Preferred number
en.wikipedia.org
en.wikipedia.org
edit - on second thoughts, twelfth root of ten isn't particularly helpful for musical notes, where the factor between semitones is the twelfth root of two. So... carry on.
def series(x, precision=2)
root = 10 ** (1 / x.to_f)
return x.times.map {|i| (root ** i).round(precision)}
end
> series(2)
=> [1.0, 3.16]
> series(5)
=> [1.0, 1.58, 2.51, 3.98, 6.31]
> series(10)
=> [1.0, 1.26, 1.58, 2.0, 2.51, 3.16, 3.98, 5.01, 6.31, 7.94]
[0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, 500]
The USD is close, too. Bills are in 1, 2, 5, although the $2 bill is rare. For some reason our coins are irregular, although people in my experience find all coins a bother.
The "ideal" set of 3 denominations would be 1, 2.15, and 4.64.
Contrast with the Euro where the highest coin is €2 and bills start at €5.
Also prices tend to be inclusive of sales tax / VAT in Europe, so a €9.99 widget can be purchased with a €10 bill and you get 1 single €0.01 coin in change. Whereas your $9.99 widget will be $10.79 (at 7% tax), so you'll get 3 dimes and a nickel (4x the amount of change) assuming you handed over $11. Most likely, you had a $20 so you also get lots of $1 bills with your coins. :)
I would only use them routinely if forced.
In some US stores they set prices so that the total is an even total after tax.
25-Bills get wet with sweat and insulate your legs. I kinda prefer 25-coins over that.
: user@debian:~; python
Python 2.7.3 (default, Mar 13 2014, 11:03:55)
[GCC 4.7.2] on linux2
Type "help", "copyright", "credits" or "license" for more information.
>>> import numpy as np
>>> (10**(1+np.arange(24)/24.0)).reshape((2,3,4)).round()
array([[[ 10., 11., 12., 13.],
[ 15., 16., 18., 20.],
[ 22., 24., 26., 29.]],
[[ 32., 35., 38., 42.],
[ 46., 51., 56., 62.],
[ 68., 75., 83., 91.]]])
>>> (10**(2+np.arange(48)/48.0)).reshape((2,3,8)).round()
array([[[ 100., 105., 110., 115., 121., 127., 133., 140.],
[ 147., 154., 162., 169., 178., 187., 196., 205.],
[ 215., 226., 237., 249., 261., 274., 287., 301.]],
[[ 316., 332., 348., 365., 383., 402., 422., 442.],
[ 464., 487., 511., 536., 562., 590., 619., 649.],
[ 681., 715., 750., 787., 825., 866., 909., 953.]]])
>>>
: user@debian:~; irb
irb(main):001:0> require 'narray'
=> true
irb(main):002:0> (10*10**(NArray.float(24).indgen!/24)).round.reshape(4,3,2)
=> NArray(ref).int(4,3,2):
[ [ [ 10, 11, 12, 13 ],
[ 15, 16, 18, 20 ],
[ 22, 24, 26, 29 ] ],
[ [ 32, 35, 38, 42 ],
[ 46, 51, 56, 62 ],
[ 68, 75, 83, 91 ] ] ]
irb(main):003:0> (100*10**(NArray.float(48).indgen!/48)).round.reshape(8,3,2)
=> NArray(ref).int(8,3,2):
[ [ [ 100, 105, 110, 115, 121, 127, 133, 140 ],
[ 147, 154, 162, 169, 178, 187, 196, 205 ],
[ 215, 226, 237, 249, 261, 274, 287, 301 ] ],
[ [ 316, 332, 348, 365, 383, 402, 422, 442 ],
[ 464, 487, 511, 536, 562, 590, 619, 649 ],
[ 681, 715, 750, 787, 825, 866, 909, 953 ] ] ]
irb(main):004:0>
(Related joke: How do we know the Babylonians were idiots? Because the day doesn’t have 83160 seconds.)
