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Preferred number (wikipedia.org)
272 points by twic on July 30, 2015 | hide | past | favorite | 40 comments

Had always assumed there must be some method behind the madness of standard resistor sizes (why the obsession with 4.7?) but never knew exactly what it was - and the explanation here is excellent. The fact they fall along a twelfth-root scale is an intriguingly direct mapping to musical semitones; I wonder if that was ever significant in enabling the creation of early electronic instruments?

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.

The twelfth root of ten contains a twelfth root of two, but it also has a pesky twelfth root of five. It might still be possible to use series/parallel constructions to cancel out the 5 and still save resistors vs' some other numbering scheme.

The idea here is to get a smooth logarithmic division of each order of magnitude (power of 10). To get x steps that perfectly divide the order of magnitude, just use the xth root of 10 as your factor. In Ruby:

    def series(x, precision=2)
      root = 10 ** (1 / x.to_f)
      return x.times.map {|i| (root ** i).round(precision)}

    > 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]
And of course, if you want more aesthetically pleasing increments (like for axis markers on a logarithmic graph), just round each increment as desired. I usually find that the [1, 2, 5] series is sufficient.

That happens to be the series used for the Euro currency:

[0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, 500]

Yep, that's a good way to minimize both the number of denominations offered and the number of bills/coins needed to make change for a given amount of money.

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.

Is it at all surprising? Coins don't fit in wallets, represent small denominations, and are therefore not worth the trouble.

That's because the highest US coin in practice is $0.25—even though $1 coins exist, they are not widespread and overlap with the $1 bill.

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 tried using $1 coins for a while. I found then too bulky and heavy with zero advantage over paper.

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.

I actually found the contrary. 25 coins hold surprisingly well in a pocket, much easier than a thick set of 25-bills IMO.

25-Bills get wet with sweat and insulate your legs. I kinda prefer 25-coins over that.

I think most currency uses that. Dollars are a weird exception.

You don’t have to make it so complicated!

    : 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.]]])
Hmm, Ruby has a numpy-like thing too, no? NArray, it’s called? There are a few minor differences, but you can do the same thing:

    : 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 ] ] ]
Presumably 12, 24, and 48 were chosen because they are https://en.wikipedia.org/wiki/Highly_composite_numbers, and 96 and 192 for compatibility with 48 (since the HCNs in that neighborhood — 60, 120, and 180 — aren’t divisible by 48.)

(Related joke: How do we know the Babylonians were idiots? Because the day doesn’t have 83160 seconds.)

I have often wondered if there are any subtle emergent properties that tend to occur due to humans tendency to round interacting with a given system of measurement.

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.

One thing (which is mentioned in the wikipedia page) is that in the metric system where e.g. meters and kilometers are related by a power of 10, a length which is a preferred number in one unit will also be a preferred number in the other unit. This does not work out for e.g. feet and miles.

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.

What a wonderful example of two opposing hypotheses from same premises, both providing value. I choose both.

I don't really buy that any particular ratios are more or less aesthetically pleasing. Seems like designer received wisdom.

This section was especially interesting: https://en.wikipedia.org/wiki/Preferred_number#Buildings

Basically re-creating the nice aspect of imperial units within the metric system.

You could also switch to base 12.

All of our human measurement systems would be much friendlier if we had a consistently applied base 12 system. Breaking things into 3, 4, or 6 parts is much more common than breaking them in fifths. Along the way, we could mostly straighten out our measurements for dates/times, angles including longitude/latitude, etc.

(Though I must say, I’m also partial to the Babylonian base 60 system.)

I noticed as a kid that while 60 seconds/minutes seemed odd when we live in a base-10 number system, it was incredibly convenient when dividing into parts: 60 is divisible by 2,3,4,5,6,8,10,12,15,20,30. Always thought that was sort of clever.

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.

Our measures for angles and for time descend from the base 60 Babylonian system, whereas base 10 descends from the ancient Egyptian system.

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...

60 is divisible by ... 8?

Are you serious?

I also thought he was full of it, thinking he was advocating for a base-12 system of units. But upon further thought, it's apparent that he's advocating for a base 12 number system, and then a base-12 system of units would come naturally from that.

Base 12 could be a great number system, but it's impossible to switch now.

That's what they said when we switched from the 12 inch foot to the meter!

Switching measurement systems is tough, but switching number systems would be about a million times harder. (Or about 12^5.5 times harder, if you prefer.) I have trouble imagining where you'd even start....

True, but my point was that imperial had those benefits of divisibility, yet we switched away from it...

Because it conflicted with our number system, and we (mostly...) decided that harmony with the number system was better than divisibility. If both were base 12, there wouldn't be any conflict.

Absolutely serious. Using base 12 instead of base 10 would be a huge advantage, if only we had done it 500 years ago. To start, it would make the relationship between fractions and duodecimals much clearer and easier to learn. But it would also make pretty much every type of routine measurement task at least slightly more convenient.

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.

>It would cost trillions of dollars and require redoing our whole measurement system, rewriting all our technical books, and changing all of our manufacturing standards.

That's the easy part. The hard part is making laymen accept and understand the new system.

Can you link to or write more about what other kinds of things need to be fixed? What is wrong with the sign of an electron's charge?

I guess it doesn’t make all that much difference which sign: it’s an arbitrary choice. It’s just a bit confusing for students to define the electron charge as -e instead of +e, or draw diagrams with arrows pointing the opposite direction of electron movement. http://www.allaboutcircuits.com/textbook/direct-current/chpt...

On the other hand this guy seems to appreciate the definition: http://www.austincc.edu/wkibbe/truth.htm

Of course this is serious. Why do you think there were 240 pence in a Pound Sterling? Divisible by 1,2,3,4,5,6...

Is there something particular about base-10 that makes it a better system?

We have ten fingers, so it's pretty intuitive.

I had the same idea when I was a kid. There were other neat aspects to it, like how temperatures worked out really nicely. I don't know really how to link hooniverse comments, maybe this takes you to my last comment about the matter and then scroll-up, or search for SNL and scroll up:


As an applied math person who does design and art off-hours, I found this article very pleasant to read, since it brings the two worlds together. Good find!

Can someone update the title to convey a little more context?

Interesting -- thanks for posting!

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