-note border due to inflation and practicability

Partial source and more interesting stuff to read: http://ec.europa.eu/economy_finance/euro/cash/index_en.htm

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    $\begingroup$ Also: 4) Should the lowest-value note or coin be removed because the extremely low value makes it effectively worthless for trade and the materials it's made of are worth more than the coin itself? $\endgroup$ – Nzall Jun 10 '16 at 13:47
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    $\begingroup$ @Nzall That's an argument I've seen made with the Euro. The 1 cent and 2 cents coins are only used because we have dumb 9.99 prices. Nobody wants to have to deal with those coins, some merchants are rounding their prices to 5 cents (e.g. 9.95) for that very reason. Still, that will likely remain legal tender since you indeed have to be able to pay any amount, 9.99 included. $\endgroup$ – AmiralPatate Jun 10 '16 at 13:59
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    $\begingroup$ @AmiralPatate you should check out how the Netherlands are handling it. They don't use 1 and 2 cent coins, instead all merchants may legally round to the nearest 5 cents, even for electronic payments. 1 and 2 are rounded to 0, 3, 4 6 and 7 are rounded to 5, 8 and 9 are rounded to 10. it averages out for the merchant, and the civilian doesn't have to mess around with anything beneath 5 cents. Also check out the history of the half cent of the Dollar. $\endgroup$ – Nzall Jun 10 '16 at 14:03
  • $\begingroup$ @Nzell AFAIK, all coins are always legal tender in the Eurozone, though I guess a merchant could look at you sideways if you tried dumping your foreign 1 cent coins. Some countries have argued small coins are expensive for no reason, some countries have argued you should be able to set any price you want. If you're worldbuilding, you'll have to make a hard cut one way or the other. $\endgroup$ – AmiralPatate Jun 10 '16 at 14:27
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    $\begingroup$ Muffin. Check out CGPGrey on YouTube. He has a video that explains that every penny takes 2.4 pennies to make and that the USA has already discontinued use of the half penny about 150 years ago due to devaluation from inflation. $\endgroup$ – Nzall Jun 10 '16 at 15:24
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First off all, currency denomination is not always in easily divided numbers. For example, there are a number of fictional systems that divide their currency in odd denominations. Harry Potter has knuts, sickles (29 knuts) and Galleons (17 sickles), with currency calculations happening through magic. ASOIAF has copper pennies, silver stags (56 pennies) and gold dragons (210 stags). the Kingkiller Chronicles have 3 different systems, one of which involving 10 different different denominated coins with divisions ranging from 2 to 2.5 to 8 of the lower coin.

Even in our history, the GBP had 20 shillings to the pound and 12 pence to the shilling. And the USD has had coins anywhere from 1/2 cent to 50 dollars, even with 3 and 4 USD coins. Even for bank notes, the US has had 5000 and 10,000 USD notes.

So there is not really much of a logic behind what denominations can be made available. For example, there is no reason why you can't have a bank note with denomination 28.46 faekbuks. If you can justify for your story why that number is chosen like that, readers will be willing to accept that.

All in all, it comes down to what the organization behind the currency decides. You can just as much have a system with just 1, 12 and 144 faekbuk notes as a system with one of every note between 1 and 12 and every whole value from 12 to 144 in a base-12 system. If the organization behind the currency is trusted by the people, they will accept nearly everything.

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If it has to be realistic and practical, the denomination should be such that it can add to any value. For example with denominations of 1, 2, 5 you can add to any integer value. If you need partial of integer you can have subcurrency (orwhatever is called). Also another meaningful denomination can be added. For example if 3 is widely used although it can be the addition of 1 and 2 it may be convenient to have it as denomination.

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