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# Currency that Fits

Solve three problems at once with Base Four
 (+5, -4) [vote for, against]

Note: This Idea is being presented in detail from the perspective of the US, but there is no reason why the arithmetic presented herein (in the solution to existing US-centric problems) cannot work for any national currency.

Also note: I think this idea was posted here a while back, but it was only as an annotation to someone else's idea, and I cannot find it, so I suppose it (annotation or whole idea) has been deleted (I suspect the latter). So, because I think my annotation was unique enough to stand on its own, here goes:

One of the problems with the US currency system is that there are too many denomination types to fit in cash register drawers. At various times there have been \$1, \$2, \$5, \$10, \$20, \$50, and \$100 bills in wide use, but most cash register drawers only have 5 slots (one usually reserved for personal checks). As a result, \$2 bills have generally been rejected from circulation, and \$50 and \$100 bills usually get hidden UNDER the cash drawer, to reduce incentive for theft. OK.

But there is another problem. The US Mint printed \$2 bills for a reason, and that was simply to reduce the total number of \$1 bills in circulation (they have an average lifespan of less than a year, so the Mint is ALWAYS printing lots of them, just to replace the worn-out ones -- expensive, see?). If only the \$2 bill was not rejected, then fewer total bills would be in circulation, yet the total number of \$ in circulation would be the same (and printing costs would go down, saving tax dollars). But where to put them in the cash register drawer?

AND there is another problem: Inflation is causing production of \$100 bills to be ramped up, and the Treasury Department really does NOT want to put \$500 bills into wide circulation (it forces drug dealers to carry around larger volumes of cash, which makes them more easily notice-able, and capture-able). But inflation is going to cause average folks to want to have something more convenient than a wad of \$100 bills (on those occasions when such is warranted).

Do note that the \$20 bill exists for the same reason that the \$2 was created: When the next-higher denomination is broken into change, would need four \$10 bills or four \$1 bills. Having the intermediate denomination means counting change can be faster, involving fewer bills (and requiring fewer in overall circulation). Also note that there is no \$200 bill, which might be a numerically-consistent answer to the inflation problem.

Well folks, there is an alternate and pretty simple solution to all those problems. It will involve getting people to exercise their arithmetic skills a little more (but counting money is how most people learn simple arithmetic in the first place, heh), and I'm sure that all who hate math will now fishbone this idea and read no further. So be it.

Let the Mint STOP making \$5, \$10, \$20, \$50, and \$100 bills altogether, and replace them with \$4, \$16, \$64, and \$256 bills. This works much more smoothly than you might expect, when making change from a larger denomination:
Break a \$4 bill into three \$1 bills, plus coins.
Break a \$16 bill into three \$4 and three \$1 bills, plus coins.
Break a \$64 bill into three \$16, three \$4, and three \$1 bills, plus coins.
Break a \$256 bill into three \$64, three \$16, three \$4, and three \$1 bills, plus coins.
(Of course, if one does not want to break a larger denomination all the way down to receive \$1 in coins, then alter any single word "three" above to become "four", and stop the sequence.)

In the cash register drawer would easily fit \$1, \$4, \$16, and \$64 bills, with that fifth slot still reserved for personal checks. Remember that the same inflation which will encourage the use of bills larger than \$100 will also make both \$50 and \$64 amounts less desire-able for theft, so eventually having them in the cash drawer will become reasonable. Meanwhile, of course, the larger \$256 bills would still go "securely" under the drawer. (The Mint could let \$500 bills go extinct, knowing that the next "natural" denomination in this system would be for \$1024 -- and actually printing such a bill may not be necessary. A \$1000 under this system would be broken down into three \$256, three \$64, TWO \$16, and two \$4 bills (without bothering with \$1 bills or any coins.) --and of course the Mint would want to keep \$1000 bills out of circulation, anyway.)

Next, the Mint benefits by not needing to print so many \$1 bills, because fewer are needed to make change from \$4 bills than from \$5 bills. \$2 bills can be welcomed into extinction -- and the equivalent problem involving \$10, \$20 and \$50 bills disappears because of that intermediary \$16 denomination.

Finally, when someone answers a \$64 Question, payment will be easy!

 — Vernon, Jan 07 2004

newser, you get three \$16, three \$4, and three \$1 bills.
 — Vernon, Jan 07 2004

This will just make me feel stupid when I find myself with six \$16-bills and three \$8 bills and I have to use a pencil and paper to figure out how much I’ve got.
 — AO, Jan 07 2004

+ The next logical progression will be to count large amounts of currency in kilo-, mega-, tera-, dollars.
 — Worldgineer, Jan 07 2004

I thought it was "the million dollar question." And ditto what [AO] said. Math is not my strong suit.
 — k_sra, Jan 07 2004

Which is easier, chopping off two fingers or adding six?
 — theircompetitor, Jan 07 2004

Now you're thinking!
 — k_sra, Jan 07 2004

It would help (?) if store prices were also written in base 4.
 — phundug, Jan 07 2004

yeah, I wouldn't have such a time with money if those stores would just factor in the tax already, so as to make round numbers of payment
 — benlevi7, Jan 07 2004

 Folks, one aspect of this Idea that I did not go into involves printing Base-Four denomination numbers on the bills (discretely, so as not to be confused with their official Base-Ten values). Remember that the secret to a numerical Base is "positional notation", where each POSITION of a digit indicates its value. So, while in Base Ten we have a "ones column", a "tens column", a "hundreds column", and so on, in Base Four we have a "ones column", a "fours column", a "sixteens column", and so on.