For example, perhaps cars designed in America tend to be more|less aesthetically pleasing because American designers will round up to an inch whereas European designers are (hypothetically) unlikely to feel the need to round up to 2.54 cm instead of say 2.5 cm.
So imagine you have a landscape architect designing the interior of a park and a civil engineer designing the city around it. In the metric system, their preferred placements automatically agree. In imperial units they have to coordinate that.
Basically re-creating the nice aspect of imperial units within the metric system.
(Though I must say, I’m also partial to the Babylonian base 60 system.)
24 is similarly divisible by 2,3,4,6,8,12, whereas 25 is only evenly divisible by 5. So those odd-looking "constants" are good for something after all.
For dealing with heterogeneous bases, as with measuring in imperial units, take a look at the fraction notation in Fibonacci’s book. https://en.wikipedia.org/wiki/Liber_Abaci#Fibonacci.27s_nota...
Base 12 could be a great number system, but it's impossible to switch now.
Obviously it’s not feasible at this point to switch our number system, as a social/political question. It would cost trillions of dollars and require redoing our whole measurement system, rewriting all our technical books, changing all of our manufacturing standards, [edit: and retraining the whole society]. But our current systems are by no means perfect.
Within the metric system, there are inconsistencies all over the place, especially where we still accommodate important prior systems, like the calendar and time measures. Day -> Hour -> Minute -> Second and especially Day -> Week -> Month -> Year are quite annoying systems to interact with in an otherwise base 10 world.
Angle measures in terms of either 2pi or 360° -> arcminutes/arcseconds are also pretty bad, bearing little relation to the metric system.
Most of the basic metric system units would surely be chosen differently if designed from scratch knowing what we know now. We could also definitely pick better names for them than names of dead men. For one thing, derived units should have a more obvious relation to the base units, in their names. For one thing, we could finally fix the sign for the electron’s charge. Personally I’d recommend defining time using the day as a base unit, and then measuring distance with respect to the speed of light divided by 12^13, or similar (but that’s just me; perhaps a smart committee could come up with something better).
* * *
Much of our standard mathematics notation could also be improved if we put some thought into it. Fractions are a pretty excellent notation. Positional (decimal) notation is alright, but making some kind of floating point/scientific notation more basic would improve understanding in many ways I think. Basic infix operator notation works reasonably well, though I’m not the biggest fan of the way we handle subtraction or division operators or inverses in general. I like the way we use parentheses. Our notation for exponents is okay but not great, and having two notations for the exponential function is especially confusing. Our notation for square roots, logarithms, and trigonometric functions are all absolutely awful: terribly named, symbolically heavyweight, and revealing nothing about their structural properties. The way we handle vectors and complex numbers is pretty bad (we should use so-called geometric algebra instead, and develop a good notation for it). Our common notations for derivatives are confusingly varied and overloaded, and I don’t at all like our notations for summation and integration. In general, we should measure exterior rather than interior angles, since continuing straight is most natural to define as zero angle.
At a more basic level, our languages should have a highly unique single-syllable word for each basic numeral (from 0–9 or 0–11 or whatever), combined with a highly unique and fast to write glyph for each. We should have a single syllable for making a number negative, even “minus” isn’t a good word, especially since it’s overloaded to also mean subtraction, and we should have a single-syllable word for rotating a vector through a right angle. In speaking, we should compose multi-digit numbers by just saying the digit names, or saying a bunch of numeral names then some marker word, then a numeral name for the exponent. Phrases like “fourteen thousand, seven hundred and twelve” are horribly confusing and wasteful, and do immense damage to fluency with basic arithmetic. Don’t even get me started on the way numbers are spoken in French.
That's the easy part. The hard part is making laymen accept and understand the new system.
On the other hand this guy seems to appreciate the definition: http://www.austincc.edu/wkibbe/truth.htm
http://hooniverse.com/2015/05/12/review-2015-toyota-prius-c/...