# simpler coin denominations

 Coin denomination-steps of 4x-5x are simpler than 2x-2.5x, and easier, and business-righter

In the mid-1990's I proposed a Roosevelt "New Dollar" in white gold (silver+gold) to replace the nearly-useless Roosevelt-dime of the day ... It wasn't Roosevelt's fault, It was the notion of binary coinage that never completely worked: Intended to minimize coin change-exchange handling, it was single-endedly fine (think of how many coins are needed to represent each possible total: on average 0.50 in each binary denomination), But that's not how business is done: Smaller totals are more common, to smaller average in the largest denomination, and manifest a trickle-down effect, And transactions are two-sided (think of how many coins are needed to do one transaction in binary-with-change-returned: only 0.25 each denomination: cf pay-one-receive-half accomplishes pay-half, and accomplishes its obtaining), And sorting and binning coins is half the labor and cost for near-quaternary decimal-conforming 4'-5'ary (alternations)-- in our ideal: pennies, 5, nickels, 5, quarters, 4, in coin drawers; dollar bills, 5, '5's, 4, '20's, 5, '100's, 5, in bill drawers, and '500's ad indefinitum: and spanning all the 5xD's (5x5, 5x4, 5x?) and both unique 100-cycles (5x5x4, 5x4x5), for a general public needing no two-thousand-dollar-bill which is not a bill but a note (and transitions in counting practice to thousands not 20-hundreds). The remnant oddity is the dollar-unit transition in which way it should go: coin or bill, or both as requires an extra bin ... and whence my golden 'New-Dollar' approach, intending that gold dollar coins would be middle-of-the-rowed yankeekeepsakes, not separately binned change, not a general transaction coin: but one 'non-commemorative' commemorative 'proof' of dimes past.

Next on the agenda is neutron-detection (or nuclear-proton, or NMR-resonance autc.) metal-spectrum-total-content-verification of coins in machines .... back to 'weighing gold' ....

PS. Mathematically speaking, Binary coinage results in denominations differing in size by steps of 26-36%, a quarter-or-third and feeling nearly identical for handsorting; but Quaternary doubles that in steps of 59-71%, 2-thirds feeling very nonidentical.

How many dimes go into a quarter? - cannot be answered wholely: But five nickels do go into a quarter. Our decimal-based arithmetic puts simple constraints on expectations for our denominational coinage. Binary is not quite possible, though it offers the minimum coin-count [and bill-count] in one-way transfer of currency (cf all possible payments, in binary). And binary has the most coin denominations-sizes, -double what quad has,- and the denominational coins are nearer equal size.

And then-again, coin transactions are two-way: payment and change - with the same total handling per each coin, albeit in opposite directions {taking-out, and putting-in}, and the change approaches the same coin-moving, albeit with the further vantage that there tends to be a common balance exchange of coins themselves, not just value: For example:

0 0 0 0:0 0
.25 .25 .25 1 s:1 s 1:1
both .50-.25 1.00-.75 2 var. (1 m-1s): 2 var (1 b-3 s) 0:-2
.50 .50 .25+.25 1 m:2 s 1:2
both 1.00-.50 1.00-.50 2 v (1 b-1 m):1 b-2 s 0:-1
.75 .50+.25 .25+.25+.25 (1 m+1 s):3 s coins
both 1.00-.25 1.00-.25 1 b-1 s: 1 b-1 s 0:0
1.00 1.00 1.00 1 b:1 b 1:1

Trinary should be near optimal in sum-total exchange of coins ... but again its not strictly convenient in/to our decimal notation, and, it wastes information possibilities - for example, to give 2/3rds of a coin, the giver may give 2 coins, each a 3rd-part, or give 1 coin, the full, and receive a 3rd-part: the same coin-count, two coins exchanged in the transaction.

Qui-Quater-Quinary, bases four and [twice] five, to reach a hundred, reduce the number of overlapped possibilities, thus making more efficient use of same-said - as well as keeping coinage within the double-decade [hundred] range. The few overlapped possibilities are when giving 3/5ths of a coin, the choice is how to give 3 coins: either 3 coins, 5th-parts, or, give the whole, and receive 2/5ths. In 4ths, 2 coins can give 2/4ths, or, give 1 whole, and receive 1/4th, effecting 3/4ths - very efficient. In fact, compared to binary, 1 or 2 coins suffice the incremental steps upto 5x being

• 1 for 1x
• 2 for 2x
• 2 for 3x = '4' - '1'
• 1 for 4x = '4'
• 2 for 5x = '4' + '1'
whereas in binary, 1 or 2 coins reaches 6x - slightly more, but while doubling the choices of coins and bills by a factor of two: quarters are simpler by this factor of two. Also, odd-denominations offer statistically more nearly even recovery of change: returning as many coins of each denomination as loosing - so the purchaser does not need to obtain loose change when replenishing [his] pocket with larger bills. Thus quinary is odd and useful, and quaternary is similar, and together quin-quater-quinary reaches a hundred: the second power of our counting-base ten.

I propose denominating only

• 1¢ penny
• 5¢ nickel
• 25¢ quarter
• \$1 bill and 'commemorative' coin
• \$5 bill
• \$20 bill
• \$100 bill
• \$500 bill
• \$2000 note not bill
• etc. notes
It has been interesting to see that the other coins we used to have, the half-cent, the 3-cent-piece, the two-dollar-bill, have disappeared from usage. And the other 'in-between-ie's, the dime and the half-dollar, find lessening usage.

I have proposed the 'Roosevelt' dime be replaced by a golden-silver 'Roosevelt' new-dollar. [cf 'New-Deal'er; pun] The gold-and-silver blend is established metalurgy, and its size will be neither too small nor large.