 In Base Four, some numerical values are: one = 1 (normal ones column) two = 2 three = 3 four = 10 (one in the fours column) five = 11 (above plus one in the ones column) six = 12 seven = 13 eight = 20 (two in the fours column) twelve = 30 (three in the fours column) fifteen = 33 sixteen = 100 (one in the sixteens column) seventeen = 101 twenty = 110 twenty-four = 120 (above plus two in the fours column) thirty-two = 200 (two in the sixteens column) forty-eight = 300 sixty-four=1000 (one in the sixty-fours column)

 As you can see, this would take some getting-used-to! Anyway, the point of doing that is not really to confuse people, but to stretch their minds to more easily handle different Bases for counting. See, in these days of the computer age, it is very handy to know Base Two and Base Sixteen -- and Base Four is actually a natural relative of those!!! Thus will future students become technically competent that much more quickly, by growing up with familiarity with a Base that converts easily to computer work.

However, for [AO], [k_sra], and others not hugely interested in computer tech, using the ordinary Base Ten values printed on the bills should work just fine, after some practice, as I previously mentioned. How long will it take to remember that three times sixteen is forty-eight? How hard is it to add two \$4 bills to that, to get fifty-six? How hard would it be after you've done it many times? AND- how often do you actually re-count the cash in your pocket, anyway? Once per shopping trip? Twice? After any such counting, you proceed as usual, knowing what limit there is to your purchasing.
 — Vernon, Jan 07 2004

I feel a numbers headache coming on....
 — k_sra, Jan 07 2004

Actually, besides the \$2 bills (which were IIRC issued in 1976 as a Bicentenial commemorative; had a common denomination been issued in a special 1976 version it would have messed with automated equipment. I don't know any particular reason for the 1995 reissue) it seems the lesser-used demoninations are the \$10 and \$50. It seems that the progression \$1/\$5/\$20/\$100 works pretty well for most people.
 — supercat, Jan 07 2004

 This will create a need for more bills in every scenario you describe.

 In your system, \$63.99 takes twice as many bills as we currently use, and \$255.99 requires THREE TIMES as many bills as the current system. That's just nuts.

I don't often fishbone you, but this deserves one.
 — waugsqueke, Jan 07 2004

 waugsqueke, you may be right, but this is actually a natural consequence of having fewer denominations! What do you expect? Now suppose we took supercat's abbreviated list of \$1/\$5/\$20/\$100, and compare the number of bills it takes to make a whole lot of different dollar amounts, to the number of bills it takes to get those same dollar amounts using a set of \$1/\$4/\$16/\$64. (Ignore the \$.99 in coins; I was not suggesting altering their denominations in this Idea.) Suppose we counted the first 200 dollar amounts. Here is the first 25 (T for Total)

 1/5/20/100.T..1/4/16/64.T 1/0/00/000.1..1/0/00/00.1 2/0/00/000.2..2/0/00/00.2 3/0/00/000.3..3/0/00/00.3 4/0/00/000.4..0/1/00/00.1 0/1/00/000.1..1/1/00/00.2 1/1/00/000.2..2/1/00/00.3 2/1/00/000.3..3/1/00/00.4 3/1/00/000.4..0/2/00/00.2 4/1/00/000.5..1/2/00/00.3 0/2/00/000.2..2/2/00/00.4 1/2/00/000.3..3/2/00/00.5 2/2/00/000.4..0/3/00/00.3 3/2/00/000.5..1/3/00/00.4 4/2/00/000.6..2/3/00/00.5 0/3/00/000.3..3/3/00/00.6 1/3/00/000.4..0/0/01/00.1 2/3/00/000.5..1/0/01/00.2 3/3/00/000.6..2/0/01/00.3 4/3/00/000.7..3/0/01/00.4 0/0/01/000.1..0/1/01/00.2 1/0/01/000.2..1/1/01/00.3 2/0/01/000.3..2/1/01/00.4 3/0/01/000.4..3/1/01/00.5 4/0/01/000.5..0/2/01/00.3 0/1/01/000.2..1/2/01/00.4

 Subtotal the two T columns, and we get (DRUM ROLL...) 87 bills on the left and 79 on the right. So, while you can find places where my Idea uses more bills, I can also find places where my idea uses fewer bills. Over the long haul, they may run neck-and-neck (but at the moment I have a slight efficiency advantage).

Do you care to do the next 25?
 — Vernon, Jan 08 2004

 Vernon: The use of alternating powers of 5 and 4 may use a few more notes than would using strictly powers of four, but with four denominations gets up to \$100 rather than \$64. While having a bill larger than \$100 would be good, that would interfere with the government's ability to control people's money.

