

Preferred number - twic
https://en.wikipedia.org/wiki/Preferred_number

======
jameshart
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.

~~~
nitrogen
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.

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baddox
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 _x_ th 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)}
        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]
    

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.

~~~
athenot
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]

~~~
baddox
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.

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

~~~
athenot
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.
:)

~~~
ars
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.

~~~
dragontamer
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.

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mninm
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.

~~~
semi-extrinsic
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.

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

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blt
This section was especially interesting:
[https://en.wikipedia.org/wiki/Preferred_number#Buildings](https://en.wikipedia.org/wiki/Preferred_number#Buildings)

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

~~~
alricb
You could also switch to base 12.

~~~
jacobolus
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.)

~~~
philtar
Are you serious?

~~~
jacobolus
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.

~~~
superdude264
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?

~~~
jacobolus
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...](http://www.allaboutcircuits.com/textbook/direct-
current/chpt-1/conventional-versus-electron-flow/)

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

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croisillon
Hence this great tumblr
[http://thingsfittingperfectlyintothings.tumblr.com/](http://thingsfittingperfectlyintothings.tumblr.com/)

~~~
CarVac
[https://www.reddit.com/r/Perfectfit/](https://www.reddit.com/r/Perfectfit/)

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akeck
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!

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

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mdunn
Interesting -- thanks for posting!