Is it not rather ironic, is it not, that as the value of a dollar has over the last 50 years, the dollar-denominated value of the largest circulating banknote has been dropped tenfold? The government and banks used to circulate \$1,000 bills. Now the largest bill, the \$100, is only worth about as much as an old \$10 or \$20.
 — supercat, Jan 08 2004

 supercat, if it wasn't for the sheer quantity of text, I'd add the \$500 bill to your list, and the \$256 bill to my list, and work that "total number of bills" list up to \$1000. I'm pretty sure that in the long run the Base Four system will use fewer bills. Next, it is exactly because inflation has made the \$100 bill no more valuable than a 1930's \$20 bill -- and because inflation continues (although unusually slowly these days) -- that the Mint is going to have to either let the \$500 bill out again, or start issuing some intermediate size, like a \$200 bill -- or pray, of course, that everyone switches to totally electronic banking, making cash altogether obsolete, heh heh. It is just a matter of time. The thing is, WHEN that happens in a still-using-cash economy, the merchants will have one more denomination to keep in their cash register drawers. I continue to submit that the Base Four system solves the problems neatly, even if it would take a bit of getting-used-to-it.

---------------------
(Added due to newser's comment below) That would be two \$64 bills and coins
 — Vernon, Jan 08 2004

 If a \$500 note comes back into circulation, storing large sums of money with \$500 notes will take only slightly over half as many notes as using \$256 notes.

Also, have you evaluated the possibility of a \$1/\$5/\$15/\$60 sequence? Might be numerically somewhat nicer to deal with than \$1/\$4/\$16/\$64 since us crazy humans have five fingers on each hand.
 — supercat, Jan 08 2004

 supercat, you seem to be forgetting that a major reason for doing Base Four was to shrink the gap between the \$1 bill and the next denomination. Sure, those other numbers are very nice, but that first gap is still ugly, as far as the Mint is concerned.

 Heh, just to be even more off-the-wall, consider a \$1, \$3, \$9, \$27, \$81, and \$243 sequence! Those last two would go under the cash register drawer, just like current \$50 and \$100 bills, but this range is very nicely divided, and total quantities of bills, for most ordinary amounts, is lower than even what the Base Four system offers (a \$27 bill breaks into two \$9, two \$3, and two \$1 bills, and coins). And it handles inflation just as well, since it offers a denomination between \$100 and \$500 (even similar in size to that other proposal, the \$256 bill). However, Base Three is even more alien to humans than Base Four, partly because it doesn't correlate to any other application (the way Base Four is good mental preparation for computer work). Ah, well....

 As a compromise, what of a \$1, \$3, \$12, \$60, and \$240 sequence? Or is the whole world so metric these days that nobody thinks about dozens any more? (However, that 5:1 gap has merely been moved between two of the larger denominations. But this may be OK, since the Mint is bothered by total quantites of bills to print, and the ratio goes down for each higher denomination.)

Folks, please remember that people can and do get used to the strangest things -- especially if they grow up with them. Not so long ago, England changed its coinage from a really irregular assortment ("tuppence", "shillings", "farthings", etc. --and no, I don't know how many different denominations they had) to Base Ten, and people got used to it. I bet kids in England these days scratch their heads at how their forefathers got any accurate business done with that system....
 — Vernon, Jan 08 2004

You know, this title would serve much better to address the various world currencies that don't actually fit into a wallet.
 — theircompetitor, Jan 08 2004

cripes! this is too much math for the average american consumer! let alone me! if you want to cut down on the bills we make we need to make stronger bills. maybe teflon laced or something...or maybe custard laced...
 — Space-Pope, Jan 08 2004

Let's get rid of everything but the \$1. Sure, this means more printing at first, but people will soon be so inconvienenced that we'll all quickly switch to electronic payments.
 — Worldgineer, Jan 08 2004

Maybe someone should post an idea for a draw with more slots. Could well be cheaper.
 — RobertKidney, Jan 08 2004

A better solution is to use a credit card. This eliminates the need for actual cash altogether.
 — Wayne Scotting, Mar 29 2011

 and of course gives a third party an opportunity to get involved in you and your store's commercial habits.

Not to say it isn't a stupid idea [-]
 — FlyingToaster, Mar 29 2011

Vernon, \$2 bills were made for betting on the ponies, from what I've heard. Please don't take those away.
 — Zimmy, Mar 30 2011

 The \$20 bill was likely a conversion of a double eagle gold coin- \$20 to paper.

I don't think exponents in currancy would work very well. I'm sorry, vernon.
 — Zimmy, Mar 30 2011

"I'll have two cheeseburgers and a large fries."
"That'll be \$100, sir."
"WHAT? WHATRE YOU TALKING ABOUT YOU STUPID WHAT THE GRRR BETTER WATCH IT OR I'M GONNA MESS YOU UP YOU STUPID"... ohhhhhhh, Base 4, sorry about that..."
 — phundug, Mar 30 2011

 [annotate]

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